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We developed a new technique that conducts dynamic light scattering (DLS) under a microscope with high spatial resolution. This technique dramatically extends the range of DLS application from transparent to opaque samples. The total scattered electric field contains both electric field generated from the samples and time-independent reflected electric field. These two components are decomposed by applying a partial heterodyne method. By using this technique, we successfully calculate the characteristic size distribution of both multiple-scattering samples and strong light-absorbing samples. This is the first study to observe the collective motion of particles in a highly concentrated solution by using DLS.
©2013 Optical Society of America
1. Introduction
Dynamic light scattering (DLS) is a technique for obtaining the size distribution of particles in solution. This technique is especially powerful for the characterization of polymer solutions, colloids, emulsions, etc. However, it cannot be applied to opaque samples. There are two types of opaque samples [1, 2]. One is a black sample, which is a strong light-absorbing material such as ink. In this case, scattered light is completely absorbed by the sample itself and one cannot obtain the signal. The other type is milky samples, which are strong light-scattering materials such as dense suspension of polystyrene latex. In this case, multiple scattering is inevitable although we want to obtain is single scattering. In practice, opaque samples need to be diluted for DLS measurement. However, there is no evidence that highly diluted samples have the same size distribution as that of a neat solution. This disadvantage has been a long-standing issue for the application of the DLS technique to opaque samples such as paint and food. In addition, poor spatial resolution sometimes also becomes a problem. In typical DLS, the irradiated volume is of the order 100 μm on a side. However, there is a growing demand to investigate the dynamic behavior of samples with a higher spatial resolution. One such example is the tracking of material transportation in biological cells. Another example is the investigation of spatial inhomogeneity of soft materials such as gels [3]. To realize such applications, the spatial resolution needs to be near the diffraction limit (~1 μm).
Many studies have been conducted on how to overcome these disadvantages. A cross correlation spectroscopy is a technique to extract singly scattered light by using two-beam, two-detector light scattering spectrometer [4, 5]. In this experiment, a cross correlation was used to remove the effect from the multiply scattered light. Another example is a low-coherence DLS [6]. The unique thing about this technique is the use of a superluminescent diode as the light source. Its short coherence length was efficiently utilized to remove multiple scattering. The different approach for the measurement of dense media is the evaluation of the intensity correlation function in consideration of multiple scattering explicitly, called diffusing wave spectroscopy [7, 8]. From a viewpoint of the application of DLS to biological cells, novel technique was proposed by Dzakpasu and Axelrod [9]. They successfully created spatial maps of fluctuation decay rates with a high spatial resolution (μm order) by a combination of confocal microscopy and a streak camera. However, no studies have been reported on the application of DLS to opaque media with high spatial resolution.
In this paper, we propose a new DLS technique, a DLS microscope that mitigates both the above-mentioned disadvantages. Although there are many reports on how to perform DLS under a microscope [10–12], all of them utilize forward scattering. This restricts the scope of its application to transparent materials. The proposed DLS microscope obtains a signal with a backscattering geometry similar to that of a Rayleigh light scattering microscope [13, 14]. This enables us to measure opaque samples with high spatial resolution. However, this technique leads to contamination of reflected light from the interface of the sample. We regard this reflected light as a local oscillator and actively utilize it for data analysis.
2. Principle and experiment
The schematic of the DLS microscope is shown in Fig. 1. An Ar-Kr ion laser (λ = 514.5 nm, Stabilite 2018, Spectra-Physics) is used as the light source. The beam is reflected by a half mirror and focused onto the sample by an objective (Olympus: UPlan FL 40x / NA 0.75). In order to minimize the effect of the laser on the sample, such as that of absorption, the laser power at the sample is kept at less than 1 mW. The sample is shielded in a hole-slide glass, covered by a plate of cover glass. Note that typical DLS needs approximately 1 mL of sample. However, because of the small irradiated volume of our DLS compared with that of typical DLS, we need only 50 μL of sample. Scattered light from the sample is collected with a backscattering geometry and guided to an avalanche photodiode, and then to the autocorrelator (ALV, Langen, Germany). To improve the performance, we adopt a confocal optical system. Concretely speaking, there are following two reasons. Firstly, the multiple scattered lights can be eliminated because these lights have changed their traveling direction and cannot pass the pinhole (PH2 shown in Fig. 1). Secondly, the reflection from the boundary of the sample and cover glass can be decreased when the focal point is sufficiently far from the boundary. Although signal is strongly affected by the reflection when the focal point is near the boundary, this can be precisely evaluated by using a partial heterodyne analysis, shown in the latter of this section. The irradiated volume achieved by this confocal optical system is approximately 1 μm in a lateral direction and 10 μm in a vertical direction with the loose setting of PH2 in mind. High spatial resolution originated from this small irradiated volume is also experimentally shown in the latter of this section.
Fig. 1 Schematic of the proposed DLS microscope: VND, variable neutral density filter; PH1, pinhole, ϕ = 25 μm; HM, half mirror; BD, beam diffuser; PH2, pinhole, ϕ = 50 μm; APD, avalanche photodiode.
What we obtain thorough the DLS experiment is a time-averaged normalized intensity correlation function, . This function shows an exponential decay whose initial amplitude is 1. Practically, the initial amplitude is a little smaller than 1, mainly due to the finite size of the irradiated volume, which is called the smearing effect. This reduction is represented by a coherence factor. However, the smearing effect is considered to be a minor factor in case of the DLS microscope because the irradiated volume is very small. In addition to the smearing effect, the initial amplitude can be reduced when some time-independent electric field is mixed to the signal from the sample. In the case of the DLS microscope, this time-independent electric field is the reflection from the boundary. Quantitative relationship between the initial amplitude and the proportion of time-independent component was formulated as follows:
where A is the initial amplitude of the time-correlation function of the scattered intensity, , is the intensity from the sample, and is the total intensity. This analysis is based on a partial heterodyne method, which is originally developed for the analysis of the nonergodic media such as gels [15–17]. Since we obtain A and from one experiment, we can calculate from each experiment. In response to the variation of the initial amplitude, the apparent diffusion constant, DA, is also changed as follows:where D is the genuine diffusion coefficient.The sample is polystyrene latex suspensions (3000 Series Nanospheres, Duke Scientific Cooperation) and Chinese ink. The nominal diameter of the polystyrene latex particles used is 50 nm and 100 nm. These samples were used as purchased or after the appropriately dilution.
3. Results and discussion
3.1 Polystyrene beads: multiple scattering media
To see the performance of the DLS microscope, we measure the polystyrene latex suspension whose diameter of the particles is 50 nm. To compare the DLS microscope to a typical DLS system, we measure the sample without dilution by the DLS microscope and 103 times-diluted sample by the typical DLS system. Figure 2 shows some intensity correlation functions obtained by the proposed DLS microscope and the typical DLS system. The decay times obtained from the DLS microscope and typical DLS are different in principle because they have different λ (wavelength of the incident light) and θ (scattering angle). We find the initial amplitude obtained from the DLS microscope to be significantly below 1. In this experiment, we obtain values of A in the range 0.19 - 0.86. In addition to the variation in A, the values of the diffusion constants obtained from the decay time also vary.
Fig. 2 Intensity correlation functions for a polystyrene latex suspension, 1 wt%. The nominal diameter of the polystyrene latex particles is 50 nm. Solid lines: Several data sets obtained from the DLS microscope (λ = 514.5 nm, θ = 180°) at different points within the suspension. Dashed line: Data obtained from a typical DLS system (DLS/SLS 5000 compact goniometer, ALV, λ = 632.8 nm, θ = 90°). The inset is a plot of DA vs. . The solid line is the best fit result based on Eq. (2).
The inset of Fig. 2 indicates a plot of DA vs. . Each point lies on a straight line from the origin, as expected from Eq. (2). This result strongly supports the validity of the assumption that the total intensity is decomposed into the components from the samples and that from the boundary. This linearity also verifies that we can evaluate the genuine diffusion constant, D, from a single measurement. From the slope of the straight line in the inset of Fig. 2, D is obtained as 8.22 x 10−12 m2 s−1. From this value, the hydrodynamic diameter, dh, is obtained using the Stokes-Einstein equation. In this experiment, the calculated diameter is 58 nm, which is in good agreement with the value determined by the typical DLS system (57 nm).
By using the partial heterodyne method, we measure a 1 wt% polystyrene latex suspension having a nominal diameter of 100 nm. This suspension is so dense that it has a milky liquid appearance because of multiple scattering (Fig. 3(a)). Taking advantage of the high spatial resolution, we measure the intensity correlation functions by scanning the focus along the vertical (z) axis to show the effect of reflection explicitly. The definition of z is shown in Fig. 3(b). Figure 3(c) shows the position dependence of the scattered intensity of the polystyrene latex suspension. Near the boundary of the sample and glass (0 - 40 μm, 170 - 180 μm), the scattered intensity is strongly enhanced. The intensity decrease in the region 40 < z < 170 μm is mainly due to multiple scattering; the greater the amount of medium the light passes through, the more the light is scattered and the intensity attenuated. We experimentally determined the coherence factor as 0.97, which is the maximum value of the initial amplitude in the data set for this experiment. This large value corresponds to the small irradiated volume of this DLS microscope. The observed intensity, , is decomposed into its two components; , the time-averaged intensity from the sample and , the intensity of reflection from the boundary of the sample and glass. Note that we do not have to take the time-average for IR because this quantity is time-independent. We calculate the proportion of these components from the initial amplitude of each intensity correlation function by using Eq. (1). The result is also shown in Fig. 3(c). This figure shows that these two components are successfully decomposed into two components. It is experimentally demonstrated that the source of the irregular intensity enhancement near the boundary region is enhancement, while the decrease in intensity between the boundaries is due to . In addition, the position dependence of the diameter is calculated at each point by using Eq. (2). The result is shown in Fig. 3(d). The apparent diameter (dashed line) shows that the hydrodynamic diameter increases near the edge. However, by using partial heterodyne analysis, the hydrodynamic diameter shows almost the same value regardless of the focus point. In other words, we do not observe the effect of the interface in this case, as previously reported [18].
Fig. 3 (a) The appearance of the sample. The cover glass is fixed with manicure. (b) The definition of z. (c) Position dependence of (thin solid line), (dashed line), and (thick solid line) for polystyrene latex suspension, 1 wt%. The nominal diameter of the polystyrene latex particles is 100 nm. The intensity is expressed in s−1, the photon counts in a second. (d) Position dependence of the diameter of the polystyrene latex suspension. The diameter calculated from the typical DLS is 116 nm, which is indicated in the figure. Solid lines: The diameter corrected using Eq. (2). Dashed line: The apparent diameter calculated by using apparent diffusion constant, DA.
3.2 Ink: strong light-absorbing media
So far, we have focused on using the polystyrene latex suspension as the opaque sample. However, as mentioned before, there is another source of opaqueness: absorption. Our DLS microscope can also be used with strong light-absorbing materials such as ink, where the scattered light travels a very short path. This assertion is supported by the results of a vertical scan experiment using Chinese ink (Fig. 4(a)). We measure the intensity itself by scanning the focus along the vertical axis. Although the intensity correlation functions are also obtained, these functions do not show single exponential decay, implying the polydisperse nature. This polydispersity is discussed in the next section. The value of is extracted at each point by using Eq. (1). The result is shown in Fig. 4(b). The experimental values for show an exponential decay as a function of z, which corresponds to the Lambert-Beer law. The semi-log plot in Fig. 4(c) visualizes this explicitly. By using the volume fraction as the concentration, the absorption coefficient is calculated as 3.10 x 103 cm−1. This value implies that the light intensity becomes 1/e when the light passes in the neat ink approximately 14 μm. Interestingly, this value shows good agreement with the value determined independently by a common absorption spectrophotometer, 3.23 x 103 cm−1. Note that we had to dilute the neat ink approximately 104 times to perform this measurement.
Fig. 4 (a) The appearance of the sample. (b) Position dependence of the scattered intensity from Chinese ink (10 wt%). Dashed line is the exponential fit. (c) The semi-log plot implying the Lambert-Beer law.
3.3 Concentration dependence of the size distribution
Although we cannot see the difference between the neat ink and the diluted ink from the viewpoint of the absorption coefficient, this does not mean that the microscopic morphology is invariant with dilution. To see the concentration dependence of the microscopic morphology, we measure the intensity correlation function with various concentrations. Highly concentrated solutions are measured with the DLS microscope and low-concentration solutions are measured with the typical DLS system. At intermediate concentrations, we measure the sample with both DLS instruments. From these data, the characteristic decay time distribution functions are obtained with an inverse Laplace transform program (a constrained regularization program, CONTIN provided by ALV). These functions are converted into the characteristic size distribution, G(dh), by using the Stokes-Einstein equation.
Figure 5(a) shows the results for the polystyrene latex suspension, and Fig. 5(b) shows that for Chinese ink. Since Fig. 5(a) shows peaks at almost the same position in all concentrations, we can conclude that the polystyrene latex suspension shows no morphology change through varying the concentration. This is consistent with the following molecular interpretation: each particle has a negative charge on the surface, and the particles repel each other keeping their form unchanged and showing Brownian motion. This result corresponds to the result obtained by using diffusing wave spectroscopy [19].
Fig. 5 (a) Concentration dependence of the size distribution of a polystyrene latex suspension. The nominal diameter of the polystyrene latex particles is 100 nm. The 1 - 0.01 wt%, as measured by the DLS microscope, is represented by the solid lines. The 0.01 - 0.0001 wt%, as measured by the typical DLS system, is represented by the dashed lines. (b) Concentration dependence of the size distribution of Chinese ink. The 10 - 0.05 wt%, as measured by the DLS microscope, is represented by the solid lines. The 0.05 - 0.001 wt%, as measured by the typical DLS system, is represented by the dashed lines.
In contrast, initial observations seem to show that the average particle size increases and the size distribution broadens at higher concentrations of Chinese ink. However, as Chinese ink is a protective colloidal solution, the particles are considered not to cluster. One of the possible explanations for our result is the existence of attractive interactions between colloidal particles. In other words, colloidal particles show collective motion rather than Brownian motion at higher concentrations. This consideration is supported by the fact that the viscosity of neat ink is approximately five times higher than that of water. To the best of our knowledge, this is the first observation of such collective motion within a dense opaque media by using DLS.
4. Conclusion
We have developed a new DLS technique that allows us to obtain intensity correlation functions from opaque samples under a microscope. The intensity correlation functions show the initial amplitude as much less than 1 because of the mixing of the time-independent reflected electric field with the electric field from the sample. The data are analyzed by the partial heterodyne method, and the size of the polystyrene latex suspension particles is determined precisely without being affected by multiple scattering. In addition, the intensity for the sample is extracted from the total intensity by using the value of the initial amplitude. We also apply the DLS microscope to Chinese ink and observe the collective motion in highly concentrated samples. In addition, we find that this instrument can also measure the absorption coefficient of the ink without dilution. Taking advantage of its high spatial resolution, this technique can also be readily applied to other media such as biological cells and gels.
Acknowledgment
This study is supported by “Advanced Leading Graduate Course for Photon Science, Program for Leading Graduate Schools, Japan Society for the Promotion of Science granted to TH.
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