Letter pubs.acs.org/JPCL
Position-Dependent Three-Dimensional Diffusion in Nematic Liquid Crystal Monitored by Single-Particle Fluorescence Localization and Tracking Seonik Lee, Koushi Noda, Shuzo Hirata, and Martin Vacha* Department of Organic and Polymeric Materials, Tokyo Institute of Technology, Ookayama 2-12-1-S8-44, Meguro-ku, Tokyo, Japan S Supporting Information *
ABSTRACT: Anisotropic mass diffusion in liquid crystals (LCs) is important from the point of both basic LC physics and their applications in optoelectronic devices. We use superresolution fluorescence microscopy with astigmatic imaging to track 3D diffusion of quantum dots (QDs) in an ordered nematic LC. The method allowed us to evaluate the diffusion coefficients independently along the three spatial axes as well as to determine the absolute position of the QD with respect to the cell wall. We found variations of the diffusion coefficient along the different directions across the cell thickness and explained these as being due to changes of a tilt angle of the LC director. Close to the surface, the diffusion is slowed down due to the confinement effect of the cell wall. Overall, the QD diffusion is much slower than expected for a corresponding particle size. This phenomenon is suggested to originate from reorientation of the LC director in the vicinity of the particle.
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relation to the LC orientation.15 Experiments on both nematic and smectic LCs provided diffusion anisotropy values consistent with other methods, that is, in the range between 1.2 and 1.6.16−18 In contrast, fluorescence correlation spectroscopy (FCS) tends to provide much higher values.19,20 Apart from single-molecule fluorescence, anisotropic diffusion in LCs has been studied by monitoring larger colloidal particles,21,22 with the obtained anisotropy value in nematic LCs of 1.6. Even though the single-molecule techniques remove the spatial averaging, they have so far provided information on diffusion in only two dimensions (in a plane parallel to the substrate). With respect to the third dimension, that is, perpendicular to the cell substrate, the position is either unspecified or the information is averaged over the cell thickness. Here we use single-particle localization and tracking combined with astigmatic imaging23,24 to study the anisotropy of the diffusion in nematic LCs in all three dimensions. The technique allows us to determine the absolute position of the tracked particle with respect to the cell wall and thus to systematically study the 3D diffusion in different regions along the cell thickness. Apart from contributing to the basic knowledge of LC physics, the results are also important from the application point of view because they can shed light on the diffusion of particulate impurities in LC displays, which are the cause of local display darkening. The LC used is nematic at room temperature, with nematic− isotropic transition at 77.7 °C. At the experimental temperature, the LC is far away from phase transitions, and we assume
iquid crystals (LCs) combine long-range order typical of crystals with fluidity characteristic of liquids. LCs occur naturally and can also be prepared synthetically. As such, they have found many applications, the most important one being in LC displays. Characteristic physical phenomena occurring in thermotropic LCs are phase transitions and phenomena related to the structural ordering, such as optical anisotropy and anisotropic mass diffusion. Self-diffusion is accessible mainly by nuclear magnetic resonance (NMR) methods.1,2 Anisotropy of diffusion coefficients D∥/D⊥ along and perpendicular to the orientation of the LC (director) for nematic LCs has been found to vary between 1.5 and 2.5 for a wide range of temperatures.3,4 Confinement of nematic LC in pores of a diameter down to 30 nm does not change the diffusion behavior observed by NMR.5 The diffusion processes can also be studied also optical methods. For that purpose, the LCs are doped with optical contrast agents such as dye molecules. Forced Rayleigh scattering (FRS) is based on the measurement of the decay of a transient grating due to the mass diffusion.6 Applications of FRS in nematic LCs revealed the diffusion anisotropy values on the order of 1.3 to 1.6.7−11 An alternative optical method is fluorescence recovery after photobleaching (FRAP), which monitors diffusion of fluorescent dye molecules into a spot previously darkened by a strong laser.12 All of the previously described methods probe the diffusion in LCs on a macroscopic scale, and the obtained values are thus spatial averages over potentially heterogeneous microscopic areas of the sample. Single-molecule and single-particle detection and tracking techniques remove such spatial averaging13 and are suitable for studies of diffusion on submicrometer scales.14 The methods are also straightforward in revealing the diffusion anisotropy and in providing direct © 2015 American Chemical Society
Received: March 8, 2015 Accepted: April 1, 2015 Published: April 1, 2015 1403
DOI: 10.1021/acs.jpclett.5b00488 J. Phys. Chem. Lett. 2015, 6, 1403−1407
Letter
The Journal of Physical Chemistry Letters that it is structurally homogeneous. The LC is oriented by surface rubbing, as shown in Figure 1a.
Figure 1. (a) Orientation of the liquid crystal cell with respect to the rubbing direction and cell walls. (b,c) Cross-polarized transmission images of the liquid-crystal cell. Black and white cross arrows indicate the directions of the polarizers; the yellow arrows show the rubbing direction. (d) Astigmatic fluorescence microscopic images of a single quantum dot at different times. (e) Positions of a single quantum dot in the z direction obtained from 2D Gaussian fits.
Doping of the LC with CdSe quantum dots (QDs) at the concentration of 10−10 M does not disrupt its macroscopic alignment on the scale of tens of micrometers, as seen from the cross-polarized transmission images in Figure 1b,c. Examples of astigmatic fluorescence image snapshots of a single diffusing QD are shown in Figure 1d and Figure S1 in the Supporting Information (SI). The change of the pattern elongation from horizontal to vertical indicates diffusion along the z direction. The single QD images were used to determine the timedependent 3D position of the QDs, with an example of z positions shown in Figure 1e. Localization error of the 3D positions was determined by analyzing an ensemble of immobilized QDs with the same spatial orientation. In all three spatial directions, the localization error is better than 20 nm. The positions along a specific direction were used to calculate the mean-square displacement (MSD), which was then used to calculate the diffusion coefficients Dx, Dy , and Dz along the spatial directions and the 3D diffusion coefficient D3D. The limited lengths of the diffusion traces (five positions for each QD) did not enable us to distinguish between possible non-Brownian modes of diffusion. Details are described in the SI. To study the position-dependent diffusion, we divided the LC cell into virtual 0.5 μm thick horizontal sections along the z direction. We stress that these sections are not related to the structure of the LC. In each section, the measurement was done for an ensemble of 160−380 QDs. Examples of histograms of the diffusion coefficients obtained from one section are shown in Figure 2a−d. In Figure 2d, a simulated distribution of diffusion coefficients corresponding to a single-mode diffusion in a homogeneous environment is plotted as the solid line. The large width and long tail of the simulated distribution originate from the fact that only a limited number of data points are analyzed for each QD, that is, that each particle is observed only for a short time interval. Because the simulated distribution reasonably replicates the measured one, we do not assume that
Figure 2. (a−c) Distributions of diffusion coefficients along the x (a), y (b), and z (c) directions in the section closest to the surface (0 to 0.5 μm). (d) Distribution of the 3D diffusion coefficient in the section closest to the surface (0 to 0.5 μm). The solid line is an expected distribution for single-mode diffusion in homogeneous environment calculated according to ref 26. (e) Average values of the diffusion coefficients calculated as numerical means of the distributions in different sections. The section boundaries are indicated by dotted vertical lines. The solid lines are linear fits to the data. (f) Schematic orientation of the director in different sections and decomposition of the diffusion along the director into components along the y and z directions. (g) Tilt angle of the director in individual sections calculated according to eq 3. The error analysis was based on refs 27−29 and is described in detail in the SI.
effects such as the LC structural inhomogeneities or QD size distributions would contribute to the shapes and widths of the diffusion coefficient distributions. Overall diffusion coefficients D̅ x, D̅ y, and D̅ z along specific directions were calculated as numerical averages of the histograms. Over all sections, these average diffusion coefficients range between 0.05 and 0.13 μm2 s−1. Compared with that, self-diffusion in nematic LC probed by NMR gives the diffusion coefficients typically on the order of 10 μm2 s−1.2 Diffusion coefficients measured by FRS are in the range 13−21 μm2 s−1.7 In contrast, FRAP and single-molecule tracking yield diffusion coefficient values almost one order smaller, in the range 5−9 μm2 s−1.12,18 Still, the values found here are further more than an order of magnitude lower. The main reason for this difference is the different nature of the fluorescent dopant. Compared with small dye molecules,12,18 the QDs are much bulkier (with a diameter of 3.4 nm) and are further stabilized with hexadecylamine ligand coating as surface treatment, resulting in a diameter of 6 nm. Apart from the size effect, 1404
DOI: 10.1021/acs.jpclett.5b00488 J. Phys. Chem. Lett. 2015, 6, 1403−1407
Letter
The Journal of Physical Chemistry Letters
cdf treatment is that it can clearly reveal multimodal diffusion processes14 that are not obvious from the distributions of diffusion coefficients and that it works more reliably with fewer data points. The cdf C(dz,Δt) of the diffusivity dz in the z direction for the time interval Δt was calculated according to ref 17. (See the SI for details.) Examples of the C(dz,Δt) are plotted in Figure 3a, and all data are shown in the Figure S2 in
the difference is partially attributable to the interaction of the QDs with the LC, specifically to reorientation of the LC in the vicinity of the QD. This effect that changes the local viscosity around the QD and its effective size has been studied in detail for large colloidal particles in LCs.25 The average diffusion coefficients obtained in each section were plotted as a function of distance z0 of the section from the cell wall in Figure 2e. Averaging further the diffusion coefficient along the direction y (along the LC director projection) over all sections and along x and z over all sections provides the diffusion anisotropy D∥/D⊥ of 1.88, which is in good agreement with the NMR and FRS results for nematic LCs.3,4,7−11 If the director of the LC was parallel to the substrate there should be no difference between diffusion in the x and z directions; however, for all sections the observed D̅ z are larger than D̅ x. The most plausible explanation for this difference is the existence of a tilt angle θ of the director with respect to the surface of the cell. In the LC, the particles are diffusing along the director with an intrinsic diffusion coefficient D∥ and perpendicular to it with a coefficient D⊥.The tilt angle θ leads to decomposition of these intrinsic coefficients into components parallel with the surface, D∥ cos θ and D⊥ sin θ, and perpendicular to it, D∥ sin θ and D⊥ cos θ (Figure 2f). The component D∥ sin θ contributes to the apparent diffusion in the z direction by adding to the component originating from the perpendicular diffusion D⊥ cos θ. Because in the current geometry D⊥ = D̅ x, the observed diffusion in the z direction can be expressed as Dz̅ = D sin θ + Dx̅ cos θ
Figure 3. (a) Complementary cumulative distribution function of the diffusivity along the z direction analyzed at different distances from the cell surface (thick lines) and the corresponding two-exponential fits (thin lines) according to eq 4. (b) Average diffusion coefficient (symbols) at different distances from the surface calculated using the fitting parameters and eq 5. The solid line is an expected distance dependence of the relative diffusion coefficient Dz/D0 calculated for the quantum dot radius of 3 nm using eq 6. The dashed line represents a fit of the first four experimental data points using eq 6
(1)
In a similar way, the observed diffusion along the y direction is determined mainly by D∥ cos θ but also contains the component originating from the D⊥ = D̅ x, as follows Dy̅ = D cos θ + Dx̅ sin θ
(2)
the SI. For the analysis, the section closest to the surface was further divided to narrower subsections. To examine the diffusion in the nearest proximity to the surface, we newly analyzed the motion of those QDs that were eventually adsorbed on the surface and immobilized. These QDs provided the diffusivities in the 0−100 nm region. As seen from Figure 3a, the cdf C(dz,Δt) deviates from straight lines. The experimental data were fitted with a sum of two exponential functions (Figure S3 in the SI)
For zero tilt angle, these equations give D̅ y = D∥ and D̅ z = D̅ x = D⊥, as expected. Equations 1 and 2 can be combined into an equation for the tilt angle θ as Dx̅ (sin 2 θ − cos 2 θ) − Dy̅ sin θ + Dz̅ cos θ = 0
(3)
Using this equation and the experimental values of D̅ x, D̅ y, and D̅ z we obtain for the section closest to the surface (0 to 0.5 μm) a tilt angle value of 18.0°. With increasing distance from the surface, the average diffusion in the x direction D̅ x is approximately constant, the diffusion D̅ y along y increases, and the diffusion D̅ z along z decreases. Following the previous discussion, this observation can be explained by a decrease in the tilt angle from the surface toward the center of the cell. Because the D∥ is larger than D̅ x, the changes of D̅ y and D̅ z are determined mainly by the D∥ components (Figure 2f). Using eq 3, the tilt angle can be estimated for all sections, as shown in Figure 2g. Although the values are accompanied by a large error due to the error propagation of the D̅ x,D̅ y, and D̅ z values in eq 3, there is an indication of an overall trend toward smaller tilt angles with increasing z0. The 3D average diffusion coefficient D̅ 3D does not show any significant position dependence, as also seen in Figure 2e. To study in detail the diffusion in the z direction in the proximity of the cell surface, we reanalyzed the positiondependent diffusion data in terms of complementary cumulative distribution function (cdf). The advantage of the
⎛ d ⎞ ⎛ d ⎞ C(dz , Δt ) = A exp⎜ − z ⎟ + B exp⎜ − z ⎟ ⎝ Dz1 ⎠ ⎝ Dz 2 ⎠
(4)
where Dz1 and Dz2 are components of the diffusion coefficient in the z direction. Using the fitting results, the average diffusion coefficient was calculated for each section as A B ⟨Dz ⟩ = Dz1 + Dz 2 (5) A+B A+B and plotted as a function of position in Figure 3b. From the plot it can be seen that the ⟨Dz⟩ increases from the center of the cell toward the cell wall until the distance of 0.5 μm. This result confirms the observation made by the analysis of the diffusion coefficient distribution in Figure 2e and is attributed to the change in the tilt angle of the director. For distances closest to the wall (
