Article pubs.acs.org/Langmuir
High-Precision Tracking of Brownian Boomerang Colloidal Particles Confined in Quasi Two Dimensions Ayan Chakrabarty,† Feng Wang,† Chun-Zhen Fan,†,§ Kai Sun,‡ and Qi-Huo Wei*,† †
Liquid Crystal Institute and Department of Chemical Physics, Kent State University, Kent, Ohio 44242, United States Department of Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States
‡
ABSTRACT: In this article, we present a high-precision image-processing algorithm for tracking the translational and rotational Brownian motion of boomerang-shaped colloidal particles confined in quasi-two-dimensional geometry. By measuring mean square displacements of an immobilized particle, we demonstrate that the positional and angular precision of our imaging and image-processing system can achieve 13 nm and 0.004 rad, respectively. By analyzing computer-simulated images, we demonstrate that the positional and angular accuracies of our image-processing algorithm can achieve 32 nm and 0.006 rad. Because of zero correlations between the displacements in neighboring time intervals, trajectories of different videos of the same particle can be merged into a very long time trajectory, allowing for long-time averaging of different physical variables. We apply this image-processing algorithm to measure the diffusion coefficients of boomerang particles of three different apex angles and discuss the angle dependence of these diffusion coefficients.
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INTRODUCTION Colloids with anisotropic particle shapes and interactions are an emerging topic of many studies owing to their potential applications as building blocks for self-assembling new structures and materials.1−4 Innovative chemical synthesis and microfabrication methods have enabled the production of colloids with a variety of exotic shapes such as Janus, patchy, and lock−key particles.5−9 Janus particles with half hydrophobic and half hydrophilic surfaces aggregate into highly ordered structures,10 and triblock Janus particles self-assemble into more complex structures such as Kagome lattices.11,12 Patchy colloidal particles enable site-specific directional interactions, allowing for the assembly of complex structures such as colloidal micelles13 and mimicking the valence bonding of molecular systems.14 By using complementary particle shapes and depletion forces, lock−key-type colloidal systems can form flexible dimeric, trimeric, and tetrameric colloidal molecules and colloidal polymers.15 The ability to engineer geometric shapes and patterned surface chemistry of individual colloidal particles has opened a new horizon in programmable and directed colloidal self-assembly. One topic related to anisotropic colloidal particles is how these geometric shapes affect their diffusion and transport behaviors. Since Einstein’s seminal work in 1905, Brownian motion has been an active topic of research in various fields for the past century.16,17 Although hydrodynamic theories for the Brownian motion of irregularly shaped bodies were developed by Brenner and others over 50 years ago,18−23 experimental studies of anisotropic particles appeared only recently.24−27 For example, video microscopic and theoretical studies of ellipsoidal © 2013 American Chemical Society
particles have demonstrated that the translational−rotational coupling makes the displacement probability distribution functions (PDF) non-Gaussian, although such coupling disappears in the body-moving frame.24 The Brownian motion of more complex geometric shapes has been realized by using aggregates of spherical particles.28−33 It has also been shown that the hydrodynamics of anisotropic particles is important to the motion of self-propelled particles and can be designed to yield different types of motion trajectories.34−36 In a recent work, we studied the Brownian motion of boomerang-shaped colloidal particles under quasi-two-dimensional (2D) confinement and demonstrated that the center of hydrodynamic stress (CoH), defined as the point fixed to the particle where the coupled diffusion coefficients are zero, is important to understanding the diffusion behavior of nonskewed anisotropic particles.37 When the point used for motion tracking is not coincident with the CoH, the Brownian motion exhibits a crossover between short-time and long-time diffusion. The short-time diffusion coefficients vary with the point used for motion tracking, and the long-time diffusion coefficients correspond to those with the CoH as the tracking point.37 Given that the bent-core molecules exhibit a variety of liquidcrystal phases,38 the boomerang colloidal suspensions may also serve as an interesting model system for studying these liquidcrystal phases. Received: September 4, 2013 Revised: October 30, 2013 Published: October 30, 2013 14396
dx.doi.org/10.1021/la403427y | Langmuir 2013, 29, 14396−14402
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Because their mean displacements for fixed initial orientation are normally nonzero and biased toward the CoH, highprecision tracking and longtime trajectories are essential to characterizing the Brownian motion of the boomerang-shaped particles carefully.37 The normal tracking algorithm based on the intensity-weighted center of mass does not yield high precision for the boomerangs because some out-of-plane rotational motion causes uneven intensities for the two arms and affects the accuracy in locating the center of mass. In this article, we present an image-processing algorithm for tracking the position and orientation of the boomerang colloidal particles confined in a quasi-two-dimensional geometry. The image processing first finds points on the central axes of two arms by fitting the intensity profiles with Gaussian functions and then determines the cross point and the angle bisector line of these two arms (i.e., the position and orientation of the particles). By analyzing the images of an immobilized particle, we show that our optical microscopy system and image-processing algorithm can yield very high precision: ±13 nm in position and ±0.004 rad in orientation. By analyzing computer-generated images with known positions and angles, we show that our image-processing algorithm can yield ±32 nm positional and ±0.006 rad angular accuracies. We use this image-processing algorithm to measure the diffusion coefficients of boomerang particles and discuss their dependence on the apex angles.
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Imaging and Image Processing. The Brownian motion of the boomerang particles was observed under an inverted transmission bright-field optical microscope. Videos of single boomerang particles were taken by using an electron-multiplying charge-coupled device (EMCCD, Andor Technologies). The time interval τ between neighboring frames was set at τ = 0.05 s for all of the videos. A representative optical microscopy image of a boomerang particle is shown in Figure 2a. To determine the position and orientation of the boomerang particle, image processing undergoes the following steps: (i) the grayscale image is inverted (Figure 2b); (ii) the image is smoothed using a Gaussian filter, and the background intensity is subtracted (Figure 2c); (iii) the intensity profiles along different directions are scanned to find the points on the central axes of two arms and the cross point of the two central axes (Figure 2d); and (iv) on the basis of the results from the last step, intensity profile scanning is repeated to refine the particle location and orientation (Figure 2e). For steps iii and iv, a region of interest around the particle is chosen (e.g., Figure 2c), and our calculations are restricted to this region to reduce the processing time substantially. As for the image smoothing in step ii, we used the Gaussian filter, which is especially effective at eliminating high-frequency noises that have normal distributions.40 The intensity value I0(x1, x2) at pixel (x1, x2) is reset as the average of its neighboring pixel intensities weighted by a zero-mean discrete Gaussian function g(i, j) = exp[−(i2 + j2)/ 2σ2]. The reset value I(x1, x2) can be expressed as n
I(x1, x 2) =
m
∑i =−n ∑ j =−m I0(x1 + i , x 2 + j) g (i , j) n
m
∑i =−n ∑ j =−m g (i , j)
Here g(i, j) is a (2m+1) × (2n+1) matrix. The degree of smoothing is determined by the standard deviation σ. The filter matrix size (i.e., 2m +1 and 2n+1) is usually chosen to be 2 to 3 times σ for reasonable clipping of the Gaussian curve. For our images, we found that m = n = 2 and σ = 2 give optimal results. The effectiveness of the smoothing can be seen from the representative pictures before and after the filtering (Figure 2b,c). In step iii, we first find a direction that is approximately parallel to the angle bisector of the boomerang arms. The image is scanned along 10 different directions that are evenly distributed between 0 and 180°. For each scanning direction, the intensities on parallel lines at equally spaced subpixel points are calculated through interpolation. Here the spacings between both the scanning lines and the subpixel points on these lines are set at a 0.5 pixel size. For each scanning line in a given direction, the number of points with intensities above a set threshold is counted as the cross-section length; the maximal cross-section length among these parallel scanning lines is recorded as the cross-section length for that scanning direction. The scanning direction that has the smallest cross-section length is the direction closest to the bisector of the two arms. With this approximate bisector direction, we proceed to find approximate central axes of the arms. Along equally spaced lines parallel to this direction, the intensities at equally spaced positions are calculated again through linear interpolation (schematic in Figure 2c); the intensity profiles along these lines are fitted with a Gaussian functions (Figure 2f). The peak loci of these Gaussian fittings give the points on the central axes. The spacings between the scanning lines and between the points on these lines are set at a 0.5 pixel size. These central points close to the arm tips and the apex point are eliminated, and the rest are fitted with linear functions to obtain the central axes of these two arms and then their cross point (i.e., center of the body, CoB) (Figure 2d). In step iv, the bisector line is calculated on the basis of the central axes found in step iii. The image is split into two parts by this bisector line, and then the intensity profiles across each arm are scanned separately using the direction perpendicular to the arm axis found in step iii. Gaussian fittings are used to obtain points on the central axes of each arm. Similar to step iii, these central points close to the arm tips and the CoB are eliminated, and the rest are linearly fitted to find the central axes of two arms.
EXPERIMENTAL SECTION
Particle and Cell Fabrication. To fabricate boomerang-shaped colloidal particles, we developed a fabrication procedure based on photolithography as reported in the literature.39 A 17 nm sacrificial layer (Omnicoat, a dissolvable polymer, Microchem Inc.) and a ∼500 nm UV-curable epoxy photoresist (SU8, Microchem Inc.) were sequentially spin-coated on Si wafers, and then boomerang-shaped particles with different sizes and apex angles were patterned by using an autostepper projection photolithography system with a 5-fold size reduction. These boomerang particles made on Si wafers were released by submerging them in the Omnicoat stripper (PG remover, Microchem Inc.). Sonication is used to reduce the release time. The stripper solvent was then replaced with deionized water through centrifugation. Anionic surfactants (SDS, 1 mM) were added to the aqueous particle suspensions to prevent the particles from aggregating. Representative scanning electron microscopy (SEM) pictures of the boomerang particles used in this article are shown in Figure 1a−c.
Figure 1. SEM pictures of the boomerang particles fabricated on a silicon wafer. (a) Vertex angle, 90°; arm length, 2.33 μm; width, 0.7 μm. (b) Vertex angle, 110°; arm length, 2.4 μm; width, 0.8 μm. (c) Vertex angle, 120°; arm length, 2.45 μm; width, 0.83 μm. The scale bars are 4 μm.
To confine the particles in quasi-2D geometries, sample cells composed of two parallel glass slides were used. The cell thicknesses were controlled to about 2 μm by using glass beads as spacers. A dilute dispersion of the colloidal particles was placed in the cells, and particle separations were larger than 100 μm so that no hydrodynamic interactions exist between these particles. 14397
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Figure 2. Image-processing steps: (a) an optical microscopy image of a boomerang particle with a 110° apex angle; (b) inverted image of image a; and (c) image after Gaussian filter smoothing and background subtraction. The green box indicates the area of interest; the yellow dashed lines indicate three parallel scanning lines in the scanning direction. (d) Central points of each arm determined from intensity profile scanning along different directions. The cross point between the central axis of each arm gives the center of the body (CoB). Yellow lines are schematic intensity scanning lines for step iv as described in the text. (e) Refined processing results of the central axes of the arms. (x1−x2), the laboratory frame coordinate system; (X1−X2), the body frame coordinate system. θ is the particle orientation. (f) Exemplary Gaussian fitting to a typical scanned intensity profile. depending on the dimensionality of the system. This so-called “longtime tail” originates from hydrodynamics, (i.e., the diffusive transportation of momentum of the particle through the fluid). For a 1-μmdiameter colloidal particle suspended in water, the decay time of the long-time tail is of the order of microseconds. The time interval τ = 0.05 s in our experiment is at least 3 orders of magnitude larger than the decay time of the hydrodynamic long-time tail; therefore, the correlations between the velocities or between the displacements in two neighboring time intervals are negligible. In experiments, we verified that the correlations between translational or angular displacements in next-nearest time intervals are indeed zero within statistical error. Diffusion Coefficients of Anisotropic Particles. The drag force and torque experienced by an anisotropic particle are linearly proportional to both translational and rotational velocities, and the proportionality is described by the Stokes law with a hydrodynamic resistance matrix
With the central axes and the CoB position as inputs, we repeat step iv. We find that a single iteration is sufficient to achieve the optimized precision and that further iterations result in less than 10 nm variations in the results. As shown in Figure 2e, the coordinates of the CoB are used to represent the particle position, and the angle made by the angle bisector with the horizontal axis defines the angular orientation of the boomerang. For the body frame coordinate system, the angle bisector of the arms is set as the X1 axis, and the X2 axis is defined on the basis of X1 (Figure 2e). This method of intensity profile fitting with Gaussian functions is based on the Gaussian point spread functions and has been previously used in image processing to find the central axes of ellipsoidal particles41 and in super-resolution molecular and spherical particle imaging and tracking.42,43 This Gaussian profile fitting has also been used to analyze the bending dynamics of fluctuating filaments.44 Trajectory Merging. As limited by the computer memory, each video taken with the EMCCD contains 3000 frames. Because a sufficient length of trajectories is essential to obtaining the mean displacements and displacement probability distribution functions with good averaging, we have taken 35 videos for each of the three particles. After all videos were image processed, the trajectories of the same particle were merged into a single long trajectory. To merge the trajectories of two videos, the particle positions of the second video are shifted such that the position of the first frame in the second video matches that of the last frame in the first video. The trajectory of the second video then undergoes a rotational transformation to make the particle orientation of the first image match that of the last image in the first video. Random sequences of merging these trajectories provide additional averaging. This video merging has enabled us to measure mean displacements and mean square displacements over 4 decades of time scales with excellent averaging.37 The process of merging multiple videos of motion trajectories into a single trajectory is based on the fact that the motion of the particle in the nearest-neighbor time intervals is uncorrelated, an assumption that is physically justified. As shown by Alder and Wainwright and others,45−49 for a hard sphere suspended in fluid, the velocity correlation function decays algebraically with time with a power
⎛ ζ T ζ C ′⎞ P ⎟ ζ = ⎜⎜ C R⎟ ζ ζ ⎝ P P ⎠ where ζT and ζR represent the translational and rotational resistance submatrixes and ζC, which is generally asymmetric, represents the coupling matrix. The diffusion coefficient matrix is related to this resistance matrix through the generalized Einstein−Smoluchowski relationship: D = kBTζ−1. The resistance matrix varies with the reference point (i.e., the tracking point used in experiments). Brenner and others have shown that rigid bodies with certain symmetries possess two unique points, one named as the center of reaction at which the coupling resistance submatrix is symmetric and the other named as the center of diffusion where the coupling diffusion submatrix becomes symmetric.18−22 When the coupling submatrix ζC is zero, these two hydrodynamic centers coincide at one unique point that is called the center of hydrodynamic stress (CoH); particles with zero ζC are called nonskewed. In other words, at the CoH, both coupling diffusion 14398
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Figure 3. Results from real videos: (a−c) Position and orientation of an immobilized particle vs time. (d−f) Probability functions of the variations, Δx1, Δx2, and Δθ, calculated from the trajectories in a−c. Red lines represent best Gaussian fittings with the standard deviations σx1 = 13 nm, σx2 = 11 nm, and σθ = 0.0042 rad.
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and resistance submatrixes go to zero and the translational and rotational motion are decoupled. Brenner showed that in three dimensions, particles with at least three mutually perpendicular planes of symmetry are nonskewed whereas particles with two or fewer planes of symmetry are skewed and do not have the CoH.18,19 The conditions for the existence of CoH in two dimensions are not given in prior theory. The boomerang particles possess C2v symmetry. When the CoB is used as the tracking point, translational motion along the X2 axis leads to particle rotation due to the nonzero drag torque with respect to the CoB. Therefore, the coupled diffusion coefficient D2θ is nonzero, characterizing the coupling between rotation and translation along the X2 axis. The diffusion coefficient matrix of the boomerang particles at the CoB in two dimensions can thus be written as
RESULTS AND DISCUSSION Imaging and Tracking Precision. The precision in determining the particle’s position and orientation is limited by several physical factors, including intensity fluctuations of the illuminating light, mechanical vibrations of the microscope stage, and electronic noises from the EMCCD detector. To determine the precision of our imaging and image-processing system, a cell containing the boomerang particles was dried and then videos of an immobilized boomerang particle were taken for 150 s (3000 frames). The trajectory of this immobilized particle shows that the variations in the particle positions and orientations are within approximately 30 nm and 0.01 rad, respectively (Figure 3a−c). We calculated the variation probability distribution functions (PDFs) for both positions and orientation and fitted these PDFs with Gaussian functions. The standard deviations of these Gaussian fittings or the precision of our optical microscopy system and imageprocessing algorithm is ±13 nm for particle position and ±0.004 rad for particle orientation. To illustrate further the precision of our imaging and imageprocessing system, we set the motorized stage of the optical microscope to move by ∼100 nm in steps and record a video of the immobilized particle. The image-processed positions of the particle as shown in Figure 4 demonstrate that the precision of our system does go far below 100 nm. Here the step size of the stage movement varies a little around 100 nm, which is ascribed to some hysteresis in the motorized stage. To determine the accuracy of the image-processing algorithm, we generated images with known particle positions and orientations with a computer program. We define the CoB and nine additional equally spaced points on each boomerang arm and then create a root image of the boomerang particle by summing Gaussian point-spread functions that peak at these points. By adding random Gaussian noises with set variance onto this root image, a sequence of images can be generated, mimicking the video of a fixed particle. Similar to the experiments, the mean background intensity is set at 0.32 (the maximum intensity is set as unity). To see the effect of
⎛ D11 0 0 ⎞ ⎜ ⎟ D = ⎜ 0 D22 D2θ ⎟ ⎜ ⎟ ⎝ 0 D2θ Dθ ⎠ Mean Square Displacements in the Body Frame. After image processing, we obtain the positions of the CoB, x(tn) = [x1(tn), x2(tn)] and the orientation θ(tn) of the boomerang as functions of instantaneous time tn = nτ. The translational displacements in the laboratory frame are calculated as Δxi(t, tn) = xi(tn + t) − xi(tn), where i = 1, 2. To measure the anisotropic diffusion coefficients, the laboratory frame displacements need to be transformed into those in the body frame through a rotation transformation: ΔXi(tn) = R(θn) Δxi(tn) where
⎛ cos θn sin θn ⎞ ⎟⎟ R(θn) = ⎜⎜ ⎝− sin θn cos θn ⎠ As shown in our previous paper, for low-symmetry particles such as boomerangs, θn needs to be chosen carefully because the Brownian motion is biased toward the CoH as a result of translational−rotational coupling, which may lead to nonzero drift and nonlinear MSDs in the body frame.37 To eliminate the nonlinear behaviors, all of our calculations use θn = [θ(tn) + θ(tn + 1)]/2, which defines the continuous body frame.37 The body frame trajectories are constructed by accumulating displacements Xi(tn) = ∑kn= 0ΔXi(tk). 14399
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Similar analyses for the particle orientations are performed to estimate the angular accuracy. The deviations of the mean particle orientations from the set values are quite small (∼0.001 rad) and independent of the set orientations (Figure 5b). As for the angular accuracy, we use the summation of the standard variations of the mean angles from the set values and the standard variations around the mean positions (Figure 5d). For noises equivalent to the experimental conditions, the accuracy of our image-processing algorithm is ∼0.006 rad. Diffusion Coefficients. As discussed above, the diffusion coefficients are obtained from the displacements in the body frame through ⟨ΔXiΔXj⟩ = 2Dijt where i = 1, 2, θ. The measured mean square displacements (MSDs) and translational−rotational displacement correlations for the CoB are shown in Figure 6 for three particles with different apex angles. The MSDs grow linearly with time; linear fittings yield diffusion coefficients D11, D22, and Dθ (Figure 6a,b). Because of the particle symmetry, displacements of the CoB along the X1 axis do not induce rotations because the drag torques on the two arms cancel out. The experimental data verify that the correlations between displacements in X1 and θ are zero (Figure 6d). In contrast, displacements of the CoB along X2 cause rotation as the drag torques of two arms add up. The experimental data show that the correlations between displacements along X2 and θ grow linearly with time, and the best linear fitting gives coupled diffusion coefficient D2θ. Figure 6f shows the variation in the measured diffusion coefficients for the CoB and also for CoH on the apex angle of the boomerangs. For the CoB, the translational, rotational, and coupled diffusion coefficients show the same trend: decreasing with the increasing apex angle. This result can be understood qualitatively. When the apex angle is increased, the effective cross-section of the particle along X1 increases or D11 should decrease. Similarly, Dθ should decrease as the effective particle length is increased. Considering that the translational− rotational coupling is zero for 0° and 180° apex angles, the coupled diffusion coefficient D2θ should be maximized at some angle between 0° and 180°. This is in qualitatively agreement with theoretical calculations for bent rods in three dimensions.22 In the previous paper, we demonstrated that the diffusion coefficients D22 and D2θ are affected by the position of the tracking point used in the measurements.37 For tracking points on the angle bisector line, D11 and Dθ remain the same whereas D22 and D2θ vary with the position of the tracking point. The boomerang particles confined in quasi-2D geometry possess the CoH that is located on the symmetry line. At this CoH, the coupled diffusion coefficient D2θ becomes zero, and D22 reaches 37 2 its minimal as given by DCoH 22 = D22 − D2θ /Dθ. In comparison CoH to D22, D22 exhibits a smaller θ dependence and seems to flatten out when the apex angle is above 110° (Figure 6f). Intuitively, the cross-sectional area along the X2 axis is maximal for 0 and 180° apex angles, as is the Stokes drag force. CoH Therefore, D22 should decrease when the apex angle approaches 0 and 180°, and the measured angle dependence of measured DCoH 22 is in qualitative agreement with this picture. On the basis of our previous work, the distance between CoH and CoB can be expressed as d = D2θ/Dθ. It can be obtained that for the apex angle at 90°, 110°, and 120°, d is equal to 1.05, 0.93, and 0.88 μm, respectively. This monotonic decrease in d with the apex angle can be understood by considering the following facts. For a 180° apex angle, the CoH and CoB coincide, whereas for a 0° apex angle, the CoH
Figure 4. Position of the immobilized particle vs time. The motorized microscope stage moves in the x direction with an ∼100 nm step size.
particle orientations on image processing, we generated videos of boomerang particles with different set orientations. For each particle orientation, we created five videos corresponding to five different levels of Gaussian noises. The level of Gaussian noises is determined by the variance, σnoise2, of the background intensity. For convenience, we take the set CoB position as the origin and present results only for the x1 position (results for x2 are similar). We analyzed these simulated videos using our imageprocessing algorithm and observed that the mean positions x1̅ of the CoB deviate from the origin and vary with the set particle orientations (Figure 5a). This deviation results from the
Figure 5. Results from computer-simulated images: (a, b) Mean position and orientation calculated from two set particle orientations (60 and 30°) vs the background noise. The error bars represent standard variations around the mean positions/angles. (c, d) Mean position and orientation averaged over different set orientations vs the background noise. The error bars represent the summation of the standard variations of the mean positions/angles from the set values at different set angles and the standard variations around the mean positions/angles.
anisotropy of the pixel grids and decreases with the increase in the image magnification. The standard variations around the mean positions show little angle dependence and grow with the noise level. When the noise level is equivalent to that in the experiments (σnoise2 = 6.25 × 10−10), the standard variation around the mean position is around ±11 nm, which is in good agreement with the experimental result of ±13 nm. As for the positional accuracy, we use the summation of the standard variations of the mean positions for different set orientations and the standard variations of the positions around the mean positions (Figure 5c). For noises equivalent to the experimental condition, the positional accuracy of our image-processing algorithm is around ±32 nm. 14400
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Figure 6. Measured mean square displacements for the CoB: (a) MSDs along X1 vs time; (b) MSDs along X2 vs time; (c) MSDs for θ vs time; (d) correlation function between displacements along X1 and θ; and (e) correlation function between displacements along X2 and θ. The types of data points represent different particle apex angles: □, 90°; ○, 110°; and Δ, 120°. (f) Measured diffusion coefficients D11, D22, Dθ, and D2θ for CoB (○) 2 and DCoH 22 (□) for the CoH as a function of the vertex angle of the boomerang particles. The units on the vertical axis are μm /s for D11, D22, and CoH 2 D22 , rad /s for Dθ, and μm·rad/s for D2θ.
Notes
approaches the middle point of the arm. With the increase in the apex angle, we can expect that the distance d between the CoH and CoB should decrease.
The authors declare no competing financial interest.
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CONCLUSIONS In this work, we have developed a high-precision imageprocessing algorithm for tracking the translational and rotational motions of boomerang-shaped colloidal particles confined in quasi-2D geometries. By studying the mean square displacements of an immobilized particle, we demonstrated that the precision of our imaging and image-processing algorithm can achieve 13 nm for position and 0.004 rad for orientation. By analyzing computer-simulated images with known positions and orientations, we observe that the mean particle position deviates from the set position as a result of the anisotropic nature of the pixels. We show that the positional and angular accuracies of our image processing can achieve 32 nm and 0.006 rad, respectively. On the basis of zero correlations between the displacements in neighboring time intervals, we merged trajectories of different videos of the same particle into a long-time trajectory, allowing for measurements of mean displacements and mean square displacements with excellent averaging. By using this imaging-processing algorithm, we measured the diffusion coefficients of boomerang particles with three different apex angles, and the angle dependences of these diffusion coefficients agree qualitatively with expectations.
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ACKNOWLEDGMENTS
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REFERENCES
We acknowledge valuable discussions with Oleg Lavrentovich, Andrew Konya, Jonathan V. Selinger, and Ji-Ping Huang. The work was supported by a Farris Family Award and partially supported by NSF ECCS-0824175.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Present Address §
School of Physical Science and Engineering, Zhengzhou University, Zhengzhou 450052, China. 14401
dx.doi.org/10.1021/la403427y | Langmuir 2013, 29, 14396−14402
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