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Tetsuro
Tsuji‡
*,
Ryoji
Nakatsuka
,
Kichitaro
Nakajima
,
Kentaro
Doi
and
Satoyuki
Kawano
*

Graduate School of Engineering Science, Osaka University, Toyonaka 560-8531, Japan. E-mail: tsuji.tetsuro.7x@kyoto-u.ac.jp; kawano@me.es.osaka-u.ac.jp

Received
15th December 2019
, Accepted 21st January 2020

First published on 24th January 2020

We experimentally and theoretically characterize dielectric nano- and microparticle orbital motion induced by an optical vortex of the Laguerre–Gaussian beam. The key to stable orbiting of dielectric nanoparticles is hydrodynamic inter-particle interaction and microscale confinement of slit-like fluidic channels. As the number of particles in the orbit increases, the hydrodynamic inter-particle interaction accelerates orbital motion to overcome the inherent thermal fluctuation. The microscale confinement in the beam propagation direction helps to increase the number of trapped particles by reducing their probability of escape from the optical trap. The diameter of the orbit increases as the azimuthal mode of the optical vortex increases, but the orbital speed is shown to be insensitive to the azimuthal mode, provided that the number density of the particles in the orbit is same. We use experiments, simulation, and theory to quantify and compare the contributions of thermal fluctuation such as diffusion coefficients, optical forces, and hydrodynamic inter-particle interaction, and show that the hydrodynamic effect is significant for circumferential motion. The optical vortex beam with hydrodynamic inter-particle interaction and microscale confinement will contribute to biosciences and nanotechnology by aiding in developing new methods of manipulating dielectric and nanoscale biological samples in optical trapping communities.

Given such growing interest in the field, this paper addresses techniques for improving particle manipulation technologies and validates them both experimentally and theoretically. The novel aspect is that the technique is not based on optics but rather on mechanics. We systematically investigate the optical-vortex-induced motion of dielectric micro- and nanoparticles with diameters ranging from 0.2 μm to 2 μm, which includes biological scales from viruses to cells. The two key ideas for the successful optical manipulation of nanoparticles are hydrodynamic inter-particle interaction and microscale confinement of slit-like fluidic channels. We find that the number of particles trapped in an orbit is important in the dynamics of orbital motion: the orbiting speed increases with the number of nanoparticles. This acceleration is schematically described in Fig. 1 and the ESI (Movie S1†). A theoretical model including hydrodynamic interaction^{8,10–12,20} explained the qualitative characteristics of microparticle motion observed in experiments. The diameter of the orbit increases as the azimuthal mode of the optical vortex increases, but the orbital speed is shown to be insensitive to the azimuthal mode, provided that the number density of the particles in the orbit is same. Such a trend is shown to be closely related to the above-mentioned hydrodynamic interaction. It should be noted that when a perfect vortex beam^{31,32} is used, the diameter of the orbit is independent of the azimuthal mode, and the orbital speed increases linearly as the azimuthal mode increases. For the stable orbital motion of the smallest particle investigated in this paper, which has a diameter of 0.2 μm, hydrodynamic inter-particle interaction is required to accelerate its speed enough to overcome the inherent thermal fluctuation, which is significant for nanoscale targets. Evaluating the orbital motion of such tiny particles is facilitated by using microchannels with a small height, high-precision flow control, and a mixture of non-fluorescent and fluorescent particles. In particular, channels with a small height that confine the motion to two dimensions reduce the chance that nanoparticles will escape the optical trap in the beam propagation direction, leading to a larger number of orbiting particles and thus enhancing orbiting speed.

Micro- and nanofluidic devices can be used to analyze target biological nanoparticles and/or molecules dispersed in fluids.^{33} Various methods to control target motion have been proposed to overcome the difficulties of controlling nanoscale targets.^{34} Among them, optical forces have attracted much attention because they are a non-contact and feasible method available to researchers in various fields.^{35–41} In fact, optical manipulation of tiny particles or biomolecules has been a fundamental tool in state-of-the-art nanoscience and bioengineering ever since the discovery of optical trapping by Ashkin.^{42} The method proposed in this paper enhances the controllability of nanoscale dielectric targets and contributes to the development of micro- and nanofluidic devices. Optical manipulation, electrophoresis,^{43} and thermophoresis^{44–46} are useful transport phenomena in tiny devices and have different physical mechanisms, so combining them will yield a selective manipulation method.^{47,48}

Fig. 2 Schematic of the experimental setup. AF: absorption filter to cut the laser with a wavelength of 1064 nm. BS: beam splitter. CD: condenser. DM1 and DM2: dichroic mirrors. DVI: digital visual interface. HW: half-wave plate to adjust the direction of linear polarization to that required by the liquid-crystal-on-silicon spatial-light-modulator (LCOS-SLM). L1 and L2: plano-convex lenses to obtain desired beam diameters. OL: objective lens. PC: personal computer. S1 and S2: shutters. TLS: transmission light source. The inlet and outlet reservoirs are used to control the channel flow rate. The photograph of the microfluidic device on the microscope stage is presented in Fig. S1.† |

d (μm) | Fluorescent particle (Molecular Probes, Eugene, USA) | Non-fluorescent particle (Merck, Darmstadt, Germany) | ||
---|---|---|---|---|

Product name | Concentration | Product name | Concentration | |

0.2 | F8811 | 2.0 × 10^{−4} wt% |
K1-020 | 2.0 × 10^{−2} wt% |

0.5 | F8813 | 2.0 × 10^{−4} wt% |
K1-050 | 1.5 × 10^{−2} wt% |

1.0 | F8823 | 2.0 × 10^{−4} wt% |
K100 | 1.0 × 10^{−2} wt% |

2.0 | F8827 | 2.0 × 10^{−4} wt% |
L200 | 1.0 × 10^{−2} wt% |

The wavelength of the laser used in this study is 1064 nm, because this wavelength is standard and frequently used in biosciences owing to the low absorption rate of the laser beam by water solutions. The optical vortex in our experiments is the Laguerre–Gaussian (LG) beam, which is one of the higher-order modes of a Gaussian beam. The LG beam is obtained by converting a continuous-wave (CW) Gaussian beam with a wavelength of 1064 nm (ASF1JE01, Fitel, Furukawa Electronics, Tokyo, Japan) using a liquid-crystal-on-silicon spatial-light-modulator (LCOS-SLM; X13138-03, Hamamatsu Photonics K.K., Hamamatsu, Japan), as shown in Fig. 2. More specifically, the PC sends a digital-visual-interface (DVI) signal to the LCOS-SLM to show a specific image on the LCOS-SLM display. If the Gaussian beam irradiates the display, the image produces a spatially inhomogeneous phase difference from the incident beam, and then the reflected beam becomes the optical vortex. The azimuthal mode m of the optical vortex may be changed using another specific image. Before the Gaussian beam impinges on the LCOS-SLM, the direction of linear polarization of the beam must be modulated using a half-wave (HW) plate such that it is consistent with the LCOS-SLM specification. The LG beam is combined with the excitation light and introduced in the microscope. The optical power P_{laser} is measured at the top of the OL using a power meter (3A-QUAD, Ophir Optronics, Jerusalem, Israel).

The microchannel pattern is schematically shown in Fig. 3(a). The channel height in the z direction is denoted by h, as shown in the side view of Fig. 3(b). The center of the irradiated optical vortex is at the center of the test section. The radial coordinate of the particle is expressed as (r_{p}, θ_{p}), as shown in Fig. 3. The azimuthal mode m of the optical vortex is defined so that dθ_{p}/dt > 0 when m > 0, where t is a time variable. The PDMS roof may collapse owing to the small microchannel height. To avoid collapse, square columns with an area of 100 × 100 μm^{2} are fabricated in the channel with a spatial interval of w = 100 μm in both the x and y directions.

The reason for using a mixture of 2.0 × 10^{−4} wt% dilute fluorescent particles and ≤2.0 × 10^{−2} wt% dense non-fluorescent particles is to facilitate particle tracking (see Movie S1†). We optically trap both types of particles in the circular path so that a single fluorescent particle is contained. In this way, only the fluorescent particle is recognized in the particle tracking algorithm. Using such a mixture of fluorescent and non-fluorescent particles is especially important when d_{p} is small and the number of particles in the circular path, N, is large. For the particle tracking from the obtained video images, we use an image analysis software ImageJ and its plugin Particle Track and Analysis (PTA).

We consider the two-dimensional motion of N particles with mass m_{p} and radius a_{p} = d_{p}/2 in the focal plane, which is the xy plane of the Cartesian coordinate system as shown in Fig. 3. The motion is induced by an optical vortex. The position and velocity of the nth particle (n = 1, …, N) are denoted by r_{n}(t) = (x_{n}(t), y_{n}(t)) and v_{n}(t) = (v_{x,n}(t), v_{y,n}(t)). We also introduce the polar coordinates (x,y) = (rcosθ,rsinθ). The unit vectors in the r and θ directions are e_{n,r} = (x_{n}/r, y_{n}/r) and e_{n,θ} = (−y_{n}/r, x_{n}/r), respectively. The circumferential component of the particle velocity is then written as v_{n,θ} = v_{n}·e_{n,θ}. The Reynolds number Re is a non-dimensional parameter in fluid mechanics that is the ratio of inertial to viscous effects. In this setting, we have Re ≡ ρ(d_{p}/2)v_{n,θ}/η = 5.8 × 10^{−7} for representative values v_{n,θ} = 1 μm s^{−1} and d_{p} = 1 μm, where the viscosity of water is η = 8.55 × 10^{−4} Pa s at 300 K. The value of 300 K is slightly larger than that in the experiment, 297.7 K. The relative difference is less than 0.8% and we do not think the results are affected by this difference. Because Re ≪ 1, we can safely neglect the inertial effect and assume Stokes drag acts on the particle. However, we need additional justification for applying Stokes drag to this problem because the sphere is orbiting around the origin with a radius r_{trap}; that is, the sphere experiences an effective shear flow and not a uniform flow in which the Stokes drag is justified. We discuss this point in Appendix B and describe the Stokes dynamics next.

The characteristic time scale τ of the motion is given by τ = m_{p}/(6πa_{p}η) = 68 ns for a_{p} = 0.5 μm, and is much smaller than the time scales of the simulation and experiment in the aqueous solution of this study. Therefore, we can neglect the inertial term of the equation of motion. The motion of the nth particle is hydrodynamically affected by that of the others, which have the index n′ ≠ n, and thus the over-damped equations are written as

(1a) |

(1b) |

(2a) |

(2b) |

r_{nn′}(t) = r_{n} − r_{n′}, | (2c) |

(3) |

The force f_{n} is the superposition of the optical scattering force f^{scat},^{51} the gradient force f^{grad},^{52} the short-range repulsive force between particles f^{rep},^{20} and the thermal fluctuation f^{rand}:

(4a) |

(4b) |

(4c) |

(4d) |

(4e) |

(5) |

P_{laser} = (n_{f}cε_{0}/4)(πw_{0}^{2})E_{0}^{2}, | (6) |

The above equations are used to investigate the effect of the number of particles N on the particle dynamics, especially the average angular frequency , where t_{0} = 1 s and t_{1} = 10 s determine the sampling interval in the simulation. The effect of the initial configuration disappears for t > 1 s. The time step of the simulation is taken to be Δt = 1 × 10^{−7} s, and Δt is small enough to avoid Δt-dependence of the results.

(7) |

We remark that the scattering force f^{scat}(r_{n}) with |r_{n}| = r^{(m)}_{trap} acts on the particles trapped in the orbit. Substituting in eqn (4b), we obtain |f^{scat}| = C_{0}A(m), where C_{0} is a constant independent of m and . We can check that A(m) is a monotonically increasing function with respect to m and ; that is, |f^{scat}| can be considered to be almost independent from m:

|f^{scat}| ≈ C_{0}A(1), (C_{0}: independent from m). | (8) |

Before going into detail, we remark on the θ_{p}-dependence of the dynamics observed in the experiment. The created LG beam intensity has slight inhomogeneity in the azimuthal direction, that is, the imperfection of the optical vortex seems to exist in our experiment. This can be seen from Movie S1:† the orbital speed for the case of N = 8 in the movie seems to be not constant during a single orbiting, i.e., the speed is higher when θ_{p} ≈ π/2 than when θ_{p} ≈ 3π/2, where the definition of θ_{p} is found in Fig. 3(a). However, the effect of θ_{p}-dependence can be reduced by considering the time-averaged angular frequency, as discussed in Appendix C with more detailed data analysis on the dynamics during the orbital motion. Therefore, we mainly discuss in this paper the time-averaged orbital speed.

Fig. 4 shows the snapshots of the orbital motion of nano- and microparticles with diameters of 0.2, 0.5, 1.0, and 2.0 μm, using an optical vortex with m = 1. Note that the length of the scale bar is different for d_{p} = 0.2 μm. As shown in Fig. 2, only the excitation light of wavelength 488 nm is used for d_{p} = 0.2 and 0.5 μm, while both the excitation and transmission lights are used for d_{p} = 1.0 and 2.0 μm. The absence of the transmission light for d_{p} = 0.2 and 0.5 μm is to increase the signal-to-noise ratio of the fluorescence intensity of the tracked particle, where the fluorescent particle can only be detected. We see that the particle with d_{p} = 0.2 μm does about half a revolution in 0.40 s, while the particles with d_{p} = 0.5 and 1.0 μm do about one revolution in 0.40 s. The particle with d_{p} = 2.0 μm is the fastest and completes about one and a half revolutions in 0.20 s. In the ESI (Movie S2†), we show that the particle cannot undergo the orbital motion without the microscale confinement.

Next, we investigate the orbital characteristics more systematically. The particle tracking is carried out using a video with 50 fps. This frame rate is considered enough to analyze orbital motion with a frequency of O(1) Hz, which is typical of the experimental conditions of this study. To further validate the frame rate, we compare the orbital frequencies obtained with 50 fps and 1000 fps for d_{p} = 0.2 μm, where the thermal fluctuation is the most significant among the cases investigated here. Fig. 5 shows the radial and circumferential positions r_{p} and θ_{p} as a function of time for a laser power P_{laser} = 313 mW and azimuthal mode m = 1. The linear-fit curves agree very well for 50 fps and 1000 fps. The rotational frequency ω, which is the slope of the linear fits, is computed for both cases to compare them quantitatively. We obtain ω = 2.68 × 10^{−1} Hz for both 50 fps and 1000 fps, and the deviation is less than 0.4%. For the radial direction, the difference in variance σ_{r}^{2} of the resulting position distribution for 50 Hz and 1000 Hz is less than 4%. Therefore, we conclude that a frame rate of 50 fps is enough to analyze orbital motion with thermal fluctuation.

In the following, the azimuthal mode is set to be m = 1 for d_{p} = 0.2 μm and m = 1, 3, 5, and 7 for d_{p} ≥ 0.5 μm. It is difficult for our experimental apparatus to trap a particle with d_{p} = 0.2 μm and m ≥ 2 because of the strong thermal fluctuation and weak optical gradient force. Stronger thermal fluctuation for smaller d_{p} can be seen theoretically from eqn (4e), which leads to the situation that the contribution of the random force to the velocity, _{nn}f^{rand}, is proportional to a_{p}^{−1/2}. A weaker optical gradient force for a smaller d_{p} is theoretically predicted as discussed previously below eqn (8). Therefore, the results for d_{p} = 0.2 and d_{p} ≥ 0.5 μm are next presented separately.

(i) Case of d
_{
p
}
= 0.2 μm

Fig. 6 shows the result of particle tracking for d_{p} = 0.2 μm and P_{laser} = (a) 408, (b) 367, (c) 313, (d) 295, (e) 223, and (f) 148 mW. The left (red) axis is the distance r_{p} from the beam center and the right (blue) axis the circumferential coordinate θ_{p}. The particles undergo thermal fluctuation near an equilibrium position around r_{p} ≈ 1 μm. Note that we are considering a Brownian particle under an external optical force, and thus its motion is different from ordinary diffusion in a free solution, where the diffusion coefficient D can be described as D = σ_{r}^{2}/(2t). The data for P_{laser} = 223 and 148 mW in Fig. 6(e) and (f), respectively, terminate at t ≈ 18 s and 5 s because the trapped particles escape at these times owing to insufficient optical power to overcome the thermal fluctuation. We see from the figure that r_{p} is fluctuating but constant on average in time, and θ_{p} linearly increases with time as long as the particle is trapped. Therefore, we treat r_{p} as the radius r_{trap} of the orbit of trapped particles. The details of the radial force balance will be discussed in the next paragraph. We conclude that the nanoparticle with the diameter of 0.2 μm, which is much smaller than that previously reported,^{4–12} is driven to stable orbital motion by the optical vortex. The angular frequency for P_{laser} = 408 mW is ω = 3.31 rad s^{−1}. This success of nanoparticle optical trapping and orbiting is due to the hydrodynamic inter-particle interaction, as will be described later with the simulation results. Fig. 6(g) shows the circumferential force f_{θ}, obtained using eqn (3), acting on the particles for various laser powers. If the optical force is sufficiently larger than the magnitude of the thermal fluctuation, f_{θ} is expected to be proportional to P_{laser}. Although Fig. 6 does not clearly show a linear relationship between f_{θ} and P_{laser} owing to the strong thermal fluctuation, it does show that f_{θ} increases with P_{laser}. One may think a larger P_{laser} would lead to the data with a higher signal-to-noise-ratio. However, the use of larger P_{laser} than those presented in this paper may break the microfluidic channel, possibly because of the temperature increase of the PDMS block, and thus, is avoided.

Fig. 6 Time development of the orbital radius r_{p} and circumferential coordinate θ_{p} for a nanoparticle with diameter d_{p} = 0.2 μm with laser powers of (a) 408, (b) 367, (c) 313, (d) 295, (e) 223, and (f) 148 mW, where the azimuthal mode is set to be m = 1. (g) Circumferential force f_{θ} obtained using eqn (3) and the data of panels (a)–(f) for those laser powers. The value of v_{θ} is also given in panel (g) as a reference. |

We estimate the trapping stiffness K as follows. If we use a typical parameter set in the experiment such as m = 1, d_{p} = 0.2 μm, P_{laser} = 313 mW, and r_{trap} = 1.07 μm, the trapping stiffness K is obtained both experimentally and theoretically as O(10^{−7}) N m^{−1} with a single significant digit. Here, K in the experiment is obtained using the law of energy equipartition: Kσ_{r}^{2}/2 = k_{B}T/2, where σ_{r}^{2} is the variance of the radial position distribution around equilibrium. However, theory yields about triple this value of K, maybe owing to the thermal fluctuation. The thermal fluctuation gives the particles the chance to escape the trap, which softens the effect of the trapping potential. The experimental and theoretical values of the trapping stiffness estimated using the above simple approximation agree well. To validate the treatment of r_{p} ≈ r_{trap}, r_{p} = r_{trap} + Δr_{trap} is considered in a mechanically strict manner, where Δr_{trap} is a displacement that should satisfy the equation |f^{grad}(|r_{n}| = r_{trap} + Δr_{trap})| = f^{cent}(= m_{p}r_{p}ω^{2}) and the force is the origin of realizing the orbital motion. Using experimentally obtained values of f^{cent} = 5.12 × 10^{−23} N, Δr_{trap} must be O(10^{−16}) m for the present order of K. Therefore, Δr_{trap} can be negligible and is not detectable in the present measuring system mainly due to the spatial resolution of image acquisition and thermal fluctuation. Although the theory may have limitations due to the complexity of phenomena, the detailed mechanics can be clarified. It is also confirmed that the trapping effect represented by K certainly governs the stable orbital motion of nanoparticles.

Furthermore, to quantify the effect of thermal fluctuation, we use the experimental displacement distribution to estimate the diffusion coefficient in the radial and circumferential directions for 50 fps and 1000 fps with P_{laser} = 313 mW. If we assume Wiener diffusion, the displacement forms a Gaussian distribution, and therefore the diffusion coefficient is evaluated by dividing the ensemble average of the mean square displacement by the sampling interval. The mean square displacement is equivalent to the variance of the displacement distribution. We obtain a radial diffusion coefficient of D_{r} = 1.05 × 10^{−12} m^{2} s^{−1} for 50 fps and D_{r} = 3.51 × 10^{−12} m^{2} s^{−1} for 1000 fps. One may think that the diffusion coefficient in the radial direction is affected by the presence of the optical force. This point will be discussed later in this paragraph. Using the same procedure, we obtain a circumferential diffusion coefficient of D_{θ} = 1.90 × 10^{−12} m^{2} s^{−1} for 50 fps and D_{θ} = 4.00 × 10^{−12} m^{2} s^{−1} for 1000 fps. The reason why the diffusion coefficients are underestimated for 50 fps is as follows. On the basis of the overdamped Langevin equation,^{53} the characteristic time of a phenomenon is τ_{0} = ξ/K ≈ 10 ms, where K = 2.42 × 10^{−7} N m^{−1} is the trapping stiffness theoretically obtained as described above. Note that τ_{0} is the time in which a nanoparticle is accelerated by the optical force. In other words, the particle reflected by an optical potential barrier does not collide with the other barrier at least during τ_{0} in the case of overdamping. The time resolution for 1000 fps is 1 ms (<τ_{0}), and thus the nanoparticle may diffuse freely during the time interval of 1 ms if the optical force is much weaker than the random force. Therefore, it is reasonable to estimate the diffusion coefficient using the overdamped regime in free solutions for the case of 1000 fps. This is the reason why we can evaluate D_{r} even under the confining optical force in the radial direction. However, the time resolution for 50 fps is 20 ms (>τ_{0}), and the diffusion coefficient is possibly affected by the optical force. Thus, we suspect that the high frame rate of 1000 fps is better for obtaining a correct diffusion coefficient upon further consideration based on molecular fluid dynamics.^{54} In fact, it was reported that the diffusion coefficient obtained from the displacement distribution under an external force was underestimated for data acquired with a large frame interval,^{55} where the external force acting on a particle was caused by the confining particle–wall interaction of two parallel walls. The values of D_{r} and D_{θ} for 1000 fps slightly overestimate the diffusion coefficient obtained via the Stokes–Einstein relationship D = k_{B}T/(6πa_{p}η) = 2.59 × 10^{−12} m^{2} s^{−1}, which may be because the other particles in the orbit enhance the self-diffusion.^{56} However, the results show that the particle diffusivity in these experimental conditions is similar to that in bulk solutions. We conclude that the diffusion coefficient may be roughly estimated from the displacement distribution even when the Brownian particle is under an optical external force and has motion characteristics different from those of ordinary diffusion.

(ii) Case of d
_{
p
}
≥ 0.5 μm

For d_{p} ≥ 0.5 μm, f_{θ} clearly correlates with P_{laser} linearly because the orbital motion under the optical force is stronger than the thermal fluctuation. Fig. 7 shows the results for the larger diameters 0.5, 1.0, and 2.0 μm for the azimuthal modes (a) m = 1, (b) m = 3, (c) m = 5, and (d) m = 7. The number of particles in the orbit, N, is also indicated in the figure for d_{p} = 1.0 and 2.0 μm. We have chosen N so that the line density ρ_{L} along the orbit has similar values for different modes: that is, ρ_{L} ≡ 2πr_{trap}/d_{p} ≈ 0.64 ± 0.03. In these figures, the circumferential force f_{θ} obtained using eqn (3) is plotted for various laser powers. First, we find that f_{θ} linearly correlates with P_{laser} for all m, as expected from eqn (4b) and (6). As d_{p} increases, f_{θ} also increases owing to the larger scattering cross section. In contrast, the figure does not show a clear dependence of f_{θ} on the mode m. Therefore, we rearrange the data in Fig. 7 for each d_{p} in Fig. 8, which shows f_{θ} for (a) d_{p} = 0.5, (b) 1.0, and (c) 2.0 μm for various azimuthal modes. Fig. 8 shows no clear qualitative m-dependence, and thus the linear-fit lines are drawn using the data for all the m cases. More specifically, in the case of d_{p} = 0.5 μm (panel (a)), f_{θ} tends to be larger for m = 1 but f_{θ} takes similar values for m = 3, 5, and 7; in the case of d_{p} = 1.0 μm (panel (b)), f_{θ} tends to be smaller for m = 1 but f_{θ} takes similar values for m = 3, 5, and 7; in the case of d_{p} = 2.0 μm (panel (c)), f_{θ} takes similar values for m = 1, 3, 5, and 7. The absence of clear m-dependence is due to the fact that |f^{scat}| is almost independent of m in eqn (8). However, one may think that the experiment shown in Fig. 8 is not well controlled, because m = 1 has the highest f_{θ} value for d_{p} = 0.5 μm (panel (a)) and the lowest value for d_{p} = 1.0 μm (panel(b)). This quantitative inconsistency may be due to the fact that f_{θ} is a function of N. For d_{p} = 1.0 μm, we will show more quantitatively that the lower f_{θ} for m = 1 than for m = 3, 5, and 7 in Fig. 8(b) is due to the N-dependence of f_{θ} by comparing the experimental results with simulation.

Fig. 9 Relationship between the radius change rate r_{ratio} in eqn (7) and the azimuthal mode m for d_{p} = 0.5, 1.0, and 2.0 μm. The solid curve represents eqn (7), which is obtained with the Rayleigh approximation. |

We carry out a numerical simulation based on our model in eqn (1b)–(6) to reveal the origin of the N-dependence of angular frequency observed in the experiments. The parameters in the simulation are similar to those in the experiments: P_{laser} is set to be 500 mW, and w_{0} is obtained from the relationship with r_{trap} = 2.1 μm, which is obtained experimentally with m = 1 and the OL with a magnification of 20×. In preliminary computations, we find that eqn (4b) overestimates the experimental values of ω by a factor of approximately 10. This overestimation by the simulation may be due to the overestimation of the scattering force and/or the increase of viscosity η in the experiment, which arises because of the wall effect and/or the addition of surfactant. Therefore, we multiply the scattering force by S_{c} = 0.1 to obtain values of ω that are comparable to the experimental values in Fig. 10. This inconsistency will be closely investigated in future work by including the effects of the wall and surfactant in the model. One may think that S_{c} should also multiply the gradient force in eqn (4c). However, doing so results in the situation where some particles cannot be kept in the orbit when two or more of them are close to each other. This behavior is not observed in our experiments, and thus multiplying the gradient force by S_{c} is even qualitatively inconsistent with observation. Therefore, we only multiply the scattering force by S_{c}. The value S_{c} = 0.1 is used throughout the paper because it yields moderate agreement between theory and experiment for different particle diameters and azimuthal modes.

Let us describe the result for d_{p} = 1 μm. Fig. 11(a) shows the simulation results for ω, where m = 1, 2, 3, 5, and 7. First, we focus on the profiles for m = 1 and 2, which correspond to the experimental results in Fig. 10(a) and (b), respectively. Fig. 11(b) shows the comparison between the simulation and the experiment. Because we introduced the fitting parameter S_{c} = 0.1, the experimental values of ω agree well with the simulated values of ≈10–30 rad s^{−1}. The remarkable agreement to be noted is that both the simulation and the experiment show the increase of ω for increasing N up to a critical value N ≤ N_{c}. Moreover, ω shows a different trend for N > N_{c} in both the simulation and the experiment: the rate of increase becomes low. Therefore, the N-dependency observed in the experiment in Fig. 10 is well explained qualitatively by our simulation model. However, for more a quantitative comparison, we remark that the effects of the imperfection of the LG beam or of the presence of particles on the electromagnetic field should be carefully taken into account. Because these are extremely difficult problems and complicate the discussion, we leave them for the constitutive investigation in future research and focus mainly on the effect of N in this paper. Given a qualitative agreement between the simulation and experiments, we conclude that the hydrodynamic inter-particle interaction, which becomes strong as N increases, significantly affects the orbital speed. Fig. 11(a) shows that the effect of this hydrodynamic interaction is weaker for higher m. We attribute this to the increase of for higher m. More precisely, the distance |r_{nn′}| between particles in eqn (2) increases when r_{trap} increases, leading to lower |f^{hydr}_{n}|.

Fig. 11 (a) Simulation results for ω as a function of the number of particles N for m = 1, 2, 3, 5, and 7 with d_{p} = 1.0 μm and P_{laser} = 500 mW. The orbital frequency ω increases as N increases owing to the hydrodynamic inter-particle interaction. (b) Comparison between the simulation results for m = 1 and 2 in panel (a) and the experimental results in Fig. 10. (c) Simulation results for ω for m = 1 with d_{p} = 0.2 μm, including the results without thermal fluctuation. |

Next, we compare the resulting f_{θ} values for m = 1, 3, 5, and 7 between the simulation in Fig. 11(a) and the experiment in Fig. 8(b). The goal of this comparison is to understand the m-dependence of f_{θ} in Fig. 8(b). The results are summarized in Table 2, where f_{θ} is normalized by dividing its values for m = 3, 5, and 7 by that for m = 1. The qualitative behavior is consistent between theory and experiment. Therefore, the subtle m-dependence of f_{θ} in Fig. 8(b) is due to the different values of N.

m = 1 | m = 3 | m = 5 | m = 7 | |
---|---|---|---|---|

N = 8 | N = 12 | N = 15 | N = 17 | |

Experiment (Fig. 8(b)) | ||||

ω (rad s^{−1}) |
16.9 | 14.9 | 14.2 | 10.9 |

f
_{
θ
} (fN) |
275 | 385 | 450 | 385 |

Normalized f_{θ} (—) |
1.00 | 1.40 | 1.63 | 1.40 |

Simulation (Fig. 11(a)) | ||||

ω (rad s^{−1}) |
18.0 | 13.2 | 11.1 | 9.53 |

f
_{
θ
} (fN) |
291 | 367 | 399 | 406 |

Normalized f_{θ} (—) |
1.00 | 1.26 | 1.37 | 1.39 |

Finally, Fig. 11(c) presents the simulation result for d_{p} = 0.2 μm. In the experiment, the orbital motion of these particles is the least stable owing to the weak optical force. Recall that the scattering cross sections are σ^{2} = 8.53 × 10^{−1} μm^{2} for d_{p} = 1 μm and σ^{2} = 4.31 × 10^{−4} μm^{2} for d_{p} = 0.2 μm. The weakness of the scattering force may be why there are no previous studies on dielectric particles with a diameter of 0.2 μm or less. In our experiments, we clearly observe the orbital motion of dielectric particles with d_{p} = 0.2 μm using a microfluidic channel with no biased background fluid flow and a mixture of fluorescent and non-fluorescent particles, as shown in Fig. 4(a1)–(a3) and 6. The corresponding simulation is carried out to show that the hydrodynamic effect on ω is also significant for d_{p} = 0.2 μm. The beam waist w_{0} is obtained from the relationship with an experimental value of r_{trap} = 1.0 μm. Fig. 11(c) shows the relationship between ω and N. The effect of thermal fluctuation is significant for d_{p} = 0.2 μm, and thus ω has larger stochastic fluctuation. Ensemble averages over 10 runs are taken to reduce stochastic noise, and the error bars in Fig. 11(c) show the standard deviation. The results without thermal fluctuation are also plotted for reference in Fig. 11(c), and a little quantitative difference between the plots with and without thermal fluctuation is observed. We find that single-particle orbital motion, N = 1, is difficult to observe because ω is overwhelmed by the thermal fluctuation. As N increases, ω also increases and eventually overcomes the fluctuation, leading to ω ≈ 1.5 rad s^{−1} at N = N^{theory}_{c}, which means about 0.25 Hz. The experimental value of ω is ω = 3.31 rad s^{−1}, which is on the same order as the simulation value. Therefore, we conclude that the hydrodynamic interaction between particles increases the orbital speed even for d_{p} = 0.2 μm. This is the key to inducing the orbital motion of small dielectric particles.

The significance of the hydrodynamic effect is explained theoretically as follows. Considering N particles as a single aggregate, as shown in Fig. 12(a) for N = 4, the net circumferential component of optical forces acting on the aggregate is proportional to N. However, the net drag force acting on the aggregate can be analytically obtained as C_{N}f^{drag} if we approximate the aggregate as a line moving with speed v, as shown in Fig. 12(b).^{57} We use f^{drag} = 6πηav as the Stokes drag. The analytical form of C_{N} depends on N and the distance between particles, and is given in the literature for N up to 4.^{57} More specifically, C_{N} is in the range 1 ≤ C_{N} ≤ N, as shown in Fig. 12(b); that is, the net drag force C_{N}f^{drag} is less than the superposition of the drag forces acting on N single particles, Nf^{drag}. Therefore, assuming the linear aggregate in Fig. 12(b), we obtain the analytical form of v_{θ} as v^{(anal)}_{θ} = Nf^{scat}_{n} (C_{N}6πηa)^{−1}. The values of v^{(anal)}_{θ} for = 2a are presented in Fig. 12(c). We carry out the simulation without thermal fluctuation to compare the simulation results with these analytical values. We use a harmonic potential to fix the distances between the particles at ≈ 2a as in the analytical result. Fig. 12(c) also presents the values of the simulated speed v^{(sim)}_{θ}, under the condition /2a < 1.05. We find that v^{(anal)}_{θ} and v^{(sim)}_{θ} agree well, as do their rates of increase with respect to N. Therefore, even though the net optical and drag forces acting on the aggregate both increase with N, the rate of increase for the net optical force overtakes that of the net drag force. The above estimate shows that the hydrodynamic inter-particle interaction is significant in the orbital motion of nanoparticles.

Fig. 12 Comparison between simulation and analytical results. (a) An N-particle system under the optical vortex is approximated as a single aggregate. The cases up to N = 4 are illustrated. (b) Schematic of a linear aggregate.^{57} Because d_{p} = 0.2 μm is smaller than the orbital radius r_{trap} = 1.07 μm, we assume that the aggregate in (a) is approximately linear, as shown in (b). The drag force acting on the linear aggregate is a modified Stokes drag C_{N}(6πηav), where C_{N} is a numeric constant depending on N whose analytical form is given in ref. 57. The analytical form of the velocity v of the linear aggregate is obtained as v = Nf(6πηaC_{N}), where f is an external force acting on each particle. (c) Analytical result v_{θ}^{(anal)} approximated by v = Nf/(6πηaC_{N}) in panel (b) and the simulation result v_{θ}^{(sim)} of panel (a) for N = 1,…4, d_{p} = 0.2 μm, P_{laser} = 313 mW, and m = 1. The rate of increase, dv_{θ}/dN, obtained with the second-order finite-difference scheme is also given. |

The microchannel has contraction parts with a length scale of 3 μm. These contraction parts are prepared to increase the hydraulic resistance R_{h} over the channel and to increase the flow control resolution. Note that the unexpected flow speed u in the x-direction in the channel is proportional to ΔP/R_{h}, where ΔP is the pressure difference. Therefore, higher R_{h} is preferred to achieve lower u. In our experiments, u should be as low as possible to eliminate the effect of the mean flow on the orbiting motion of particles.

The mold has a micrometer-scale pattern with an aspect ratio approximately equal to unity. It is difficult to demold a cured PDMS block with such a fine mold structure because the PDMS sticks to the fine mold pattern and breaks during demolding. To avoid the sticking of the PDMS to the fine pattern, the mold is covered with a self-assembled monolayer (SAM) of 1H,1H,2H,2H-perfluorodecyltriethoxysilane. The process of SAM coating is as follows. First, the mold is carefully cleaned using ultra-sonication in acetone and 2-propanol for 3 minutes. Then, the mold is baked at 120 °C to remove the residual organic solvents. The mold is exposed to ultraviolet light (PL16, Sen Light Corp., Osaka, Japan) for 10 minutes for further cleaning. The cleaned mold is put in a perfluoroalkoxy alkane (PFA) container. In this container we also put a smaller PFA container including a few drops of 1H,1H,2H,2H-perfluorodecyltriethoxysilane solution. After tightly closing the larger PFA container, we put it in an oven (Do-300, RB Technology, Fukuoka, Japan) and heat it at 120 °C for 3 hours. The evaporated solution forms the SAM layer on the mold. The presence of the SAM layer, which is highly hydrophobic, is verified using the droplet method. The contact angle of a water droplet on the mold is improved from 66° before treatment to 106° after treatment, and the formation of the SAM layer is confirmed. Now, we can safely demold the PDMS without breaking the fine pattern.

The θ_{p}-dependence may be caused by the characteristics of our optical system, but also may be due to the effects of the presence of charged particles and induced solvent flow on the electromagnetic field. These are extremely difficult problems and beyond the scope of the present paper, we leave them for constitutive investigation in future research. Nonetheless, we carry out a similar experiment using a different objective lens with a higher NA value (NA = 1.4, 40× magnification, oil immersion, UPLXAPO40XO, Olympus) to see whether the θ_{p}-dependence can be reduced in a different optical setup. The result is given in Fig. 14, where the panel (a) shows the snapshots of the orbital motion using the parameters P_{laser} = 93 mW, m = 7, and d_{p} = 1 μm with N = 1. It is seen that the single particle orbiting is observed, and Fig. 14(b) presents the θ_{p}-dependence of the orbital angular frequency ω for Fig. 14(a). The standard deviation σ_{dev} of the locally averaged data from the mean value is σ_{dev} = 8.26 rad s^{−1} and is smaller than those in Fig. 13. However, the complete elimination of the θ_{p}-dependence is difficult in the present experimental setup. Therefore, we discuss the averaged quantities (blue-bold solid lines) in the main text.

- L. Allen, M. Beijersbergen, R. Spreeuw and J. Woerdman, Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes, Phys. Rev. A, 1992, 45, 8185 CrossRef PubMed.
- S. Barnett and L. Allen, Orbital angular momentum and nonparaxial light beams, Opt. Commun., 1994, 110, 670–678 CrossRef.
- J. Courtial, K. Dholakia, L. Allen and M. Padgett, Gaussian beams with very high orbital angular momentum, Opt. Commun., 1997, 144, 210–213 CrossRef CAS.
- J. Curtis and D. Grier, Structure of optical vortices, Phys. Rev. Lett., 2003, 90, 133901 CrossRef PubMed.
- S. Tao, X. Yuan, J. Lin, X. Peng and H. Niu, Fractional optical vortex beam induced rotation of particles, Opt. Express, 2005, 13, 7726–7731 CrossRef CAS PubMed.
- Y. Roichman, D. G. Grier and G. Zaslavsky, Anomalous collective dynamics in optically driven colloidal rings, Phys. Rev. E, 2007, 75, 020401 CrossRef PubMed.
- Y. Sokolov, D. Frydel, D. G. Grier, H. Diamant and Y. Roichman, Hydrodynamic pair attractions between driven colloidal particles, Phys. Rev. Lett., 2011, 107, 158302 CrossRef.
- Y. Sassa, S. Shibata, Y. Iwashita and Y. Kimura, Hydrodynamically induced rhythmic motion of optically driven colloidal particles on a ring, Phys. Rev. E, 2012, 85, 061402 CrossRef PubMed.
- H. Nagar and Y. Roichman, Collective excitations of hydrodynamically coupled driven colloidal particles, Phys. Rev. E, 2014, 90, 042302 CrossRef PubMed.
- S. Okubo, S. Shibata, Y. S. Kawamura, M. Ichikawa and Y. Kimura, Dynamic clustering of driven colloidal particles on a circular path, Phys. Rev. E, 2015, 92, 032303 CrossRef PubMed.
- Y. Kimura, Hydrodynamically induced collective motion of optically driven colloidal particles on a circular path, J. Phys. Soc. Jpn., 2017, 86, 101003 CrossRef.
- K. Saito, S. Okubo and Y. Kimura, Change in collective motion of colloidal particles driven by an optical vortex with driving force and spatial confinement, Soft Matter, 2018, 14, 6037–6042 RSC.
- H. He, M. Friese, N. Heckenberg and H. Rubinsztein-Dunlop, Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity, Phys. Rev. Lett., 1995, 75, 826 CrossRef CAS PubMed.
- A. T. O'neil and M. J. Padgett, Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner, Opt. Commun., 2000, 185, 139–143 CrossRef.
- A. O'neil, I. MacVicar, L. Allen and M. Padgett, Intrinsic and extrinsic nature of the orbital angular momentum of a light beam, Phys. Rev. Lett., 2002, 88, 053601 CrossRef PubMed.
- M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss and K. Dholakia, Optical vortex trap for resonant confinement of metal nanoparticles, Opt. Express, 2008, 16, 4991–4999 CrossRef PubMed.
- P. Figliozzi, N. Sule, Z. Yan, Y. Bao, S. Burov, S. K. Gray, S. A. Rice, S. Vaikuntanathan and N. F. Scherer, Driven optical matter: Dynamics of electrodynamically coupled nanoparticles in an optical ring vortex, Phys. Rev. E, 2017, 95, 022604 CrossRef PubMed.
- K. Gahagan and G. Swartzlander, Optical vortex trapping of particles, Opt. Lett., 1996, 21, 827–829 CrossRef CAS PubMed.
- V. Garcés-Chávez, K. Volke-Sepulveda, S. Chávez-Cerda, W. Sibbett and K. Dholakia, Transfer of orbital angular momentum to an optically trapped low-index particle, Phys. Rev. A, 2002, 66, 063402 CrossRef.
- M. Reichert and H. Stark, Circling particles and drafting in optical vortices, J. Phys.: Condens. Matter, 2004, 16, S4085 CrossRef CAS.
- J. Ng, Z. Lin and C. Chan, Theory of optical trapping by an optical vortex beam, Phys. Rev. Lett., 2010, 104, 103601 CrossRef PubMed.
- M. Tamura, T. Omatsu, S. Tokonami and T. Iida, Interparticle-interaction-mediated anomalous acceleration of nanoparticles under light-field with coupled orbital and spin angular momentum, Nano Lett., 2019, 19, 4873–4878 CrossRef CAS PubMed.
- R. Nagura, T. Tsujimura, T. Tsuji, K. Doi and S. Kawano, Coarse-grained particle dynamics along helical orbit by an optical vortex irradiated in photocurable resins, OSA Continuum, 2019, 2, 400–415 CrossRef CAS.
- J. A. Briggs, K. Grünewald, B. Glass, F. Förster, H.-G. Kräusslich and S. D. Fuller, The mechanism of HIV-1 core assembly: insights from three-dimensional reconstructions of authentic virions, Structure, 2006, 14, 15–20 CrossRef CAS PubMed.
- N. M. Bouvier and P. Palese, The biology of influenza viruses, Vaccine, 2008, 26, D49–D53 CrossRef CAS PubMed.
- K. Toyoda, K. Miyamoto, N. Aoki, R. Morita and T. Omatsu, Using optical vortex to control the chirality of twisted metal nanostructures, Nano Lett., 2012, 12, 3645–3649 CrossRef CAS PubMed.
- F. Takahashi, K. Miyamoto, H. Hidai, K. Yamane, R. Morita and T. Omatsu, Picosecond optical vortex pulse illumination forms a monocrystalline silicon needle, Sci. Rep., 2016, 6, 21738 CrossRef CAS PubMed.
- M. Watabe, G. Juman, K. Miyamoto and T. Omatsu, Light induced conch-shaped relief in an azo-polymer film, Sci. Rep., 2014, 4, 4281 CrossRef PubMed.
- D. Barada, G. Juman, I. Yoshida, K. Miyamoto, S. Kawata, S. Ohno and T. Omatsu, Constructive spin-orbital angular momentum coupling can twist materials to create spiral structures in optical vortex illumination, Appl. Phys. Lett., 2016, 108, 051108 CrossRef.
- J. Lee, Y. Arita, S. Toyoshima, K. Miyamoto, P. Panagiotopoulos, E. M. Wright, K. Dholakia and T. Omatsu, Photopolymerization with light fields possessing orbital angular momentum: generation of helical microfibers, ACS Photonics, 2018, 5, 4156–4163 CrossRef CAS.
- M. Chen, M. Mazilu, Y. Arita, E. M. Wright and K. Dholakia, Dynamics of microparticles trapped in a perfect vortex beam, Opt. Lett., 2013, 38, 4919–4922 CrossRef PubMed.
- M. Chen, M. Mazilu, Y. Arita, E. M. Wright and K. Dholakia, Creating and probing of a perfect vortex in situ with an optically trapped particle, Opt. Rev., 2015, 22, 162–165 CrossRef.
- E. K. Sackmann, A. L. Fulton and D. J. Beebe, The present and future role of microfluidics in biomedical research, Nature, 2014, 507, 181 CrossRef CAS PubMed.
- U. F. Keyser, Controlling molecular transport through nanopores, J. R. Soc., Interface, 2011, 8, 1369–1378 CrossRef CAS PubMed.
- D. G. Grier, A revolution in optical manipulation, Nature, 2003, 424, 810 CrossRef CAS PubMed.
- K. C. Neuman and S. M. Block, Optical trapping, Rev. Sci. Instrum., 2004, 75, 2787–2809 CrossRef CAS.
- T. Sugiyama, K.-i. Yuyama and H. Masuhara, Laser trapping chemistry: from polymer assembly to amino acid crystallization, Acc. Chem. Res., 2012, 45, 1946–1954 CrossRef CAS PubMed.
- O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe and A. C. Ferrari, Optical trapping and manipulation of nanostructures, Nat. Nanotechnol., 2013, 8, 807 CrossRef PubMed.
- A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. Tong and M. Kall, Laser trap ping of colloidal metal nanoparticles, ACS Nano, 2015, 9, 3453–3469 CrossRef CAS PubMed.
- F. Nito, T. Shiozaki, R. Nagura, T. Tsuji, K. Doi, C. Hosokawa and S. Kawano, Quantitative evaluation of optical forces by single particle tracking in slit-like microfluidic channels, J. Phys. Chem. C, 2018, 122, 17963–17975 CrossRef CAS.
- T. Tsuji, Y. Matsumoto and S. Kawano, Flow with nanoparticle clustering controlled by optical forces in quartz glass nanoslits, Microfluid. Nanofluid., 2019, 23, 126 CrossRef.
- A. Ashkin, J. M. Dziedzic, J. Bjorkholm and S. Chu, Observation of a single-beam gradient force optical trap for dielectric particles, Opt. Lett., 1986, 11, 288–290 CrossRef CAS PubMed.
- S. Tanaka, M. Tsutsui, H. Theodore, H. Yuhui, A. Arima, T. Tsuji, K. Doi, S. Kawano, M. Taniguchi and T. Kawai, Tailoring particle translocation via dielectrophoresis in pore channels, Sci. Rep., 2016, 6, 31670 CrossRef CAS PubMed.
- T. Tsuji, K. Kozai, H. Ishino and S. Kawano, Direct observations of thermophoresis in microfluidic systems, Micro Nano Lett., 2017, 12, 520–525 CrossRef CAS.
- T. Tsuji, S. Saita and S. Kawano, Dynamic pattern formation of microparticles in a uniform flow by an on-chip thermophoretic separation device, Phys. Rev. Appl., 2018, 9, 024035 CrossRef CAS.
- T. Tsuji, Y. Sasai and S. Kawano, Thermophoretic manipulation of micro-and nanoparticle flow through a sudden contraction in a microchannel with near-infrared laser irradiation, Phys. Rev. Appl., 2018, 10, 044005 CrossRef CAS.
- L. Lin, J. Zhang, X. Peng, Z. Wu, A. C. Coughlan, Z. Mao, M. A. Bevan and Y. Zheng, Opto-thermophoretic assembly of colloidal matter, Sci. Adv., 2017, 3, e1700458 CrossRef PubMed.
- K. Setoura, T. Tsuji, S. Ito, S. Kawano and H. Miyasaka, Opto-thermophoretic separation and trapping of plasmonic nanoparticles, Nanoscale, 2019, 11, 21093–21102 RSC.
- J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys., 1969, 50, 4831–4837 CrossRef CAS.
- H. Yamakawa, Transport properties of polymer chains in dilute solution: hydrodynamic interaction, J. Chem. Phys., 1970, 53, 436–443 CrossRef CAS.
- J. Zhang, S.-J. Huang, F.-Q. Zhu, W. Shao and M.-S. Chen, Dimensional properties of Laguerre–Gaussian vortex beams, Appl. Opt., 2017, 56, 3556–3561 CrossRef PubMed.
- Y. Harada and T. Asakura, Radiation forces on a dielectric sphere in the Rayleigh scattering regime, Opt. Commun., 1996, 124, 529–541 CrossRef CAS.
- W. Qian, K. Doi and S. Kawano, Effects of polymer length and salt concentration on the transport of ssDNA in nanofluidic channels, Biophys. J., 2017, 112, 838–849 CrossRef CAS.
- T. Li, S. Kheifets, D. Medellin and M. G. Raizen, Measurement of the instantaneous velocity of a Brownian particle, Science, 2010, 328, 1673–1675 CrossRef CAS PubMed.
- I. Hanasaki, Y. Kazoe and T. Kitamori, Time resolution effect on the apparent particle dynamics confined in a nanochannel evaluated by the single particle tracking subject to Brownian motion, Microfluid. Nanofluid., 2018, 22, 56 CrossRef.
- K. Zahn, J. M. Méndez-Alcaraz and G. Maret, Hydrodynamic interactions may enhance the self-diffusion of colloidal particles, Phys. Rev. Lett., 1997, 79, 175 CrossRef CAS.
- J. Happel and H. Brenner, Low Reynolds number hydrodynamics: with special applications to particulate media, Springer Science & Business Media, 2012, vol. 1 Search PubMed.
- D. A. Drew, The force on a small sphere in slow viscous flow, J. Fluid Mech., 1978, 88, 393–400 CrossRef.

## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/C9NR10591C |

‡ Present address: Graduate School of Informatics, Kyoto University, Kyoto 606–8501, Japan. |

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