Quantitative Evaluation of Optical Forces by Single Particle Tracking in

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C: Plasmonics; Optical, Magnetic, and Hybrid Materials

Quantitative Evaluation of Optical Forces by Single Particle Tracking in Slit-like Microfluidic Channels Fumika Nito, Tetsuya Shiozaki, Ryo Nagura, Tetsuro Tsuji, Kentaro Doi, Chie Hosokawa, and Satoyuki Kawano J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b02701 • Publication Date (Web): 25 Jun 2018 Downloaded from http://pubs.acs.org on June 30, 2018

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The Journal of Physical Chemistry

Quantitative Evaluation of Optical Forces by Single Particle Tracking in Slit-like Microfluidic Channels Fumika Nito,† Tetsuya Shiozaki,† Ryo Nagura,† Tetsuro Tsuji,† Kentaro Doi,∗,† Chie Hosokawa,‡,¶ and Satoyuki Kawano∗,† †Department of Mechanical Science and Bioengineering, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan ‡Biomedical Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Ikeda, Osaka 563-8577, Japan ¶Advanced Photonics and Biosensing Open Innovation Laboratory (PhotoBIO-OIL), National Institute of Advanced Industrial Science and Technology (AIST), Suita, Osaka 565-0871, Japan E-mail: [email protected]; [email protected]

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Abstract Optical trapping and manipulation techniques have attracted significant attention in various research fields. Optical forces divided into two terms, such as a scattering force and gradient one, work to push forward and attract objects, respectively. This is a typical property of optical forces. In particular, a tool known as optical tweezers can be created when a laser beam is converged at a focal point, causing strong forces to be generated so as to trap and manipulate small objects. In this study, we propose a novel method to build up cluster structures of polystyrene particles by using optical trapping techniques. Recording trajectories of single particles, the optical forces are quantitatively evaluated using particle tracking velocimetry. Herein, we treat various particle sizes whose diameters are ranging from 1 to 4 µm and expose them to a converged laser beam of 1064 nm wavelength. As a result, both experimental and theoretical results are in good agreement. The behavior of particles is understood in the framework of Ashkin’s ray optics. This finding clarifies optical force fields of microparticles distributed in a slit-like microfluidic channel and will be applicable for effectively forming ordered structures in liquids.

Keywords: Optical trapping, Geometrical optics, Langevin dynamics, Optical gradient force, Slit-like microfluidic channel

Introduction Optical technologies have been recognized as a powerful tool for fabrication, measurement, and manipulation of objects. These techniques have been widely available not only for laboratories but also for industrial and medical applications. The use of these techniques also extends across various research fields, including physics, chemistry, materials science, biology, and medicine. Some review articles have appeared which introduce the fundamentals and applications of laser trapping techniques. 1–4 At various scales, optical forces have important roles to play in manipulating objects ranging from ions to biological cells and drugs. 1–4 The 2


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theoretical framework of optical forces differs depending on the relationship between the wavelength of light and the size of the target objects. For the case that a particle size a is much smaller than a light wavelength λ, such that a ≪ λ, dielectric or charged particles are exposed to time-dependent electromagnetic fields. In such situations, the transport of dipoles and/or electrical charges is directly influenced by the electromagnetic fields. 5–7 It was found that nanoparticles could be assembled in optical force fields, which sheded light on molecular assemblies and soldification processes. 8,9 Furthermore, various methods have also been proposed to manipulate nanoobjects by using structured light, 1,10 optical resonance, 11,12 surface plasmon, 13,14 plasmonic-nanopores, 15–17 photonic-crystal slabs, 18 and so on. Laser trapping techniques, which utilize amplified optical forces, can expand the trapping size range. 13,14,19–21 In contrast, for the case of a ≫ λ, the phenomena of rays that are reflected and refracted at a particle surface is treated as momentum transfer. In this regime, optical forces are evaluated based on geometrical optics. 22 Since Ashkin 23,24 demonstrated optical tweezers for microparticles, many researchers have contributed to develop novel methods utilizing the optical manipulation of microobjects. 25–28 For instance, laser trapping techniques are applied to microfluidic devices, where micrometer-sized objects were applied to pumps or valves in the channels and were used for a switch to select a channel branch. 1,29 Recently, assembled structures of microparticles were formed at solid-liquid interfaces using polarized laser beams. 30 The optical scattering force was numerically analyzed at a solid-liquid interface by using ray optics theory. 31 Electrostatic repulsive forces between microparticles were analyzed by using optical trapping at heterogeneous liquid-liquid interfaces, 32,33 where optical forces were also numerically estimated. 34 Furthermore, attractive electrostatic forces between microparticles were also evaluated by using a Coulomb crystal in which ionic flows around microparticles were suggested to cause the attractive forces. 35 On the other hand, detailed procedures to evaluate optical forces in liquids have not yet been established. To do so, many-body problems have to be solved based on the optical

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theories, taking into account the behavior of the surrounding environment. Ashkin 23 was one of the pioneers who experimentally and theoretically demonstrated acceleration and trapping of microparticles. He also developed a theoretical model 24 to quantitatively evaluate the optical forces on single particles, in which the momentum transfer from an incident ray to objects was considered to be the origin of optical forces. This model is acceptable in the case that the particle size is much larger than the wavelength of light. The trapping process is characterized by reflective and refractive indices and the incident angle of rays. On the other hand, the nonequilibrium process of optical tweezers in liquids has actually included difficulties in revealing the nature of phenomena because the surrounding environment is also involved in the dynamics. Generally, irradiation by a laser beam causes temperature increases and thermal effects, such as heat conduction, convection, and thermophoresis, which cause disturbances so as to isolate optical forces. In this study, we carry out observations of single particle transport driven by optical forces in a slit-like microfluidic channel and analyze the behavior by using particle tracking velocimetry (PTV). 36,37 Confining the transport in the microchannel makes it easy to focus on the radial component of optical forces. The velocity and acceleration of single particles are expressed as a function of distance from a focal point. By using a slit-like microchannel, the effects of heating on the transport phenomena, convection, and thermophoresis are considerably suppressed. Consequently, the gradient force is accurately evaluated from the PTV results based on the equation of motion of particles in viscous liquids. Furthermore, the gradient force is quantitatively calculated for a particle whose size is larger than the wavelength of light according to Ashkin’s theory, 22–24 reproducing experimental conditions. Based on previous studies, 38–41 the Langevin dynamics simulation is performed to treat a many-body problem, where dielectric microparticles are exposed to a converged laser beam. As a result, the particle velocities are represented as a function of positions and the evaluation of the viscous drag force that is equilibrated with external forces clarifies the optical force fields. Finally, we find assembled microparticle structures formed at a solid surface in

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(a) Single Particle Tracking Optical axis r PDMS wall surface PSt particle

Cover glass Objective lens

Longer time observation (b) Assembled Structure Formtaion r

Objective lens Figure 1. Schematic illustrations of optically trapped microparticles in narrow spaces: (a) trapping process of single particles, where Brownian motions let a particle be occasionally irradiated with light. The particle exposed to light is forced to move toward a focal point and is finally trapped at the center. (b) In the observation over a longer period of time, assembled structures are obtained which are much larger than a focal spot of 1 µm.

the optical force fields, which are well explained by the computational models.

Experimental methods Figure 1 shows the experimental setup, including an experimental device and optical apparatus. In this experimental device, a microfluidic channel fabricated in a polydimethylsiloxane (PDMS) block is sealed with a cover slip. A silicone elastomer base and its curing agent R (Silgard⃝ 184, Dow Corning Corp., MI, USA) are mixed with a mass ratio of 10:1 and the

mixture is poured on a Si substrate molded into a slit-like microfluidic channel having a height of 7.1 ± 0.1 µm. A flow channel pattern is replicated in a PDMS block with heating at 343 K for more than 4 h. After that, 1 mm diameter inlet and outlet holes are placed at 5



the channel ends in the PDMS block. Next, both surfaces of the PDMS block are bonded with a cover slip by using a laboratory corona treater (BD20ACV, Electro-Technic Products Inc., IL, USA). These holes are used for the entrance of polystyrene particle dispersions. Herein, as listed in Table I, we prepare polystyrene spheres of 4 different diameters, ϕ =1.01 R (Estapor⃝ , Merck KGaA, Darmstadt, Germany), 2.02 ± 0.02 (Duke StandardsTM , Thermo R Fisher Scientific Inc., Tokyo, Japan), 2.93 (Estapor⃝ , Merck KGaA, Darmstadt, Germany),

and 4.00 ± 0.04 µm (Duke StandardsTM , Thermo Fisher Scientific Inc., Tokyo, Japan). Dispersions are prepared for the volume fractions of 1.0 × 10−1 and 1.0 × 10−2 vol% for ϕ = 1.01, 2.02 µm and ϕ = 2.93, 4.00 µm, respectively. The particle sizes range in diameter from 1 to 4 µm so that the optical gradient force can be evaluated, considering the framework of geometrical optics and its limitation. For a ϕ = 1.01 µm particle, the wavelength of near infrared light becomes comparable and the limitation of geometrical optics is suspected to appear. The size of ϕ = 4.00 µm is sufficiently large as to allow the gradient force to be solely evaluated, excluding the other effects. These dispersions are injected into the flow channel through silicone tubes connected to the inlet and outlet. In this study, polystyrene particles are used for the target because of the higher index of refraction, n2 = 1.60, compared with that of water, n1 = 1.33. That is, the effective index results in n = n2 /n1 = 1.20 > 1. It is important that the n should be greater than 1 to attract a particle to the optical axis by the gradient force. 24 As shown in Figure 2, a λ =1064 nm continuous wave laser beam is irradiated from a light source (Fitel ASF1JE01, Furukawa Electric Co., Ltd., Tokyo, Japan) whose maximum output power is 2 W. This wavelength is used to avoid the absorption spectrum of water. A laser beam is expanded with a custom-made collimator lens (Sigmakoki Co., Ltd., Tokyo, Japan) and is directed to samples via a custom-made dichroic mirror whose cut-off wavelength is 880 nm (Sigmakoki Co., Ltd., Tokyo, Japan). The irradiation of samples is turned on and off by using a mechanical shutter with a controller (SSH-25RA-W and SSH-C2B, Sigmakoki Co., Ltd., Tokyo, Japan). A laser beam is focused on a sample via an objective of 100× IR (LCPLN100×IR, Olympus Corp., Tokyo, Japan) whose numerical

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aperture (NA) is 0.85. Transmission images are observed under illumination of a halogen lamp with a condenser lens of NA 0.55, where a dichroic mirror and a heat absorbing filter (W3110, Sigmakoki Co., Ltd., Tokyo, Japan) are inserted to cut off the back-scattering light of the trapping laser beam. The optical system above is combined with an inverted microscope (IX-73, Olympus Corp., Tokyo, Japan) and images are recorded by using an sCMOS high-speed camera (Zyla 5.5, Andor Technology Ltd., Tokyo, Japan) with frame rates of 479 and 501 fps and an exposure time of 0.001 s. A laser beam passing through the objective along the z-axis is converged at the upper surface of a slit-like microchannel. The total recording time is approximately 10 s, sufficient for observing particle trapping processes, in which the shutter is opened at 0.5 s from the beginning. The shutter controller is synchronized with a trigger signal from the sCMOS camera modulated by a function generator (WF1973, NF Corp., Kanagawa, Japan). The intensity of the laser beam is suspected to decay during travel through the optical system, and thus the laser power is again measured in front of the objective lens with a laser power meter (3A-QUAD, Ophir Ltd., Saitama, Japan). In this study, the laser power is determined exiting the objective lens and set to 10 mW. Trajectories of single particles are recorded and time series of snapshots are analyzed by using the PTV method. The center of a particle is determined from the luminance data, where a single particle is focused in each frame and the distance from a focal point, velocity, and acceleration are evaluated from the differences between snapshots. First, we obtain the displacement of particles as a function of time. The transport velocities and accelerations are evaluated by numerical derivatives for the particle centers. Next, we make plots for the velocity and acceleration as a function of the radial position from a focal point. Considering Newton’s equation of motion, the superposition of inertial force and viscous drag is equilibrated with external forces, and therefore the optical forces are quantitatively evaluated by observing the transport of particles.

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Halogen lamp Experimental device (Side view) Condenser lens (NA 0.55) PSt dispersion Device 7 μm 30 mm Objective lens (NA 0.85, 100×IR)

Shutter Collimator lens Laser λ = 1064 nm

Dichroic mirror Heat absorbing filter

sCMOS camera

PC

Figure 2. Illustration and photograph of experimental setup. A continuous wave of 1064 nm wavelength is irradiated from a light source and is converged at the upper surface of a slit-like microfluidic channel made of PDMS. Another light source illuminates a sample focused with a condenser lens and the transmitted light refracted by particles is detected with a high-speed sCMOS camera connected to the inverted microscope. Velocities and accelerations of particles are evaluated from the trajectories by using the PTV method.

Theoretical methods and numerical solutions Optical forces based on Ashkin’s ray optics In an actual experimental system, a laser beam is sufficiently collimated and converged through an objective lens. Herein, we consider multiple rays that hit a sphere and then the net force on the particle is computed by integrating all incident rays on the surface. To avoid complexity, detailed procedures are described in Appendix A. Setting a light source power, single rays are irradiated from the source divide the power of the laser beam. When we consider N individual rays, the incident momentum per second is expressed as n1 Pˆ /(N c). According to Ashkin’s theory, 24 optical forces are evaluated by tracing individual rays based on eqs 17–19 in the Cartesian coordinate system, as shown in Appendix A. Herein, for manybody problems as discussed later, incident rays are assumed to sufficiently decay after hitting a particle. Therefore, multiple scattering effects are ignored for the first step of this study. 8


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Langevin dynamics simulation Using the equations above, the optical forces acting on single particles can be evaluated in three-dimensional space. The velocity v of a single particle transport in liquid can be expressed by a Langevin equation for over-damped situations as follows: 40,41

6πµav(t) = Fopt (r) + Fth (t),

(1)

where µ and a are the viscosity of the liquid and the radius of the particle, respectively. Fopt is the optical force expressed by eqs 17–19 shown in Appendix A. Here, it is assumed that the inertial term is immediately decayed and negligible due to the high viscosity against the external force. The validity of this assumption is verified in experiments later. Fth is the random force caused by thermal fluctuations of solvent molecules, which satisfies Einstein’s fluctuation-dissipation theorem, such that

⟨Fth,i (t)⟩ = 0, ⟨Fth,i (t)Fth,j (t′ )⟩ = 2ξkB T δij δ(t − t′ ),

(2) (3)

where {i, j} = {x, y, z}, ξ = 6πµa, kB is the Boltzmann constant, and T is temperature. Solving eq 1, we can follow the trajectory of a particle driven by optical forces. In the simulation, Fth is generated by using a random number generator of the Gaussian distribution. By integrating eq 1, the position vector r(t) is successively calculated as

r(t + ∆t) = r(t) + [Fopt (r) + Fth (t)]∆t.

(4)

Computational conditions For the quantitative evaluation of optical forces in comparison with experiments, a computational model is developed, where a laser beam is irradiated from the bottom surface of

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a rectangular box of 35.0 × 35.0 × 7.0 µm3 (width × length × height) and converged at the top surface. Considering the experimental conditions, a 22.58 µm diameter area of the bottom surface is exposed to a laser beam. The optical axis is fixed at the center of the xy cross-section. The incident rays are mapped on the grid points of 0.35 × 0.35 µm2 meshes in the irradiated area. Polystyrene particles are dispersed in pure water, and thus their indices of refraction are set to 1.60 and 1.33, respectively. The power of the laser beam is fixed at 10 mW, corresponding to the experimental condition in which the laser power is measured just as it leaves the objective. The particle radius a is varied, ranging from 1.00 to 4.00 µm, considering the experimental conditions. Firstly, the computed optical forces are mapped on the yz cross-section by fixing the position of a single particle. Next, Langevin dynamics simulations are performed for various initial positions, varying the radial positions ranging from 3.00 to 7.00 µm, and the height is set at 1.05a above the bottom surface. The time step of the Langevin dynamics simulation is set to 5.0×10−7 s and the computational time is taken sufficiently long for the particle to become steadily trapped at a focal point.

Results and Discussion Experimental observations In experiments, trajectories of single polystyrene particles were recorded to analyze the velocities and accelerations, which were induced by optical forces. Figure 3 shows typical experimental results of the radial position r of a particle as a function of time t for each particle diameter 1.01, 2.02, 2.93, and 4.00 µm. In each time expansion, it was found that a particle gradually translocated and was quickly attracted to a focal point. Figure 4 shows the radial velocities v as a function of t, where each data point resulted from the numerical time difference of r shown in Figure 3. In each case, sharp spikes of v were found when a particle quickly approached the focal point. Figure 5 also shows the acceleration α in the radial direction by the second time derivative of r, using data in Figure 3. Responses 10


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Figure 3. Typical experimental results of particle displacement in the radial direction r driven by optical forces for various diameter particles: (a) ϕ = 1.01, (b) 2.02, (b) 2.93, and (b) 4.00 µm.

were obtained when a particle approached the focal point, although the data plot became very noisy. Using these data, we analyzed v and α as a function of r. From this result, a viscous drag and inertial force which were equilibrated with external forces were evaluated. Carrying out 10 trials for the preliminary and 30 for the main experiments for each condition, more than 30 experimental data of v and α were organized at each data point, with outliers excluded by applying the Smirnov-Grubbs test for the set of the most prominent negative peaks of v.

v and a as a function of r Figure 6 shows v − r curves for a variety of particle sizes, where error bars denote standard errors. These curves were evaluated from Figures 3 and 4, using the experimental data of

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Figure 4. Typical results of particle velocity v in the radial direction, which were derived from numerical derivatives of experimental data in Figure 3, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

t > 0.5 s after the irradiation of laser beam. The trajectories of single particles represent the stochastic nature because the particles that moved different pathways to a focal point were averaged. In the case of a ϕ = 1.01 µm particle, as shown in Figure 6a, a particle moved very slowly when it was far from the focal point, and the velocity quickly increased in the negative direction within r = 0.4 µm from the focal point converged at r = 0 µm. A peak value at rp = 0.23 µm reached vp = −16.0 ± 1.3 µm/s, where we define the peak velocity as vp and its radial position as rp . Deviations observed near the peak were suspected to be caused by the variety of trajectories in the optical force field. After that, the speed decayed to v = 0 µm/s because the repulsive gradient forces acted on when a particle crossed over the optical axis. Finally, the particle was trapped at the focal center by these gradient forces and the scattering forces. A particle is also subject to the scattering forces that are in the same direction of irradiating laser beam, which fix a particle on a channel wall 12


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Figure 5. Typical results of particle acceleration α in the radial direction, which were derived from numerical second derivatives of experimental data in Figure 3, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

surface. Particles seemed to be affected by the gradient force limited very near the focal point. Especially, the velocity negatively increased near 0.4 µm from the focus, where single particles were strongly attracted to the focal center. Figure 6b shows the v − r curve for a ϕ = 2.02 µm one. In this case, the velocity was weakly negative over r = 2 µm and gradually increased the slope within the inner region. The negatively increased velocity reached a peak of vp = −87.8 ± 4.2 µm/s at rp = 0.68 µm. Within a distance of rp , the velocity decreased and converged to v = 0 µm/s. Finally, the particle was trapped at the focal point. Figure 6c shows a result from a ϕ = 2.93 µm particle, which presented a similar trend with the previous one shown in Figure 6b. Single particles far from the focal point were weakly attracted, gradually increasing the transport velocity toward the focus. The slope drastically increased near r = 3 µm and the velocity reached a peak of vp = −103 ± 2.8 µm/s at rp = 1.35 µm. The particle was forced to decrease in velocity within rp and was

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trapped at the center. Figure 6d shows the case of a 4.00 µm diameter particle. In this case, single particles were also attracted to the focal point in the same manner as the other cases. A peak of vp = −83.9 ± 2.0 µm/s appeared at rp = 1.84 µm. Beyond the peak, the transport velocity successively decreased and was converged to v = 0 µm/s at the focal point. As summarized in Table II, it was found that vp tended to absolutely increase with the particle diameter and that rp also increased with the size, which approximately agreed with particle radii. According to Ashkin’s theory, rays irradiating a dielectric sphere basically increase in strength as a particle size increases, because the particle receives incident rays across its whole surface. In the present experimental results, the increased velocities decreased near the focal point of the converged laser beam, because the gradient force worked in the counter direction when a particle of a finite size crossed over the optical axis. Figure 7 shows α − r curves corresponding to each size of particle, which was made in the same procedure as Figure 6 using Figures 3 and 5. In each case, sharp peaks were caused by the drastic change in velocity.

Evaluation of optical gradient force Using Figures 6 and 7, we can evaluate the net force Fext (r) working on a particle, which may be expressed as follows:

Fext (r) = 6πµav(r) +

4π 3 ρa α(r), 3

(5)

where ρ is the density of a particle. Figure 8 presents the force curves along the direction of r for various a, separately for the components of Stokes drag and inertial force. As a result, it was clarified that the forces increased with increasing the particle diameters. Furthermore, it was found that 4πρa3 |α|/3 ≪ 6πµa|v|. Thus, Fext is almost equilibrated with the Stokes drag, such that Fext = 6πµav. Therefore, the additional mass and Basset’s historical term were also neglected, which appeared in unsteady conditions of liquid flows. 42 This estimation

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Figure 6. Transport velocity v as a function of radial position r, evaluated from v − t curves in Figure 4, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm. Each data point is the average of segments (a) ∆r = 0.04, (b) 0.15, (c) 0.10, and (d) 0.16 µm, which were properly determined to clarify the peak point.

also supports the assumption in the Langevin dynamics mentioned in the theoretical part. With a multiplier of 6πµa, Figure 6 directly represents a force field caused by optical forces, thermal fluctuations, and other factors. Although it was believed that Fext was mainly dominated by optical forces, contributions of other factors should be quantitatively evaluated. Based on Ashkin’s theory, 24 the converged laser beam is treated as a ray whose wavelength was much shorter than the particle diameters. For a 1064 nm wavelength corresponding to a frequency on the order of 1 × 1014 Hz, the amplitude of forced oscillation of a particle to the ac electric field is reciprocally proportional to the frequency and that for the size of ∼ 1 µm is negligibly small. Therefore, any response to electromagnetic fields was not detected in the present experiments. Effects of thermal fluctuations Fth , thermophoresis Ftp , and convection Fcnv were also involved as well as optical forces Fopt , such that Fext = Fopt +Fth +Ftp +Fcnv .

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α [mm2/s]

α [mm2/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 42

0 -5

ɸ=4.00 µm

5 0 -5

-10 -15

-10 0

1

2

3 4 r [µm]

5

6

0

1

2

3 4 r [µm]

5

6

7

Figure 7. Acceleration α as a function of r, evaluated from α − t curves in Figure 5, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm. Each data point is the average of segments (a) ∆r = 0.04, (b) 0.15, (c) 0.10, and (d) 0.16 µm, which were properly determined to clarify the peak point.

Here, focusing on the behavior of microparticles in the radial direction, the effects of gravity and buoyancy were excluded. The exposure time of 0.001 s was too long from the viewpoint of molecular collisions to detect detailed Brownian motions, and thus such an averaged random force caused by thermal fluctuations Fth was negligible on the net force. Taking sample averages in equilibrated liquids, natural convective flows in arbitrary directions were canceled. The Rayleigh number and temperature gradients were estimated to discuss the possibility of convective flow and thermophoretic transport of particles, as shown in Appendix B. In this study, such thermal effects caused by the laser irradiation were denied because of a slight temperature increase and its weak spatial gradient in the slit-like microchannel. This result means that the present condition sufficiently suppressed the thermal convective effect Fcnv and the thermophoresis Ftp . Therefore, in our experimental conditions, Fext is mainly

16


Page 17 of 42

×10−12

×10−16 0.05

1

×10−12

-0.10

-0.10

-0.15 0.0

2

0.2

0.4 0.6 r [µm]

-0.15 1.0

0.8

×10−12

-1

-2

-2

-3

-3 0

1

2

3 4 r [µm]

5

-1.0 0

1

ɸ=2.93 µm

-0.5

-2

1 0

-1

ɸ=2.02 µm

2

0

0.0

-1

×10−16 2

(c)

1

0

1

2 3 r [µm]

×10−12

4

5 ×10−16 2

(d)

1

0

0

-1

-1

ɸ=4.00 µm

-2

-2

-3

6

F=mα [N]

-0.05

ɸ=1.01 µm

F=6 µav [N]

-0.05

F=6 µav [N]

0.00

F=mα [N]

(b)

0.00

F=mα [N]

F=6 µav [N]

(a)

×10−16 0.5

F=mα [N]

0.05

F=6 µav [N]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-3 0

1

2

3 4 5 r [µm]

6

7

Figure 8. External forces on single particles divided into the two terms of Stokes drag (left axis) and inertial force (right axis) as a function of r, which is expressed by eq 5, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

dominated by Fopt . These conditions enable us to evaluate the optical gradient force, using trajectories of single particles.

Numerical evaluation of optical forces Considering particle diameters from 1.00 to 4.00 µm, optical forces were evaluated based on Ashkin’s theory. 24 Figure 9 shows numerical evaluations of optical forces mapped on the yz plane. The optical force vectors were computed when the mass centers of single particles were individually located at each site of the vectors. Herein, taking into account the experimental conditions, we set the net power of a laser beam to Pˆ = 10 mW, which fully irradiated ˆ = 0.5 are shown to explain the particles in water. First of all, results from the case of Tˆ = R overview of optical forces numerically evaluated, although a variety of the coefficients are compared later. It was clarified that the optical forces were usually directed to a focal point 17


4 2 0 2

4

6 r [µm]

8

10

12

Focal point

15 10 5 0

2

4

6 r [µm]

8

10

Focal point

12

7 6 5 4 3 2 1 0

20 (b) ɸ=2.00 µm

15 10 5 0

0 20

(c) ɸ=3.00 µm

0

7 6 5 4 3 2 1 0

F [pN]

Optical force

z [µm]

6

F [pN]

8

(a) ɸ=1.00 µm

z [µm]

7 6 5 4 3 2 1 0

10

Page 18 of 42

2

4

6 r [µm]

8

10

12

Focal point

20 (d) ɸ=4.00 µm

15 10

F [pN]

Focal point

F [pN]

7 6 5 4 3 2 1 0

0

z [µm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

z [µm]


5 Light cone 0 0

2

4

6 r [µm]

8

10

12

Figure 9. Numerical results of optical forces mapped on the yz plane resulting from eqs 17–19 for individual particles whose mass centers are placed at the origin of each vector, for (a) ϕ = 1.00, (b) 2.00, (c) 3.00, and (d) 4.00 µm, where ˆ = 0.5. Normalized a power of 10 mW is irradiated for the particles of Tˆ = R arrows and gray scale present the direction and magnitude of optical forces, respectively. The boundary of the light cone is shown with a dashed line.

and that the magnitude increased closer to the center. Depending on the situation, rays sometimes irradiated only a part of the particle surface when the center of mass was close to the edge of a light cone. In such a case, the particle tended to be swept away from the light, which was obvious as the size increased. In a slit-like microchannel, a particle pushed away from the light cone possibly touched the upper face of the channel wall. Successively receiving the gradient force, a particle approached the focal center along the wall surface. As shown in Figure 9, the range in which the center of mass could move was confined in the z direction with increasing the particle size. In particular, a 4.00 µm diameter particle that was considerably distant from the light cone was also influenced by the optical forces. Using the Langevin dynamics expressed by eqs 1–3, trajectories of single particles were traced. For initial conditions, a single particle was located at r = 3, 4, 5, 6, and 7 µm with v = 0 µm/s. The computations were carried out assuming fully elastic wall surfaces. For comparison with experimental results, trajectories were recorded with a sampling rate of 500 Hz and classified

18


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for the same initial conditions.

Trajectories of single particles Figure 10 shows computational results of mass center trajectories of individual particles, which were projected on the rz plane. As shown in Figure 10a, 1.0 µm diameter particles were transported to the focal point, although their behavior thermally fluctuated in the early stages. A particle initially located at r = 7 µm seemed to be swept away from the light cone in the distance, but came back as long as the surface was partly irradiated with rays and the gradient force was stronger than thermal fluctuations. Similar trends were found in the other cases, as shown in Figure 10b–d. Furthermore, the travelling direction tended to be further away from the focal point as the particle size increased and the location came nearer to the edge of the light cone. This was obvious for the cases of 3.00 and 4.00 µm diameter particles, as shown in Figure 10c,d, respectively. Averaging these trajectories, external force fields were evaluated in the same manner as in the experiments.

v and Fext as a function of r Figure 11 shows v − r curves, resulting from 10 samples for each particle diameter. As mentioned in the previous section and Appendix A, to take into account the differences in the properties of particles, we examined three pairs of reflection and transmission coefficients. Analysis showed that the transmission coefficient Tˆ dominantly causes the gradient force to ˆ increases the scattering force, as expressed by eqs 10 and increase and the reflection one R ˆ = 0.7), (0.5, 0.5), 11 in Appendix A. Figure 11a–c correspond to the cases of (Tˆ = 0.3, R and (0.7, 0.3), respectively. It was clear that a greater fraction of Tˆ resulted in a higher ˆ = 0.5, peak velocities and corresponding positions are also velocity. For the case of Tˆ = R summarized in Table III. The peak velocity tends to monotonously decrease with increasing the diameter. This trend is a little bit different from the experimental results in which the peak velocities of 1.01 and 2.02 µm diameter particles were lower than the others. 19



7

Focal point Collision at top

6

(a) ɸ=1.00 µm z [µm]

z [µm]

7

Upper limit position

5 4 3 2

6


5

(b) ɸ=2.00 µm

4 3

0

0 2

4

Focal point

6 r [µm]

8

10

12

0 7

(c) ɸ=3.00 µm Upper limit position

6

z [µm]

5 4 3 2

2

4

Focal point

6 r [µm]

8

(d) ɸ=4.00 µm

5


12

4 3 Initial position

t=0 s

1

0

10

6

2

Initial position

t=0 s

1

Initial position

t=0 s

1

Initial position

t=0 s

0

7

Focal point

2

1

z [µm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 42

Light cone

0 0

2

4

6 r [µm]

8

10

12

0

2

4

6 r [µm]

8

10

12

Figure 10. Trajectories of microparticles driven by optical forces, resulting from Langevin dynamics simulations according to eqs 1–4: (a) ϕ = 1.00, (b) 2.00, (c) 3.00, and (d) 4.00 µm. A power of 10 mW is irradiated for the particles of ˆ = 0.5. Positions of the mass centers are presented with closed circles, Tˆ = R where single particles initially located at r = 3, 4, 5, 6, and 7 µm move upward until they touch the upper surface, approaching the focal point. Gray colored circles indicate the size of particles. The boundary of the light cone is shown with a dashed line.

External forces that were equilibrated with Stokes drag were evaluated, as shown in Figure 12a–c corresponding to Figure 11a–c, respectively. The particles receive optical forces far from the focus with increasing the radius. Furthermore, the peak value of the gradient force tends to show clear differences with increasing Tˆ. Especially for the case of a ϕ = 1.00 µm diameter particle, the peak value is obviously lower than the others. This result means that the gradient force is strongly caused by refraction of the light that transmits a particle. In comparison with experimental results shown in Figure 8, the computational results are on the same order and well represent the characteristics of optical forces, although computational results seem to overestimate the force for the ϕ = 1.00 and 2.00 µm diameter cases.

20


Page 21 of 42

50

100

(a) Tˆ =0.3, Rˆ =0.7

0

100

(b) Tˆ =0.5, Rˆ =0.5

(c) Tˆ =0.7, Rˆ =0.3

0

0

-100

-100 -150

ϕ [µm] 1.00 2.00 3.00 4.00

-200 -250 0

1

2

3 4 r [µm]

-100

v [µm/s]

v [µm/s]

-50 v [µm/s]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-200 ϕ [µm] 1.00 2.00 3.00 4.00

-300 -400 5

6

0 1 2 3 4 5 6 7 r [µm]

-200 -300

ϕ [µm] 1.00 2.00 3.00 4.00

-400 -500 0

1

2

3 4 r [µm]

5

6

Figure 11. Numerical results of v as a function of r for various values of transmission ˆ (a) (Tˆ = 0.3, R ˆ = 0.7), (b) (0.5,0.5), and (c) coefficient Tˆ and reflection one R: (0.7,0.3).

Comparison between experimental and computational results For comparison between experimental and theoretical results, the normalized gradient forces are shown in Figure 13. In each data, in the radial direction, r and Fext are normalized by a and 6πµavp , respectively. In the computational result, as shown in Figure 13a, the maximum peaks of each particle appeared close to the corresponding radius. The force on the mass center reached the maximum at the distance of the radius as long as the attractive force to the center was mainly caused by the optical forces, because the gradient force repulsively worked within the radius. When a particle partly crosses the optical axis, rays from the front cause it to be repelled in the opposite direction. In this case, the peak position tends to increase as the radius decreases. On the other hand, in the experimental result, although the force peak position rp corresponded to the radius a in the cases of ϕ = 2.93 and 4.00 µm, that is rp /a = 1, the peak position particularly tended to appear at less than 1 in the cases of ϕ = 1.01 and 2.02 µm, as shown in Figure 13b. Although the reason has not yet been clarified, the size of ϕ = 1.01 µm was almost equal to the wavelength of 1064 nm and such a small particle whose size is comparable with or less than the wavelength implies the limitation of geometrical optics and consideration of Mie and Rayleigh theories. 5–7 The investigation of this difference between the experimental results and theoretical models is a 21



1

(a) Tˆ =0.3, Rˆ =0.7

0 Fext [pN]

-1 -2 ϕ [µm] 1.00 2.00 3.00 4.00

-3 -4 0

1

2

3 4 r [µm]

0

0

-1

-2

-2 -3 ϕ [µm] 1.00 2.00 3.00 4.00

-4 -5 -6 5

6

2

(b) Tˆ =0.5, Rˆ =0.5

Fext [pN]

1

Fext [pN]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 42

(c) Tˆ =0.7, Rˆ =0.3

-4 -6 ϕ [µm] 1.00 2.00 3.00 4.00

-8 -10 -12

0 1 2 3 4 5 6 7 r [µm]

0

1

2

3 4 r [µm]

5

6

Figure 12. External force on particles evaluated from Fext = 6πµav, where a = ϕ/2, as expressed by eq 5 with the approximation that the inertial term and the other factors are negligibly small in the r direction, for the cases corresponding to Figure 11.

subject for future work.

Assembled structures of microparticles at a solid surface in liquid The optical force evaluation in the previous section is expected to contribute to the understanding of assembly of microparticles. Based on the quantitative evaluation of the gradient force on single particles, we carried out longer time observations of multiple particles. When the power of irradiation was increased to 19.3 mW in the front of the objective, microparticles were successively attracted to the optical axis and formed a closely packed structure. Figure 14 shows a time series of an experimental result in which microparticles successively approached the focal point, and after the first one was trapped at the center, the next particle approached and settled as the neighbor of the central one. Finally, we could find a hexagonal shape of assembled particles. It is well known that spherical particles usually formed a closely packed structure in the stable conditions. 2,43–45 This process was also reproduced by applying the Langevin dynamics simulation for many-body problems. Here, we additionally consider interactions between individual particles. Microparticles are known to be aggregated in aqueous solutions, because their polarized 22


Page 23 of 42

0.2

(a)

0.2

Computational

0.0

0.0

-0.2

-0.2 Fext/6 µavp

Fext/6 µavp

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


-0.4 -0.6 ϕ [µm] 1.00 2.00 3.00 4.00

-0.8 -1.0 -1.2 0

1

2 r/a

3

(b)

Experimental

-0.4 -0.6 ϕ [µm] 1.01 2.02 2.93 4.00

-0.8 -1.0 -1.2 4

0

1

2 r/a

3

4

Figure 13. Comparison of normalized external forces on single particles between (a) computational and (b) experimental results, where the external force is normalized by Stokes drag resulting from the peak velocity. For ϕ smaller than 2.02 µm, the peaks in both (a) and (b) tend to deviate from r/a = 1. In (a), the peak position of ϕ = 1.00 µm shifts to greater than 1 and on the other hand, in (b), that of ϕ = 1.01 and 2.02 µm shift to less than 1.

surfaces attract counterions which are shared by other particles. To simulate assembly of colloidal particles, we assume weak attractive interactions between the particles with excluded volumes and then a simple shifted Lennard-Jones potential ψLJ (rj , rj ) is applied as follows 43,46 ( ) ( )6  12 σ σ  , − ψLJ (ri , rj ) = 4Eb 

(6)

rij = |ri − rj | − 2a.

(7)

rij

rij

where

Eb denotes the binding energy and σ represents effective particle size, corresponding to the particle diameter. Here, Eb was defined as the degree of 0.1kB T . 38 We obtain the conservative force by taking the derivative of eq 6 at each position of particle. Using this method, simulations were performed for 20 randomly placed particles irradiated with a converged

23



(a) t=0.90 s

(b) t=2.18 s

Focal point

5 µm

(c) t=5.40 s

(d) t=13.52 s

Figure 14. Time series of cluster formation process resulting from observation, where snapshots at (a) t = 0.90, (b) 2.18, (c) 5.40, and (d) 13.52 s are shown.

beam of 20 mW in a space of 80.0 × 80.0 × 7.0 µm3 . Here, we demonstrated for the case of ˆ = 0.3, which were the same previously used in the force evaluation of single Tˆ = 0.7 and R particles and seemed to be suitable to reproduce the experimental results. Time series of the computational result are also shown in Figure 15. It was found that the computational result well reproduced experimental observations. In the computational result, it took a longer time until a hexagonal shape consisting of 7 particles was formed than the experimental results. It depends on the probability of existence of particles near the focal point in the light cone. To reduce the time for assembling, a laser beam should be focused on the denser area of particles.

Conclusions In this study, optical forces on single polystyrene particles, which were irradiated with a converged laser beam whose wavelength was 1064 nm, were experimentally and theoretically evaluated by using a single particle tracking analysis. The experimental and theoretical results were in reasonably good agreement. It was a fundamental analysis to extend the optical 24


Page 24 of 42

Page 25 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(a) t=0.30 s s

(b) t=1.95 s

Focal point

5 µm

(c) t=18.95 s

(d) t=23.08 s

Figure 15. Time series of cluster formation process resulting from Langevin dynamics simulation, where snapshots at (a) t = 0.30, (b) 1.95, (c) 18.95, and (d) 23.08 s are shown.

trapping of single particles to assembled structure formations in liquids. Tracking motions of single particles, optical force fields were clarified, which depended on the particle diameter varied from 1 to 4 µm. Using a slit-like microfluidic channel, the behavior of particles was confined in the radial direction properly for microscope observations. It was elucidated that the drag force was mainly equilibrated with optical forces in liquid, where the inertial force for such small particles was much weaker than the drag force. As the particle diameter increased, the attractive force to the optical axis tended to become stronger. In particular, the force fields drastically changed within the distance of corresponding particle diameters and reached a peak near the distance of the particle radii. Partly crossing the optical axis, the particle was decelerated due to the repulsive force and finally trapped at the focal center. The microchannel effectively reduced convection and thermophoresis, and thus the optical gradient force on a single particle was solely and quantitatively evaluated. The present results provided a useful method to evaluate the optical force in viscous liquids. Furthermore, assembled structures were also observed at a solid surface in a slit-like microchannel and also well replicated by applying the Langevin dynamics simulation, which reproduced 25



a particle transport phenomenon driven by optical forces in a many-body system. Further applications of optical forces are expected to design and optimize self-assembly structures of microparticles and equivalent dielectric objects such as biological cells. On the other hand, small particles whose size is comparable to the wavelength showed a disagreement in the framework of geometrical optics, and therefore Mie and Rayleigh theories should be taken into account in future work.

Acknowledgment This study was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (JSPS KAKENHI) grant no. JP16H06504 in Scientific Research on Innovative Areas ‘Nano-Material Optical-Manipulation’.

Appendix A: Optical forces of Ashkin’s ray optics Based on Ashkin’s methods, 23,24 optical forces on single polystyrene particles in water are quantitatively calculated and validated by comparing with experimental results. According to Ashkin, when an incident ray of power Pˆ hits a dielectric sphere with an incident angle θ, the momentum per second n1 Pˆ /c transfers to the particle, where n1 is the index of refraction of the liquid and c is the speed of light. As shown in Figure A1, a reflected incident ˆ ray imparts a force on a sphere that is expressed with the Fresnel reflection coefficient R, ˆ Pˆ . The remains pass through the sphere and are successively reflected and such that R ˆ TˆPˆ , ..., TˆR ˆ n TˆPˆ , where Tˆ is the transmission transmitted, decreasing the power TˆTˆPˆ , TˆR coefficient. Then, these scattering rays make angles relative to the direction of the incident ray of π + 2θ, α, α + β, ..., α + nβ, respectively. The optical force is divided into two components: FZ along the optical axis and FY the vertical one. Integrating this optical process, the net force is evaluated as follows

26


Page 26 of 42

Page 27 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Rˆ Pˆ n

Incident ray Pˆ θ

θ 2(θ−ɸ)

ɸ Tˆ Rˆ Rˆ TˆPˆ

n θ

θ

O

"+2(θ−2ɸ) Tˆ TˆPˆ

"−2ɸ Z

Y

Dielectric particle "−2ɸ Tˆ Rˆ TˆPˆ

Figure A1. Schematic diagram of an incident ray that is reflected and refracted at the surface of a dielectric particle. 24 An incident ray of power Pˆ is reflected at the surface and is transmitted in the dielectric particle with the fractions of ˆ and Tˆ, respectively, where n presents the normal vector of the surface. R ]

[

FZ FY

∞ ∑ n1 Pˆ ˆ n cos(α + nβ) , ˆ cos(π + 2θ) − Tˆ2 R = 1−R c n=0 ] [ ∞ ∑ n1 Pˆ 2 n ˆ sin(α + nβ) . ˆ sin(π + 2θ) + = Tˆ R −R c n=0

(8) (9)

After continuous computations, we obtain [

FZ FY

]

ˆ2 ˆ n1 Pˆ ˆ sin 2θ − T [cos(2θ − 2ϕ) + R cos 2θ] , = 1+R ˆ 2 + 2R ˆ cos 2ϕ c 1+R [ ] ˆ sin 2θ] Tˆ2 [sin(2θ − 2ϕ) + R n1 Pˆ ˆ R sin 2θ − , = ˆ 2 + 2R ˆ cos 2ϕ c 1+R

(10) (11)

where θ and ϕ are the incidence and refraction angles, respectively. The components of FZ and FY are hereafter represented as scattering force Fs and gradient force Fg , respectively. As shown in Figure A2a, this is a situation when the light is converged at a point F(0, h) in the hemisphere of a dielectric particle located at a point G(y ′ , z ′ ). An incident ray is refracted at a point W apart from the origin O by the distance R, where we define an angle α between the ray and the refracting interface and β measured from the y ′ -axis. The extension of a ray and the beam axis are crossing at W′ and B on the equatorial plane, respectively. The ray hits a point V with the incident angle θ, where the extension reaches a point P on the equatorial 27



R O

Ray

Ray

Beam Axis

(a)

(b)

n

y′ Fg

h V F

α

x′

P y′

W

Ray

W α

β

n

Page 28 of 42

θ

θ

V

n

V′

Fg

η γ

V′ η

P

G(y′,z′) Fs

α P

G(y′,z′)

β′

V V′

F

d

W′

γ B

θ

a η

η C

γ B

W′ y′

G(y′,z′)

β′

x′ z′

z′ y′cosβ β y′

B (z′−h)/tanα P

z′

V′ y′sinβ G(y′,z′)

β′

η

y′

Fg

d β−β′ d=(z'−h)cosβ'/tanα+y'cos(β−β')

G

V′

Fg cosη

Fg sinη

C

P

α

Fg cosη α Fg cosηcosα Fg cosηsinα

C

z′

Figure A2. Schematic of geometrical relations between an incident ray and a dielectric particle, for the case of (a) focal point inside the particle body. 24 Details are magnified in (b).

̸

plane. n define the normal vector on the particle surface. Additionally, we define angles WPG and ̸ W′ PG as γ and β ′ , respectively. As shown in Figure A2b, a perpendicular line from G to line WP is denoted as V′ and the torsional angle between plane WPW′ and plane V′ PG is defined as η. In this case, Fg is along line GV′ and Fs is perpendicular to it. According to these relations, the length of line PG denoted as d is expressed as follows cos β ′ + y ′ cos(β − β ′ ), d = (z − h) tan α

(12)

y ′ sin β . y ′ cos β + (z ′ − h)/ tan α

(13)

′

where β ′ is expressed by tan β ′ =

From a relation of cos γ = cos α cos β ′ ,

28


(14)

Page 29 of 42 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Beam Axis

(a)

Ray

(b)

O

R

β

α θ

β

α P γ β′ G

η

W

V

y′

n

θ

Fg

V

B

Ray

Ray n

W

V' W′

n a

θ

P γ

Fg V η

d V' C P η

G(y′,z′) Fs

x′

β

B

W′ y′

G y′

x′ z′

z′

F (h−z′)/tanα B

z′

β y′

P

V′

β′

d β′−β

G

d=y'cos(β'−β)−(h−z')cosβ'/tanα

η

y′

Fg

G

V′

Fg cosη

Fg sinη

C

P

α

Fg cosη α Fg cosηcosα Fg cosηsinα

C

z′

Figure A3. Schematic of geometrical relations between incident ray and dielectric particle, for the case of (a) a focal point outside the particle body. 24 Details are magnified in (b). we get

R sin θ = d sin γ.

(15)

Finally, we obtain

cos µ =

tan α . tan γ

(16)

Even when the focal point is outside of the sphere, we can compute the geometric relations of eqs 12–16 by the same procedure, as shown in Figure A3. Therefore, optical forces are fully computed for arbitrary focal points. By rotating the angle β, the ray in the first quadrant contributes to the force represented in the Cartesian coordinate system as follows

Fx = −Fs cos α sin β + Fg cos η sin α sin β − Fg sin η cos β,

(17)

Fy = −Fs cos α cos β + Fg cos η sin α cos β + Fg sin η sin β,

(18)

29



Fz = Fs sin α + Fg cos η cos α,

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(19)

where Fs and Fg correspond to eqs 10 and 11, respectively. The other rays that pass through the other quadrants are also calculated by transforming the coordinate system. Herein, for many-body problems, incident rays are assumed to sufficiently decay after hitting a particle. Therefore, multiple scattering effects are ignored for the first step of this study.

Appendix B: Estimation of convective flows and thermophoresis Based on the Rayleigh number that represents the influence of convective flows, such that Ra = βwater g|∆T |h3 /(ναwater ), the parameters in the present experimental condition corresponded to a coefficient of thermal expansion βwater = 2.41 × 10−4 K−1 , an acceleration of gravity g = 9.81 m/s2 , a kinematic viscosity ν = 8.6 × 10−7 m2 /s, a thermal diffusivity αwater = 1.47 × 10−7 m2 /s at T = 300 K, and a height of the microchannel h = 7.1 ± 0.1 µm. The temperature difference ∆T was also measured by using a digital thermometer (TM-300, AS ONE, Osaka, Japan) and resulted in ∆T = 1.0 K. Consequently, we obtained Ra = 6.7, which is much less than the critical value of thermal convective flows in a slit-like microchannel, Rac ≈ 1700. 47 This means that the convective flow was found to be negligibly weak. Erwin et al. 48 stated that heating caused a temperature increase of 8 K/W, which was mainly caused by the absorption of water. For the case of Pˆ = 10 mW, the temperature increase was evaluated to be 0.08 K, which did not seriously induce convection. This level of increase was in agreement with others. 49 Ito et al. 50 reported 24 K/W of water absorption, which corresponded to a 0.24 K increase for 10 mW irradiation. It was also reported that temperature gradients on the order of 106 K/m were required to induce thermophoretic transport. 51 Actually, measuring the temperature in our experimental system in the same

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way as described in the literature, 51 the increase was about 1 K and the temperature gradient was estimated as on the order of 105 K/m localized near the focus. This result indicates that thermophoretic transport does not affect the optical force measurement.

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Table I. Properties of particles. Diameter [µm] 1.01 2.02 2.93 4.00

Material Polystyrene microsphere Polystyrene microsphere Polystyrene microsphere Polystyrene microsphere

Concentration [vol%] 1.0×10−1 1.0×10−1 1.0×10−2 1.0×10−2

Specific weight 1.05 1.05 1.05 1.05

Table II. Experimental data of peak velocities and corresponding radial positions observed as single particles approached a focal point. Particle diameter [µm] 1.01 2.00 2.93 4.00

Peak point: rp [µm] 0.23 0.68 1.35 1.84

Peak velocity: vp [µm/s] −16.0 ± 1.3 −87.8 ± 4.2 −103 ± 2.8 −83.9 ± 2.0

Table III. Numerical results of peak velocities and corresponding radial positions ˆ = 0.5. observed as single particles approached a focal point for the case of Tˆ = R Particle diameter [µm] 1.00 2.00 3.00 4.00

Peak point: rp [µm] 0.75 1.05 1.32 1.65

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Peak velocity: vp [µm/s] −264 ± 67 −249 ± 16 −194 ± 9 −150 ± 5



Figure and Table Captions Figure 1. Schematic illustrations of optically trapped microparticles in narrow spaces: (a) trapping process of single particles, where Brownian motions let a particle be occasionally irradiated with light. The particle exposed to light is forced to move toward a focal point and is finally trapped at the center. (b) In the observation over a longer period of time, assembled structures are obtained which are much larger than a focal spot of 1 µm.

Figure 2. Illustration and photograph of experimental setup. A continuous wave of 1064 nm wavelength is irradiated from a light source and is converged at the upper surface of a slit-like microfluidic channel made of PDMS. Another light source illuminates a sample focused with a condenser lens and the transmitted light refracted by particles is detected with a high-speed sCMOS camera connected to the inverted microscope. Velocities and accelerations of particles are evaluated from the trajectories by using the PTV method.

Figure 3. Typical experimental results of particle displacement in the radial direction r driven by optical forces for various diameter particles: (a) ϕ = 1.01, (b) 2.02, (b) 2.93, and (b) 4.00 µm.

Figure 4. Typical results of particle velocity v in the radial direction, which were derived from numerical derivatives of experimental data in Figure 3, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

Figure 5. Typical results of particle acceleration α in the radial direction, which were derived from numerical second derivatives of experimental data in Figure 3, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

Figure 6. Transport velocity v as a function of radial position r, evaluated from v − t curves 38


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in Figure 4, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm. Each data point is the average of segments (a) ∆r = 0.04, (b) 0.15, (c) 0.10, and (d) 0.16 µm, which were properly determined to clarify the peak point.

Figure 7. Acceleration α as a function of r, evaluated from α − t curves in Figure 5, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm. Each data point is the average of segments (a) ∆r = 0.04, (b) 0.15, (c) 0.10, and (d) 0.16 µm, which were properly determined to clarify the peak point.

Figure 8. External forces on single particles divided into the two terms of Stokes drag (left axis) and inertial force (right axis) as a function of r, which is expressed by eq 5, for (a) ϕ = 1.01, (b) 2.02, (c) 2.93, and (d) 4.00 µm.

Figure 9. Numerical results of optical forces mapped on the yz plane resulting from eqs 17–19 for individual particles whose mass centers are placed at the origin of each vector, for (a) ϕ = 1.00, (b) 2.00, (c) 3.00, and (d) 4.00 µm, where a power of 10 mW is irradiated for ˆ = 0.5. Normalized arrows and gray scale present the direction and the particles of Tˆ = R magnitude of optical forces, respectively. The boundary of the light cone is shown with a dashed line.

Figure 10. Trajectories of microparticles driven by optical forces, resulting from Langevin dynamics simulations according to eqs 1–4: (a) ϕ = 1.00, (b) 2.00, (c) 3.00, and (d) 4.00 ˆ = 0.5. Positions of the mass µm. A power of 10 mW is irradiated for the particles of Tˆ = R centers are presented with closed circles, where single particles initially located at r = 3, 4, 5, 6, and 7 µm move upward until they touch the upper surface, approaching the focal point. Gray colored circles indicate the size of particles. The boundary of the light cone is shown with a dashed line.

39



Figure 11. Numerical results of v as a function of r for various values of transmission coeffiˆ (a) (Tˆ = 0.3, R ˆ = 0.7), (b) (0.5,0.5), and (c) (0.7,0.3). cient Tˆ and reflection one R:

Figure 12. External force on particles evaluated from Fext = 6πµav, where a = ϕ/2, as expressed by eq 5 with the approximation that the inertial term and the other factors are negligibly small in the r direction, for the cases corresponding to Figure 11.

Figure 13. Comparison of normalized external forces on single particles between (a) computational and (b) experimental results, where the external force is normalized by Stokes drag resulting from the peak velocity. For ϕ smaller than 2.02 µm, the peaks in both (a) and (b) tend to deviate from r/a = 1. In (a), the peak position of ϕ = 1.00 µm shifts to greater than 1 and on the other hand, in (b), that of ϕ = 1.01 and 2.02 µm shift to less than 1.

Figure 14. Time series of cluster formation process resulting from observation, where snapshots at (a) t = 0.90, (b) 2.18, (c) 5.40, and (d) 13.52 s are shown.

Figure 15. Time series of cluster formation process resulting from Langevin dynamics simulation, where snapshots at (a) t = 0.30, (b) 1.95, (c) 18.95, and (d) 23.08 s are shown.

Figure A1. Schematic diagram of an incident ray that is reflected and refracted at the surface of a dielectric particle. 24 An incident ray of power Pˆ is reflected at the surface and is ˆ and Tˆ, respectively, where n transmitted in the dielectric particle with the fractions of R presents the normal vector of the surface.

Figure A2. Schematic of geometrical relations between an incident ray and a dielectric particle, for the case of (a) focal point inside the particle body. 24 Details are magnified in (b).

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Figure A3. Schematic of geometrical relations between incident ray and dielectric particle, for the case of (a) a focal point outside the particle body. 24 Details are magnified in (b).

Table I. Properties of particles.

Table II. Experimental data of peak velocities and corresponding radial positions observed as single particles approached a focal point.

Table III. Numerical results of peak velocities and corresponding radial positions observed ˆ = 0.5. as single particles approached a focal point for the case of Tˆ = R

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Optical axis

r PDMS wall surface PSt particle

Cover glass Objective lens

t=0.90 s

t=13.52 s

Focal point

5 µm

TOC graphic

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Quantitative Evaluation of Optical Forces by Single Particle Tracking in

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