Laser induced dynamics of interacting small particles

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(1)Laser induced dynamics of interacting small particles F. Claro, P. Robles, and R. Rojas. Citation: Journal of Applied Physics 106, 084311 (2009); doi: 10.1063/1.3243308 View online: https://doi.org/10.1063/1.3243308 View Table of Contents: http://aip.scitation.org/toc/jap/106/8 Published by the American Institute of Physics.

(2) JOURNAL OF APPLIED PHYSICS 106, 084311 共2009兲. Laser induced dynamics of interacting small particles F. Claro,1,a兲 P. Robles,2 and R. Rojas3 1. Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile Escuela de Ingeniería Eléctrica, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile 3 Departamento de Física, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile 2. 共Received 8 July 2009; accepted 9 September 2009; published online 22 October 2009兲 We study the translational motion of two interacting polarizable nanospheres in the presence of a laser field. Dependences of the resulting paths on geometry, viscosity of the medium, polarization, and wavelength of the incident field are discussed. It is found that in general clustering trajectories are more probable thus favoring agglomeration, and that viscosity and circular polarization of the applied field increase further the probability of clustering. © 2009 American Institute of Physics. 关doi:10.1063/1.3243308兴 I. INTRODUCTION. Manipulation of neutral atoms, molecules, and nanoparticles by means of an external electromagnetic field has been widely explored both theoretically and experimentally.1–5 The particles may have no permanent dipole moment but due to quantum fluctuations or by being subjected to an external electric field, they can acquire an induced electric dipole moment. As a consequence, electric forces and torques are produced resulting in changes in the translation and rotation of the particles, phenomena known generically as ac electrokinetics. Within these phenomena, dielectrophoresis corresponds to the motion of particles in the presence of a nonuniform electric field,6,7 while electrorotation refers to the rotational motion of suspended dielectric particles interacting with a rotating ac electric field.8 Understanding the dynamics of a system of particles due to the above phenomena is important in several applications, such as the control of agglomeration or separation of proteins or living cells in suspension.9–11 An essential issue is the determination of the laser-induced electric forces between the particles. In Ref. 6, the time-averaged force was obtained in the dipole approximation. It was found to be nonconservative and noncentral, and that for frequencies near resonance its magnitude may become greatly enhanced. In the present work, we study the motion of two polarizable nanoparticles interacting in the presence of a laser field. For simplicity, we take them to be spherical and identical, a model that save for details captures the main physical response of the pair. As long as the particles are not bounded, their rotation is independent of the translational motion and we will ignore the former. In Sec. II, we describe the equations governing the dynamics, in Sec. III we discuss the numerical results, and in Sec. IV we present our discussion.. that the long wavelength limit applies. The particles are immersed in a viscous fluid that damps motion with a force proportional to velocity. It is assumed that forces due to the intensity gradient and Rayleigh scattering are small and may be neglected.12,13 Furthermore, electromagnetic retardation and radiation pressure effects are not included, considering that wavelengths are assumed much greater than nanosphere sizes and their separation.14 Motion is referred to the center of mass 共c.m.兲, while light is incident along the x-direction with the z-axis chosen along the electric field direction. Since the particles move on a plane, we choose polar coordinates 共r , ␪兲 and 共r , ␪ + ␲兲 for the position of the particles, as shown in Fig. 1. The radius of each sphere is a. Due to the external field, the spheres become polarized and multipole moments are induced on them. As a consequence, electric forces appear, with time-averaged radial and angular components given by6 Fr共rជ兲 =. F␪共rជ兲 = −. 0021-8979/2009/106共8兲/084311/4/$25.00. 冋. 共1兲. 册. sin 2␪ 共E0a兲2 , 4 Re 96␴ 共nA − u兲共nR − u兲ⴱ. 共2兲. y r θ CM. z. r E 0 e−iωt. We consider two identical electrically neutral nanoជ 0e−i␻t. The spheres of mass m in the presence of a laser field E particles are small compared to the wavelength of light so Electronic mail: fclaro@uc.cl.. 册. where the dipole approximation has been used. Multipoles are expected to matter when the particles are less than a. II. EQUATIONS OF MOTION. a兲. 冋. 2 cos2 ␪ 共E0a兲2 sin2 ␪ , 4 2 − 96␴ 兩nR − u兩兩nA − u兩2. FIG. 1. 共Color online兲 Two polarizable spheres excited by an oscillating electric field along the z-direction. The origin is at the c.m. 106, 084311-1. © 2009 American Institute of Physics.

(3) 084311-2. J. Appl. Phys. 106, 084311 共2009兲. Claro, Robles, and Rojas. y. diameter apart, a rather small fraction of the phase space in typical trajectories.15 The quantity ␴ = r / a is a dimensionless radial coordinate, such that 2␴ represents the center to center particle separation. The material properties appear in the spectral variable u = 关1 − ␧共␻兲兴−1 = u⬘ + iu⬙ only, with ␧共␻兲 the complex dielectric function of the spheres. The attractive and repulsive mode depolarization factors are6 nA =. nR =. 冉. 冊. 共3兲. 冊. 2. + ␭r. 冋. 1 sin2 ␪ d␴ = F0 + 4 dT ␴ 共nR − u⬘兲2 + 共u⬙兲2 −. ␴. 册. 2 cos2 ␪ , 共nA − u⬘兲2 + 共u⬙兲2. d 2␪ d␪ d␴ d␪ + ␴␭␪ +2 dT2 dT dT dT =−. Clustering z 5. ∆t = 1.6τ ∆t = 0.6τ. 共4兲. Notice that the radial force in Eq. 共1兲 is attractive up to a critical angle ␪c共␴ , ␭兲 whose value in the limit of large separation is found to be 54.74°. We restrict our discussion to translational motion and use for its description Lagrange’s equations, including a Rayleigh dissipation function.16 In terms of the variables ␴ and ␪, one obtains the coupled equations. 冉冊. θmax. −5. 1 1 1+ 3 . 3 8␴. d␪ d 2␴ − ␴ dT dT2. 5. −5. 1 1 1− 3 , 3 4␴. 冉. Unbounded. 再. 共5兲. 冎. sin 2␪ cos共␣2 − ␣1兲 1 . ␴4 关共nA − u⬘兲2 + 共u⬙兲2兴1/2关共nR − u⬘兲2 + 共u⬙兲2兴1/2 共6兲. We have added an intrinsic radial force component F0 that may be due to a van der Waals interaction or to other effects. We assume the long range value of such force to be negligible in the presence of a strong laser field, however, and take only a contact repulsive interaction of the form F0 = As共1 − ␴兲, where s共x兲 is the step function and the amplitude A is given the value 100 for convenience. The time scale is ␶ = 冑96ma / E0a so that in such units the time is denoted by a dimensionless variable T = t / ␶. Damping is characterized by the parameters ␭r = kr␶ / m and ␭␪ = k␪␶ / m, with kr = k␪ = 6␲a␩, where ␩ is the viscosity coefficient of the medium surrounding the spheres. Finally, ␣1 and ␣2 are the polar angles of the complex quantities 共nA − u兲 and 共nR − u兲, respectively.. III. NUMERICAL RESULTS. We have performed a numerical evaluation of Eqs. 共5兲 and 共6兲 assuming the spheres to be of radius 10 nm and made of gold. Taking an electric field amplitude of 10 kV/m, the time unit becomes ␶ = 3 ns. The dielectric response of the spheres is represented by a Drude model:17. FIG. 2. 共Color online兲 Regions of diverging 共shaded兲 and clustering 共unshaded兲 initial positions of trajectories in the absence of viscosity. One particle is at the origin while the center of the second particle can be anywhere save for the black circle, excluded by the contact repulsive interaction. The axes are in units of the particles radius.. ␧共␻兲 = ␧b −. ␻2p , ␻2 + i␥␻. 共7兲. with ␧b = 9.9, ␻ p = 8.19 eV, and ␥ = 0.027 eV.18 The particles are released from rest and followed in their motion on the y-z plane, resulting in trajectories as those in Fig. 2. For convenience, the coordinate system is shown fixed at the center of one of the spheres, around which a dark circle marks the region excluded to the center of the other sphere by the strong short range repulsive interaction. Dots along the path followed by the second particle are at equal time intervals ⌬t of values included in the figure. Acceleration 共deceleration兲 has taken place when dots separate further 共get closer兲 along the path followed. For the results in this figure, the laser field wavelength is 1050 nm and the viscosity is null. Shaded 共unshaded兲 regions correspond to initial positions leading to unbounded 共clustering兲 trajectories, with a boundary placed at about ␪max = 63°. In determining this boundary, a trajectory was considered unbounded if after a time greater than T = 5000 共units of ␶兲 the radial velocity was still positive, meaning a diverging path. For comparison, the slightly smaller critical angle for the boundary of attraction, ␪c = 54.74°, is also shown in Fig. 2 as a dashed straight line. Two trajectories are included for illustration, one bounded 共full line, initial point at ␴0 = 2.5 and ␪0 = 45°兲 and one unbounded 共dashed line, initial point at ␴0 = 2.5 and ␪0 = 75°兲. Our calculations show that when viscosity is turned on, the boundary moves to larger angles ␪max. Figure 3 shows a trajectory in the presence of a rather large viscosity ␩ = 0.05 N s / m2, otherwise under the same conditions as in Fig. 2. The particle starts at a point of large angle ␪0 = 89.5° in spite of which the motion, initially diverging, turns around and touching occurs after a time t = 2430␶, or about 7 ␮s for the parameters chosen. The dependence of ␪max on viscosity is shown in Fig. 4 for two values of the laser wavelength. Its value saturates in the limits of low and large viscosity, with a rather abrupt growth in between. While the large viscosity limit is independent of wavelength, the low viscosity angle decreases.

(4) 084311-3. J. Appl. Phys. 106, 084311 共2009兲. Claro, Robles, and Rojas. y. 5. ∆t = 2430 τ. 10. Log Log [Tc] (Tc). 4. 5. 3 2 5*10^-4 −4 5×10 Ns/m2 −3 5*10^-3 5×10 Ns/m2 −2 5*10^-2 5×10 Ns/m2. 1. −10. 5. −5. 10. z. 0 0. 20. FIG. 3. 共Color online兲 Trajectory of a particle starting at ␴0 = 1 and ␪0 = 89.5° in a medium with ␩ = 0.05 N s / m2 and under illumination with light of wavelength 1050 nm. The axes are in units of the particles radius.. toward the infrared. At room temperature, air 共␩ ⬃ 2 ⫻ 10−5 N s / m2兲 lies near the low viscosity limit, while water 共␩ ⬃ 9 ⫻ 10−4 N s / m2兲 is at the other end, where diverging trajectories have very low phase space. If the incident laser field has circular polarization, then the polar angle ␻t may be averaged out owing to the fact that particle motion is comparatively much slower. The time average of Eqs. 共1兲 and 共2兲 gives. 册册. Fr共r兲 =. 2 1 共E0a兲2 , 4 2 − 96␴ 兩nR − u兩兩nA − u兩2. F␪共r兲 =. 1 共E0a兲2 . 4 Im 48␴ 共nA − u兲共nR − u兲ⴱ. 冋. 共8兲. 共9兲. As can be easily checked using Eqs. 共3兲 and 共4兲, the bracket in Eq. 共8兲 is always negative, yielding an attractive radial component of the force that makes all trajectories bounded. Our numerical results for this case show that at all wavelengths and viscosities, the particles released from rest end up touching, after following essentially straight paths. Because the sphere radius appears differently in the various terms of Eqs. 共5兲 and 共6兲, we have calculated the dependence of the clustering time on sphere size, finding a smooth increase by about an order of magnitude when the sphere radius grows from 10 to 100 nm. This is shown in Fig. 5, 90. θmax (degrees). 85 80 75 70 520 nm 1050 nm. 65 60 -6. 80. 100. FIG. 5. 共Color online兲 Dependence of the clustering time Tc on sphere size for three values of the viscosity coefficient: ␩ = 5 ⫻ 10−2 共dash-dotted line兲, 5 ⫻ 10−3 共dashed line兲, and 5 ⫻ 10−4 共solid line兲, all in N s / m2. The unit of time Tc is ns.. −10. -8. 60. radius (nm). −5. 冋. 40. -4. -2. 0. Log(viscosity coefficient) Log (viscosity coefficient). FIG. 4. 共Color online兲 Effect of viscosity and wavelength over the angular dimension of the clustering region. The viscosity coefficient is in N s / m2.. where such dependence is exhibited for three values of the viscosity. The result is what would have been expected based merely on inertia. IV. DISCUSSION. In summary, we have studied the motion of a pair of small particles in a laser field under the force of the induced polarization. Our main finding is that for particles starting from rest, the size of the phase space is larger for initial positions leading to touching than for diverging trajectories, thus favoring clustering. In the limit of zero viscosity, at which the boundary separating bounded from unbounded trajectories is approximately 63°, the probability that a particle lies within the clustering region in a spherical volume surrounding the first particle may be estimated as 0.71. This value, exact for vacuum, is still appropriate for low viscosity fluids such as air. However, when the viscosity grows, the boundary moves to higher angular values increasing the probability for clustering trajectories, saturating to essentially 100% binding when the viscosity reaches a value of about 10−2 N s / m2. If light is circularly polarized, then all particle initial positions lead to clustering. We conclude that light-induced forces favor in general cluster formation of neutral particles in suspension. Evidence for such behavior has been experimentally found in several works.4,19–21 Our numerical analysis was done using a Drude model for gold particles. Since the dielectric response enters in the denominators in the force expressions 共1兲 and 共2兲 giving rise to resonant conditions, we expect that some details will be different for other materials or dielectric models. For instance, the dielectric response may be more structured exhibiting several resonances, in which case the wavelength dependence of the boundary angle is expected to follow a similar structure. The main result, however, that light favors agglomeration is robust and does not depend on such structure. ACKNOWLEDGMENTS. This work was supported by the FONDECYT under Grant No. 1060650, the Dirección de Investigación, Universidad Técnica Federico Santa María, under Grant No..

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