Arrested Dimer's Diffusion by Self-Induced Back-Action Optical Forces

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Arrested dimer's diffusion by self-induced back-action optical forces Jorge Luis-Hita, Juan Jose Saenz, and Manuel I Marqués ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.6b00259 • Publication Date (Web): 03 Jun 2016 Downloaded from http://pubs.acs.org on June 7, 2016

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Arrested dimer’s diffusion by self-induced back-action optical forces Jorge Luis-Hita,∗,†,‡ Juan José Sáenz,∗,‡,¶ and Manuel I. Marqués∗,§ †Departamento de Física de la Materia Condensada, Universidad Autónoma de Madrid, 28049 Madrid, Spain ‡Donostia International Physics Center (DIPC), Paseo Manuel de Lardizabal 4, 20018 Donostia-San Sebastian, Spain ¶IKERBASQUE, Basque Foundation for Science, 48013 Bilbao, Spain. §Departamento de Física de Materiales, Condensed Matter Physics Center (IFIMAC) and Instituto “Nicolás Cabrera”, Universidad Autónoma de Madrid, 28049 Madrid, Spain E-mail: [email protected]; [email protected]; [email protected]

Abstract The diffusion of a dimer made out of two resonant dipolar scatters in an optical lattice is theoretically analyzed. When a small particle diffuses through an optically induced potential landscape, its Brownian motion can be strongly suppressed by gradient forces, proportional to the particle’s polarizability. For a single lossless monomer at resonance, the gradient force vanishes and the particle diffuses as in absence of external fields. However, we show that when two monomers link in a dimer, the multiple scattering among the monomers induces both a torque and a net force on the dimer’s center of mass. This “self-induced back-action” force leads to an effective potential energy landscape, entirely dominated by the mutual interaction between monomers, which strongly influences the dynamics of the dimer. Under appropriate illumination,

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single monomers in a colloidal suspension freely diffuse while dimers become trapped. Our theoretical predictions are tested against extensive Langevin molecular dynamics simulations.

Keywords Resonant dimer, Self-induced back-action optical forces, Arrested diffusion, Potential energy landscapes, Langevin Molecular Dynamics

Understanding Brownian motion in spatially periodic and random landscapes has long been of great interest from a fundamental and practical point of view. (1 –5 ) The ability of light to exert significant forces on small particles (6 ) offers the opportunity to sculpt potential energy profiles enabling the study of Brownian dynamics in complex landscapes. Extended periodic landscapes, known as optical lattices, can be generated by the periodic intensity maxima arising in the interference pattern of several crossed laser beams (7 –9 ) or by an array of optical tweezers (10 ,11 ) generated by holographic techniques. (12 ) Tracking the Brownian motion of particles in a single optical trap can be used to study non-equilibrium diffusion processes, (13 –15 ) to detect acoustic vibrations (16 ) or to map the accessible space by Photonic Force Microscopy. (17 ) Optical forces have been widely exploited to understand, manipulate and control the statistical properties (18 ,19 ) and dynamics (20 ,21 ) of colloidal particles by optically induced potential energy landscapes (PEL). (22 ) Randomly modulated intensity patterns, so-called speckle patterns, can also be used to create a random PEL. (23 ,24 ) Counter intuitive phenomena like the giant diffusion induced by an oscillating periodic potential (25 ) or kinetically locked-in states in driven diffusive transport (26 ) have been realized by using optically induced PEL, (27 ,28 ) demonstrating optical guiding and sorting of particles in microfluidic flows. (29 –31 ) The intriguing properties of Brownian motion under 2


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periodic landscapes of non-conservative “curl forces” (optical vortex lattices (32 –34 ) ) is also a subject of increasing interest (35 –40 ) . Although the Brownian motion of non-interacting particles through PEL is relatively well understood, the influence of multiple scattering effects on the Brownian dynamics has been much less studied. Active superdiffusion can be realized in colloidal suspensions where the particles move in a time-dependent, self-generated, speckle intensity pattern induced by multiple scattering of light. (41 ) Multiple scattering effects can induce significant interactions between particles leading to optical binding (8 ,42 ) with peculiar non-conservative effects. (43 ,44 ) Attractive or repulsive Casimir-like forces can be controlled by using rapidly fluctuating random light fields (45 ) . Optical binding manifest itself in the extent of Brownian fluctuations of particles optically linked. (46 ) Multiple scattering not only leads to the formation of aggregates but also modifies the Brownian motion of the aggregates themselves. Multiple scattering effects are behind the complex behavior of the driven diffusion of dimers and other optically bound structures experimentally observed in a “tractor beam” set-up. (47 ) Our main goal here is to understand the role of multiple scattering in the diffusion of aggregates in an optical lattice. In order to understand the interplay between the optically induced PEL and multiple scattering in the diffusion of dimers or other aggregates we have analysed a simple system consisting of a dimer made out of two resonant particles diffusing on a two-dimensional optical lattice. Among other possible physical realisations, direct fabrication of metal nanoparticle dimers of a given length could be feasible by using DNA-origami and related techniques. (48 –50 ) In an optical potential energy landscape, the electromagnetic forces on small particles are conservative gradient interactions (33 ,51 ) proportional to the real part of the particle’s polarizability. At resonance, the real part of the polarizability is negligible and, as a consequence, a single resonant nanoparticle does not see the underlying optical lattice and undergoes free thermal Brownian motion. However, as we will show, when two resonant monomers link into a dimer, multiple scattering among them induces both a torque and a

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as sketched in Fig. 1. The external illuminating electric field is given by E (0) (x, y, z) = i2E0 (sin kx + sin ky) eˆz = iE0 Ψ(x, y) eˆz

(1)

where k = nω/c = 2π/λ is the wave number, λ is the wavelength in water, n the refractive index, E0 is the amplitude of the electric field, Ψ is a real function and eˆz is the unitary vector in the z-direction. For a single electric dipolar particle moving through a field E = Ez eˆz , linearly polarized along z, the time averaged optical force is given by (51 ) ǫǫ0 Re {α∗ Ez∗ ∇Ez } 2 ǫǫ0 α′ 2 ′′ ∗ ∇|Ez | + α Im {Ez ∇Ez } . = 2 2

F =

(2) (3)

At resonance we have α = i 6π , i.e. α′ = 0, and the total force on a single monomer moving k3 through an external field given by Eq. (1) is identically zero: there are no forces coming from the gradient force term (proportional to α′ ) and the “scattering force” (proportional to o n (0)∗ (0) = 0) vanishes. Im Ez ∇Ez

However, when a dimer is formed, the actual field that polarises each monomer comes

not only from the external field but also from the radiation scattered by the other dipole . This scattered light contributes to the optical force and may alter some dynamical properties of the system such as the diffusion. The optical polarising fields on particles 1 and 2 are expressed as E(r1 ) = E (0) (r1 ) + k 2 G(r1 , r2 )αE(r2 )

(4)

E(r2 ) = E (0) (r2 ) + k 2 G(r2 , r1 )αE(r1 ) where G(ri , rj ) is the Green dyadic function and E (0) the external field. Since the motion of the dimer in our system is restricted to the xy-plane and the external electric field is along 5


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the z-axis, the only relevant component of the Green dyadic function is eikL Gzz (ri , rj ) = Gzz (L) = 4πL being L = |r1 − r2 | =

p

1 1 1+i − 2 2 kL k L

(5)

(x1 − x2 )2 + (y1 − y2 )2 the distance between the particles. Defining,

G12 ≡ Gzz (|r1 − r2 |) = G21 , the system of equations (4) admits a simple analytic solution given by (0)

(0)

Ez (r1 ) =

Ez (r1 ) + ik 2 α′′ G12 Ez (r2 ) 1 + [k 2 G12 α′′ ]2

Ez (r2 ) =

Ez (r2 ) + ik 2 α′′ G21 Ez (r1 ) , 1 + [k 2 G12 α′′ ]2

(0)

(6)

(0)

where we considered the resonant condition α′ = 0. Then, the field polarising the particles always points along z and, from Eq. (2), the force on the resonant monomers “1” and “2” is given by ǫǫ0 ′′ α Im {Ez∗ (r1 )∇1 Ez (r1 )} 2 ǫǫ0 ′′ α Im {Ez∗ (r2 )∇2 Ez (r2 )} F2 = 2 F1 =

(7)

where Ez∗ (r1 ) and Ez∗ (r2 ) are given by Eq. (6), ∇1 ≡ {∂/∂x1 , ∂/∂y1 } and ∇1 Ez (r1 ) = ∇1 Ez(0) (r1 ) + iα′′ k 2 ∇1 G12 Ez (r2 )

(8)

∇2 Ez (r2 ) = ∇2 Ez(0) (r2 ) − iα′′ k 2 ∇1 G12 Ez (r1 ) (where we took into account that ∇2 G12 = −∇1 G12 ). These expressions will be used to perform molecular dynamics simulations discussed below. Notice that they are exact only within the dipolar approximation which, for resonant nanoparticles, is known to be valid for L larger than approximately three times the monomer radius. (57 ) At smaller inter-particle separations, higher order multipoles become relevant in the computation of the forces. 6


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Analytical approach In order to get some insight into the problem, let us consider Eqs. (6) and (8) in the limit k 2 G12 α′′ = 3λG12 ≪ 1 ( a good approximation for L & λ/2), Ez (r1 ) ≈ Ez(0) (r1 ) + ik 2 G12 α′′ Ez(0) (r2 ) + · · · ∇1 Ez (r1 ) ≈ ∇1 Ez(0) (r1 ) + iα′′ k 2 ∇1 G12 Ez(0) (r2 ) + · · · , and analogous expressions for monomer “2”. At lowest order in [k 2 G12 α′′ ], the forces [Eq. (7)] are given by n o ǫǫ0 2 ′′ 2 k (α ) |E0 |2 [Ψ(r1 )Ψ(r2 )] ∇1 Re {G12 } − [Ψ(r2 )∇1 Ψ(r1 )] Re {G12 } + · · · (9) 2 n o ǫǫ0 2 ′′ 2 k (α ) |E0 |2 − [Ψ(r1 )Ψ(r2 )] ∇1 Re {G12 } − [Ψ(r1 )∇2 Ψ(r2 )] Re {G12 } + · · · F2 = 2

F1 =

and we can readily calculate the force on the center of mass:

Fcm = F1 + F2 n o ǫǫ0 2 ′′ 2 2 ∼ − k (α ) |E0 | [Ψ(r2 )∇1 Ψ(r1 )] + [Ψ(r1 )∇2 Ψ(r2 )] Re {G12 } + · · · 2

(10)

It is interesting to consider Fcm when the external field is identical on both nanoparticles, (0)

(0)

i.e. Ez (r1 ) = Ez (r2 ). In this particular case, Eqs. (6) become (0)

Ez (r1 ) = Ez (r2 ) =

Ez (r1 ) , 1 − ik 2 G12 α′′

(11)

and it is possible to obtain a simple, and exact, closed expression for Fcm :

Fcm = −

n o ǫǫ0 ∗ |E0 |2 Re {αeff } Ψ(r1 ) ∇1 Ψ(r1 ) + ∇2 Ψ(r2 ) 2

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where

αeff ≡

iα′′ 1 − ik 2 G12 α′′

(13)

can be understood as an effective dimer’s polarizability. The force could then be understood as a result of a shift in the dimer resonance that leads to a real part in the effective polarizability. However, this description is not strictly valid to describe the actual forces at arbitrary dimer positions and orientation. We shall then use Eq. (10) to obtain information about the dynamical behavior of the dimer: First, the force on the center of mass is equal to zero if Re {G12 } vanishes. For this particular situation, the diffusion of the dimer is purely Brownian and the existence of an external electromagnetic field is not going to modify the dynamics. The real part of the Green function is written as 1 1 1 cos kL 1 − 2 2 − sin kL . Re{G12 } = 4πL k L kL

(14)

First two zeros of equation (14) are given by L = 0.72λ and L = 1.2λ. So, for dimers of those lengths, we expect to find free Brownian dynamics. In second place, we can also obtain information about the behavior of the dimer when the force on the center of mass is different from zero. We will describe the dimer configuration in terms of its center-of-mass coordinates (x, y) and an angular coordinate θ describing the angle of the dimer with respect to the x-axis. The relationship between the monomer’s coordinates and the center of mass is given by

r1 = r + ∆rθ

,

r2 = r − ∆rθ

∆rθ = (

L L cos θ, sin θ). 2 2

if we consider now a fixed orientation of the dimer with respect to the electromagnetic field 8


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Langevin molecular dynamics simulations. First, we must write the equation of motion for the dimer by transforming the coordinates of the monomers(x1 , y1 , x2 , y2 ) into the dimer coordinates (x, y, θ). Then, the equations of motion are given by d2 x dx = −2γ + ξx1 + ξx2 + Fx d2 t dt d2 y dy 2m 2 = −2γ + ξy1 + ξy2 + Fy dt dt 2 d θ dθ mL2 2 = −γ L2 + dt dt 2m

+ L(ξx2 + Fx2 − ξx1 − Fx1 ) sin(θ) + L(ξy1 + Fy1 − ξy2 − Fy2 ) cos(θ) being γ the friction coefficient (γ = 6πaη, η is the viscosity, η = 0.89 × 10−3 kg m−1 s−1 for water at T = 298K and we consider particles with a = 50nm radius) and ξ(t) a thermal force with zero mean and variance given by the fluctuation-dissipation theorem

hξi (t)ξj (t′ )i = 2γkB T δij δ(t − t′ )

(18)

with i and j equal to (x1 , y1 , x2 , y2 ) and T being the system’s temperature. F , F1 and F2 are the optical forces on the center of mass and on each particle respectively obtained through the exact expressions for the fields and their derivatives [with λ = 387.585nm wavelength in water. The refractive index is n = 1.8]. Notice that for the parameters used in the Langevin simulations the condition that the dimer length is larger than three times the particle radius is fulfilled (dipolar limit condition). When no optical force is applied the dimer follows diffusive dynamics with mean squared displacement given by < r2 (t) >= 4D0 t

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(19)

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dimer’s dimension. Note that, since each isolated monomer feels no optical force at all, the decrease in the value of the diffusion constant is due only to the non symmetric scattering interaction among the monomers. That is, the interaction force exerted on particle 1 by 2 is not reciprocal to the interaction force exerted on particle 2 by 1. We already found in the literature a situation in which action-reaction in optical forces is not fulfilled (44 ) for a dimer made up of two different particles. In our case both particles are identical and the asymmetry in the interactions comes from the external field.

(a)

(b)

Figure 4: (a) Typical trajectories of the dimer’s center of mass in x, y ∈ [0, λ) obtained from Langevin molecular dynamics simulations for L = 0.6λ and P = 0.3 × 105 W/cm2 . Polar plots of the dimer orientation probability distribution at specific points “A” to “E” are also shown. (b) The same as in (a) but for for L = 0.84λ and P = 0.8 × 105 W/cm2 . The mean square displacement, the reduction of the diffusion constant and the strong dependence of the main statistical features of the dimer’s dynamics on the dimer’s length L, can be understood in terms of the potential energy landscape obtained in Eq. (17). First, we have performed statistics of the data obtained from the Langevin molecular dynamics simulations to obtain the distribution function for the dimer orientation with respect to the field (θ). Figure 4 shows typical Brownian trajectories of the dimer’s center of mass together with a polar plot of the dimer orientation at specific points for two different dimer lengths. 12


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(a)

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(b)

Figure 6: (a) Probability distribution for the dimer’s center of mass in x, y ∈ [0, λ) obtained from Langevin molecular dynamics simulations for L = 0.6λ and P = 0.3 × 105 W/cm2 . Darker regions correspond to a higher probability (the integral of the distribution over the area of the figure is normalized to 1). Typical Brownian trajectories from “A” to “E” are shown in Fig. 4a. The contour map shows the underlying field intensity landscape [contour lines 2 correspond to 0.15, 0.30, 0,45, 0.60, 0.75 and [red line] 0.90 E (0) max ]. (b) Potential energy landscape −Uθ (x, y)/(kB T ) from Eq. (17) for a fixed angle θ = 0 or θ = π/2 corresponding to the preferential angles observed in Fig. 5a. Potential minima are located in the darker regions.

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It is interesting to note that the relative positions of the dimer, i.e. the potential minima, are not trivially linked to the underlying field intensity map (see Fig. 6a). We should emphasise that this potential landscape is felt by the system only when the two particles are linked together in a dimer. For the isolated dipoles there is no energy landscape at all.

(a)

(b)

Figure 7: (a) Probability distribution for the dimer’s center of mass in x, y ∈ [0, 2λ) obtained from Langevin molecular dynamics simulations for L = 0.84λ and P = 0.8 × 105 W/cm2 . Darker regions correspond to a higher probability. Typical Brownian trajectories from “A” to “E” [or “A” to “C ”] are shown in Fig. (4b). The contour lines show the underlying field intensity landscape as in Fig. 6a. (b) Potential energy landscape −Uθ (x, y)/(kB T ) from Eq. (17) for fixed angles θ = π/4 and θ = π/4 + π/2 corresponding to the preferential angles observed in Fig. 5b. Although the translational Brownian motion can be understood as a diffusive motion in an effective potential energy landscape, this simplified description does not capture the interplay between forces and exerted torques in the Brownian motion of the dimer. The coupling between rotational and translational diffusion processes can be understood by analysing the orientation distributions along the typical paths between trapping positions [indicated by the A,B,C,D,E spots in Figs. 4a and 6a]. Our results show that rotational and translational diffusion processes are clearly locked since dimer’s orientation changes as it moves along the path as shown by the orientation distributions in B,C and D. The results for a different length L = 0.84λ are sumarized in Figs. 4b and 7. Again, 15


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the effective potential landscape minima, with θ = π/4 + nπ/2, capture the most probable locations of the center of mass. However, in contrast with the shorter dimer, each trapping position corresponds now to a single preferred orientation (which is not captured by the simplified effective potential). In this case, the rotational and translational locking is clear since the dimer has to change its orientation as it moves from a trapping position to the next one (see trajectories A-B-C and A-D-E in Fig. 4b).

Conclusions In conclusion, two resonant lossless particles, moving free on an optical lattice, become arrested when they couple to form a dimer. Although our discussion focused on dimers made out of resonant lossless monomers, we expect a similar behaviour for actual dimers of plasmonic nanoparticles at resonance, where the real part of the polarizability can be neglected. The predicted dramatic change on the dynamical properties is due to the self induced back action forces (SIBA forces) arising as a consequence of the non symmetric multiple scattering between the monomers. In first approximation, this interaction results in a potential energy landscape that only prevails when the two particles are linked. The properties of this landscape, including wells depth and position, strongly depend on dimer’s length, making it suitable as an active sorting tool.

Acknowledgement The authors thank financial support from Spanish Ministerio de Economía y Competitividad ( MICINN) project FIS2012-36113-C03 and FIS2015-69295-C3-3-P and the “María de Maeztu” Program Ref: MDM-2014-0377.

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Arrested Dimer's Diffusion by Self-Induced Back-Action Optical Forces

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