Sorting Nanoparticles with Intertwined Plasmonic and Thermo

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Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces Aurelien Cuche, Antoine Canaguier-Durand, Eloïse Devaux, James A. Hutchison, Cyriaque Genet, and Thomas W. Ebbesen Nano Lett., Just Accepted Manuscript • DOI: 10.1021/nl401922p • Publication Date (Web): 08 Aug 2013 Downloaded from http://pubs.acs.org on August 13, 2013

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Sorting nanoparticles with intertwined plasmonic and thermo-hydrodynamical forces A. Cuche ISIS & icFRC, University of Strasbourg and CNRS (UMR 7006), 8 allée Gaspard Monge, 67000 Strasbourg, France and CEMES, University of Toulouse and CNRS (UPR 8011), 29 rue Jeanne Marvig, BP 94347, 31055 Toulouse, France A. Canaguier-Durand, E. Devaux, J. A. Hutchison, C. Genet,∗ and T.W. Ebbesen ISIS & icFRC, University of Strasbourg and CNRS (UMR 7006), 67000 Strasbourg, France (Dated: July 8, 2013)

Abstract We exploit plasmonic and thermo-hydrodynamical forces to sort gold nanoparticles in a microfluidic environment. In the appropriate regime, the experimental data extracted from a Brownian statistical analysis of the kinetic motions are in good agreement with Mie-type theoretical evaluations of the optical forces acting on the nanoparticles in the plasmonic near field. This analysis enables us to demonstrate the importance of thermal and hydrodynamical effects in a sorting perspective. Key Words: Optical sorting, metallic nanoparticles, Mie theory, thermo-hydrodynamical forces, plasmofluidics, Brownian motion.

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The involvement of surface plasmon (SP) modes when manipulating objects in micro/nanofluidic systems [1] has allowed for totally new functionalities, stimulating a new area of research, so-called plasmofluidics [2]. In contrast with evanescent fields above dielectric surfaces [3–6], SP modes excited on a metal film give rise to very efficient momentum transfers enabling direct object manipulation, as recognized recently [7–10]. Of particular interest is the pressing issue of being able to perform dynamical and non-contact size sorting of nanoparticle populations in fluids. This is one of the most challenging targets in conventional microfluidics, as witnessed by the huge amount of work in designing various techniques in order to achieve such a goal [11–19]. We demonstrate in this Letter that SP fields represent an outstanding tool to perform size sorting of metallic nanoparticles. We thus take here one step further with respect to recent work that have shown the propulsion capacity of SP modes in fluidic environments [7, 10]. Our method is based on exploiting the size-dependent optical resonances of the metallic nanoparticle as a mean of motional control through the enhanced near-field coupling between the nanoparticle and a resonant SP mode propagating along a metal-fluid interface. In this context, it is crucial to fully understand the forces at play. Although all our evaluations will be multipolar, the forces can be most easily described in the Rayleigh regime of point-like nanoparticles of complex polarizability α where the total SP force (time-averaged) acting on the particles in a fluid of refractive index n can be split into two contributions: a reactive one (known as the gradient force) and a dissipative one (the radiation pressure) which is essentially related to scattering processes through the extinction cross-section Cext = kIm [α]. One thus writes Re [α] hFiT = ∇|ESP |2 + ωµ0 Im [α] hSiT (1 + ∆) 2 n 4

(1)

The gradient force contribution is directed towards the metal surface due to the exponential decay of the plasmonic field ESP away from the interface. It attracts the particles towards the metal surface, maintaining them in the near field of the SP mode. The radiation pressure sets the particles into motion along the time-averaged Poynting vector hSiT of the SP field with two components, one dominating and directed along the metal surface and a second one pointing towards the metal surface. The ∆ correction to the radiation pressure corresponds to the well-known contribution due to the spin density of the field [20, 21]. Because the SP mode is associated with strong electromagnetic field confinement, optical forces are enhanced, in particular when the SP mode is tuned close to the nanoparticle intrinsic 2


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FIG. 1: (a) Schematic of the setup with the two lasers at wavelengths λ1 = 594 nm and λ2 = 980 nm. (b) Scanning Electron Microscope image of two gold nanoparticles with respective radius of 50 and 125 nm, lying on a 50 nm thick gold layer. Scale bar: 200 nm. (c) Experimental (dots) and theoretical (solid line) reflectivity, for the same 50 nm thick gold layer under Kretchmann illumination at λ1 (black) and λ2 (red), as a function of the incident angle θinc . The metal film is vertically oriented in the experiment, with the gravity vector g along the y < 0 axis.

resonances. We exploit the size-dependency of the spectral shape of the scattering cross-section Cext of the nanoparticle to modulate these forces. We thereby show an efficient method for separating two populations of Au nanoparticles with radii r = 50 nm and r = 125 nm, immersed in a fluidic cell. The wavelength λ tunability of the SP mode not only enables coupling to the nanoparticles at different values of Cext but also allows different regimes of forces to be revealed. A multipolar evaluation of the optical forces acting on the nanoparticle turns out to be crucial in this context in order to understand the motional evolutions at all r/λ ratios. The central aspect of this study lies in the fact that these motional effects must be discussed in relation with thermal and hydrodynamic processes. Indeed, and in particular when involving SP excitations, these effects constitute a challenge in microfluidics and are usually considered an unwanted source of kinetic motion [22]. They have triggered recent work that aimed at avoiding as much as possible such effects when discussing SP-based optical trap setups [23]. Taking the opposite view, we see thermal and hydrodynamic processes as an opportunity for additional motional control of nanoparticles in micro- and nanofluidic environments, in a spirit close to thermal nonequilibrium transport in molecular and colloidal sciences [24, 25]. This goes along with current 3


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FIG. 2: (a) Experimental trajectories of single r = 125 nm particles under TM (red curve) and TE (blue curve) illumination of the gold film at λ2 = 980 nm. Scale bar is 5 µm. (b) The ballistic shift µ is extracted, in the Langevin representation, from the location of the mean value of the displacement distribution in the (∆x,∆y) plane with respect to free Brownian motion (centered at ∆x=∆y= 0) under TM (red dot) and TE (blue dot) illumination. The distribution is zoomed-in from the whole distribution (inset).

approaches exploiting thermal effects in lab-on-a-chip platforms to control fluid dynamics [26]. We demonstrate, and therefore emphasize here, the importance of intertwined sources of motion in the context of dynamical control of nanoparticles in plasmofluidics. Our experiments were performed using a total internal reflection setup (TIR) sketched in Fig. 1(A). The sample consists of a water cell (thickness h = 70 µm) on top of a semi-transparent 50 nm gold film. The film is evaporated on a glass coverslide, and different colloidal suspensions of gold nanoparticles (radius of 50 and 125 nm (see Fig. 1(B)) are injected sequentially inside the cell. SP modes are launched along the metal/water interface from TIR with a TM polarized laser beam. The modes are resonantly excited at different wavelengths and corresponding illumination angles θinc , as illustrated in Fig. 1(C). Particles immersed in the cell will undergo Brownian motion induced by the stochastic √ Langevin force Fth. = 2kB T γ with thermal energy kB T and viscous damping in the fluid characterized by the Stokes drag γ = 6πηr. For an Au nanoparticle of mass m in water of viscosity η, friction prevails over inertia by several order of magnitudes (γ/m ∼ 107 s−1 ). In this regime of low Reynolds numbers, the ballistic motion associated with the SP radiation pressure exerted 4


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FIG. 3: (a) Experimental wavelength dependency of the efficiency factors Qext = Cext /πr2 for gold nanospheres with radii r = 50 nm (black line) and r = 125 nm (red line) in solution [27]. (b) Mie theoretical extinction cross-sections for the same diameters. Relative amplitudes account for the difference in particle concentrations in solution. The two laser wavelengths λ1 = 594 nm and λ2 = 980 nm are indicated by the vertical black dashed lines. (c) Multipolar evaluation of the x component of the total SP force as a function of the particle radius r, at λ1 (blue curve) and λ2 (red curve). (d) Theoretical evaluation of the evolution of the force difference exerted on nanoparticles by two SP counter-propagating modes as a function of r. The mode excited at λ1 propagates leftward, while the other one at λ2 propagates rightward.

on the nanoparticles can be detected over the intrinsic Brownian motion as clearly shown in Fig. 2(A). The highly confined SP field, excited under TM polarization, is a source of strong radiation pressure that propels efficiently the particles in the cell. This contrasts with the case of a TE polarization where the non-plasmonic evanescent field at the metal-fluid interface hardly perturbs the natural Brownian motion of the particles. As we have shown recently [10], the effective strength of the radiation pressure moving the nanoparticles can be quantified from the measured ballistic shift µ of the probability distribution associated with the ensemble of elementary particle displacements that compose a trajectory along the in-plane orientation of the Poynting vector -Fig. 2(B). The experimental resolution in terms of force determination is obtained from the noise level fixed by the smallest averaged value of the distribution shift µ ¯TE obtained under TE polarization. This provides an expectation value for the actual shift µeff. = |¯ µTM − µ ¯TE | from which is evaluated the effective plasmonic force FSP = (µeff. γ)/∆t that conveys the particles over the sampling time interval ∆t. 5


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To validate an event, we choose the following criterion: the amplitude of the barycenter shift of the displacement distribution has to be larger than the resolution of our setup given by the absolute value of the maximal shift under TE illumination µeff. > |µmax TE |. The statistical approach provides √ confidence intervals σµ = σ/ N evaluated for each data set, which, added to experimental error bars, correspond to a statistical resolution criterion for the quantitative results. Remarkably, the methodology allows the measure of optical forces smaller than the intrinsic Langevin force at play r=50 r=125 in the fluid (evaluated in water at room temperature as Fth = 2.79 fN and Fth = 4.42 fN for

the two radii considered). Note that the optical field decays exponentially from the metal-fluid interface, over ` ∼ 200 and 500 nm at λ1 = 594 and λ2 = 980 nm respectively. Because it thus depends on the particle height z, FSP is not a constant of motion and we can only measure an effective driving strength of the extinction force, with a lowest estimate given through the Stokes limit for the viscous damping term γ. We present in Fig. 3(A) the extinction cross sections measured for colloidal assemblies of Au nano particles of radii r = 50 nm and r = 125 nm. As clearly seen, the cross-sections evolve with the radii, with a red-shift and a broadening of the resonance as the radius increases. These evolutions are related to the progressive onset of scattering multipoles as seen from the very good match between the experimental cross-section and the one given from Mie theory -Fig. 3(B). Since Cext characterizes the optical response of the particle to a broad-band excitation, it immediately appears interesting to choose two different SP excitation wavelengths in relation with the radii, coupling to spectral regions respectively close and far from the nanoparticle resonance. As explained above, this will lead to different regimes of radiation pressure strengths and we accordingly chose the two values (λ1 = 594 nm, λ2 = 980 nm). These regimes are fully described by evaluating the SP force exerted on a particle using Mie theory when the particle is touching the metal film (see Supplementary Information and [28]). The x-component of the plasmonic force, directing the motion along the metal surface, is displayed in Fig. 3(C) as a function of the particle radius, for each wavelengths λ1,2 (more details and z component are given in the Supplementary Information section). This evaluation immediately brings to the fore the relevant choice of radii, based on the cumulative effects of the particle resonances on the force amplitude as r increases. At λ1 = 594 nm, fundamental and first order shape resonances of the nanoparticles are excited for r = 50 nm and r = 125 nm respectively. This contrasts strongly with the situation at λ2 = 980 nm which is only in tune with the first particle 6


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FIG. 4: Experimental distribution shift µeff. in the (∆x, ∆y) plane under λ2 = 980 nm (a) and λ1 = 594 nm excitation (b) for r = 50 nm (labeled as a small yellow circle) and r = 125 nm (large yellow circle) ax for r = 50 nm (black dashes) and r = 125 nm nanoparticles. Vertical dashed lines correspond to µM TE (blue dashes). Error bars are related to experimental reproducibility. Statistical confidence intervals are also displayed as colored surfaces for σµ (dark gray) and 2σµ (light gray). The number of data points for each measured ensemble of trajectory displacements is indicated. (c) Comparison between experimental (red dot) and theoretical (blue square) ratios Fxr (λ2 )/Fxr (λ1 ) for the two sphere sizes. (d) Same for Fxr=125 /Fxr=50 at λ1 and λ2 .

resonance for r = 125 nm. These differential effects can be exploited most efficiently in the counter-propagating scheme proposed recently on a non-plasmonic TIR setup [5]. By evaluating theoretically in Fig. 3(D) the evolution, as a function of radius, of the force difference exerted on nanoparticles by two SP modes excited respectively at λ1 and λ2 and counter-propagating along the metal film, an optimal SP sorting strategy immediately appears close to the two chosen radii. The strong difference, expected from Fig. 3(C), in SP radiation pressures at λ2 = 980 nm between the two radii is indeed clearly observed in our experiment, as shown in Fig. 4(A). The large r=125 particles with r = 125 nm are propelled along the x direction with a measured force FSP,λ ∼ 6.77 2 r=50 fN much stronger than for r = 50 nm with FSP,λ ∼ 0.13 fN. This value, below the noise level set 2

by the TE polarization, clearly reveals that at this wavelength, the smaller particles are basically transparent to the plasmonic near field. This situation must be contrasted with the second regime at λ1 = 594 nm. This time, the r = 50 nm nanoparticles are conveyed along the surface within the SP near field with an effective 7


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FIG. 5: (a),(b) Theoretical planar Fx and (d),(e) vertical Fz components of the normalized plasmonic force (in units of 10−15 ) calculated in the dipolar (red curve) and multipolar (black curve) regimes, with 10 modes taken into account, and exerted on a r = 50 nm (labeled as a small yellow circle) and a r = 125 nm (large yellow circle) gold nanospheres. The film-particle distance is fixed at 20 nm, i.e. typically the Debye length of the solution. The results of the calculations performed in the dipolar regime are presented here to emphasize the importance of a multipolar approach when aiming at precisely evaluating SP radiation pressure. r=50 force FSP,λ clearly above the TE noise value (Fig. 4(B)). But the large particles turn out almost 1 r=125 insensitive to the in-plane SP force with a measured FSP,λ as small as ∼ 1.13 fN, a surprise when 1

comparing the theoretical expectations giving Fxr=125 (980)/Fxr=125 (594) ∼ 1.4 (Fig. 5(B)). Also, we observe in Fig. 4(B) a strong reduction of the number of data points (Nλr=125 ∼ Nλr=50 /15) 1 1 despite the fact that theory predicts that the larger particles should be better held in the near field than the smaller with Fzr=125 (594)/Fzr=50 (594) ∼ 2 (Fig. 5(C) and (D) and Supplementary Information). These startling results are in discrepancy with theory and show that the sorting efficiency cannot be solely induced by optical effects. We emphasize that Brownian motion does also not help: with the diffusion coefficient D = kB T /γ scaling inversely with the radius r, the smaller spheres should exit more rapidly the near field than the larger ones. This would yield a reduction of Nλr=50 , a conclusion not verified experimentally. 1 These observations reveal the presence of an additional contribution that pushes the nanoparticles away from the Au film surface explaining the significant reduction in Nλr=125 . This contri1 bution must work along the z < 0 axis against Fz in order to shorten the averaged length of the

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measured trajectories and, hence, reduce the in-plane effective drive measured by FSP . We believe that it is induced by thermal ∇T and fluid velocity ∇v gradients (schematized in the insert in Fig. 4) that derive from the efficient heating of the surrounding fluid by the metallic structures. To support this contention, we need to show that these gradients (i) are much stronger at λ1 = 594 nm than at λ2 = 980 nm and (ii) induce effects that transfer preferentially the larger nanoparticles away from the Au film surface. The first condition is clearly assessed from a heat transfer analysis. We estimate that the local heating of the fluid in the vicinity of the nanoparticle due to its own absorption is below 1 ◦ C (for the laser excitation density used in our experiments) and is thus negligible [29]. In contrast, the heat transferred to the fluid by the plane film can be important at the SP resonance and is given from the Joule heat power density as [23] q(r) ∝

ω Im [εAu (ω)] ATM (ω) Iinc 2

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where εAu is the permittivity of the metal. Iinc is the intensity (i.e. in W · cm−2 ) of the incident laser illuminating the finite thickness film under TIR with ATM (ω) the associated wavelengthdependent absorption coefficient. The most direct consequence of the local heating is the onset of thermal gradients and the build-up of Rayleigh-Bénard convection cells that impact the motion of the nanoparticles within the fluid. This is clearly observed when measuring the laser power and wavelength dependence of the force Fxr=125 . As the incident power is increased, thermal and convection effects end up changing the whole motional dynamics of the nanoparticles, as seen in Fig. 6(A). The linear increase, expected from Eq.(1) of the effective force as a function of incident power, is disrupted above a certain threshold. Remarkably, the wavelength-dependent positions of the thresholds correspond to a power ratio Pλthres. > Pλthres. in agreement with the estimated heat 2 1 power density ratio qλ1 > qλ2 by the Au film (see Supplementary Information). The highest heat transfer efficiency is thus reached at λ1 = 594 nm (i.e. close to Au interband transitions) and sets Pλthres. = Pexp. = 18 mW as the chosen power for all sorting experiment presented in this work. 1 The second condition that could explain the better transfer of the larger particles away from the Au film surface requires a full discussion of the phenomena at play in the system. Because the vertical component Fz scales quasi-linearly with the radius r since Fzr=125 (594)/Fzr=50 (594) ∼ (125/50), we must seek thermo-hydrodynamical forces that are opposed to the z > 0 component of the total SP force while scaling more than linearly with r (Fig. 5 and Supplementary Information). 9


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Obviously, thermal and hydrodynamical effects jointly contribute to the particle motions, such as thermophoresis and convection. Thermal gradients, on the one hand, can move the particles along the z < 0 axis with a velocity v = −DT ∇T where DT is the thermophoretic mobility. From hot (metal surface) to cold regions (center of the cell), this displacement is oriented away from the metal surface. Buoyancy, on the other hand, determines convective cells in the x − y plane with main streamlines along the y > 0 axis, given the vertical orientation of our sample (see Fig.1). The associated drag force Fdrag = γv yˆ moves the particles along the y axis and scales linearly with the incident power as perfectly seen in Fig. 6(B). This force too can progressively remove the particles from the plasmonic near field which extension is defined by the surface S = 200 × 100 µm2 of the optical excitation spot [30]. However, DT cannot scale more than linearly with the radius of the particle [31, 32] and Fdrag scales linearly with r through the drag coefficient η. Therefore, both thermophoretic and convective sources of motion can hardly explain the strong r-dependent reduction in the length of the particle trajectories reported in Fig.4(B). Interestingly though, convection induces velocity gradients ∇v along the z axis due to the v(z = 0) = 0 no-slip condition at the metal surface. We have checked in Fig. 6(B) that such gradients, depending on the laser power, extend through the whole fluidic cell, over much larger distances than the SP near field, with a maximal velocity reached at its center (z = −35 µm). This allows us to consider the possibility that nanoparticles, immersed in such a shear flow, will experience a lift force acting normally to the drag force, in the z < 0 direction [33]. In our experiments, the particles are nearly touching the metal surface (up to the Debye layer), bounding therefore the shear flow. In this situation, the lift force is Flift ∝ −ρ|∇v|2 r4 zˆ, with ρ the fluid density [34]. The strong non-linearity in r appears consistent enough with the observations of Fig. 4(D) to suggest the lift as a good candidate for the more efficient decoupling mechanism of the larger spheres. A simple evaluation, estimating ∇v from the data in Fig. 6(B), predicts that |Flift | & |Fzr=125 | as soon as the nanosphere is located 1 µm away from the metal surface (see Supplementary Information). The vertical resultant force |Flift − Fz | turns smaller than the r=125 Langevin force Fth. evaluated above but it certainly has a cumulative effect on the particle axial

displacements. This implies that the larger spheres, as they vertically diffuse, will on average spend less time close to the metal surface, consistently with the measured reductions of both Nλr=125 and 1 FSP . A quantitative evaluation of the lift accounting for the superimposed Brownian motion of the particles lies beyond the scope of this work, although it can certainly be tackled by adapting to 10


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vertical motions the same methodology used for analyzing plasmonic in-plane driven motions. Here, we rather want to emphasize the important observation that the combined influence of these thermo-hydrodynamical contributions, sketched in Fig. 6(C), leads to an enhancement of the SP force difference exerted on the larger particles at λ1 and λ2 . This effect thus favors the efficiency of the whole sorting strategy. To conclude: our experimental and theoretical study has demonstrated that sorting Au nanospheres can either be performed through purely plasmonic coupling or via thermal and hydrodynamical effects. Careful theoretical evaluations in the multipolar regime of the plasmonic forces at play have led us to show that the kinetic motions at λ1 = 594 nm can only be understood by enlarging the motional spectrum to the contributions from the thermal and hydrodynamical forces. By doing so, it is important to stress that a simple dipolar approach for evaluating the near-field forces is inappropriate given the specific r/λ ratios involved in our experiments. Our data have revealed how a subtle interplay taking advantage of the high field confinement of SP modes and the size-dependent thermo-hydrodynamics forces eventually moves with high efficiency a specific population of nanoparticles with respect to another. Acknowledgments - We acknowledge support from the ERC (Grant 227557) and from the Agence Nationale de la Recherche (ANR, Equipex Union).

∗

Electronic address: [email protected]

[1] D. Erickson, X. Serey, Y.-F. Chen, and S. Mandal, Lab Chip 11, 995 (2011). [2] J. Kim, Lab Chip 12, 3611 (2012). [3] S. Kawata and T. Sugiura, Opt. Lett. 17, 772-774 (1992). [4] S. Gaugiran, S. Gétin, J. M. Fedeli, and J. Derouard, Opt. Express 15, 8146-8156 (2007). [5] M. Ploschner, T. Ci˜zmár, M. Mazilu, A. Di Falco, and K. Dholakia, Nano Lett. 12, 19231927 (2012). [6] M. Tamura and T. Iida, Nano Lett. DOI: 10.1021/nl302716c (2012). [7] K. Wang, E. Schonbrun, and K.B. Crozier, Nano Lett. 9, 2623-2629 (2009). [8] M.L. Juan, M. Righini, and R. Quidant, Nat. Photon. 5, 349 (2011). [9] A. Cuche, O. Mahboub, E. Devaux, C. Genet, and T. W. Ebbesen, Phys. Rev. Lett. 108, 026801(2012). [10] A. Cuche, B. Stein, A. Canaguier-Durand, E. Devaux, C. Genet, and T.W. Ebbesen, Nano Lett. 12, 4329-4332 (2012).

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FIG. 6: (a) Experimental effective force FSP and planar velocity along x of r = 125 nm particles under TM illumination as a function of the incident power at λ1 (black points) and λ2 (red points) in the near field (h < 0.5 µm). At λ1 , experimental values of the force are multiplied by a factor 5 for sake of clarity. A linear fit of the linear regime of FSP is presented as a solid straight grey line. In both figures, the vertical dashed line indicates the power Pexp. used systematically in all experiments. (b) Experimental drag force Fdrag and planar velocity along the y axis of r = 125 nm nanoparticles under TM illumination as a function of the incident power at λ1 for sphere-surface distance h ≈ −1 µm (green dots) and h = −35 µm (blue dots). A linear fit is presented as a solid straight line with the appropriated color code. Note that the x − y plane being vertically oriented, the negative values observed for Fdrag and v are a signature of gravity that leads to sedimentation of nanoparticles for all sizes. (c) Sketched representation of the setup with the illumination beam totally reflected at the glass-metal interface launching the SP mode along the metal-fluid interface and generating thermal ∇T and convective velocity ∇v gradients along the z axis. The different forces (other than the Brownian force) that have intertwined motional effects on the particle are schematized: the plasmonic force FSP in the (x, z) plane directed towards the metal surface, the drag force Fdrag opposed to gravity in the y direction, and the combination of the lift and thermophoretic forces in the −z direction, away from the metal surface. [11] J.P. Novak, C. Nickerson,S. Franzen,D.L. Feldheim, Anal. Chem. 73, 5758-5761 (2001). [12] C. Contado and R.J. Argazzi, J. Chromatogr. A 1216, 9088-9098 (2009). [13] T. Kaneta, Y. Ishidzu, N. Mishima, and T. Imasaka, Anal. Chem. 69, 2701 (1997).

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Nano Letters

[14] M.P. MacDonald, G.C. Spalding, and K. Dholakia, Nature 426, 421 (2003). [15] M. Pelton, K. Ladavac, and D.G. Grier, Phys. Rev. E 70, 031108 (2004). [16] T. Ci˜zmár, M. Siler, M. Sery, P. Zemanek, V. Garcés-Chávez and K. Dholakia, Phys. Rev. B 74, 035105 (2006). [17] R.F. Marchington, M. Mazilu, S. Kuriakose, V. Garcés-Chávez, P.J. Reece, T.F. Krauss, M. Gu and K. Dholakia, Opt. Express 16, 3712 (2008). [18] P. Jakl, T. Ci˜zmár, M. Sery and P. Zemanek, Appl. Phys. Lett. 92, 161110 (2008). [19] K. Xiao and D.G. Grier, Phys. Rev. Lett. 104, 028302 (2010). [20] J. Arias-Gonzales and M. Nieto-Vesperinas, JOSA A 20, 1201 (2003). [21] S. Albaladejo, M.I. Marqués, M. Laroche and J.J. Sáenz, Phys. Rev. Lett. 102, 113602 (2009). [22] V. Garcés-Chávez, R. Quidant, P. J. Reece, G. Badenes, L. Torner and K. Dholakia, Phys. Rev. B 5, 5457 (2011). [23] J.S. Donner, G. Baffou, D. McCloskey and R. Quidant, ACS Nano 5, 5457 (2011). [24] A. Würger, Rep. Prog. Phys. 73, 126601 (2010). [25] R. Golestanian, Phys. Rev. Lett. 108, 038303 (2012). [26] G. Baffou and R. Quidant, Laser Photonics Rev. 7, 171 (2013). [27] H. C. van de Hulst, Light scattering by small particles (Dover Publications, New York, 1981). [28] Y. G. Song, B. M. Han, and S. Chang, Opt. Commun. 198, 7 (2001). [29] Y. Seol, A. E. Carpenter and T. T. Perkins, Opt. Lett. 31, 2429 (2006). [30] Note that although present at all powers, the force has no effect at Pexp. = 18 mW on the FSP measurements since it is perpendicular to the orientation of the SP force given by Eq.(1). [31] S. Duhr and D. Braun, Phys. Rev. Lett. 96, 168301 (2006). [32] M. Braibanti, D. Vigolo, and R. Piazza, Phys. Rev. Lett. 100, 108303 (2008). [33] J. Happel and H. Brenner, Low Reynolds number hydrodynamics (Kluwer editions, The Hague, 1983). [34] D. Leighton and A. Acrivos, J. Appl. Math. Phys. 36, 174 (1985).

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Sorting Nanoparticles with Intertwined Plasmonic and Thermo

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