Nanovortex‐Driven All‐Dielectric Optical Diffusion Boosting and Sorting Concept for Lab‐on‐a‐Chip Platforms

Abstract The ever‐growing field of microfluidics requires precise and flexible control over fluid flows at reduced scales. Current constraints demand a variety of controllable components to carry out several operations inside microchambers and microreactors. In this context, brand‐new nanophotonic approaches can significantly enhance existing capabilities providing unique functionalities via finely tuned light−matter interactions. A concept is proposed, featuring dual on‐chip functionality: boosted optically driven diffusion and nanoparticle sorting. High‐index dielectric nanoantennae is specially designed to ensure strongly enhanced spin−orbit angular momentum transfer from a laser beam to the scattered field. Hence, subwavelength optical nanovortices emerge driving spiral motion of plasmonic nanoparticles via the interplay between curl−spin optical forces and radiation pressure. The nanovortex size is an order of magnitude smaller than that provided by conventional beam‐based approaches. The nanoparticles mediate nanoconfined fluid motion enabling moving‐part‐free nanomixing inside a microchamber. Moreover, exploiting the nontrivial size dependence of the curled optical forces makes it possible to achieve precise nanoscale sorting of gold nanoparticles, demanded for on‐chip separation and filtering. Altogether, a versatile platform is introduced for further miniaturization of moving‐part‐free, optically driven microfluidic chips for fast chemical analysis, emulsion preparation, or chemical gradient generation with light‐controlled navigation of nanoparticles, viruses or biomolecules.


Expressions for the total angular momentum of an electromagnetic field
In this first section we provide useful, well-known expressions regarding angular momentum which were utilized in the derivations of section 2 in the main text. The angular momentum of a circularly polarized plane wave can be written using the expressions for paraxial waves 1 where inc E is the electric field, 0  is the vacuum permittivity, L is the orbital angular momentum operator 2 defined explicitly in Equation (S15) , and   The first term in the right-hand side of (S1) corresponds to the OAM carried by inc E .
Additionally, without any loss of generality, the total angular momentum surface density of the field scattered by an object can be expressed as 2 : where s S denotes the time-averaged scattered Poynting vector, and r is the position vector.

Excited components of the magnetic quadrupole under circular plane wave illumination
We now proceed to a rigorous derivation of Equations (2)-(3) of the main text. For that purpose, we now express the amount of power extracted from the field by the nanocube (referred to as the extinction power) 2 as   ik z E z E e   and  the helicity or spin of the incident field. Following the steps outlined in Ref. 4 , we consider the current inside the scatterer is primarily driven by the magnetic quadrupole, and therefore has the functional form 1 ( )), ( ) (  3. Higher order multipolar near-field vortices: magnetic quadrupoles We now show a rigorous analytical proof illustrating the direct link between the excited magnetic quadrupole and the singular optical force vortex induced in the near field of the nanocube. We then make a comparison between the near fields of the optimized magnetic quadrupole nanocubes in two different host environments: liquid water (the one utilized in the main text) and air. In the latter case, an even more pronounced magnetic quadrupolar response can be obtained.
We start with the analysis of the multipole fields. In Ref. 5 , the authors clearly showed the equivalence between the multipolar expansions in the Cartesian and Spherical representations.
While the first one allows a more straightforward physical interpretation of the results, the second one can be more convenient for calculations. In what follows, we will make use of the spherical representation. It is thus necessary for completeness to introduce the relations between the excited components of the Cartesian and Spherical 2 nd rank tensors in our system, given in e.g. Ref. 6  Substituting Equation (S12) into (S9) it is straightforward to show that the only excited spherical multipole component will be of the form 2 T  , i.e. m   . This is in agreement with angular momentum conservation 6 .
Neglecting reflections from the glass substrate, the scattered electric fields outside an arbitrary confined source under plane wave illumination with amplitude 0 E can be written as a superposition of vector spherical harmonics 7 : The total field outside the source is then the sum of the incident and scattered fields s inc  E E E . Nevertheless, since we are interested in the near and mid-range fields, in a first approach we can approximate the total field as s  EE . The total scattering forces (considering both spin and Poynting vector contributions) exerted on a dipolar particle by an arbitrary field can be grouped together in 3 We have now all the necessary tools to investigate analytically the near field forces of the nanocube presented in the main text. Specifically, by virtue of Equations (S9) and (S12) and discussions therein, the problem reduces to analyzing the behavior of the forces induced by a magnetic quadrupole source of the form 2 b  , since it corresponds to the resonant contribution to scattering from the nanocube. Setting 2, lm   for the magnetic coefficient in (S13) (corresponding to an incident circularly polarized wave) and considering all the other terms to be zero leads to a simplified expression for the magnetic quadrupole field where 1    for left-and right-hand polarization, respectively. Finally, substituting (S17) into (S16) we obtain the near field optical non-conservative forces exerted by a magnetic quadrupole to a dipolar particle. E.g. for 2 confirming the vortex is confined in the near and mid-field regions. As demonstrated numerically in Figure 3, in the near-field of a multipole source, the azimuthal scattering forces are not necessarily proportional to the Poynting vector, since both spin and radiation pressure contributions have relevant roles. We can easily test this statement via Equation (S18). In fact, we note that at 0 z  ( /2    ), the scattered azimuthal force has a maxima, while the azimuthal component of the scattered Poynting vector of the magnetic quadrupole (Equation (4) in the main text) is zero. This means that the optical force vortex is entirely driven by spin forces.
The resulting force fields of a left-handed magnetic quadrupole 21 b are plotted in Figure S1(a).
Our analytical formulae are contrasted with the numerical results obtained for the nanocube, after a careful optimization of its size in order to maximize their magnetic quadrupole response in air (Figure S1(b)) and water environments (Figure S1(c)).  Table   S2. Both (b,c) are plots in the near-and mid-field zones of the scatterer.
Interestingly, stronger azimuthal forces can be obtained in air. This is because the magnetic quadrupole resonance has a higher quality factor as a result of a larger refractive index contrast between the nanocube and air 4 , while in water the contrast is 2.996 . The multipole decomposition of the optimized nanocube in air is shown in Figure S2. The parasitic electric quadrupole contribution is also lower in this case. This can be appreciated by comparing Figure   S1(b,c) with Figure S1a. The numerical results in air show a better defined vortex, while the optimized nanocube in water has small inward force components at some positions due to the electric quadrupole contribution. Nevertheless, the near field is still dominated by the magnetic quadrupole. We remark that the considerations in this section are only valid when the influence of the substrate can be neglected. Otherwise, Equation (S13) must be modified in order to take into account the multipole reflections from the substrate. Unfortunately, no closed analytical solution exists for this problem, except for extreme cases when the substrate can be assumed a perfect electric or a perfect magnetic conductor 3 .

Orbital torque induced by the scattering forces
A particle scattering as an electric dipole in an arbitrary electromagnetic field will be affected by an orbital torque entirely driven by the scattering forces. The z component of the optical with r  the distance to the z-axis. Equation (S19) clearly illustrates that the amount of orbital torque transmitted to the particles depends on the radiation pressure and the helicity spatial distribution of the scattered field by means of S  and s L , as well as the optical response of the particle itself by means of "  . In particular, Equation (S19) is utilized to obtain the results shown in Figure 3.

Dielectric function of Au nanoparticles
The calculations of the Au nanoparticle polarizabilities in Figure 4 are performed for the well-known optical dispersion properties of bulk Au 8 ( Au b  ), taking into account the Drude size corrections due to the limitation of the electron mean free path in small metallic particles 9 where pl  , b  and F v are the plasma resonant frequency, the damping constant from the free electron Drude model, and the Fermi velocity, respectively.

Validity of the dipole approximation
The additional figures provided in this section are given in order to justify the validity of the dipole approximation considered in Equation. (9) for the analytical treatment of the optical forces induced by the dielectric scatterer in relatively large Au nanoparticles (20-50 nm radii).
In figure S1(A) we have plotted the exact contribution of electric (TM) and magnetic (TE) Mie coefficients to the total scattering cross section of a single 40 nm radius Au spherical nanoparticle submerged in water (i.e. environment refractive index is taken to be 1.34). The dispersion relation of Au is the same as in the main text. As it can be appreciated, radiation in the visible range is entirely described by the first electric Mie coefficient, corresponding to the total electric dipole contribution. Thus, this first result confirms that the investigated Au nanoparticles effectively radiate as dipoles. However, the derivation of Equation    Table S1. Information regarding materials involved in the hydrodynamical simulations.

Analytical formulas taking into account particle-wall interactions: viscosity tensor
Accurate predictions of the particle dynamics require a quantitative understanding of particlewall interactions. In a general situation the fluid flow around the nanoparticles will be affected by the walls of the nanocube, the glass substrate or the lateral walls of the microchamber. We will take the first two into account in our model and neglect the effect of the third ones due to the reduced size of the optical nanovortex in comparison with the microchamber. Hereafter, we introduce the linearized Navier-Stokes equations and their solution in the presence of two perpendicular walls. Based on these formalism we are able to accurately reproduce the effective viscosity experienced by the nanoparticles in the cases of particle-substrate and particle-cubesubstrate interactions.
We model the Au nanoparticles as point forces (Stokeslets) arising in the calm fluid due to the optical field and Brownian forces. In the viscous regime, the governing equations for the fluid motion are the linearized Navier-Stokes equations 18 where F is a point force corresponding to the external forces acting on a Au nanoparticle, u is the fluid velocity and p the pressure. We further impose 0  u at infinity and at the positions of any wall in the system (no-slip). The general approach 19 consists in rewriting both u and p in terms of another vector function which we call ψ : with the parameter / pp q R z  and  is the ratio of the distance between the closest wall of the nanocube d and the radius of the nanoparticle p R (see Figure S3). The effective viscosity is now a tensor quantity no longer corresponding to the viscosity of water, defined as: The i k functions are directly related to the integrals of Equation (S25). In our approach we simply interpolate their numerical values which are given in Table I and Table II  showing that the effect of the walls of the nanocube is already negligible at this distance, and the glass substrate is the leading contribution to the effective viscosity tensor. Since 1 q  (i.e. the nanoparticles are almost in contact with the glass substrate due to radiation pressure in the z direction, as explained in the main text), they are at the onset of the lubrication regime 26  However, this force is orders of magnitude lower than the conventional drag force 19 , and consequently it is fully compensated by the radiation pressure of the incident beam, keeping the movement restricted in the x-y plane.

Expressions for viscous and Brownian forces in the laminar regime
For small spherical geometries, the Brownian and viscous forces can be expressed as 10 :

Calculations of the MSAD and MSD
The MSAD and MSD shown in the main text were calculated by averaging the azimuthal angle