Experimental setup

We investigate the dynamics and probability distribution of gold nanoparticles trapped in a focused laser beam (λ = 785 nm). We employ commercially available monodisperse nanoparticles with radius a = 75 nm (Sigma Aldrich, < 12% variability in size). Although often referred to as nanospheres, these nanoparticles feature a crystalline structure that distinguishes them from an ideal sphere, as can be seen in the SEM image in Fig. 1a.

Fig. 1

Nanoparticles and their driving mechanism.

a SEM image of the gold nanoparticles employed in this work. From this image, it can be appreciated how they are approximately spherical, but also feature characteristic crystalline facets. The scale bar is 150 nm. b A nanoparticle (radius a) is trapped in a harmonic potential by a focused laser beam (magenta shading, propagating upwards in the direction of the red arrow, beam width σ). The particle is confined in a quasi-two-dimensional space in the xy-plane near the sample upper cover glass at distance d by the competing effects of the optical scattering force and the electrostatic repulsion by the glass. Depending on its distance from the trap center, the nanoparticle is irradiated by different intensities and, therefore, reaches different temperatures T. If T exceeds the critical temperature T_c, a concentration gradient is locally induced around the nanoparticle (green ring surrounding the particle), thereby leading to a drift velocity away from the trap’s center.

As schematically shown in Fig. 1b, the trapping beam propagates upwards and is focused near the top cover glass surface of the sample cell. The nanoparticle is confined along the vertical z-direction at distance d from the cover glass by counteracting actions of the radiation pressure pushing it towards the cover glass and of the short-range electrostatic repulsion pushing it away from the glass surface²². Therefore, the nanoparticle is effectively confined in a quasi-two-dimensional space in the xy-plane parallel to the cover glass, where it is trapped by an optical tweezers in a harmonic optical potential, i.e., $$ V\left(r\right)=-V_0{e}^{-\frac{1}{2}{r}^2/{\sigma }^2}$$, where $$ r=\sqrt{x^2+y^2}$$, σ is the beam waist and where the prefactor V₀ = KP is proportional to the power P by the proportionality constant K.

A schematic of the experimental setup is shown in Supplementary Fig. 1. The nanoparticle motion is captured via digital video microscopy at 719 frames per second.

Non-equilibrium state

We start by trapping the particle in water to establish a baseline in a standard medium²³. The trajectories and the resulting probability density histograms at laser power P = 4.4 and 7.3 mW are shown in Fig. 2a. The data are fitted with the Boltzmann probability density $$ {\rho }_{\mathrm{eq}}\propto \exp \left(-\frac{V}{k_{\mathrm{B}}T}\right)$$. The particle is confined at the center of the beam, where the potential may be replaced by its harmonic approximation V_h = V₀r²/σ². Indeed, the data in Fig. 2a are very well described by a Gaussian profile. Since the stiffness of the potential increases with laser power, the distribution function is narrower at larger P; consequently, an even narrower distribution function is expected at larger laser powers (e.g., P = 10.16 mW).

Fig. 2

Nanoparticle trajectories and probability density distributions.

Trajectories (top) and probability density ρ(r) (bottom) for a nanoparticle with radius a = 75 nm held by an optical tweezers, a in water at powers P = 4.36 (gray), and 7.25 mW (black), and, b in a critical mixture of water-2,6-lutidine at P = 4.36 (blue), 7.25 (yellow), and 10.16 mW (orange). Sample trajectories in the xy-plane are shown for 200 ms, while the background shading indicates the counts (darker colors indicate higher counts); the scalebar is 1 μm. The probability densities ρ(r) are calculated from data acquired for 1 s. a In water, the data are well described by the Boltzmann distribution $$ {\rho }_{\mathrm{eq}}\left(r\right)\propto \exp \left(-\frac{V}{k_{\mathrm{B}}T}\right)$$ (solid lines), which becomes narrower as the laser power P and, therefore, the optical trap potential depth increase. b In water-2,6-lutidine, the particle features an out-of-equilibrium distribution, which broadens with increasing laser power. Here, the solid lines are given by Eq. (5). All data are taken at environment temperature T₀ = 3 °C, i.e., ≈ 30 K below T_c. Due to absorption, the particle’s surface temperature increases by 6 K mW⁻¹ such that, at P = 7.25 and 10.16 mW, T exceeds T_c and the solution locally demixes. The radial distance has been normalized by the beam waist σ = 340 nm.

We then study a nanoparticle in a near-critical mixture of water and 2,6-lutidine at a critical lutidine mass fraction c_c = 0.286 with a lower critical point at the temperature T_c ≈ 34 °C (see phase diagram in Supplementary Fig. 2)²⁴. At a temperature T below T_c the mixture is homogeneous and behaves as a standard viscous fluid (just like water). When T approaches T_c density fluctuations emerge, leading to water-rich and lutidin-rich regions. Finally, when T exceeds T_c the solution demixes into water-rich and lutidin-rich phases.

Absorption of part of the laser light of the trapping beam heats the nanoparticle and results in a temperature profile in its vicinity. If the surface temperature exceeds T_c, a critical droplet with a modified water content Φ(r) forms around the nanoparticle. Its excess surface temperature is proportional to the laser power. By choosing the critical temperature T_c, attained at the critical power P_c, as a reference point, the excess temperature can be written as

In Fig. 2b, we show the probability densities for a nanoparticle trapped at three different laser powers in a near-critical mixture kept at T₀ = 3 °C via a heat exchanger coupled to a water bath (i.e., about 30 K below T_c). At low laser power (P = 4.36 mW, T = 31 °C < T_c), the nanoparticle position distribution is qualitatively similar to that of the nanoparticle in water (Fig. 2a) and features only very small deviations from a Gaussian profile. As we raise the laser power (P = 7.25 mW, T = 45 °C > T_c), the nanoparticle position distribution acquires a distinctively non-Gaussian shape. Finally, as we raise the laser power even further (P = 10.16 mW, T = 63 °C ≫ T_c), the nanoparticle position distribution develops a peak at a finite radial distance r from the trap center, which is also observed in the form of a ring in the histogram of the trajectories. These non-Gaussian distributions cannot be ascribed to a harmonic potential at higher effective distribution and are clear signatures of the out-of-equilibrium nature of this system.

Self-propulsion of near-spherical particles

Figure 3 shows the velocity profile v(r) as a function of the distance from the beam axis, as well as its radial and azimuthal components v_r and v_θ. We have determined the local average velocity of the particle by dividing the distance between two subsequent positions by the time separation Δt = 1.39 ms. This local average velocity consists of an active contribution u(r) depending on the beam intensity and thus on position, and a diffusive contribution v_D that accounts for Brownian motion as well as other random motion components,

Fig. 3

Particle velocity dependence on radial position and laser power.

a The particle’s total velocity v follows the intensity profile of the laser beam indicated by solid lines given by Eq. (2), where u₀ = 23.7, 60.5 and 131.6 μm s⁻¹ for P = 4.36 (blue, square markers), 7.25 (yellow, x markers), and 10.16 mW (orange, triangle markers), respectively, taken from fits of ρ(r) in Fig. 2. Similarly, b the absolute radial velocity ∣v_r∣ and, c the absolute azimuthal velocity ∣v_θ∣ follow the intensity profile of the beam. Data is taken from a single 1-s trajectory sampled at 719 Hz. Each data point is an average over the times the particle passes through that value of r/σ. Error bars are the standard error of the mean.

With increasing power, the particle’s surface temperature exceeds the lower critical point T_c of water-2,6-lutidine (see Supplementary Information), causing a local modification of the composition according to the spinodal line of the phase diagram. Then, the particle is surrounded by a droplet of modified water content, $$ \varphi \left(\mathbf{r}\right)-{\varphi }_{\mathrm{c}}\propto \pm \sqrt{T\left(\mathbf{r}\right)-T_{\mathrm{c}}}$$, where the sign of the excess term depends on the wetting properties of the surface. Within this droplet, isotherms correspond to iso-composition surfaces. For non-uniformly heated particles, the resulting composition gradient parallel to the surface, ∇_∥ϕ, drives self-diffusiophoresis. Indeed, active motion above T_c has been reported for both laser-heated Janus particles^25,26 and silica colloids with iron-oxide inclusions¹⁵. This mechanism was worked out in detail by analytical theory²⁷ and simulations²⁸.

Yet, the usual mechanism of self-diffusiophoresis does not apply to homogeneous colloidal spheres, since their symmetry does not allow for a composition gradient along the surface. Therefore, we propose self-propulsion that arises from the non-spherical shape of our nanoparticles, visible in Fig. 1a. The large thermal conductivity of gold imposes an isothermal surface, yet the temperature gradient varies with the local curvature. Thus above the critical point, the composition ϕ(r) varies at a constant distance along the particle surface, and the parallel component of the gradient ∇_∥ϕ induces creep flow and self-propulsion of the particle. This is schematically shown in Fig. 4, which shows the isothermals (gray lines) surrounding an asymmetric nanoparticle. Moving at a finite distance away from the surface close to an edge (black dashed line, Fig. 4c), multiple isothermals are crossed, indicating a tangential concentration gradient responsible for the nanoparticle motion. For a spherical particle (black dashed line, Fig. 4c) isolines follow the shape of the particle and no tangential concentration gradient is produced. Similar observations have been made for a Leidenfrost ratchet²⁹.

Fig. 4

Isothermals around a non-spherical particle.

a Composition profile ϕ(r) in the vicinity of an axisymmetric particle with a surface temperature above the critical value T_c of water-2,6-lutidine. ϕ is constant at the isothermal surface and decreases with distance; the dark blue area indicates the range where T ≤ T_c and where the composition takes the bulk value ϕ_c. The gray lines in the critical droplet (T ≥ T_c) indicate iso-compositon surfaces. b The curvature of the top of the particle is larger than that of its bottom; as a consequence, ϕ varies more rapidly close to the top and the iso-composition lines are denser. The dashed line, at a constant distance from the particle surface, crosses iso-composition lines; thus there is a composition gradient ∇_∥ϕ parallel to the surface, which induces a diffusio-osmotic creep velocity and results in self-propulsion of the particle. Our detailed analysis relates the particle velocity to the Fourier series of the particle shape R(θ). c Instead, the bottom of the particle is almost spherical with roughly constant curvature and zero creep velocity.

Starting from an axisymmetric profile R(θ) = a(1 + χ(θ)) with $$ \chi ={\sum }_n{\alpha }_n{P}_n\left(\cos \theta \right)$$, with the polar angle θ and Legendre polynomials P_n, and evaluating the temperature profile in the vicinity of the isothermal surface of a gold particle, we obtain self-diffusiophoresis at a velocity $$ u\propto {\alpha }^2={\sum }_{n=2}^{\infty }\frac{3n+2}{2n+3}{\alpha }_n{\alpha }_{n+1}$$. Thus, motion arises from the superposition of odd and even Fourier components of the particle shape. The series starts at n = 2, since the dipolar term n = 1 corresponds to an irrelevant displacement. For our fits, we assume that less than eight modes contribute with α_n ~ 0.1 and thus find α² ~ a few percent. For later convenience, we rewrite the self-propulsion velocity as

As alternative mechanisms, we have also evaluated (and excluded) diffusiophoresis due to the intensity gradient of the laser beam, and spontaneous symmetry breaking due to a small molecular Péclet number. Spontaneous symmetry breaking is excluded since it works only if activity and mobility, as defined in ref. ³⁰, carry opposite signs. This condition can be met by chemically active particles producing a solute that is repelled from the surface, but not by phase separation above a lower critical point because the particle motion tends to diminish the composition gradient along its surface, independently of the wetting properties, while the spontaneous symmetry breaking would require that the moving particle enhances the gradient in the interaction layer. As to motion driven by the intensity gradient, it is not compatible with the fast orbital motion shown by the trajectories in Fig. 2, nor with the fast motion at the beam center where the gradient vanishes. Details are given in the SI.

Finally, we briefly discuss the anisotropy of the velocity data shown in Figs. 3b and c (∣v_θ∣ > v_r), which is also visible in the trajectories in Fig. 2b. Qualitatively, this is accounted for by the quadrupolar order parameter Q (see methods, Eq. (23)). Retaining only the dominant term results in the estimate

Probability density and polarization

The observed probability densities in water–2,6-lutidine shown in Fig. 2b cannot be described by the Boltzmann distribution. In order to relate these deviations to the particle’s activity, we have investigated the dynamical behavior in terms of the steady-state distribution Ψ(r, n), accounting for the gradient diffusion − D ∇ Ψ with Einstein coefficient D, the optical tweezers force F = − ∇ V, and the self-propulsion velocity u = un. Since the direction of the latter is given by the nanoparticle axis n, the distribution function Ψ(r, n) depends both on the nanoparticle position r and on its orientation n, and the Fokker–Planck equation (see methods, Eq. (13)) accounts for rotational diffusion, with coefficient D_r, and eventually for spinning motion due to an external torque.

Following previous work on the dynamics of Janus particles^31,32, we resort to a moment expansion Ψ = ρ + n ⋅ p + . . . , where the probability density ρ(r) = 〈Ψ〉_n and the polarization density p(r) = 〈nΨ〉_n are orientational averages with respect to n. When truncating higher-order terms, one readily integrates the steady state

Such fits have been done for three different particles at five values of the laser power P. Their propulsion speed u₀, plotted in Fig. 5, agrees well with Eq. (3). The three particles have the same radius a and absorption coefficient β; accordingly, they experience the same optical tweezers potential and reach the critical point at the same laser power P_c (obtained from the fit of u₀ using Eq. (5)). Not surprisingly, the values of the slope C differ significantly, which can be related to the fact that C is proportional to the nonsphericity parameter α², which varies from one particle to another (see Fig. 1a).

Fig. 5

Propulsion velocity as a function of the laser power.

The values of the propulsion velocity u₀ as a function of the laser power P are obtained from fits like those shown in Fig. 2b, using Eq. (5). The solid line is given by u₀ = C(P − P_c), with P_c = 2.5 mW and C = 39.5, 30.9, and 54.7 ms⁻¹ mW⁻¹ for particles 1 (yellow), 2 (red), and 3 (blue), respectively. The data of Figs. 2 and 3 are for particle 1.

The quantity $$ \mathcal{D}$$ has been calculated with a diffusion coefficient D fitted at low power P = 4.36 mW and on time scales larger than the inertial regime (where τ ≪ 1 μs) of the trajectory mean-squared displacement between 1–20 ms which we found to be linear (see Supplementary Fig. 4). Its value (D = 1.09 μm² s⁻¹) is smaller than the bulk value in water–2,6-lutidine (D₀ = 2.3 μm²s⁻¹, with viscosities taken from ref. ²⁴). Similarly, the rotational diffusion coefficient used for the fitted curves of Figs. 2 and 5 is smaller than the theoretical value. There are two physical mechanisms which are probably at the origin of this discrepancy: hydrodynamic coupling close to a solid boundary and the confining effect of the critical droplet surrounding an active particle heated above T_c. The former reduces the drag coefficient of a sphere moving parallel to a wall³³. For the latter, the critical droplet formed locally around the particle does not follow its motion but lags behind thus slowing down the particle’s diffusion. A more detailed discussion is found below.

Controlling the direction of orbital rotation

Transfer of angular momentum from circularly polarized laser light to plasmonic nanoparticles is an efficient means for fueling nanoscopic rotary motors at high-spin rates³⁴. It has already been shown theoretically and experimentally verified that, even in a tightly focused Gaussian beam with circular polarization, spin-to-orbital light momentum conversion occurs and can lead to effects such as orbit splitting^35–37. Here, we show that the spinning motion of an active particle results in orbital trajectories whose preferred handedness is imposed by the polarization of the beam. These measurements are taken with gold nanoparticles of a = 100 nm, at P = 1 mW, and at room temperature, thus leading to an increase in surface temperature of about 40 K, corresponding to 30 K above T_c.

We have investigated the azimuthal component of the velocity depending on the polarization of the beam (Fig. 6a–c). For linearly polarized light, v_θ is approximately zero, as expected (Fig. 6b). For circularly polarized light, however, we find v_θ to be different from zero: left-handed polarization results in a positive azimuthal velocity, corresponding to anti-clockwise rotation (Fig. 6a); and right-handed polarization, to negative v_θ corresponding to clockwise rotation (Fig. 6c).

Fig. 6

Controlling the direction of orbital rotation through light polarization.

The particle orbital rotation is biased toward the direction of the polarization of the trapping beam (laser power P = 1 mW, nanoparticle radius a = 100 nm). a–c Experimental values (red symbols) and theoretical fits (black lines) of the azimuthal velocity v_θ: a for left-handed circular polarization, v_θ > 0 showing counter-clockwise orbital rotation of the particle; b for linear polarization (see also Figs. 1 and 2), v_θ ≈ 0 showing no preferred direction of rotation; and, c for right-handed circular polarization, v_θ < 0 showing clockwise orbital rotation. Error bars are the standard error of the mean. The solid line is calculated from Eq. (7) with Ω the same as in a–c, and taking D_r = 70 s⁻¹, K = 1.27 × 10⁻¹⁶ J W⁻¹ (corresponding to about 30 k_BT_c per 1 mW), u₀ = 120 μm s⁻¹, and P_c = 0.2 mW. d–f Experimental (red lines) and theoretical fits (black lines) of the scattering autocorrelation C(τ) as a function of lag time τ. The absolute value of the spinning frequency Ω is d ∣Ω∣ = 2.7 Hz for left-handed circular polarization, e ∣Ω∣ = 0 Hz for linear polarization, and f ∣Ω∣ = 2.7 Hz for right-handed circular polarization. The absolute value of the decay constant is d τ₀ = 0.37 s for left-handed circular polarization, e τ₀ = 0.11 s for linear polarization, and f τ₀ = 0.41 s for right-handed circular polarization.

This effect can be explained as follows: Due to spin angular momentum transfer from the laser light, the particle spins about its axis at frequency Ω (Fig. 6d–f). The particle’s spinning motion under circular polarization is recorded via a photomultiplier. By placing a linear polarizer in front of the photomultiplier, the intensity of the scattered light changes with its orientation due to its non-sphericity. An active particle in a trap self-propels most of the time in outward direction, as rationalized by the finite polarization density p = −∇ (uρ)/D_r (Eq. (22)); the spinning motion then turns the particle axis in the azimuthal direction, $$ \dot{\mathbf{p}}=\mathrm{\Omega }\times \mathbf{p}$$. Solving the corresponding Fokker-Planck equation (see methods, Eq. (13)) with a finite spinning frequency, we find the azimuthal polarization p_θ given in Eq. (22) and the velocity

Experimental details

We consider a suspension of gold nanoparticles (radius a = 75 ± 9 nm, Sigma Aldrich) in a critical mixture of water and 2,6-lutidine at critical lutidine mass fraction c_c = 0.286 with a lower critical point at a temperature of T_c ≈ 34  °C²⁴ (see Supplementary Fig. 2). As shown by their SEM image in Fig. 1a, these nanoparticles possess clear crystalline faces determining their non-sphericity. The suspension is confined in a sample chamber between a microscopic slide and a coverslip with an approximate height of 100 μm.

A schematic of the experimental setup is shown in Supplementary Fig. 1. The nanoparticle’s translational motion is captured via digital video microscopy at 719 Hz, whereas its spin rotation under spherical polarization is recorded by a photomultiplier (by placing a linear polarizer in front of the photomultiplier, the intensity of the scattered light changes with the particle’s orientation due to its non-sphericity). An example image and trajectory of the particle is shown in Supplementary Fig. 3. The corresponding scattering intensity autocorrelation reveals oscillations with spinning frequency Ω depending on the polarization of the beam, as shown in Fig. 6d–f.

Fokker–Planck equation

In this section, we develop the theory for the non-equilibrium behavior observed for hot gold nanoparticles in an optical tweezers potential. We consider an active particle subject to the force F = − ∇ V deriving from the optical tweezers potential

The equilibrium density of passive particles is determined from the steady-state condition, where motion induced by the optical tweezers force and gradient diffusion cancel each other,

The motion of an active particle in a trap arises from the gradient diffusion, the optical tweezers force F, and the self-propulsion velocity u = un. The direction of the latter is given by the orientation of the particle axis n. The probability current reads accordingly

The continuity relation for the former is given by

Note that the advection term uρ in Eq. (18) generates the polarization density p, and the advection up in Eq. (20) generates the quadrupolar order parameter Q. For small particles, rotational diffusion exceeds the derivatives of terms involving p and Q.

Accordingly, we discard the current $$ {\mathcal{J}}_p$$ except for the source term uρ, and thus find

The main approximation of the above hierarchy may be viewed as an expansion in inverse powers of the rotational diffusion coefficient. Because of its variation with particle size, D_r ∝ a⁻³, this is justified for small enough particles.

Non-equilibrium probability density

The formal expression of the probability density ρ is obtained from the steady-state condition for the radial component of the current, J_r = 0. Inserting p_r and regrouping the different terms, one finds

For an explicit evaluation, we have to determine the velocity u as a function of the laser power. Active motion requires that the power at the particle position, g(r)P, exceeds the critical value P_c, corresponding to the lower critical temperature of water–2,6-lutidine. As the simplest relation, we take

With this form, Eq. (24) is readily integrated, leading to the probability density in the active range r < r_c,

Orbital velocity

The probability and polarization densities ρ(r) and p(r) depend on the radial coordinate only, as expected from the isotropic beam profile g(r) and optical tweezers potential V(r). Yet, an applied torque (for example due to angular momentum transfer from a polarized laser beam)^34,45,46 induces a spinning motion of the nanoparticle with angular velocity Ω. Then, the polarization density p no longer points along the radial direction but acquires an azimuthal component, as shown by Eq. (22).

A finite polarization density p implies a mean velocity u(r)p(r) at position r. In the steady state, the radial component of the corresponding current up_r is compensated by the diffusion and the action of the optical tweezers force, resulting in J_r = 0. For the azimuthal component J_ϕ, however, there is no such compensation force. As a consequence, a finite p_ϕ describes a steady orbital motion of the nanoparticle around the center of the laser beam,