Driven optical matter: Dynamics of electrodynamically coupled nanoparticles in an optical ring vortex

To date investigations of the dynamics of driven colloidal systems have focused on hydrodynamic interactions and often employ optical (laser) tweezers for manipulation. However, the optical fields that provide confinement and drive also result in electrodynamic interactions that are generally neglected. We address this issue with a detailed study of interparticle dynamics in an optical ring vortex trap using 150-nm diameter Ag nanoparticles. We term the resultant electrodynamically interacting nanoparticles a driven optical matter system. We also show that a superior trap is created by using a Au nanoplate mirror in a retroreflection geometry, which increases the electric field intensity, the optical drive force, and spatial confinement. Using nanoparticles versus micron sized colloids significantly reduces the surface hydrodynamic friction allowing us to access small values of optical topological charge and drive force. We quantify a further 50% reduction of hydrodynamic friction when the nanoparticles are driven over the Au nanoplate mirrors versus over a mildly electrostatically repulsive glass surface. Further, we demonstrate through experiments and electrodynamics–Langevin dynamics simulations that the optical drive force and the interparticle interactions are not constant around the ring for linearly polarized light, resulting in a strong position-dependent variation in the nanoparticle velocity. The nonuniformity in the optical drive force is also manifest as an increase in fluctuations of interparticle separation, or effective temperature, as the optical driving force is increased. Finally, we resolve an open issue in the literature on periodic modulation of interparticle separation with comparative measurements of driven 300-nm-diameter polystyrene beads that also clearly reveal the significance of electrodynamic forces and interactions in optically driven colloidal systems. Therefore, the modulations in the optical forces and electrodynamic interactions that we demonstrate should not be neglected for dielectric particles and might give rise to some structural and dynamic features that have previously been attributed exclusively to hydrodynamic interactions.

Figure 1

Schematic of the focused optical beam and ring trap over (a) a glass coverslip and (b) a Au nanoplate. A dark-field optical microscope image of two Ag nanoparticles trapped (c) over glass and (d) over a Au nanoplate mirror. The bright line segments and polygonal shape are the edges of the Au nanoplate. Note the Au nanoplate overfills the panel. The curved arrows in (c) and (d) indicate the direction of rotation of the nanoparticles in the optical vortex ring trap. The white scale bars in (c) and (d) is 1 $\mu \mathrm{m}$.

Figure 2

Probability densities and simulated trajectories of particles in the ring traps. (a), (b) Probability densities of particles in the ring traps over glass (blue, left) or over plate (red, right). Distribution of the particle positions in (a) Cartesian coordinates and (b) polar coordinates in the laboratory frame. (c), (d) Electric field intensity (heat map) and particle trajectory from the simulation at intensity $I_0$ (c) and $4I_0$ (d) for $l=2$. Example trajectories from the simulation are shown in black. (See Supplemental Material [51], movie S1.)

Figure 3

Angular displacements vs time for experiments and simulations for glass and Au nanoplate conditions and angular drives $l=\text{0–5}$. (a) Experimental trajectories of a single Ag nanoparticle rotation over glass (see Supplemental Material [51], movies S2–S4) and (b) over the Au nanoplate (See Supplemental Material [51], movie S5) for six different drives. (c) The trajectories of a single Ag nanoparticle rotation in the simulation with intensity at $I_0$ and (d) at intensity $4I_0$ for $l=\text{0–5}$. The solid black lines are linear fits of the rotation (angular displacement) versus time.

Figure 4

MSD of experimental and simulated particle motion and forces on Ag nanoparticles from experiment and simulation. (a), (b) Angular MSD for different angular drives $l$ from the experiment trapping over the glass surface (a) or over the nanoplate mirror (b). (c), (d) Angular MSD for different angular drives $l$ from simulations with intensity $I_0$ (c) and $4I_0$ (d). The MSDs for (a)–(d) are calculated from the angular displacement vs time results from Fig 3. (e) Fit parameters of $F\left(l\right)/\gamma$ from the MSD of single nanoparticles in the optical ring vortex over glass (closed blue triangles) and over the Au nanoplate (closed red circles) using the left $y$ axis. Fit parameters of $F\left(l\right)/\gamma$ from the MSD of single nanoparticle simulations for $I_0$ (open blue triangles) and $4I_0$ (open red circles) using the right $y$ axis. Solid black lines show linear fits to the data, the solid gray line shows the fit to $F\left(l\right)/\gamma$ vs $l$ for intensity $I_0$ in the simulation. Note that the vertical axes are chosen so that the fitted lines to the Au nanoplate (and $4I_0$) results are identical.

Figure 5

Speed and force information of nanoparticles in the optical ring vortex. (a), (b) Speed as a function of position around the optical ring vortex in (a) the experiment over the nanoplate mirror at $l=4$ and (b) in the simulation at intensity $4I_0$ at $l=4$. The speeds are calculated by taking the difference between the Cartesian positions of the nanoparticles in successive frames in the experiment or time steps in the simulation and dividing by the time increment (e.g., 1/frame rate). (c), (d) Electric field intensity (heat map) and force vector map in the $4I_0$ simulation at $l=0$ (c) and $l=4$ (d).

Figure 6

Probability density plot of interparticle separations characterized as parallel or perpendicular with respect to the electric field polarization (horizontal to the ring trap). The interparticle separation is plotted with respect to the angular location of the midpoint of a pair. The radial coordinate is the interparticle separation (in $\mu \mathrm{m}$ units). The color represents the probability density of events in the radial and angular coordinates. Four of the octants of the ring trap are categorized as parallel or perpendicular with respect to the electric field. These octants, labeled $\perp$ and $\parallel$, are sections of the optical ring vortex where the trap is perpendicular or parallel to the electric field polarization. This data is from a multiparticle experiment at $l=2$ over glass. Similar results are obtained for other values of $l$ and also over the Au nanoplate.

Figure 7

Conditional probability distributions of the nearest neighbor separation grouped into categories of (a), (d) small ($l=\text{0–2}$), (b), (e) medium ($l=3$), and (c), (f) large ($l=\text{4–5}$) applied optical force for experiments over glass (blue, top row) and over the Au nanoplate (red, bottom row). In each panel there are distributions for particle pairs that are parallel (thick light colored curve) or perpendicular (thin dark colored curve) to the polarization. The separations are binned parallel or perpendicular based on the scheme in Fig. 6. Curves are plotted through the midpoints of the distribution bins.

Figure 8

Simulations of two Ag nanoparticles in an optical ring vortex with electric field intensity $4I_0$. (a) Probability density (PD) distributions of nearest neighbor separations for nanoparticle pairs in the simulation for $l=\text{0–1}$ (small $l$), $l=\text{2–3}$ (medium $l$), and $l=\text{4–5}$ (large $l$). (b) Potentials of mean force (PMF) around the first two optical binding sites calculated from the probability densities from (a). (c) Distributions of interparticle separations calculated from simulations for $l=0$, where the two particles are initialized symmetrically about the $\perp$ (0) or the $\parallel$ ($\pi /2$) positions in the ring. (d) Potentials of mean force for $l=0$ at $\perp$ (green) and $\parallel$ (purple) calculated from the probability densities in (c). All curves are plotted using a Gaussian kernel density estimator.

Figure 9

Interparticle separation probability density (PD) functions affected by periodic modulation of (a) interparticle potential or (b) drive force. 1D LD simulations with periodic boundaries of two particles using parameters extracted from the full ED-LD simulations assuming (a) constant drive and modulated interparticle potential and (b) modulated drive and interparticle potential. (See Fig. 14 and Supplemental Material [51] movie S6.)

Figure 10

Distributions of nearest neighbor separations. (a), (b) Probability densities of interparticle separation for (a) 300-nm polystyrene particles in the optical ring vortex over glass at $l=1$ for both parallel and perpendicular positions in the ring and (b) Ag nanoparticles over glass for small $l$ [same as Fig. 7]. The insets for (a) and (b) show the difference (perpendicular minus parallel) of the respective distributions.

Figure 11

Diagram showing the properties of the optical ring vortex for different $l$'s. The first column shows the phase mask used on the SLM to produce the ring trap. The second column shows the intensity of the beam for each $l$, which is the same for all $l$'s. The third column shows a schematic of the phase of the optical ring vortex for each $l$.

Figure 12

The electric field intensity of the optical ring trap over the Au nanoplate shown in the $y-z$ plane bisecting the ring trap. The surface of the Au nanoplate is at 0 nm. The maximum intensity of the first antinode occurs about 250 nm above the Au nanoplate mirror.

Figure 13

The calculated potential energy of a Ag nanoparticle in an optical ring vortex over the glass surface. The minimum is at $\sim 95$ nm, and the barrier for the particle to spontaneously jump and get stuck on the glass surface is $\sim 1.8k_{\text{B}}T$.

Figure 14

Snapshots of the LD simulations for $l=4$ for (a), (b) constant and (c), (d) modulated drive forces. Panels (a) and (b) show the positions of the particles in a constant drive force when the particles are located at $\pi /2$ (a) and $\pi$ (b). Panels (c) and (d) show the positions of the particles in a modulated drive force when the particles are located at $\pi /2$ (c) and $\pi$ (d). The angles correspond to the polar coordinate system used in the main text.