Pseudorotation and N-body Forces in an Optical Matter System

TL;DR

This work demonstrates pseudorotation in a two-dimensional optical-matter system composed of eight 150 nm nanoparticles, revealing a kite isomer with $D_2$ symmetry that undergoes smooth, collective reconfigurations without particle exchange. By combining dark-field experiments with electrodynamics-Langevin dynamics (EDLD) simulations based on Generalized Multiparticle Mie Theory (GMMT), the authors identify a $D_4$-symmetric transition state and show that the pseudorotation is driven by N-body electrodynamic interactions and field interference, not by simple pairwise forces. The intrastate coordinate, defined by the difference $d_1-d_2$ of inner-particle separations, is complemented by committor analyses and PCA mode 3 to establish a robust reaction-coordinate for pseudorotation, with ionic strength modulating both intra- and inter-state dynamics. The results highlight the importance of N-body forces in active matter and suggest that such higher-order interactions can enable exotic, collective motions in optical-m matter systems, potentially extending to three dimensions in future work.

Abstract

Isomerization in molecular systems almost invariably occurs through 3-dimensional motion due to the nature of chemical bonding. Pseudorotation is an unusual type of isomerization that occurs in some high symmetry systems that gives the appearance of rigid-body rotation yet only involves atom rearrangements. This paper demonstrates that pseudorotation occurs in 2-dimensions in an optical matter (OM) system of metal nanoparticle constituents. The difference in dimensionality of the dynamics arises from the electrodynamic field-interference nature of optical binding vs. quantum mechanical bonding in polyatomic molecules. The 8-nanoparticle OM "kite" structure we study in experiments and simulations has D2 (D2h) symmetry and a D4 symmetric transition state. The mechanism for pseudorotation involves correlated motion of all 8 nanoparticles with smooth (continuous) evolution of their interactions and without particles jumping in or out of the OM array. While the OM kite structure only occurs with 10% probability vs. other OM isomers, its rate of pseudorotation is rapid relative to transitions to other structural isomers (e.g., "teardrop"). The other isomers have structures that lie on a trigonal lattice with inter-particle separations at distances that enhance field interference and induced polarizations. Even though the kite isomer has inter-particle separations that would manifest destructive interference on a particle pair (i.e., 2-body) basis, the kite structure is the slowest to rearrange into any other isomer. We show that the unusual structure and dynamics of the kite optical matter system result from N-body interactions and forces demonstrating that N-body effects are important in this class of active matter and presumably more generally.

Figures (6)

• Figure 1: Darkfield microscopy images, pair distribution functions, and probabilities of different 8-NP OM isomers. (a-d) Dark field microscopy images of the 150 nm diameter spherical Ag nanoparticle-based OM array formed in a 0.6 mM NaCl solution. (e) Probability of the 8 particle OM structure being in the “teardrop”, “boat mast”, “kite”, “spaceport”, "sphinx", or any “other” configuration. (f) Probability density distributions of the pairwise distances of particles in the "teardrop" and "kite" isomers. The distributions were collected from trajectories of particle positions obtained from dark field microscopy videos of the 8-particle OM system ($\sim$70,000 total frames). The kite data set consists of 7,497 images and 66 transitions and the teardrop data set consists of 32,736 frames and 1230 transitions. Pairwise distances of the particles can exceed 2000 nm, but the plot axes have been limited to highlight regions of interest. The bin width for the pair distribution is 50 nm. (g) Distributions of first passage times for the isomers of the 8-particle OM system to transition to another isomer. The bin width for the first passage time distributions is 50 ms for "kite" and 10 ms for all others. The difference in bin sizes is because there are fewer kite trajectories (66) than trajectories for the other isomers (>500 each). Reducing the bin width 2-fold for the kite isomer 1st passage time data had no effect on the fitted rate constant.
• Figure 2: Experimental measurements and EDLD simulations characterizing the dynamics of the 8 Ag nanoparticle OM kite isomer. (a) - (c) Dark field microscopy images of an 8-particle OM array of 150 nm diameter spherical Ag nanoparticles. The blue and red arrows in (a) represent the two distances (labeled $d_1$ and $d_2$) between orthogonal pairs of central particles of the OM array. The three panels from consecutive frames (450 fps) display pseudorotation. The scale bars are 1 $\mu$m. (d) A single experimental time trajectory of the difference of the two distances ($d_1 - d_2$). The OM array is in the kite isomer for the duration of the trajectory and is in a 0.6 mM NaCl solution. (e) A single EDLD simulated time trajectory of $d_1 - d_2$ for an 8 Ag nanoparticle array for 100 mW beam power and 50 nm Debye screening length. The array persists in the kite isomer (state) for the duration of the trajectory. (f) Probability density of rotation-reset particle positions for the time trajectory from experiment shown in (d). (g) Probability density of rotation-reset particle positions for the time trajectory from the simulation shown in (e). Bins are 75x75 nm in size for both probability density plots. The bin size essentially matches the effective pixel size of the experimental dark field microscopy images.
• Figure 3: Ionic strength affects the stability and kinetics of the OM kite isomer. (a) - (c) Scatter plots from experiments (videos) of the center distances of 8-particle OM arrays in the kite isomer obtained from time trajectories taken in varying ionic strength solutions (18 M$\Omega$ H$_{2}$O (2,181 frames out of 50,000 total), 0.6 mM NaCl (7,496 frames out of 71,007 total), 1.2 mM NaCl (4,644 frames out of 65,446 frames). The red lines represent a linear regression of the scatter plots. The regressed lines are given by $r=(d_1+md_2)/\sqrt{m^2+1}$ where $m=-0.99$ is the slope. (d) Probability densities of the projection of the scatter plots onto their corresponding fit line. (e) Distribution of first passage times for the kite structure to switch from $d_2$ <$d_1$ (defined as the initial condition for any new kite structure) to $d_2$ > $d_1$. (f) Distribution of the first passage times of an individual OM kite structure transitioning to any other isomer for the three different ionic strength solutions. The shorter lifetimes for the 18 M$\Omega$ H$_{2}$O and the 1.2 mM NaCl cases contribute to causing the bias in the associated probability densities in (a), (c), and (d) toward the defined initial condition ($d_1$<$d_2$).
• Figure 4: The reaction coordinate for pseudorotation includes the $d_1 - d_2$ parameter. (a) Plot of committor probability for 686 configurations selected from experiment along the $d_1$ - $d_2$ coordinate. The configurations were selected from experimental video data of a single continuous kite trajectory (duration of 686 frames). (b) Scatter plot of the aspect ratio of the sides of the quadrilateral formed by the outer four particles of the kite structure vs. $d_1 - d_2$. The orange line represents a linear regression of the scatter plot, whose function is $AR = m(d_1 - d_2) + 1$, where $m = 1.96 \times 10^{-4}$. (c) Plot of rotation-reset particle positions of randomly selected timesteps of a single EDLD simulated time trajectory. (d) Rotation-reset particle positions of a single time trajectory of experimental dark field microscopy images. These color-coded scatter plots condition particle positions on the value of $d_1 - d_2$. The green (blue) points are the rotation-reset particle positions of kite isomers where $d_1 > d_2$ ($d_1 < d_2$). (e) Mode 3 of the PCA modes for the kite structure.chen_data-driven_2021chen_pseudorotation_2025 The red/blue colors indicate the phase of the collective motion and the length of the line segment is the standard deviation of the motion multiplied by 200. This mode is the collective motion of pseudorotation.
• Figure 5: Total forces and force decomposition for various configurations for the kite isomer. (a)-(d) Force calculations of total force (black arrows) on each particle for 4 different kite structures. Each structure is an average of an ensemble of kite structures from a single EDLD simulation trajectory selected with different ranges of $d_1 - d_2$ values: (a) $d_1 = d_2 \pm 10 nm$; (b) $d_1 \leq d_2$ (90 nm to 110 nm); (c) peak $d_1 < d_2$ (190 nm to 210 nm); (d) $d_1 \ll d_2$ (290 nm to 310 nm). (e)-(h) Vectors of total force and decomposed components on each nanoparticle of the structures in (a)-(d); total force (black arrows), intensity gradient force (blue arrows), the 2-interaction force (red arrows) and the N-interaction force (green arrows). Note that the scale bars are 10 fN in (a) - (d) and 25 fN in (e) - (h). See Supplementary Information for more details on the force calculations.