Electrodynamic Interference and Induced Polarization in Nanoparticle-Based Optical Matter Arrays.

Optical matter (OM) arrays are self-organizing, ordered arrangements of nanometer- to micrometer-size particles, where interparticle forces are mediated by incident and scattered coherent light. The structures that form and their dynamics depend on the properties (e.g., material, size) of the constituent particles, as well as the incident and scattered light. While significant progress has been made toward understanding how the OM arrays are affected by the phase, polarization, and intensity profile of the incident light, the polarization induced in the particles and the light scattered by OM arrays have received less attention. In this paper, we establish the roles of electrodynamic interference, many-body coupling, and induced-polarization concomitant with the coherent light scattered by OM arrays. Experiments and simulations together demonstrate that the spatial profile and directionality of coherent light scattered by OM arrays in the far field are primarily influenced by interference, while electrodynamic coupling (interactions) and the associated polarization induced in the nanoparticle constituents have a quantitative wavelength-dependent effect on the total amount of light scattered by the arrays. Furthermore, the electrodynamic coupling in silver nanoparticle OM arrays is significantly enhanced by constructive interference and increases superextensively with the number of particles in the array. Particle size, and hence polarizability, also has a significant effect on the strength of the coupling. Finally, we simulate larger hexagonal OM arrays of Ag nanoparticles to demonstrate that the electrodynamic coupling and scattering enhancement observed in small OM arrays develop into surface lattice resonances observed in the infinite array limit. Our work provides insights for designing OM arrays to tune many-body forces and the coherent light that they scatter.

Interaction between dilute water vapor and dodecane thiol ligated Au nanoparticles: Hydrated structure and pair potential of mean force.

The interaction between two ligated nanoparticles depends on whether they are isolated or immersed in a liquid solvent. However, very little is known about the influence of solvent vapor on the interaction between two ligated nanoparticles. Recent experiments yield the surprising result that the cyclic exposure of solvent free suspended monolayers of dodecane thiol ligated gold nanoparticles (AuNPs) to water vapor and dry nitrogen generates reversible cyclic decreases and increases in Young's modulus of the monolayer, implying corresponding cyclic changes in the AuNP-AuNP interaction. We examine how water vapor interacts with an isolated dodecane thiol dressed AuNP and how water vapor affects the interaction between a pair of nanoparticles, using all-atom molecular-dynamics simulations. We find that there is condensation of water molecules onto the ligand shell of an AuNP in the form of clusters of 100-2000 molecules that partially cover the shell, with most of the water in a few large clusters. A water cluster bridges the AuNPs, with a sensibly constant number of water molecules for AuNP-AuNP separations from the edge-to-edge contact up to center-to-center separations of 100 Å. The wet AuNP-AuNP interaction has a slightly deeper and wider asymmetric well than does the dry interaction, a change that is qualitatively consistent with that implied by the observed water vapor induced change in Young's modulus of a monolayer of these AuNPs. We find that macroscopic analyses of water drop-deformable surface interactions and dynamics provide both guidance to understanding and qualitatively correct predictions of the phenomena observed in our simulations.

Data-driven reaction coordinate discovery in overdamped and non-conservative systems: application to optical matter structural isomerization.

Optical matter (OM) systems consist of (nano-)particle constituents in solution that can self-organize into ordered arrays that are bound by electrodynamic interactions. They also manifest non-conservative forces, and the motions of the nano-particles are overdamped; i.e., they exhibit diffusive trajectories. We propose a data-driven approach based on principal components analysis (PCA) to determine the collective modes of non-conservative overdamped systems, such as OM structures, and harmonic linear discriminant analysis (HLDA) of time trajectories to estimate the reaction coordinate for structural transitions. We demonstrate the approach via electrodynamics-Langevin dynamics simulations of six electrodynamically-bound nanoparticles in an incident laser beam. The reaction coordinate we discover is in excellent accord with a rigorous committor analysis, and the identified mechanism for structural isomerization is in very good agreement with the experimental observations. The PCA-HLDA approach to data-driven discovery of reaction coordinates can aid in understanding and eventually controlling non-conservative and overdamped systems including optical and active matter systems.

Free Thiols Regulate the Interactions and Self-Assembly of Thiol-Passivated Metal Nanoparticles.

Thiol ligands bound to the metallic core of nanoparticles determine their interactions with the environment and self-assembly. Recent studies suggest that equilibrium between bound and free thiols alters the ligand coverage of the core. Here, X-ray scattering and MD simulations investigate water-supported monolayers of gold-core nanoparticles as a function of the core-ligand coverage that is varied in experiments by adjusting the concentration of total thiols (sum of free and bound thiols). Simulations demonstrate that the presence of free thiols produces a nearly symmetrical coating of ligands on the core. X-ray measurements show that above a critical value of core-ligand coverage the nanoparticle core rises above the water surface, the edge-to-edge distance between neighboring nanoparticles increases, and the nanoparticle coverage of the surface decreases. These results demonstrate the important role of free thiols: they regulate the organization of bound thiols on the core and the interactions of nanoparticles with their surroundings.

The influence of fractional surface coverage on the core-core separation in ordered monolayers of thiol-ligated Au nanoparticles.

We report the results of grazing incidence X-ray diffraction (GIXD) measurements from water supported Langmuir monolayers of gold nanoparticles ligated with dodecanethiol (12 carbons), tetradecanethiol (14 carbons), hexadecanethiol (16 carbons), and octadecanethiol (18 carbons). These monolayers are formed from solutions with varying concentrations of the respective thiols. We show that equilibrium between adsorbed thiol molecules and the thiols in the bulk solution implies fractional coverage of the Au nanoparticle core. We also show that the nanoparticle-nanoparticle separation and the correlation length of particles in these ordered films increases with thiol concentration in the parent solution, and that excess thiol can be found in the space between particles as well as in islands away from the particles. Using the equilibrium constant relating ligand solution concentration and nanoparticle surface coverage of the gold core by the ligand molecules, we interpret the way in which varying thiol concentration affects the nanoparticle-nanoparticle separation as a function of surface coverage of the gold core by the ligand molecules.

Pair and many-body interactions between ligated Au nanoparticles.

We report the results of molecular dynamics simulations of the properties of a pseudo-atom (united atom) model of dodecane thiol ligated 5-nm diameter gold nanoparticles (AuNPs) in a vacuum as a function of ligand coverage and particle separation in three states of aggregation, namely, the isolated AuNPs, the isolated pair of AuNPs, and a square lattice of four AuNPs. Our calculations show that the ligand density along a radius emanating from the core of an isolated AuNP has the same gross features for all values of the coverage; it oscillates around a constant value up to a distance along the chain corresponding to the position of the fourth pseudo-atom and then smoothly decays to zero, reflecting both the restricted conformations of the chain near the core surface and the larger numbers of conformations available further from the core. Interaction between two AuNPs generates changes in the ligand distributions of each. We examine the structure and general shape of the ligand envelope as a function of the coverage and demonstrate that the equilibrium structure of the envelope and the deformation of that envelope generated by interaction between the NPs are coverage-dependent so that the shape, depth, and position of the minimum of the potential of mean force display a systematic dependence on the ligand coverage. We propose an accurate analytical description of the calculated potential of mean force as a function of a set of parameters that scale linearly with the ligand coverage. Noting that the conformational freedom of the ligands implies that multiparticle induced deviations from additivity of the pair potential of mean force are likely important; we define and calculate a "bond stretching" effective pair potential of mean force for a square lattice of particles that contains, implicitly, both the three- and four-NP contributions. We find that the bond stretching effective pair potential of mean force in this cluster has a different minimum and a different well depth from the isolated pair potential of mean force. Previous work has found that the three-particle contribution to deviation from pair additivity is monotonically repulsive, whereas we find that the combined three- and four-particle contributions have an attractive well, implying that the three- and four-particle contributions are of comparable magnitude but opposite sign, thereby suggesting that even higher order correction terms likely play a significant role in the behavior of dense assemblies of many nanoparticles.

Controlling the Dynamics and Optical Binding of Nanoparticle Homodimers with Transverse Phase Gradients.

While transverse phase gradients enable studies of driven nonequilibrium phenomena in optical trapping, the behavior of electrodynamically interacting particles in a transverse phase gradient has not been explored in detail. In this Letter we study electrodynamically interacting pairs of identical nanoparticles (homodimers) in transverse phase gradients. We establish that the net driving force on homodimers is modulated by a separation-dependent interference effect for small phase gradients. By contrast, large phase gradients break the symmetry of the interaction between particles and profoundly change the electrodynamic interparticle energy landscape. Our findings are particularly important for understanding multiparticle dynamics during the self-assembly and rearrangement of optical matter.

Sequential phase transitions and transient structured fluctuations in two-dimensional systems with a high-density Kagome lattice phase.

We report studies of (i) the isothermal density dependent sequences of phases in two-dimensional systems of particles with repulsive pair potentials devised by Truskett [J. Chem. Phys. 145, 054901 (2016)] and Torquato [Phys. Rev. E 88, 042309 (2013)] to support a high-density Kagome lattice phase and (ii) transient structured fluctuations close to a transition to a Kagome lattice. The commonalities in the sequences of phases in these systems and other 2D systems suggest the existence of a universal mechanism driving all to favor similar packing arrangements as the density is increased, but the simulations also show that the only such general rule proposed, namely, the Süto theorem, is not a necessary condition for the support of multiple distinct lattice structures by a particular pair potential. The transient fluctuations in the liquid close to the liquid-to-Kagome phase transition have Kagome symmetry, whereas deeper in the liquid phase, the fluctuations have hexagonal symmetry. When the transition is string-to-Kagome phase, the transient structured fluctuations in the string phase have both six-fold and other than six-fold symmetries. The path of the string-to-Kagome transition in the Truskett system involves intermediate honeycomb configurations that subsequently buckle to form a Kagome lattice. The path of the string-to-Kagome transition in the Torquato system suggests that the Kagome phase is formed by coiled strings merging together; increasing density generates a Kagome phase with imperfections such as 8-particle rings.

A comment on the position dependent diffusion coefficient representation of structural heterogeneity.

Experimental studies of the variation of the mean square displacement (MSD) of a particle in a confined colloid suspension that exhibits density variations on the scale length of the particle diameter are not in agreement with the prediction that the spatial variation in MSD should mimic the spatial variation in density. The predicted behavior is derived from the expectation that the MSD of a particle depends on the system density and the assumption that the force acting on a particle is a point function of position. The experimental data are obtained from studies of the MSDs of particles in narrow ribbon channels and between narrowly spaced parallel plates and from new data, reported herein, of the radial and azimuthal MSDs of a colloid particle in a dense colloid suspension confined to a small circular cavity. In each of these geometries, a dense colloid suspension exhibits pronounced density oscillations with spacing of a particle diameter. We remove the discrepancy between prediction and experiment using the Fisher-Methfessel interpretation of how local equilibrium in an inhomogeneous system is maintained to argue that the force acting on a particle is delocalized over a volume with radius equal to a particle diameter. Our interpretation has relevance to the relationship between the scale of inhomogeneity and the utility of translation of the particle MSD into a position dependent diffusion coefficient and to the use of a spatially dependent diffusion coefficient to describe mass transport in a heterogeneous system.

Transient structured fluctuations in a two-dimensional system with multiple ordered phases.

We report the structure of transient fluctuations in the liquid phase of a two-dimensional system that exhibits several ordered phases with different symmetries. The density-temperature phase diagram of the system studied, composed of particles with a repulsive shouldered soft-core pair interaction, has regions with stable liquid and hexatic phases, a square solid phase, two separate hexagonal solid phases, and a quasi-crystalline phase with 12-fold symmetry. We have examined the character of the structured fluctuations by computing the same-time aperture cross correlation function of particle configurations in several fluid regions near to and far from phase transition lines. The two primary goals of our study are (1) determination if the spectrum of structures of the fluctuations in the liquid is broader than or limited to the motifs exhibited by the ordered phases supported by the system and (2) determination of the density domains in the liquid that support particular transient structured fluctuations. In the system studied, along a low-temperature isotherm in the temperature-density plane that intersects all the ordered phases we find that the liquid phase exhibits structured fluctuations with hexagonal symmetry near both liquid-hexatic transition lines. Along the same isotherm and in the stable liquid between the lower density hexatic-to-liquid and the higher density liquid-to-square solid transitions, we find that transient hexagonal ordered fluctuations dominate the liquid region near the hexatic-to-liquid transition and square ordered fluctuations dominate the liquid region near the liquid-to square solid transition, but both of these structured fluctuations occur at all densities between these transition lines. At a higher temperature, at phase points in the liquid above, but close to the density maximum of an underlying transition, there are ordered fluctuations that can be correlated with the structure of the lower temperature phase. Although it is expected that very close to a liquid-ordered phase boundary a structured fluctuation in the liquid will have the same symmetry as the ordered phase, it is not obvious that structured fluctuations in thermodynamic states deep in the liquid phase will be similarly restricted. The most striking result of our calculations is that no evidence is found in the liquid phase for structured fluctuations with other symmetries than those of the ordered phases of the system.

Direct Visualization of Barrier Crossing Dynamics in a Driven Optical Matter System.

A major impediment to a more complete understanding of barrier crossing and other single-molecule processes is the inability to directly visualize the trajectories and dynamics of atoms and molecules in reactions. Rather, the kinetics are inferred from ensemble measurements or the position of a transducer ( e. g., an AFM cantilever) as a surrogate variable. Direct visualization is highly desirable. Here, we achieve the direct measurement of barrier crossing trajectories by using optical microscopy to observe position and orientation changes of pairs of Ag nanoparticles, i. e. passing events, in an optical ring trap. A two-step mechanism similar to a bimolecular exchange reaction or the Michaelis-Menten scheme is revealed by analysis that combines detailed knowledge of each trajectory, a statistically significant number of repetitions of the passing events, and the driving force dependence of the process. We find that while the total event rate increases with driving force, this increase is due to an increase in the rate of encounters. There is no drive force dependence on the rate of barrier crossing because the key motion for the process involves a random (thermal) radial fluctuation of one particle allowing the other to pass. This simple experiment can readily be extended to study more complex barrier crossing processes by replacing the spherical metal nanoparticles with anisotropic ones or by creating more intricate optical trapping potentials.

The role of ligands in the mechanical properties of Langmuir nanoparticle films.

Langmuir monolayers of ligand-capped inorganic nanoparticles exhibit rich morphologies under lateral compression such as wrinkling, folding, and multilayer nucleation. We demonstrate that the ligands play a crucial role in the mechanical properties of nanoparticle films by probing the morphology and anisotropic stress response during lateral compression of films with systematically varied ligand concentrations. Increasing the ligand concentration of the films past a threshold value inhibits monolayer wrinkling and folding in favor of multilayer formation, and sharply reduces the compressive and shear moduli. We attribute these drastic mechanical effects to modifications to the ligand interactions between adjacent particles as well as to two-dimensional crystalline structure changes occurring on the scale of tens of particles, as determined by transmission electron microscopy.

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.

Colloid-colloid hydrodynamic interaction around a bend in a quasi-one-dimensional channel.

We report a study of how a bend in a quasi-one-dimensional (q1D) channel containing a colloid suspension at equilibrium that exhibits single-file particle motion affects the hydrodynamic coupling between colloid particles. We observe both structural and dynamical responses as the bend angle becomes more acute. The structural response is an increasing depletion of particles in the vicinity of the bend and an increase in the nearest-neighbor separation in the pair correlation function for particles on opposite sides of the bend. The dynamical response monitored by the change in the self-diffusion [D_{11}(x)] and coupling [D_{12}(x)] terms of the pair diffusion tensor reveals that the pair separation dependence of D_{12} mimics that of the pair correlation function just as in a straight q1D channel. We show that the observed behavior is a consequence of the boundary conditions imposed on the q1D channel: both the single-file motion and the hydrodynamic flow must follow the channel around the bend.

Single-pixel interior filling function approach for detecting and correcting errors in particle tracking.

We present a general method for detecting and correcting biases in the outputs of particle-tracking experiments. Our approach is based on the histogram of estimated positions within pixels, which we term the single-pixel interior filling function (SPIFF). We use the deviation of the SPIFF from a uniform distribution to test the veracity of tracking analyses from different algorithms. Unbiased SPIFFs correspond to uniform pixel filling, whereas biased ones exhibit pixel locking, in which the estimated particle positions concentrate toward the centers of pixels. Although pixel locking is a well-known phenomenon, we go beyond existing methods to show how the SPIFF can be used to correct errors. The key is that the SPIFF aggregates statistical information from many single-particle images and localizations that are gathered over time or across an ensemble, and this information augments the single-particle data. We explicitly consider two cases that give rise to significant errors in estimated particle locations: undersampling the point spread function due to small emitter size and intensity overlap of proximal objects. In these situations, we show how errors in positions can be corrected essentially completely with little added computational cost. Additional situations and applications to experimental data are explored in SI Appendix In the presence of experimental-like shot noise, the precision of the SPIFF-based correction achieves (and can even exceed) the unbiased Cramér-Rao lower bound. We expect the SPIFF approach to be useful in a wide range of localization applications, including single-molecule imaging and particle tracking, in fields ranging from biology to materials science to astronomy.

Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids.

The properties of a classical simple liquid are strongly affected by the application of an external potential that supports inhomogeneity. To understand the nature of these property changes, the equilibrium particle distribution functions of the liquid have, typically, been calculated directly using either integral equation or density functional based analyses. In this study, we develop a different approach with a focus on two distribution functions that characterize the inhomogeneous liquid: the pair direct correlation function c(r1,r2) and the pair correlation function g(r1,r2). With g(r1,r2) considered to be an experimental observable, we solve the Ornstein-Zernike equation for the inhomogeneous liquid to obtain c(r1,r2), using information about the well studied and resolved g(0)(r1,r2) and c(0)(r1,r2) for the parent homogeneous ((0)) system. In practical cases, where g(r1,r2) is available from experimental data in a discrete form, the resulting c(r1,r2) is expressed as an explicit function of g(r1,r2) in a discrete form. A weaker continuous form of solution is also obtained, in the form of an integral equation with finite integration limits. The result obtained with our formulation is tested against the exact solutions for the correlation and distribution functions of a one-dimensional inhomogeneous hard rod liquid. Following the success of that test, the formalism is extended to higher dimensional systems with explicit consideration of the two-dimensional liquid.

A model study of assisted adiabatic transfer of population in the presence of collisional dephasing.

Previous studies have demonstrated that when experimental conditions generate non-adiabatic dynamics that prevents highly efficient population transfer between states of an isolated system by stimulated Raman adiabatic passage (STIRAP), the addition of an auxiliary counter-diabatic field (CDF) can restore most or all of that efficiency. This paper examines whether that strategy is also successful in a non-isolated system in which the energies of the states fluctuate, e.g., when a solute is subject to collisions with solvent. We study population transfer in two model systems: (i) the three-state system used by Demirplak and Rice [J. Chem. Phys. 116, 8028 (2002)] and (ii) a four-state system, derived from the simulation studies of Demirplak and Rice [J. Chem. Phys. 125, 194517 (2006)], that mimics HCl in liquid Ar. Simulation studies of the vibrational manifold of HCl in dense fluid Ar show that the collision induced vibrational energy level fluctuations have asymmetric distributions. Representations of these asymmetric energy level fluctuation distributions are used in both models (i) and (ii). We identify three sources of degradation of the efficiency of STIRAP generated selective population transfer in model (ii): too small pulse areas of the laser fields, unwanted interference arising from use of strong fields, and the vibrational detuning. For both models (i) and (ii), our examination of the efficiency of STIRAP + CDF population transfer under the influence of the asymmetric distribution of the vibrational energy fluctuations shows that there is a range of field strengths and pulse durations under which STIRAP + CDF control of population transfer has greater efficiency than does STIRAP generated population transfer.

Rotation of the Orientation of the Wave Function Distribution of a Charged Particle and its Utilization.

We describe a method for selecting and sorting particles in an ion trap with respect to charge and mass. The method exploits a so-called shortcut to adiabatic passage, specifically the fast-forward field protocol, to design an electromagnetic field that rotates the spatial orientation of the wave function of the desired ion. The electromagnetic field forces ions that have different mass and electrical charge from the desired ionic species out of the trapping potential without exciting the desired ionic species, leaving the latter undisturbed in the trap.

Fast-forward assisted STIRAP.

We consider combined stimulated Raman adiabatic passage (STIRAP) and fast-forward field (FFF) control of selective vibrational population transfer in a polyatomic molecule. The motivation for using this combination control scheme is 2-fold: (i) to overcome transfer inefficiency that occurs when the STIRAP fields and pulse durations must be restricted to avoid excitation of population transfers that compete with the targeted transfer and (ii) to overcome transfer inefficiency resulting from embedding of the actively driven subset of states in a large manifold of states. We show that, in a subset of states that is coupled to background states, a combination of STIRAP and FFFs that do not individually generate processes that are competitive with the desired population transfer can generate greater population transfer efficiency than can ordinary STIRAP with similar field strength and/or pulse duration.