Acoustic vibrations of Au nano-bipyramids and their modification under Ag deposition: a perspective for the development of nanobalances

Coherent acoustic vibrations of Au nanoblocks and their modulation by Al2O3 layer deposition

Coherent acoustic phonons induced in metallic nanostructures have attracted tremendous attention owing to their unique optomechanical characteristics. The frequency of the acoustic phonon vibration is highly sensitive to the material adsorption on metallic nanostructures and, therefore, the acoustic phonon offers a promising platform for ultrasensitive mass sensors. However, the physical origin of acoustic frequency modulation by material adsorption has been partially unexplored so far. In this study, we prepared Al2O3-deposited Au nanoblocks and measured their acoustic phonon frequencies using time-resolved pump-probe measurements. By precisely controlling the thickness of the Al2O3 layer, we systematically investigated the relation between the acoustic phonon frequency and the deposited Al2O3 amounts. The time-resolved measurements revealed that the acoustic breathing modes were predominantly excited in the Au nanoblocks, and their frequencies increased with the increment of the Al2O3 thickness. From the relationship between the acoustic phonon frequency and the Al2O3 thickness, we revealed that the acoustic phonon frequency modulation is attributed to the density change of the whole sample. Our results would provide fruitful information for developing quantitative mass sensing devices based on metallic nanostructures.

Quantifying Shape Transition in Anisotropic Plasmonic Nanoparticles through Geometric Inversion. Application to Gold Bipyramids

Unraveling the nuanced interplay between the morphology and the optical properties of plasmonic nanoparticles is crucial for targeted applications. Managing the relationship becomes significantly complex when dealing with anisotropic nanoparticles that defy a simple description using parameters like length, width, or aspect ratio. This complexity requires computationally intensive numerical modeling and advanced imaging techniques. To address these challenges, we propose a detailed structural parameter determination of gold nanoparticles using their two-dimensional projections (e.g., micrographs). Employing gold bipyramids (AuBPs) as a model morphology, we can determine their three-dimensional geometry and extract optical features computationally for comparison with the experimental data. To validate our inversion model's effectiveness, we apply it to derive the structural parameters of AuBPs undergoing shape modification through oxidative etching. In summary, our findings allow for the precise characterization of structural parameters for plasmonic nanoparticles during shape transitions, potentially enhancing the comprehension of nanocrystal growth and optimizing plasmonic material design for various applications.

Low-Frequency Oscillations in Optical Measurements of Metal-Nanoparticle Vibrations

Time-resolved optical measurements of vibrating metal nanoparticles have been used extensively to probe the ultrafast mechanical properties of the nanoparticles and of the surrounding liquid, but nearly all investigations so far have been limited to the linear regime. Here, we report the observation of a low-frequency oscillating signal in transient-absorption measurements of nanoparticles with octahedral gold cores and cubic silver shells; the signal appears at the difference of two mechanical vibrational frequencies in the particles, suggesting a nonlinear mixing process. We tentatively attribute this proposed mixing to a nonlinear coupling between a vibrational mode of the nanoparticle and its optical-frequency plasmon resonance. The optimization of this nonlinear transduction may enable high-efficiency opto-mechanical frequency mixing in the GHz-THz frequency regime.

Acoustic Vibration Modes of Gold–Silver Core–Shell Nanoparticles

Bimetallic Au/Ag core–shell cuboid nanoparticles (NPs) exhibit a complex plasmonic response dominated by a dipolar longitudinal mode and higher-order transverse modes in the near-UV, which may be exploited for a range of applications. In this paper, we take advantage of the strong signature of these modes in the NP ultrafast transient optical response, measured by pump-probe transient absorption (TA) spectroscopy, to explore the NP vibrational landscape. The fast Fourier transform analysis of the TA dynamics reveals specific vibration modes in the frequency range 15–150 GHz, further studied by numerical simulations based on the finite element method. While bare Au nanorods exhibit extensional and breathing modes, the bimetallic NPs undergo more complex motions, involving the displacement of facets, edges and corners. The amplitude and frequency of these modes are shown to depend on the Ag shell thickness, as the silver load modifies the NP aspect ratio and mass. Moreover, the contributions of the vibrational modes to the experimental TA spectra are shown to vary with the probe laser wavelength at which the signal is monitored. Using the combined simulations of the NP elastic and optical properties, we elucidate this influence by analyzing the effect of the mechanisms involved in the acousto-plasmonic coupling.

Polymer dependent acoustic mode coupling and Hooke's law spring constants in stacked gold nanoplates

Metal nanoparticles are excellent acoustic resonators and their vibrational spectroscopy has been widely investigated. However, the coupling between vibrational modes of different nanoparticles is less explored. For example, how the intervening medium affects the coupling strength is not known. Here, we investigate how different polymers affect coupling in Au nanoplate-polymer-Au nanoplate sandwich structures. The coupling between the breathing modes of the Au nanoplates was measured using single-particle pump-probe spectroscopy, and the polymer dependent coupling strength was determined experimentally. Analysis of the acoustic mode coupling gives the effective spring constant for the polymers. A relative motion mode was also observed for the stacked Au nanoplates. The frequency of this mode is strongly correlated with the coupling constant for the breathing modes. The breathing mode coupling and relative motion mode were analyzed using a coupled oscillator model. This model shows that both these effects can be described using the same spring constant for the polymer. Finally, we present a new type of mass balance using the strongly coupled resonators. We show that the resonators have a mass detection limit of a few femtograms. We envision that further understanding of the vibrational coupling in acoustic resonators will improve the coupling strength and expand their potential applications.

Nanoparticle-Fluid Interactions at Ultrahigh Acoustic Vibration Frequencies Studied by Femtosecond Time-Resolved Microscopy

Liquid viscous and viscoelastic properties are very important parameters in determining rheological phenomena. Mechanical resonators with extremely high vibrational frequencies interacting with simple liquids present a wide range of applications from mass sensing to biomechanics. However, a lack of understanding of fluid viscoelasticity greatly hinders the utilization of mechanical resonators. In this paper, the high frequency acoustic vibrations of Au nanoplates with large quality factors were used to probe fluid properties (water, glycerol, and their mixtures) through time-resolved pump-probe microscopy experiments. For water, viscous damping was clearly observed, where an inviscid effect was only detected previously. Adding glycerol to the water increases the fluid viscosity and leads to a bulk viscoelastic response in the system. The experimental results are in excellent agreement with a continuum mechanics model for the damping of nanoplate breathing modes in liquids, confirming the experimental observation of viscoelastic effects. In addition to the breathing modes of the nanoplates, Brillouin oscillations are observed in the experiments. Analysis of the frequency of the Brillouin oscillations also shows the presence of viscoelastic effects in the high-viscosity solvents. The detection and analysis of viscous damping in liquids is important not only for understanding the energy dissipation mechanisms and providing the mechanical relaxation times of the liquids but also for developing applications of nanomechanical resonators for fluid environments.

Ultrafast acoustic vibrations of Au-Ag nanoparticles with varying elongated structures

Acoustic vibrations of Au and Ag elongated nano-objects with original morphologies, from Ag-Ag homodimers to Au@Ag-Ag heterodimers and Au@Ag eccentric core-shell spheroids, have been experimentally investigated by ultrafast time-resolved optical spectroscopy. Their frequencies, obtained by the analysis of time-dependent transient absorption changes, are compared with the results obtained from finite element modeling (FEM) numerical computations, which allow assignment of the detected oscillating signals to fundamental radial and extensional modes. FEM was further used to analyze the effects of morphology and composition on the vibrational dynamics. FEM computations indicate that (1) the central distance between particles forming the nanodimers has profound effects on the extensional mode frequencies and a negligible influence on the radial mode ones, in analogy with the case of monometallic nanorods, (2) coating Au with Ag also has a strong mass-loading-like effect on the dimer and core-shell stretching mode frequency, while (3) its influence on the radial breathing mode is smaller and analogous to the non-monotonic frequency dependence on the Au fraction previously observed in isotropic bimetallic spheres. These findings are significant for developing a predictive understanding of nanostructure mechanical properties and for designing new mechanical nanoresonators.

Heat-driven acoustic phonons in lamellar nanoplatelet assemblies

Colloidal CdSe nanoplatelets, with the electronic structure of quantum wells, self-assemble into lamellar stacks due to large co-facial van der Waals attractions. These lamellar stacks are shown to display coherent acoustic phonons that are detected from oscillatory changes in the absorption spectrum observed in infrared pump, electronic probe measurements. Rather than direct electronic excitation of the nanocrystals using a femtosecond laser, impulsive transfer of heat from the organic ligand shell, excited at C-H stretching vibrational resonances, to the inorganic core of individual nanoplatelets occurs on a time-scale of <100 ps. This heat transfer drives in-phase longitudinal acoustic phonons of the nanoplatelet lamellae, which are accompanied by subtle deformations along the nanoplatelet short axes. The frequencies of the oscillations vary from 0.7 to 2 GHz (3-8 μeV and 0.5-1 ns oscillation period) depending on the thickness of the nanoplatelets-but not their lateral areas-and the temperature of the sample. Temperature-dependence of the acoustic phonon frequency conveys a substantial stiffening of the organic ligand bonds between nanoplatelets with reduced temperature. These results demonstrate a potential for acoustic modulation of the excitonic structure of nanocrystal assemblies in self-assembled anisotropic semiconductor systems at temperatures at or above 300 K.

Inelastic Light Scattering by Long Narrow Gold Nanocrystals: When Size, Shape, Crystallinity, and Assembly Matter

We report the synthesis of long narrow gold nanocrystals and the study of their vibrational dynamics using inelastic light-scattering measurements. Rich experimental spectra are obtained for monodomain gold nanorods and pentagonal twinned bipyramids. Their assignment involves diameter-dependent nontotally symmetric vibrations which are modeled in the framework of continuum elasticity by taking into account simultaneously the size, shape, and crystallinity of the nanocrystals. Light scattering by vibrations with angular momenta larger than 2 is reported. It is shown to increase with the ratio of the nanocrystals diameter to the interparticle separation. It originates from the plasmonic coupling due to the self-assembly of the nanocrystals after deposition.

Plasmonic Oligomers with Tunable Conductive Nanojunctions

Engineering plasmonic hot spots is essential for applications of plasmonic nanoparticles. A particularly appealing route is to weld plasmonic nanoparticles together to form more complex structures sustaining plasmons with symmetries targeted to given applications. However, control of the welding and subsequent hot spot characteristics is still challenging. Herein, we demonstrate an original method that connects gold particles to their neighbors by another metal of choice. We first assemble gold bipyramids in a tip-to-tip configuration, yielding short chains of variable length, and grow metallic junctions in a second step. We follow the chain formation and the deposition of the second metal (i.e., silver or palladium) via UV/vis spectroscopy, and we map the plasmonic properties using electron energy loss spectroscopy. The formation of silver bridges leads to a huge red shift of the longitudinal plasmon modes into the mid-infrared region, while the addition of palladium results in a red shift accompanied by significant plasmon damping.

Mass loading effects in the acoustic vibrations of gold nanoplates

The breathing modes of single suspended gold nanoplates have been examined by transient absorption microscopy. These vibrational modes show very high quality factors which means that their frequencies can be accurately measured. Measurements performed before and after removing the organic layer that coats the as synthesized nanoplates show significant increases in frequency, which are consistent with removal of a few nm of organic material from the nanoplate surface. Experiments were also performed after depositing polymer beads on the sample. These measurements show a decrease in frequency in the region of the beads. This implies that adding a localized mass to the nanoplate hybridizes the vibrational normal modes, creating a new breathing mode which has a maximum amplitude at the bead. The nanoplate resonators have a mass sensing detection limit of ca. 10 attograms, which is comparable to the best results that have been achieved with plasmonic nanoparticles.

Making waves: Radiation damping in metallic nanostructures

Metal nanostructures display several types of resonances. In the visible and near-IR spectral regions, there are localized surface plasmon resonances (LSPRs) that involve the coherent oscillation of the conduction electrons. Extended metal nanostructures, such as nanowires or nanoplates, also exhibit propagating surface plasmon polaritons (PSPPs), which are motions of the electrons at the surface of the structure that have a well-defined momentum. In addition, the vibrational normal modes of metal nanostructures give rise to low frequency resonances in the gigahertz to terahertz range. These different types of motions/resonances suffer energy losses from internal effects and from interactions with the environment. The goal of this perspective is to describe the part of the energy relaxation process due to the environment. Even though the plasmon resonances and acoustic vibrational modes arise from very different physics, it turns out that environmental damping is dominated by radiation of waves. The way the rates for radiation damping depend on the size of the nanostructure and the properties of the environment will be discussed for the different processes. For example, it is well known that for LSPRs, the rate of radiation damping increases with particle size. However, the radiation damping rate decreases with increasing dimensions for PSPPs and for the acoustic vibrational modes.

Optical and Physical Probing of Thermal Processes in Semiconductor and Plasmonic Nanocrystals

This article reviews thermal properties of semiconductor and emergent plasmonic nanomaterials, focusing on mechanisms through which hot carriers and phonons are produced and dissipated as well as the related impacts on optoelectronic properties. Elevated equilibrium temperatures, of particular relevance for implementation of nanomaterials in devices, affect absorptive and radiative transitions as well as emission efficiency that can present reversible and irreversible changes with temperature. In noble metal or doped semiconductor/insulator nanomaterials, hot carriers and lattice heating can substantially influence localized surface plasmon resonances and yield large ultrafast changes in transmission or strongly oscillatory coherences. Transient optical and diffraction characterizations enable nonequilibrium investigations of phonon dynamics and cooling such as lattice expansion and crystal phase stability. Timescales of nanoparticle thermalization with surroundings and transport of heat within films of such materials are also discussed.

Strong vibrational coupling in room temperature plasmonic resonators

Strong vibrational coupling has been realized in a variety of mechanical systems. However, there have been no experimental observations of strong coupling of the acoustic modes of plasmonic nanostructures, due to rapid energy dissipation in these systems. Here we realized strong vibrational coupling in ultra-high frequency plasmonic nanoresonators by increasing the vibrational quality factors by an order of magnitude. We achieved the highest frequency quality factor products of f × Q = 1.0 × 10¹³ Hz for the fundamental mechanical modes, which exceeds the value of 0.6 × 10¹³ Hz required for ground state cooling. Avoided crossing was observed between vibrational modes of two plasmonic nanoresonators with a coupling rate of g = 7.5 ± 1.2 GHz, an order of magnitude larger than the dissipation rates. The intermodal strong coupling was consistent with theoretical calculations using a coupled oscillator model. Our results enabled a platform for future observation and control of the quantum behavior of phonon modes in metallic nanoparticles.

Strong vibrational coupling has not been observed in ultra-high frequency mechanical resonators. By engineering phonon dissipation pathways, the authors increase the vibrational quality factor to allow strong coupling observations in plasmonic nanostructures, which has implications for observation and control of quantum phonon dynamics.

Controlling the Quality Factor of a Single Acoustic Nanoresonator by Tuning its Morphology

The mechanical vibrations of individual gold nanodisks nanopatterned on a sapphire substrate are investigated using ultrafast time-resolved optical spectroscopy. The number and characteristics of the detected acoustic modes are found to vary with nanodisk geometry. In particular, their quality factors strongly depend on nanodisk aspect ratio (i.e., diameter over height ratio), reaching a maximal value of ≈70, higher than those previously measured for substrate-supported nano-objects. The peculiarities of the detected acoustic vibrations are confirmed by finite-element simulations, and interpreted as the result of substrate-induced hybridization between the vibrational modes of a nanodisk. The present findings demonstrate novel possibilities for engineering the vibrational modes of nano-objects.

Acoustic Mode Hybridization in a Single Dimer of Gold Nanoparticles

The acoustic vibrations of single monomers and dimers of gold nanoparticles were investigated by measuring for the first time their ultralow-frequency micro-Raman scattering. This experiment provides access not only to the frequency of the detected vibrational modes but also to their damping rate, which is obscured by inhomogeneous effects in measurements on ensembles of nano-objects. This allows a detailed analysis of the mechanical coupling occurring between two close nanoparticles (mediated by the polymer surrounding them) in the dimer case. Such coupling induces the hybridization of the vibrational modes of each nanoparticle, leading to the appearance in the Raman spectra of two ultralow-frequency modes corresponding to the out-of-phase longitudinal and transverse (with respect to the dimer axis) quasi-translations of the nanoparticles. Additionally, it is also shown to shift the frequency of the quadrupolar modes of the nanoparticles. Experimental results are interpreted using finite-element simulations, which enable the unambiguous identification of the detected modes and despite the simplifications made lead to a reasonable reproduction of their measured frequencies and quality factors. The demonstrated feasibility of low-frequency Raman scattering experiments on single nano-objects opens up new possibilities to improve the understanding of nanoscale vibrations with this technique being complementary with single nano-object time-resolved spectroscopy as it gives access to different vibrational modes.

Mechanisms of resonant low frequency Raman scattering from metallic nanoparticle Lamb modes

The low frequency Raman scattering from gold nanoparticle bimodal assemblies with controlled size distributions has been studied. Special care has been paid to determining the size dependence of the Raman intensity corresponding to the quadrupolar Lamb mode. Existing models based on a microscopic description of the scattering mechanism in small particles (bond polarizability, dipole induced dipole models) predict, for any Raman-active Lamb modes, an inelastic intensity scaling as the volume of the nanoparticle. Surprisingly experimental intensity ratios are found to be anomalously much greater than theoretical ones, calling into question this scaling law. To explain these discrepancies, a simple mechanism of Raman scattering, based on the density fluctuations in the nanoparticles induced by the Lamb modes, is introduced. This modeling, in which the nanoparticle is described as an elastic isotropic continuous medium-as in Lamb theory, successfully explains the major features exhibited by low frequency Raman modes. Moreover this model provides a unified picture for any material, suitable for handling both small and large size ranges, as well as non-resonant and resonant excitation conditions in the case of metallic species.

Phonon-Driven Oscillatory Plasmonic Excitonic Nanomaterials

We demonstrate that coherent acoustic phonons derived from plasmonic nanoparticles can modulate electronic interactions with proximal excitonic molecular species. A series of gold bipyramids with systematically varied aspect ratios and corresponding localized surface plasmon resonance energies, functionalized with a J-aggregated thiacarbocyanine dye molecule, produces two hybridized states that exhibit clear anticrossing behavior with a Rabi splitting energy of 120 meV. In metal nanoparticles, photoexcitation generates coherent acoustic phonons that cause oscillations in the plasmon resonance energy. In the coupled system, these photogenerated oscillations alter the metal nanoparticle's energetic contribution to the hybridized system and, as a result, change the coupling between the plasmon and exciton. We demonstrate that such modulations in the hybridization are consistent across a wide range of bipyramid ensembles. We also use finite-difference time domain calculations to develop a simple model describing this behavior. Such oscillatory plasmonic-excitonic nanomaterials offer a route to manipulate and dynamically tune the interactions of plasmonic/excitonic systems and unlock a range of potential applications.

Temporal Dynamics of Localized Exciton-Polaritons in Composite Organic-Plasmonic Metasurfaces

We use femtosecond transient absorption spectroscopy to study the temporal dynamics of strongly coupled exciton-plasmon polaritons in metasurfaces of aluminum nanoantennas coated with J-aggregate molecules. Compared with the thermal nonlinearities of aluminum nanoantennas, the exciton-plasmon hybridization introduces strong ultrafast nonlinearities in the composite metasurfaces. Within femtoseconds after the pump excitation, the plasmonic resonance is broadened and shifted, showcasing its high sensitivity to excited-state modification of the molecular surroundings. In addition, we observe temporal oscillations due to the deep subangstrom acoustic breathing modes of the nanoantennas in both bare and hybrid metasurfaces. Finally, unlike the dynamics of hybrid states in optical microcavities, here, ground-state bleaching is observed with a significantly longer relaxation time at the upper polariton band.

Understanding How Acoustic Vibrations Modulate the Optical Response of Plasmonic Metal Nanoparticles

Measurements of acoustic vibrations in nanoparticles provide an opportunity to study mechanical phenomena at nanometer length scales and picosecond time scales. Vibrations in noble-metal nanoparticles have attracted particular attention because they couple to plasmon resonances in the nanoparticles, leading to strong modulation of optical absorption and scattering. There are three mechanisms that transduce the mechanical oscillations into changes in the plasmon resonance: (1) changes in the nanoparticle geometry, (2) changes in electron density due to changes in the nanoparticle volume, and (3) changes in the interband transition energies due to compression/expansion of the nanoparticle (deformation potential). These mechanisms have been studied in the past to explain the origin of the experimental signals; however, a thorough quantitative connection between the coupling of phonon and plasmon modes has not yet been made, and the separate contribution of each coupling mechanism has not yet been quantified. Here, we present a numerical method to quantitatively determine the coupling between vibrational and plasmon modes in noble-metal nanoparticles of arbitrary geometries and apply it to silver and gold spheres, shells, rods, and cubes in the context of time-resolved measurements. We separately determine the parts of the optical response that are due to shape changes, changes in electron density, and changes in deformation potential. We further show that coupling is, in general, strongest when the regions of largest electric field (plasmon mode) and largest displacement (phonon mode) overlap. These results clarify reported experimental results and should help guide future experiments and potential applications.

Electron and Phonon Dynamics in Hexagonal Pd Nanosheets and Ag/Pd/Ag Sandwich Nanoplates

Pd and its hybrid nanostructures have attracted considerable attention over the past decade, with both catalytic and plasmonic properties. The electron and phonon properties directly govern conversion efficiencies in applications such as energy collectors and photocatalysts. We report the dynamic processes of electron-phonon coupling and coherent acoustic phonon vibration in hexagonal Pd nanosheets and Ag/Pd/Ag sandwich nanoplates using transient absorption spectroscopy. The electron-phonon coupling constant of Pd nanosheets, G_Pd-nanosheet (8.7 × 10¹⁷ W/(m³·K)) is larger than that of the bulk G_Pd (5.0 × 10¹⁷ W/(m³·K)). The effective coupling constant G_eff of Ag/Pd/Ag nanoplates decreases with increasing Ag shell thickness, finally approaching the bulk G_Ag. The variation of G_eff is explained in terms of reduced density of states near Fermi level of Pd nanosheets with 1.8 nm ultrathin thickness. Coherent acoustic phonon vibration in Pd nanosheets is assigned to a fundamental breathing mode, similar to the vibration of benzene. The period increases with increasing Ag shell thickness. For Ag/Pd/Ag nanoplates with 20 nm thick Ag shells, the vibrational mode is ascribed to a quasi-extensional mode. The results show that the modes of the coherent acoustic phonon vibration transform with the geometric variation of Pd nanosheets and Ag/Pd/Ag nanoplates. Our results represent an understanding of quantum-confinement related electron dynamics and bulk-like phonon kinetics in the ultrathin Pd nanosheets and their hybrid nanostructures.

The mass load effect on the resonant acoustic frequencies of colloidal semiconductor nanoplatelets

Resonant acoustic modes of ultrathin CdS and CdSe colloidal nanoplatelets (NPLs) with varying thicknesses were probed using low frequency Raman scattering. The spectra are dominated by an intense band ascribed to the thickness breathing mode of the 2D nanostructures. The measured Raman frequencies show strong deviations with respect to the values expected for simple bare plates, all the more so as the thickness is reduced. The deviation is shown to arise from the additional mass of the organic ligands that are bound to the free surfaces of the nanoplatelets. The calculated eigen frequencies of vibrating platelets weighed down by the mass of the organic ligands are in very good agreement with the observed experimental behaviours. This finding opens up a new possibility of nanomechanical sensing such as nanobalances.

Time-domain investigation of the acoustic vibrations of metal nanoparticles: size and encapsulation effects

The acoustic vibrations of single-metal and multi-material nanoparticles are studied by ultrafast pump-probe optical spectroscopy and described in the context of the continuous elastic model. The applicability of this model to the small size range, down to one nanometer, is discussed in the light of recent experimental data and ab initio calculations. Investigations of multi-material nano-objects stress the impact of the intra-particle interface on the characteristics of their vibrational modes, also yielding information on the composition and spatial distribution of the constituting materials.

Probing silver deposition on single gold nanorods by their acoustic vibrations

Acoustic vibrations of single gold nanorods coated with silver were investigated. We used single-particle pump-probe spectroscopy to monitor the silver deposition through the particle vibrations. Two vibration modes, the breathing mode and extensional mode, are observed, and the vibrational frequencies are measured as functions of the amount of silver deposited on single gold nanorods. The breathing mode frequency was found to decrease with silver deposition, while the extensional mode frequency was almost constant for silver shells up to 6 nm. The frequency changes agree with a model based on continuum mechanics and on the assumption of a uniform silver coating. The quality factors for the breathing mode and the extensional mode are hardly affected by silver deposition, indicating that the introduced interface between gold and silver contributes negligibly to the damping of the particle vibrations. Finally, we demonstrated that an atomic layer of silver can be detected using the particle acoustic vibrations.

Entropically patchy particles: engineering valence through shape entropy

Patchy particles are a popular paradigm for the design and synthesis of nanoparticles and colloids for self-assembly. In "traditional" patchy particles, anisotropic interactions arising from patterned coatings, functionalized molecules, DNA, and other enthalpic means create the possibility for directional binding of particles into higher-ordered structures. Although the anisotropic geometry of nonspherical particles contributes to the interaction patchiness through van der Waals, electrostatic, and other interactions, how particle shape contributes entropically to self-assembly is only now beginning to be understood. The directional nature of entropic forces has recently been elucidated. A recently proposed theoretical framework that defines and quantifies directional entropic forces demonstrates the anisotropic-that is, patchy-nature of these emergent, attractive forces. Here we introduce the notion of entropically patchy particles as the entropic counterpart to enthalpically patchy particles. Using three example "families" of shapes, we show how to modify entropic patchiness by introducing geometric features to the particles via shape operations so as to target specific crystal structures assembled here with Monte Carlo simulations. We quantify the emergent entropic valence via a potential of mean force and torque. We show that these forces are on the order of a few kBT at intermediate densities below the onset of crystallization. We generalize these shape operations to shape anisotropy dimensions, in analogy with the anisotropy dimensions introduced for enthalpically patchy particles. Our findings demonstrate that entropic patchiness and emergent valence provide a way of engineering directional bonding into nanoparticle systems, whether in the presence or absence of additional, non-entropic forces.