Quasi-static Limit Research Articles

Analysis of Low-Frequency Vibrational Modes and Particle Rearrangements in Marginally Jammed Amorphous Solid under Quasi-Static Shear**Supported by the National Natural Science Foundation of China under Grant Nos 11272048 and 51239006, and the European Commission Marie Curie Actions under Grant No IRSES-294976.

We present the numerical simulation results of a model granular assembly formed by spherical particles with Hertzian interaction subjected to a simple shear in the athermal quasi-static limit. The stress-strain curve is shown to separate into smooth, elastic branches followed by a subsequent plastic event. Mode analysis shows that the lowest-frequency vibrational mode is more localized, and eigenvalues and participation ratios of low-frequency modes exhibit similar power-law behavior as the system approaches plastic instability, indicating that the nature of plastic events in the granular system is also a saddle node bifurcation. The analysis of projection and spatial structure shows that over 75% contributions to the non-affine displacement field at a plastic instability come from the lowest-frequency mode, and the lowest-frequency mode is strongly spatially correlated with local plastic rearrangements, inferring that the lowest-frequency mode could be used as a predictor for future plastic rearrangements in the disordered system jammed marginally.

Resonant and nonlocal properties of phononic metasolids

We derive a general theory of effective properties in metasolids based on phononic crystals with low frequency resonances. We demonstrate that in general these structures need to be described by means of a frequency-dependent and non-local anisotropic mass density, stiffness tensor and a third- rank coupling tensor, which shows that they behave like a non-local Willis medium. The effect of non-locality and coupling tensor manifest themselves for some particular resonances whereas they become negligible for other resonances. Considering the example of a two-dimensional phononic crystal, consisting of triangular arrangements of cylindrical shells in an elastic matrix, we show that its mass density tensor is strongly resonant and anisotropic presenting both positive and negative divergent values, while becoming scalar in the quasi-static limit. Moreover, it is found that the negative value of transverse component of the mass density is induced by a dipolar resonance, while that of the vertical component is induced by a monopolar one. Finally, the dispersion relation obtained by the effective parameters of the crystal is compared with the band structure, showing a good agreement for the low-wave number region, although the non-local effects are important given the existence of some resonant values of the wave number.

On Limiting Behavior of Contaminant Transport Models in Coupled Surface and Groundwater Flows

There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The analysis of models including the transport of contaminants has, however, focused on a quasi-static Stokes-Darcy model. Herein we consider the fully evolutionary system including contaminant transport and analyze its quasi-static limits.

Limits of quasistatic approximation in modified-gravity cosmologies

We investigate the limits of applicability of the quasi-static approximation in cosmologies featuring general models of dark energy or modified gravity. We show that, at best, the quasi-static approximation breaks down outside of the sound horizon of the dark-energy, rather than the cosmological horizon as is frequently assumed. When the sound speed of dark energy is significantly below that of light, the quasi-static limit is only valid in a limited range of observable scales and this must be taken into account when computing effects on observations in such models. As an order of magnitude estimate, in the analysis of data from today's weak-lensing and peculiar-velocity surveys, dark energy can be modelled as quasi-static only if the sound speed is larger than order 1% of that of light. In upcoming surveys, such as Euclid, it should only be used when the sound speed exceeds around 10% of the speed of light. In the analysis of the cosmic microwave background, the quasi-static limit should never be used for the integrated Sachs-Wolf effect and for lensing only when the sound speed exceeds 10% of the speed of light.

Modified gravityN-body code comparison project

Self-consistent ${\it N}$-body simulations of modified gravity models are a key ingredient to obtain rigorous constraints on deviations from General Relativity using large-scale structure observations. This paper provides the first detailed comparison of the results of different ${\it N}$-body codes for the $f(R)$, DGP, and Symmetron models, starting from the same initial conditions. We find that the fractional deviation of the matter power spectrum from $\Lambda$CDM agrees to better than $1\%$ up to $k \sim 5-10~h/{\rm Mpc}$ between the different codes. These codes are thus able to meet the stringent accuracy requirements of upcoming observational surveys. All codes are also in good agreement in their results for the velocity divergence power spectrum, halo abundances and halo profiles. We also test the quasi-static limit, which is employed in most modified gravity ${\it N}$-body codes, for the Symmetron model for which the most significant non-static effects among the models considered are expected. We conclude that this limit is a very good approximation for all of the observables considered here.

Colloidal superlattices for unnaturally high-index metamaterials at broadband optical frequencies.

The recent advance in the assembly of metallic nanoparticles (NPs) has enabled sophisticated engineering of unprecedented light-matter interaction at the optical domain. In this work, I expand the design flexibility of NP optical metamaterial to push the upper limit of accessible refractive index to the unnaturally high regime. The precise control over the geometrical parameters of NP superlattice monolayer conferred the dramatic increase in electric resonance and related effective permittivity far beyond the naturally accessible regime. Simultaneously, effective permeability change, another key factor to achieving high refractive index, was effectively suppressed by reducing the thickness of NPs. By establishing this design rule, I have achieved unnaturally high refractive index (15.7 at the electric resonance and 7.3 at the quasi-static limit) at broadband optical frequencies (100 THz ~300 THz). I also combined this NP metamaterial with graphene to electrically control the high refractive index over the broad optical frequencies.

Study of energy transfer between molecules placed in the vicinity of a bimetal composite nanoparticle

In this study, the problem of energy transfer between two molecules near a bimetallic composite nanoparticle is investigated. The influence of the interaction between metal particles on the intermolecular energy is studied, because when two metal nanoparticles are placed close to each other, their plasmons coupling giving rise to new features. On the other hand, we discuss the transfer of resonance energy between donor and acceptor molecules (a single donor and a single acceptor) in the presence of a nanocomposite containing gold and silver nanoparticles based on the Maxwell-Garnett effective medium theory and within the quasistatic limit. We show that the interaction energy strongly depends on the particle size, the filling factor of metal particles, the intermolecular distance (the distance between the donor and acceptor molecules), and the dielectric constant of host matrix.

Quantitative analysis of the Clausius inequality

.In the context of driven diffusive systems, for thermodynamic transformations over a large but finite time window, we derive an expansion of the energy balance. In particular, we characterize the transformations which minimize the energy dissipation and describe the optimal correction to the quasi-static limit. Surprisingly, in the case of transformations between homogeneous equilibrium states of an ideal gas, the optimal transformation is a sequence of inhomogeneous equilibrium states.

Percolation and jamming transitions in particulate systems with and without cohesion.

We consider percolation and jamming transitions for particulate systems exposed to compression. For the systems built of particles interacting by purely repulsive forces in addition to friction and viscous damping, it is found that these transitions are influenced by a number of effects, and in particular by the compression rate. In a quasistatic limit, we find that for the considered type of interaction between the particles, percolation and jamming transitions coincide. For cohesive systems, however, or for any system exposed to even slow dynamics, the differences between the considered transitions are found and quantified.

Toroidal nanotraps for cold polar molecules

Electronic excitations in metallic nanoparticles in the optical regime that have been of great importance in surface enhanced spectroscopy and emerging applications of molecular plasmonics, due to control and confinement of electromagnetic energy, may also be of potential to control the motion of nanoparticles and molecules. Here, we propose a concept for trapping polarizable particles and molecules using toroidal metallic nanoparticles. Specifically, gold nanorings are investigated for their scattering properties and field distribution to computationally show that the response of these optically resonant particles to incident photons permit the formation of a nanoscale trap when proper aspect ratio, photon wavelength and polarization are considered. However, interestingly the resonant plasmonic response of the nanoring is shown to be detrimental to the trap formation. The results are in good agreement with analytic calculations in the quasi-static limit within the first-order perturbation of the scalar electric potential. The possibility of extending the single nanoring trapping properties to two-dimensional arrays of nanorings is suggested by obtaining the field distribution of nanoring dimers and trimers.

Accurate Modeling of Dark-Field Scattering Spectra of Plasmonic Nanostructures.

Dark-field microscopy is a widely used tool for measuring the optical resonance of plasmonic nanostructures. However, current numerical methods for simulating the dark-field scattering spectra were carried out with plane wave illumination either at normal incidence or at an oblique angle from one direction. In actual experiments, light is focused onto the sample through an annular ring within a range of glancing angles. In this paper, we present a theoretical model capable of accurately simulating the dark-field light source with an annular ring. Simulations correctly reproduce a counterintuitive blue shift in the scattering spectra from gold nanodisks with a diameter beyond 140 nm. We believe that our proposed simulation method can be potentially applied as a general tool capable of simulating the dark-field scattering spectra of plasmonic nanostructures as well as other dielectric nanostructures with sizes beyond the quasi-static limit.

Efficiency of single-particle engines.

We study the efficiency of a single-particle Szilard and Carnot engine. Within a first order correction to the quasistatic limit, the work distribution is found to be Gaussian and the correction factor to average work and efficiency only depends on the piston speed. The stochastic efficiency is studied for both models and the recent findings on efficiency fluctuations are confirmed numerically. Special features are revealed in the zero-temperature limit.

Scalar-fluid theories: cosmological perturbations and large-scale structure

Recently a new Lagrangian framework was introduced to describe interactions between scalar fields and relativistic perfect fluids. This allows two consistent generalizations of coupled quintessence models: non-vanishing pressures and a new type of derivative interaction. The implications of these to the formation of cosmological large-scale structure are uncovered here at the linear order. The full perturbation equations in the two cases are derived in a unified formalism and their Newtonian, quasi-static limit is studied analytically. Requiring the absence of an effective sound speed term in the coupled dark matter fluid restricts the Lagrangian to be a linear function of the matter number density. This leaves new potentially viable classes of both algebraically and derivatively interacting models wherein the coupling may impact the background expansion dynamics and imprint new signatures into the large-scale structure.

Plasmon Hybridization Model for a Nanoparticle above a Dielectric Interface: Dielectric Effects, Comparison with the Dimer System, Range of Applicability, and Limits

In the presence of a nearby highly reflecting dielectric substrate, the optical response of a metal nanoparticle exhibits a rich multipolar plasmonic structure. For quantifying the multimodal plasmonic response of such hybrid particle/interface systems, the hydrodynamic plasmon hybridization (PH) method has been applied in the simple case of a solid metal sphere (PI-PH model). In this work I extend the formalism described in my previous paper (J. Phys. Chem. C 2014, 118, 28118−28133) to the more general case where underlying dielectric media are present in the whole space (polarizable ionic background and matrix). In the quasistatic limit the method of images allows the PI-PH model to be closely related to the two-sphere dimer PH model. However, because the dynamics of the electron fluid and of the polarization charges in the mirror particle are enslaved to those in the real metal particle, strong differences between both PH models result from the reduction by a factor of two of the number of dynamical variables describing the system. As compared to jellium spheres in vacuum, the PH formalism is much more complex because the induced surface polarization charges in the underlying media contribute to the energetics of the system. In this work a very detailed analytical description of both PH models, allowing a direct numerical implementation, is provided. In the frame of the original and physically transparent method developed in my previous paper, particular emphasis is given to modeling the coupling of the nanosystem with an external applied field. In the presence of underlying dielectric media, I show that the taking into account of the field created by the additional polarization charge distributions induced by the bare applied field is essential for defining the effective external field that is coupled to the nanosystem. The predictions of both PH models are also compared, through absorption and extinction cross-section spectra, to exact results computed in the frame of the multipole expansion method, allowing the range of applicability and limits of both models to be determined. Finally, the PH formalism is shown to be generalizable to deal with underlying dielectric media characterized by complex (that is, absorbing) frequency-dependent permittivities.

Quasistatic stopband and other unusual features of the spectrum of a one-dimensional piezoelectric phononic crystal controlled by negative capacitance

Normal propagation of the longitudinal wave through the piezoelectric medium with periodically embedded electrodes is considered. Each pair of electrodes is connected via a circuit with capacitance C. The paper analyzes in detail the unusual features of the dispersion spectrum ω(KT) (K is the Floquet–Bloch wavenumber, T is the period) arising in the special case of a negative value of C. The solution of the dispersion equation shows explicitly the evolution of the passbands and stopbands tunable by varying C<0. One of the striking features is the existence of the poles of ImKT (infinite attenuation) and of the corresponding jumps of the phase ReKT from 0 to π in the stopbands for a certain range (C0,C1) of negative C. Besides, for C∈(C0,C∞) where C∞<C1, the spectrum possesses a low-frequency absolute stopband starting from the quasistatic limit ω=0 and including the tunable pole of ImKT inside. This stopband is related to the negative value of the quasistatic effective elastic constant in the range (C0,C∞). At C=C∞, the effective constant is infinite while the spectrum degenerates to the straight line K=0 at any ω. For C close to C∞, the spectrum consists of the branches with high group velocity and of the quasiflat branches.

Flow of wet granular materials: A numerical study.

We simulate dense assemblies of frictional spherical grains in steady shear flow under controlled normal stress P in the presence of a small amount of an interstitial liquid, which gives rise to capillary menisci, assumed isolated (pendular regime), and attractive forces, which are hysteretic: Menisci form at contact, but do not break until grains are separated by a finite rupture distance. The system behavior depends on two dimensionless control parameters, inertial number I and reduced pressure P*=aP/(πΓ), comparing confining forces ∼a2P to meniscus tensile strength F0=πΓa, for grains of diameter a joined by menisci with surface tension Γ. We pay special attention to the quasistatic limit of slow flow and observe systematic, enduring strain localization in some of the cohesion-dominated (P*∼0.1) systems. Homogeneous steady flows are characterized by the dependence of internal friction coefficient μ* and solid fraction Φ on I and P*. We also record normal stress differences, fairly small but not negligible and increasing for decreasing P*. The system rheology is moderately sensitive to saturation within the pendular regime, but would be different in the absence of capillary hysteresis. Capillary forces have a significant effect on the macroscopic behavior of the system, up to P* values of several units, especially for longer force ranges associated with larger menisci. The concept of effective pressure may be used to predict an order of magnitude for the strong increase of μ* as P* decreases but such a crude approach is unable to account for the complex structural changes induced by capillary cohesion, with a significant decrease of Φ and different agglomeration states and anisotropic fabric. Likewise, the Mohr-Coulomb criterion for pressure-dependent critical states is, at best, an approximation valid within a restricted range of pressures, with P*≥1. At small enough P*, large clusters of interacting grains form in slow flows, in which liquid bonds survive shear strains of several units. This affects the anisotropies associated with different interactions and the shape of function μ*(I), which departs more slowly from its quasistatic limit than in cohesionless systems (possibly explaining the shear banding tendency).

Spatial shape of avalanches in the Brownian force model

.We study the Brownian force model (BFM), a solvable model of avalanche statistics for an interface, in a general discrete setting. The BFM describes the overdamped motion of elastically coupled particles driven by a parabolic well in independent Brownian force landscapes. Avalanches are defined as the collective jump of the particles in response to an arbitrary monotonous change in the well position (i.e. in the applied force). We derive an exact formula for the joint probability distribution of these jumps. From it we obtain the joint density of local avalanche sizes for stationary driving in the quasi-static limit near the depinning threshold. A saddle-point analysis predicts the spatial shape of avalanches in the limit of large aspect ratios for the continuum version of the model. We then study fluctuations around this saddle point, and obtain the leading corrections to the mean shape, the fluctuations around the mean shape and the shape asymmetry, for finite aspect ratios. Our results are finally confronted to numerical simulations.

Probing Vertical and Horizontal Plasmonic Resonant States in the Photoluminescence of Gold Nanodisks

Localized surface plasmon resonance (LSPR) on gold nanoparticles and nanostructures shows its capability to modulate the spontaneous emission rate for photoluminescence. However, beyond the quasi-static limit with oblique incidence conditions, probing a retardation-based plasmonic resonant state excited by the horizontal incident wave-vector component has not been reported yet in the plasmon-mediated photoluminescence of gold nanostructures. In this paper, we investigated the photoluminescence of individual and coupled gold nanodisks with increasing size, and the peaks in the size-dependent photoluminescence spectra are shown to originate from the preferential excitation of vertical (LSPR) and horizontal (retardation-based) plasmonic resonant states. We have also proposed a three-step model to well describe the contribution of vertical and horizontal plasmonic resonant states in the photoluminescence of gold nanodisks. Our study makes a new contribution to the understanding of photoluminescence from gold nanostructures, and it paves the way toward the applications of plasmon-mediated luminescence and biological studies.

An Eulerian projection method for quasi-static elastoplasticity

A well-established numerical approach to solve the Navier–Stokes equations for incompressible fluids is Chorin's projection method [1], whereby the fluid velocity is explicitly updated, and then an elliptic problem for the pressure is solved, which is used to orthogonally project the velocity field to maintain the incompressibility constraint. In this paper, we develop a mathematical correspondence between Newtonian fluids in the incompressible limit and hypo-elastoplastic solids in the slow, quasi-static limit. Using this correspondence, we formulate a new fixed-grid, Eulerian numerical method for simulating quasi-static hypo-elastoplastic solids, whereby the stress is explicitly updated, and then an elliptic problem for the velocity is solved, which is used to orthogonally project the stress to maintain the quasi-staticity constraint. We develop a finite-difference implementation of the method and apply it to an elasto-viscoplastic model of a bulk metallic glass based on the shear transformation zone theory. We show that in a two-dimensional plane strain simple shear simulation, the method is in quantitative agreement with an explicit method. Like the fluid projection method, it is efficient and numerically robust, making it practical for a wide variety of applications. We also demonstrate that the method can be extended to simulate objects with evolving boundaries. We highlight a number of correspondences between incompressible fluid mechanics and quasi-static elastoplasticity, creating possibilities for translating other numerical methods between the two classes of physical problems.

Effective medium theory for two-dimensional non-magnetic metamaterial lattices up to quadrupole expansions

We present a formulation for deriving effective medium properties of infinitely periodic two-dimensional metamaterial lattice structures beyond the conventional static and quasi-static limits. We utilize the multipole expansions, where the polarization currents associated with the supported Bloch modes are expressed via the electric dipole, magnetic dipole, and electric quadrupole moments per unit length. We then propose a method to calculate the Bloch modes based on the lattice geometry and individual unit element structure. The results revert to well-known formulas in the traditional quasistatic limit and are useful for the homogenization of nanorod-type metamaterials which are frequently used in optical applications.