Experimental evidence of isotropic transparency and complete band gap formation for ultrasound propagation in stealthy hyperuniform media.

Following on recent experimental characterization of the transport properties of stealthy hyperuniform media for electromagnetic and acoustic waves, we report here measurements at ultrasonic frequencies of the multiple scattering of waves by 2D hyperuniform distributions of steel rods immersed in water. The transparency, for which the effective attenuation of the medium is canceled, is first evidenced by measuring the transmission of a plane wave propagating in a highly correlated and relatively dense medium. It is shown that a band gap occurs in the vicinity of the first Bragg frequency. The isotropy of both transparency and band gap are also evidenced for the case of waves generated by a point source in differently ordered and circular-shaped distributions. In other words, we thus obtain a representation of the Green's function. Our results demonstrate the huge potential of hyperuniform, as well as highly correlated, media for the design of functional materials.

Propagation of scalar waves in dense disordered media exhibiting short- and long-range correlations.

Correlated disorder is at the heart of numerous challenging problematics in physics. In this work we focus on the propagation of acoustic coherent waves in two-dimensional dense disordered media exhibiting long- and short-range structural correlations. The media are obtained by inserting elastic cylinders randomly in a stealth hyperuniform medium itself made up of cylinders. The properties of the coherent wave is studied using an original numerical software. In order to understand and discuss the complex physical phenomena occurring in the different media, we also make use of effective media models derived from the quasicrystalline approximation and the theory of Fikioris and Waterman that provides an explicit expression of the effective wave numbers. Our study shows a very good agreement between numerical and homogenization models up to very high concentrations of scatterers. This study shows that media with both short- and long-range correlations are of strong interest to design materials with original properties.

Impact of particle size and multiple scattering on the propagation of waves in stealthy-hyperuniform media.

Propagation of waves in materials that exhibit stealthy-hyperuniform long-range correlations is investigated. By using a modal decomposition of the field that takes multiple scattering into account at all orders, we study the impact of the concentration of particles on the transparency of such materials at low frequency. An upper frequency limit for transparency is defined that include both the particle size and the degree of stealthiness. We show that the independent scattering approximation is not relevant to calculate elastic mean free paths when wavelength becomes comparable to the size of particles. We find that transparency is very robust with regard to the degree of heterogeneity of the host random medium and the polydispersity of particles. Finally, it is shown that resonances can be used as the frequency filter.

Influence of the microstructure of two-dimensional random heterogeneous media on propagation of acoustic coherent waves.

Multiple scattering of waves arises in all fields of physics in either periodic or random media. For random media the organization of the microstructure (uniform or nonuniform statistical distribution of scatterers) has effects on the propagation of coherent waves. Using a recent exact resolution method and different homogenization theories, the effects of the microstructure on the effective wave number are investigated over a large frequency range (ka between 0.1 and 13.4) and high concentrations. For uniform random media, increasing the configurational constraint makes the media more transparent for low frequencies and less for high frequencies. As a side but important result, we show that two of the homogenization models considered here appear to be very efficient at high frequency up to a concentration of 60% in the case of uniform media. For nonuniform media, for which clustered and periodic aggregates appear, the main effect is to reduce the magnitude of resonances and to make network effects appear. In this case, homogenization theories are not relevant to make a detailed analysis.

Numerical simulation of two-dimensional multiple scattering of sound by a large number of circular cylinders.

The purpose of this article is to present an innovative resolution method for investigating problems of sound scattering by infinite cylinders immersed in a fluid medium. The study is based on the analytical solution of multiple scattering, where incident and scattered waves are expressed in cylindrical harmonics. This modeling leads to dense linear systems, which are made sparse by introducing a cutoff radius around each particle. This cutoff radius is deeply studied and quantified. Numerical resolution is performed using parallel computing methods designed to solve very large sparse linear systems. Comparisons with direct calculations made with another numerical software and homogenization techniques follow and show good agreement with the implemented method. The last part is dedicated to a comparison between the propagation of waves in a circular cluster made of a random distribution of cylinders and the propagation in the corresponding homogenized cluster where the multiple scattering formalism is combined with a statistical analysis to provide an effective medium.

Resonant acoustic scattering by two spherical bubbles.

The mutual interaction between two close bubbles in an acoustic field is studied. This interaction is modeled in the linear framework of the multiple scattering theory using spherical harmonics expansions and the addition theorem. In order to deal with small as well as large bubbles, viscous dissipation in the liquid, thermal dissipation in the gas, and surface tension are taken into account in the calculations of the scattering coefficients of a unique bubble. Under the assumption of the long wavelengths, the scattering coefficient of the monopolar mode is linked to the one obtained by using the Rayleigh-Plesset equation. The exact characteristic equation providing the symmetric and antisymmetric resonances of the two bubbles is established. Numerical results show that a great number of modes of vibration is required to describe the acoustic field around the bubbles. Moreover, whatever the spacing between two identical bubbles, the scattering cross section has a maximum value at the frequency of the symmetric mode while the antisymmetric mode is not detected. However, the strengthening of the scattering observed close to the symmetric resonance frequency is clearly due to the presence of the antisymmetric mode.

Propagation of coherent transverse waves: Influence of the translational and rotational subwavelength resonances.

The propagation of coherent transverse waves through a homogeneous elastic medium containing a set of spherical dense inclusions is an interesting topic. In such a material, in addition to the coherent longitudinal wave, two coherent transverse waves can propagate. The modeling used is based on the multiple scattering theory, which requires the scattering coefficients of the single scattering problem. These coefficients are calculated for moving rigid particles, leading to approximations of the two subwavelength dipolar resonances, one associated to a translational motion and the other to a rotational motion. Numerical simulations are carried out in order to compare the effective wavenumbers of the coherent elastic waves through the analysis of their phase velocity and attenuation. This comparison is performed for elastic and moving rigid spheres. It is shown that both dipolar resonances may have a great influence on the propagation of coherent transverse waves.

Random acoustic metamaterial with a subwavelength dipolar resonance.

The effective velocity and attenuation of longitudinal waves through random dispersions of rigid, tungsten-carbide beads in an elastic matrix made of epoxy resin in the range of beads volume fraction 2%-10% are determined experimentally. The multiple scattering model proposed by Luppé, Conoir, and Norris [J. Acoust. Soc. Am. 131(2), 1113-1120 (2012)], which fully takes into account the elastic nature of the matrix and the associated mode conversions, accurately describes the measurements. Theoretical calculations show that the rigid particles display a local, dipolar resonance which shares several features with Minnaert resonance of bubbly liquids and with the dipolar resonance of core-shell particles. Moreover, for the samples under study, the main cause of smoothing of the dipolar resonance of the scatterers and the associated variations of the effective mass density of the dispersions is elastic relaxation, i.e., the finite time required for the shear stresses associated to the translational motion of the scatterers to propagate through the matrix. It is shown that its influence is governed solely by the value of the particle to matrix mass density contrast.

Nonlinear acoustic propagation in bubbly liquids: Multiple scattering, softening and hardening phenomena.

The weakly nonlinear propagation of acoustic waves in monodisperse bubbly liquids is investigated numerically. A hydrodynamic model based on the averaged two-phase fluid equations is coupled with the Rayleigh-Plesset equation to model the dynamics of bubbles at the local scale. The present model is validated in the linear regime by comparing with the Foldy approximation. The analysis of the pressure signals in the linear regime highlights two resonance frequencies: the Minnaert frequency and a multiple scattering resonance that strongly depends on the bubble concentration. For weakly nonlinear regimes, the generation of higher harmonics is observed only for the Minnaert frequency. Linear combinations between the Minnaert harmonics and the multiple scattering resonance are also observed. However, the most significant effect observed is the appearance of softening-hardening effects that share some similarities with those observed for sandstones or cracked materials. These effects are related to the multiple scattering resonance. Downward or upward resonance frequency shifts can be observed depending on the characteristic of the incident wave when increasing the excitation amplitude. It is shown that the frequency shift can be explained assuming that the acoustic wave velocity depends on a law different from those usually encountered for sandstones or cracked materials.

Sound propagation in dilute suspensions of spheres: Analytical comparison between coupled phase model and multiple scattering theory.

Sound propagation in dilute suspensions of small spheres is studied using two models: a hydrodynamic model based on the coupled phase equations and an acoustic model based on the ECAH (ECAH: Epstein-Carhart-Allegra-Hawley) multiple scattering theory. The aim is to compare both models through the study of three fundamental kinds of particles: rigid particles, elastic spheres, and viscous droplets. The hydrodynamic model is based on a Rayleigh-Plesset-like equation generalized to elastic spheres and viscous droplets. The hydrodynamic forces for elastic spheres are introduced by analogy with those of droplets. The ECAH theory is also modified in order to take into account the velocity of rigid particles. Analytical calculations performed for long wavelength, low dilution, and weak absorption in the ambient fluid show that both models are strictly equivalent for the three kinds of particles studied. The analytical calculations show that dilatational and translational mechanisms are modeled in the same way by both models. The effective parameters of dilute suspensions are also calculated.

Influence of shell compressibility on the ultrasonic properties of polydispersed suspensions of nanometric encapsulated droplets.

Liquid droplets of nanometric size encapsulated by a polymer shell are envisioned for targeted drug delivery in therapeutic applications. Unlike standard micrometric gas-filled contrast agents used for medical imaging, these particles present a thick shell and a weakly compressible core. Hence, their dynamical behavior may be out of the range of validity of the models available for the description of encapsulated bubbles. In the present paper, a model for the ultrasound dispersion and absorption in a suspension of nanodroplets is proposed, accounting for both dilatational and translational motions of the particle. The radial motion is modeled by a generalized Rayleigh-Plesset-like equation which takes into account the compressibility of the viscoelastic shell, as well as the one of the core. The effect of the polydispersity of particles in size and shell thickness is introduced in the coupled balance equations which govern the motion of the particles in the surrounding fluid. Both effects of shell compressibility and polydispersity are quantified through the dispersion and absorption curves obtained on a wide ultrasonic frequency range. Finally, some results for larger gas-filled particles are also provided, revealing the limit of the role of the shell compressibility.

A model for ultrasound absorption and dispersion in dilute suspensions of nanometric contrast agents.

Ultrasound dispersion and absorption are examined in dilute suspensions of contrast agents of nanometric size, with a typical radius around 100 nm. These kinds of contrast agents are designed for targeted delivery of drugs for cancer treatment. Compared to standard contrast agents used for imaging, particles are of smaller size to pass through the endothelial barrier, their shell, made up of biocompatible polymer, is stiffer to undergo a longer lifetime, and they have a liquid core instead of a gaseous one. Ultrasound propagation in dilute suspension is modeled by combining two modes for particle oscillations. The first one is a dilatational mode assuming an incompressible shell with a rheological behavior of Kelvin-Voigt or Maxwell type. The second one is a translational mode induced by visco-inertial interaction with the ambient fluid. The relative importance of these two modes of interaction on both dispersion and absorption is quantified and analyzed for a model system and for two radii (75 and 150 nm) and the two rheological models. The influence of shell parameters (Young modulus, viscosity, and relative thickness) is finally discussed.

Effective wave numbers for thermo-viscoelastic media containing random configurations of spherical scatterers.

The dispersion relation is derived for the coherent waves in fluid or elastic media supporting viscous and thermal effects and containing randomly distributed spherical scatterers. The formula obtained is the generalization of Lloyd and Berry's [Proc. Phys. Soc. London 91, 678-688 (1967)], the latter being limited to fluid host media, and it is the three-dimensional counterpart of that derived by Conoir and Norris [Wave Motion 47, 183-197 (2010)] for cylindrical scatterers in an elastic host medium.

Multiple scattering by cylinders immersed in fluid: high order approximations for the effective wavenumbers.

Acoustic wave propagation in a fluid with a random assortment of identical cylindrical scatterers is considered. While the leading order correction to the effective wavenumber of the coherent wave is well established at dilute areal density (n0) of scatterers, in this paper the higher order dependence of the coherent wavenumber on n0 is developed in several directions. Starting from the quasi-crystalline approximation (QCA) a consistent method is described for continuing the Linton and Martin formula, which is second order in n0, to higher orders. Explicit formulas are provided for corrections to the effective wavenumber up to O (n0(4)). Then, using the QCA theory as a basis, generalized self-consistent schemes are developed and compared with self-consistent schemes using other dynamic effective medium theories. It is shown that the Linton and Martin formula provides a closed self-consistent scheme, unlike other approaches.

Propagation of elastic waves through two-dimensional lattices of cylindrical empty or water-filled inclusions in an aluminum matrix.

The layer-multiple-scattering method is developed to study wave propagation through two-dimensional lattices of cylindrical inclusions in an elastic medium. The lattices are a series of periodically spaced infinite one-dimensional periodic gratings (or rows) of inclusions. The layer-multiple-scattering method allows the analysis of the reflection and transmission properties of the two-dimensional lattice, provided those of each row are known. These are later determined by means of an exact multiple scattering formalism based on modal series developments. A new characteristic equation is obtained that describes the Bloch wave propagation into the infinite lattice. Lattices with empty and fluid-filled inclusions are compared. The comparison shows the existence of pass and stop bands due to the resonances of the fluid-filled inclusions. Resonant inclusions allow the opening of narrow pass bands inside phononic stop band, which is an interesting phenomenon for demultiplexing problems. It is worth noting that inclusion resonances have nothing to do with resonances due to defects, as they involve the whole lattice. In addition, it is shown that stop bands, at an oblique incidence, due to a strong coupling between longitudinal and transverse waves, are related to dispersive guided waves that propagate in the direction of the reticular planes of the lattices.

Influence of the slow wave on the relation between the angular resonances and the leaky guided modes properties for a poroelastic plate embedded in water.

A numerical study of the guided modes in a water-saturated poroelastic plate that obeys the Biot theory is presented. In the first part, we study the leaky guided modes and the angular resonances when the slow wave does not propagate. Two types of guided modes exist. The first ones occur from coupling of the fast longitudinal wave with the shear wave; most of them propagate whatever the frequency is, provided that it is not close to their cut-off frequencies. The leaky guided modes of the second type occur from coupling of the two longitudinal waves and the shear wave. These modes do not propagate (they are highly damped) as long as the slow wave remains diffusive. We show that the characteristics of the angular resonances can be linked to the leaky guided waves of the first type in the same way as for an elastic plate. The guided modes of the second type may not be associated to angular resonances. In the second part, we consider a thinner plate in a higher frequency range so that the slow wave can propagate. Once again its influence is studied both on the leaky guided modes and on the angular resonances.

Multiple scattering in a trabecular bone: influence of the marrow viscosity on the effective properties.

The Foldy and the Waterman and Truell approximations are used to determine the effective properties of the coherent wave that emerges after multiple scattering of a plane longitudinal fast wave by the largest pores in a trabecular bone. The unit scattering cell considered is either a single pore or two close cylindrical pores (cluster), at a fixed overall bone porosity. In the cluster case, the effective attenuation is about twice that obtained with one single pore per scatterer. It is shown that taking into account the marrow viscosity leads only to minor differences on the effective dispersion and attenuation.

Scattering by a fluid cylinder in a porous medium: application to trabecular bone.

In a trabecular bone, considered as a nondissipative porous medium, the scattering of an incident wave by cylindrical pores larger than the wavelength is studied. The goal is to know if scattering alone may cause such a high attenuation as that observed in calcaneus. The porous medium is modelized via Biot's theory and the scattering by a single pore is characterized from the definition of a scattering matrix. An approximation of weakly disordered medium is then discussed to estimate the effective attenuation and dispersion as a function of frequency. These effective properties are shown to be different of those measured on calcaneus, due to the neglect of wave conversions during the scattering process.