Acoustic and elastic waves in periodic and random media

Abstract

Wave propagation in complex media is studied in many disciplines as astronomy, meteorology, seismology, etc. A very interesting point in this study was to observe the analogy between the propagation of electron waves in solids and classical waves in microstructured materials. The discovery of photonic band gap crystals, which emerged out of the pioneering conceptual idea of Sajeev John and the proposal for the structure of Yablonovitch, aroused an upsurge in the research efforts dedicated to analogous studies on photonic (acoustic) crystals. The elastic and acoustic waves attracted the attention due to both their larger number of material parameters (such as densities and sound velocities) which can be varied in order to find the optimal combination for spectral gap appearance, and their richness of applications in many branches of science and technology, such as solid state, geophysics, medicine, metallurgy, oil exploration, nondestructive evaluation, etc. This thesis reports on theoretical work concerning the elastic and acoustic waves propagation in various periodic and random composite structures such as: bubbles in water, lamellar forming block copolymer systems, soft colloids and hard sphere colloids. The fundamental quantities and the basic laws of the theory of elasticity, which will be used in all this study, are shortly reviewed in the first chapter of the thesis. The interest among solid-state physicists in the propagation of classical waves in composite materials either periodic or random was attracted by the question of the existence or not of spectral gaps in periodic systems or localized waves in disordered systems in analogy with the case of electrons in solids. This is the reason for which, in the last section of Chapter 1, we describe the phononic crystals and compare them with the solids and the photonic crystals. The next three chapters are dedicated to the methods of calculation used in determining the properties of a phononic crystal. The simplest way to imagine a phononic crystal is spherical scatterers embedded in a host material of different elastic properties. This is the model we use in describing theoretically most of the systems presented in this thesis (with the exception of the lamellar system) and for this reason we use it also in the detailed derivation of the theoretical methods. The first method, presented in Chapter 2, is the calculation of the scattering cross section of a sound plane wave scattered by a single sphere. In special cases, when the eigenmode of vibration of one sphere in a composite is only weakly coupled with the acoustic phonon propagating through the matrix and with eigenmodes of neighboring scatterers of different nature, the information given by this single sphere sound scattering can be sufficient to describe phonons of the whole system. This kind of phonons have the density energy concentrated mostly inside or at the surface of each sphere. Therefore, in order to check this localization of the phonons, we calculate the energy density distribution by the method presented at the end of this chapter. In the case of strongly coupling between different kind of modes, hybrid modes arise which can not be easily described. In order to find these modes theoretically, one must take into account the multiple scattering effects. In Chapter 3 we present methods of band structure, transmission and density of states calculations: the plane wave method, the multiple scattering method and the finite difference time domain method. Each of these methods has advantages and disadvantages and the choice of one or the other depends on the properties of the composite. An experimental method to obtain information about the phonons in composites is the light scattering by phonons. Only the phonons which scatter strongly the incident light can be detected. In order to identify theoretically the experimentally found phonons, one needs to calculate the intensity of light scattering, and the method is given in Chapter 4. A very interesting example of phononic crystal, with wide spectral gap, formed by air bubbles in water is discussed in the next chapter. This composite is a strongly multiple scattering environment for the acoustic waves and in the case of the disordered system formed by bubbles of random size and randomly placed, analogous with the amorphous semiconductors, new eigenstates appear at the edge of the permitted frequency bands, which can be Anderson localized eigenstates. In the last chapter, phonons in composite systems, revealed experimentally by Brillouin spectroscopy, are theoretically elucidated by using the methods of calculation mentioned above. The theoretical research in this chapter was performed in close collaboration with experimental physical chemists and physicists form FO.R.T.H., Institute of Electronic Structure and Laser form Heraklion, Greece, Max Planck Institute for Polymer Research, Mainz, Germany, and Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A.. The investigated composites are: (i) block copolymer lamellar forming system made of poly(styrene-b-isoprene) symmetric copolymer in toluene; (ii) soft colloids suspensions of hairy particles made of styrene-isoprene diblock copolymer in n-decane; (iii) hard sphere colloidal crystals of silica spheres in cyclohexane/decaline or (iv) PMMA spheres in decaline/tetraline mixture; and (v) colloids of PDMS-grafted silica shperes in n-heptane/toluene. The good quantitative agreement between the theory and the experimental proves the capacity of our methods to describe real systems.