Ultrasonic investigation of the vibrational modes of a sintered glass-bead percolation system

Abstract

A physical realization of a random structural percolation system, formed from sintered glass microspheres, has allowed its elasticity and vibrational dynamics to be studied using ultrasonic techniques in the 1–55 MHz frequency range. Samples with occupied volume fractions φ between 0.18 and 0.64 were prepared, enabling the percolation threshold φ_c to be determined. From measurements of the ultrasonic velocity, the elasticity exponent τ was found to be 2.9±0.1, which falls midway between the best theoretical estimates for scalar and bond-bending elasticity. Very low values of Poisson's ratio were observed, and data for the longitudinal and shear wave velocities were shown to be consistent with the Arbabi-Sahimi suggestion that the ratio of the bulk to shear moduli approaches the value 8/z near threshold, where z is the coordination number. The frequency dependence of the scattering mean free path, l _s , was measured over an extended range of wave vectors k, corresponding to 1.6 < kl _s > 60, and spanning the transition from weak to very strong scattering. The mean free path was found to scale with a characteristic length, l _c, proportional to the percolation correlation length below which the structure exhibits fractal correlations. Comparison of the scaling behaviour for longitudinal and shear waves suggests that there may be a unique frequency for both polarizations, rather than a length scale, that characterizes the breakdown of the coherent propagation of acoustic waves due to very strong scattering.