Biot's theory of acoustic propagation in porous media applied to aerogels and alcogels

doi:10.1016/S0022-3093(98)00325-1

Journal of Non-Crystalline Solids

Volume 225, April 1998, Pages 287-292

https://doi.org/10.1016/S0022-3093(98)00325-1 Get rights and content

Abstract

Comparison of acoustic propagation in alcogels and aerogels presents an interesting difference for the high porosity: in alcogels, longitudinal wave velocity remains around the velocity in alcohol while, in aerogels, it is significantly lower than the velocity in air. In this paper, Biot's model of acoustic propagation in porous media is applied. We find a good agreement with measurements in terms of velocity that validates this theory for both alcogel and aerogel cases. Qualitative agreement is obtained for absorption but further comparisons are needed to test quantitative agreement. The theory allows explanation of behaviour differences observed by distinguishing the influence of the solid and fluid phases from that of the gel network.

Introduction

Since acoustic propagation in aerogels has been widely studied for different frequency ranges 1, 2, 3, 4, 5, 6, to our knowledge, it has never been related with acoustic propagation in alcogels. Although aerogels and alcogels exhibit approximately the same structure, they display very different acoustic behaviours, particularly for high porosity (>90%): since, for alcogels, longitudinal wave velocity is always around the velocity in alcohol, for aerogels, it is often significantly lower than the velocity of air. Is there a single theoretical model able to describe these properties?

It has been previously shown [7]that longitudinal velocity V_L in aerogels can be evaluated accurately from the classical formula in solids (dependent of E, K, μ, respectively, Young's, bulk and shear moduli, ν Poisson's ratio and ρ bulk density): $V_{L} = E(1−ν) ρ(1+ν)(1−2ν) = K+ 43 μ ρ .$

In alcogels, open porosity does not allow one to obtain elastic measurements of the whole sample (only characteristics of the gel network are measurable [8]in such a manner that no comparison with the above classical model is directly possible).

In the goal to study the effect of the fluid on propagation, it seems interesting to examine the possible application of the reference theory of propagation in porous media: Biot's theory 9, 10, 11developed in the 1950s and experimentally verified in 1980 by Plona and Johnson [12]. After having presented a synthesis of Biot's rational, we describe how the different physical parameters of this theory have been obtained. Then a comparison between theory and measurement in term of velocity and absorption is presented in two different ultrasonic frequency ranges (respectively [20–200 kHz] and [50–110 MHz]) and discussed.

Section snippets

Biot's theory

Biot considers the problem of the acoustical propagation in a porous elastic solid saturated by a viscous fluid by means of an energetic point of view. The method consists in following separately the average motions of both solid and fluid parts, i.e., in considering the elastic and inertial effects between two distinct interpenetrating `effective media' through a macroscopic energy balance. The following assumptions are implied in use of this theory: wavelength of the signal larger than the

Fluid and solid parameters

K_s for silica (269 GPa), ρ_f (1.2 kg/m³ for air and 789 kg/m³ for ethanol), K_f for air (142 kPa) and η (180 μP for air and 1.2 cP for ethanol) from the Handbook of Chemistry and Physics [13]. K_f for ethanol (1.15 GPa) has been determined from sound velocity V_eth=1207 m/s and bulk density with the help of the classical formula in fluids: K_f=ρ_fV_f².

Concerning the solid bulk density, ρ_s, He-pycnometry [14]allows this value to be obtained. The solid bulk density has been measured on neutral and base

Results

Table 1 indicates the different velocities of alcogels and aerogels calculated according to Biot's model and measured by emission–reception method. Absolute absorption measurement in the kHz frequency range remains problematic. Indeed, because no contact agent can be used, utilizing piezoelectric transducers introduces interface traps: since amplitude of the received signal depends drastically on the surface states and on the pressure applied by the transducers, usual methods cannot match in

Discussion

As noted above, the comparisons presented in this paper are almost qualitative. Nevertheless, they provide support for the good agreement of Biot's theory in the case of both aerogels and alcogels as it can easily be noticed from Table 1. This theory offers the advantage to distinguish different contributions of solid, fluid and frame. So it allows a better understanding of the different acoustical behaviours of both alcogels and aerogels by working out the pregnant parameters: in the

Conclusions

Biot's model has been shown to be appropriate for the description of sound propagation in both alcogels and aerogels, quantitatively in term of velocity and qualitatively in term of absorption. Because it distinguishes the solid, fluid and skeletal contributions, it allows prediction of acoustical behaviour if the aerogel is filled with gas other than air. Moreover, as measurements on alcogels and very low-density aerogels (ρ<15 kg/m³) are difficult to perform, it provides a good frame to

References (22)

J Gross et al.
J. Non-Cryst. Solids
(1992)
V Gibiat et al.
J. Non-Cryst. Solids
(1995)
G.W Scherer et al.
J. Non-Cryst. Solids
(1988)
T Woignier et al.
J. Non-Cryst. Solids
(1987)
A Zimmermann et al.
J. Non-Cryst. Solids
(1995)
B. Nouailhas, F. Michard, R. Gohier, A. Zarembowitch, Proceedings 11th ICA, Vol. 2 (1983) 179.Suppl....
F Michard et al.
ISA2 Montpellier
Rev. Phys. Appl. C4, Suppl. 4,
(1989)
E Courtens et al.
Phys. Rev. Lett.
(1987)
A.C Armand et al.
J. Phys. IV C1
(1992)
M Gronauer et al.
Acustica
(1986)

M.A Biot

J. Acoust. Soc. Am.

(1956)

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Biot's theory of acoustic propagation in porous media applied to aerogels and alcogels

Abstract

Introduction

Section snippets

Biot's theory

Fluid and solid parameters

Results

Discussion

Conclusions

J. Non-Cryst. Solids

J. Non-Cryst. Solids

J. Non-Cryst. Solids

J. Non-Cryst. Solids

J. Non-Cryst. Solids

ISA2 Montpellier

Rev. Phys. Appl. C4, Suppl. 4,

Phys. Rev. Lett.

J. Phys. IV C1

Acustica

J. Acoust. Soc. Am.