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Ultrasonic characterization of plastic foams via measurements with static pressure variations

Christophe Ayrault, Alexei Moussatov, Bernard Castagne`de,^a)and Denis Lafarge

Laboratoire d’Acoustique de l’Universite´ du Maine, UMR CNRS 6613, IAM Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France

~Received 4 August 1998; accepted for publication 30 March 1999!

A method for ultrasonic characterization of plastic foams by changing the static pressure of air that saturates the foam has been proposed. The method is based on high frequency asymptotic expressions of the standard Johnson–Allard equivalent fluid model. It is shown, both experimentally and theoretically, that the real part of squared acoustical refractive index and logarithm of the transmission coefficient depend linearly on the inverse of the square root of applied static pressure.

These linear relations provide a simple and convenient way to determine experimentally the constitutive parameters. The method is illustrated with industrial open-cell foams. Advantages, limitations, and perspectives are discussed. © 1999 American Institute of Physics.

@S0003-6951~99!04721-X#

Plastic foams are absorbing porous materials widely used for noise reduction. The propagation of acoustical waves in porous materials is generally described, when the frame of the porous slab is motionless, by the equivalent fluid models^1,2 ~the Biot theory should instead be used to describe additional vibrations of the frame^2,3!. In this case, the wave number k in the porous medium is written k 5(v^/c0)n, wherev is the angular frequency, n is the com- plex refractive index, and c₀is the sound velocity in the gas.

In the long wavelength limit, ~i.e., when the wavelength is much larger than typical grains and pores sizes!, the index n is well defined and is given by n5Aab^{, where}a⁽v^{) and}b (v) are the dynamic tortuosity and compressibility of the equivalent fluid, respectively.⁴ The Johnson–Allard equivalent fluid model^1,2 proposes a description of these functions with a set of geometrical parameters: the porosity f^{, the} tortuosity a_`, the permeability k₀~linked to the flow resis- tivitys5h^/k0 through viscosity of the fluidh), the viscous characteristic lengthL, and the thermal characteristic length L8^.

In this model,^1,2 as predicted by general considerations

~see Ref. 5, Appendix A!, one can see thata ^andb ^{may be} expressed in the form a⁽v⁾5F(vr0/h^{) and} b⁽v⁾5g 2(g21)G(v^Prr0/h), where F and G are functions de- pending only on the microgeometry, and r0, g^{, and Pr}~respectively, density, specific heat ratio, and Prandtl number! are constant related to the gas saturating the porous medium.

For a perfect gas, the density r0 is linearly related to the static pressure P₀by the expressionr05( M /RT) P₀, via the molar mass M, the gas constant R, and the absolute tempera- ture T. All other constants are independent on P₀. Moreover, the internal geometry can also be assumed independent on P₀ because the elastic solid constituting the frame is very stiff compared to the compressed gas. Therefore, a ^andb^, thus n, are functions of v^P0. When using low-frequency ultrasound, the expressions of a ^and b can be further simplified^4,6,7for a sufficiently largev. Thus, the square of

the propagation index n_r5Re(n) can be presented as:

n_r²~v^P0!5a_`

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with u5A²^hM^RT. ~1!

Experimentally, n_r²5(c₀/c)², where c₀5Ag^P0/r0, is pro- vided by the measurement of the phase speed c in the mate- rial.

The logarithm of the transmission coefficient through a porous slab of thickness L is not a function ofv^P0 because of the presence of factorvin k. However, it depends both on v ^{and P}0and can also be written out in the asymptotic high frequency limit in the following form:

ulnuT~v^{, P}0!uu5ln~G!1

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P^v₀

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^ha²^`^u^c^L⁰

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withG5~11f^/Aa_`!²

4f^/Aa_` ^. ^~²^! Asymptotic expressions ~1! and ~2! are written under the following conditions:

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²^P⁰^v^@¹ ^and ^L@_AP^u₀v^. ^~³^! With the traditional frequency variation method,^4,6 P₀is kept constant and equal to ambient pressure and onlyv ^var- ies. For the asymptotic model, both frequency v ^{and static} pressure P₀play similar roles. Consequently, the suggestion has been made that variation of the static pressure affects experimentally the measured values of c and T in the same way as variation of the ultrasonic frequency, thus providing a

a!Electronic mail: [email protected]

APPLIED PHYSICS LETTERS VOLUME 74, NUMBER 21 24 MAY 1999

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background for an alternative method to retrieve information on tortuosity and both characteristic lengths of the model.

From Eqs.~1!and~2!, it is straightforward to note that the functions n_r²(1/A^P0) and ulnuT(1/A^P0)uu are linear. Plot of n_r² vs 1/A^P0 directly yields the value of the tortuosity at the intersection of extrapolation of the graph line with the vertical axis. Plot ofulnuTuucannot provide this value because ln~G! is too small to be precisely determined. Slopes of the lines yield in both cases the term @1/L1(g 21)/(A^PrL8⁾#, knowing the tortuosity.

Preliminary tests performed with other porous media, i.e., cemented glass beads and Berea sandstone, were re- ported earlier by Nagy,⁸showing that the attenuation~when expressed on a logarithmic scale!is inversely proportional to the square root of the applied pressure.

To operate variation of static pressure from 0 to 7 bars, a barrocamera has been designed and built. A pair of ultrasonic wide band piezoelectric transducers were mounted on hold- ers fixed on a rail. An automatic sample holder for 1303170 mm foam pieces was placed between them. At each pressure level, a reference signal ~with the sample momentarily re- moved! and a signal provided by transmission through the sample at normal incidence are acquired and further pro- cessed. After each pressure increase/decrease, a several min- utes pause is systematically observed before acquiring the signals to be sure that equilibrium temperature within the barrocamera is reached and kept at ambient level. This parameter was further monitored by thermal probes.

Measurements have been performed on several open-cell plastic foams without internal membranes manufactured by

Recticel~Belgium!. Figures 1 and 2 present n_r²andulnuTuuvs 1/A^P0for three samples with different flow resistivities. The data points are correctly aligned, as predicted from Eqs.~1! and~2!. Table I provides comparison between values of tortuosity and characteristic lengths obtained by the proposed method and the usual frequency variation method.

Foams F1 and F2 are typical open-cell foams of 50 mm thickness with flow resistivities of 2850 and 8900 Nm²⁴s.

Results for the parametersa_` ^andLobtained by variations of static pressure are in good agreement with reference values known within 15% accuracy, while measurement errors are no more than 10%. Correlation between values of the parameters based on wavespeed and attenuation measurements also proves the feasibility of the suggested method.

Foam F3, which is much more resistive ~s538 000 Nm²⁴s, L527 mm!exhibits also a linear behavior for both types of measurements with a good correlation between them. For sample F3, at the used range of pressure ~0.5–5 bars!and at the 100 kHz frequency, the asymptotic criteria of the model are just satisfied: the first term of Eq.~3!is 12.9 at 0.5 bar~it corresponds to the minimum of the term within the range!vs 1, butLis 50 vs 10mm. For foams F1 and F2, the two criteria are well satisfied.

Pressure variation method provides an opportunity to perform all required measurements by operating at one single frequency, which is a significant advantage. The frequency can be chosen within a wide range limited by the only restriction: the corresponding wavelength has to be much smaller than the dimensions of the foam sample and larger than the average size of the pores within the material.

As illustrated by Figs. 1 and 2, an increase in operating frequency results in a change of the curves slopes on both dia- grams. Such trend is in perfect agreement with Eqs. ~1!and

~2!; this is explained by dispersion and frequency-dependent attenuation within the material.

One can notice that on transmission diagram for F1 at 200 kHz~Fig. 2!, the extrapolation of the curve into the area of infinite pressure does not correspond to Eq. ~2! which predicts the convergence of the line towards zero @theoretically: ln~G!;0.001#. Beyond the errors related to measurements, this disagreement is due to scattering phenomenon which results in additional losses in porous media and occurs when the wavelength is not any more larger than the characteristic dimensions of the porous structure.⁹With variations of static pressure, scattering remains nearly constant because the speed in the material, and thus the wavelength, does not change very much ~Fig. 1!. Additional attenuation due to scattering causes a nearly constant shift upwards of the trans-

FIG. 1. The square index n_r²(1/A^P0) of propagation for the polyurethane foams F1, F2, and F3. The solid lines are obtained each time through linear interpolation.

TABLE I. Parametersa_`andL ~reference and measurement!for the foams F1@f50.97,s52850 Nm²⁴s,L8 is equal~Ref. 5!to 750mm#, F2~f50.98,s58900 Nm²⁴s;L8is fixed to 2.5 timesL!, and F3~f50.97, s538850 Nm²⁴s;L8is fixed to 2.5 timesL). Measurements for F1 ofa_`andLare done at 30 kHz~left side!

and 200 kHz~right side!.

Parameters a` L ~mm!

Foam F1 F2 F3 F1 F2 F3

Parameter reference 1.055 1.05 1.25 300 200 60

measurement (nr

2(1/AP0)) 1.055 1.06 1.04 1.26 319 307 197 50

measurement (ulnuT(1/AP0)i) ••• ••• ••• ••• 319 314 192 50

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Appl. Phys. Lett., Vol. 74, No. 21, 24 May 1999 Ayraultet al.

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mission coefficient curve. Measured values in weak scattering regime for foam F1 are satisfactory~Table I!. Here, scattering effect is not strong enough to modify the phase speed curve. The authors believe that it could be possible to measure geometrical parameters in the weak scattering regime with this technique. However, further studies are still needed to justify this point.

Due to the similar roles played by frequencyv^{and static}

pressure P₀ in the equivalent fluid model, an alternative method for determination of the parameters of the model has been suggested. The method works well with standard industrial open-cell plastic foams and has an advantage of operating at one single frequency. Criteria for checking of the method validity have been specified. The proposed method also provides an opportunity to separate more easily scattering losses from the ones having viscothermal origin. Due to the significant decrease in attenuation when the static pressure rises above one bar, the method provides broad oppor- tunities to obtain the transmission and refraction data with highly damping air-saturated materials ~attenuation of 50 dB/cm or higher!. Obviously, for interpretation of these data, the asymptotic model is probably no more valid. Conse- quently, in such a case, another description will be required and remains a topic for future research.

1D. L. Johnson, J. Koplik, and R. Dashen, J. Fluid Mech. 176, 379~1987!.

2J. F. Allard, Propagation of Sound in Porous Media~Chapman & Hall, London, 1993!, pp. 92 and 93.

3M. A. Biot, J. Acoust. Soc. Am. 28, 168~1956!.

4M. Melon, D. Lafarge, B. Castagne`de, and N. R. Brown, J. Appl. Phys.

78, 4929~1995!.

5D. Lafarge, P. Lemarinier, J. F. Allard, and V. Tarnow, J. Acoust. Soc.

Am. 102, 1995~1997!.

6P. Leclaire, L. Kelders, W. Lauriks, M. Melon, N. R. Brown, and B.

Castagne`de, J. Appl. Phys. 80, 2009~1996!.

7M. Melon and B. Castagne`de, J. Acoust. Soc. Am. 98, 1228~1995!.

8P. B. Nagy and D. L. Johnson, Appl. Phys. Lett. 69, 2641~1996!.

9P. Leclaire, L. Kelders, W. Lauriks, J. F. Allard, and C. Glorieux, Appl.

Phys. Lett. 69, 2641~1996!. FIG. 2. The logarithm of the transmission coefficientulnuT(1/A^P0)uufor the

polyurethane foams F1, F2, and F3. The solid lines are obtained each time through linear interpolation. Points at the origin are not experimental but interpolated points.

3226 Appl. Phys. Lett., Vol. 74, No. 21, 24 May 1999 Ayraultet al.

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