Propagation of Sound in Strongly Viscous Liquids (1966) [pdf]

SOVIET PHYSICS JETP VOLUME 23, NUMBER 5 NOVEMBER, 1966 PROPAGATION OF SOUND IN STRONGLY VISCOUS LIQUIDS M. A. ISAKOVICH and I. A. CHABAN Submitted to JETP editor December 1, 1965 J. Exptl. Theoret. Phys. (U.S.S.R.) 50, 1343-1363 (May, 1966) Experimental data on anomalous absorption and dispersion of sonic, shear, and electromagnetic waves in strongly viscous liquids show that the relaxation theory of wave propagation is not ap- plicable to these liquids. In particular, with increase of frequency, the absorption coefficient per wavelength and the departure of the propagation velocity from the limiting high frequency value tend to zero as 1/£ (w is the frequency). Since such an asymptotic behavior is also characteristic of microinhomogeneous media with diffusion exchanges taking place between the components (for example, emulsions), it is suggested that a highly viscous liquid is a microinhomogeneous medium, one of the components of which is present in the form of in- clusions with an ordered molecular arrangement in the second (disordered) component. It is assumed that in the passage of the wave, the ordered part undergoes a rearrangement and the equilibrium (relative to the second component) number of holes in it changes. The disturbed equilibrium with respect to the number of holes is restored by the diffusion between the com- ponents. The phase shift of this process relative to the incident wave results in anomalous absorption and dispersion. In its simplest form, the theory for sound waves is formally identical with the theory of thermal dispersion and absorption of sound in emulsions. The theory is refined and the shear waves are considered by taking into account shear stresses due to the microinhomogeneities of the medium, it being assumed that the static displacement viscosity of the medium is due to the usual Maxwellian relaxation mechanism. In the analysis of electromagnetic waves, it is as- sumed that the mechanism related to the microinhomogeneity of the medium is superimposed on the usual Debye relaxation mechanism. ·In all cases, the velocity and the absorption of the waves can be expressed in terms of the limiting high frequency and low frequency velocity val- ues, in terms of the absorption coefficient at low frequency, and in terms of the viscosity coef- ficient of the medium. No free parameters are employed in the calculations. Specific calcula- tions of the wave velocity and absorption are performed for a number of highly viscous liquids for which the necessary data could be found in the literature (glycerine, butanediol, hexanetriol, 2-methylpentanediol-2,4); a spherical shape of the ordered inclusions is assumed. Good agree- ment has been obtained between the theory and experimental data for the indicated liquids throughout the dispersion region. The characteristic temperature for the appearance of ordered regions is presented. Some possible ways for determining the sizes of the ordered regions at various temperatures are indicated. According to a preliminary estimate, about six glycerine molecules fit into the linear dimensions of the ordered region at 22° C. The size of this region should increase with decrease in temperature. 1. INTRODUCTION As is well known, in almost all liquids the ultra- sonic absorption is larger and the hypersonic ab- sorption is much smaller than follows from the Stokes theory of viscous absorption. About 30 years ago, a relaxation theory of sound absorption in liquids explaining this anomaly was set up by Mandel 'shtam and Leontovich. [ 1 J The theory also predicted dispersion of the sound velocity which was discovered later in some liquids (first by Shpakovskil[ 2] in the ultrasonic frequency region and by Fabelinski1 and Shustin[SJ in the hyper- sonic region). Attempts at the application of re- laxation theory to a highly viscous liquid in which anomalous absorption and velocity dispersion are also observed, have not been successful. The aim of the present research was the construction of a theory of sound propagation in such liquids. Let us sum up the results of the experimental investigation of the propagation of waves in highly viscous liquids. First we make the following note. In highly viscous liquids, the static shear viscosity 7J changes markedly (by several orders of magni- 893 894 M. A. ISAKOVICH and I. A. CHABAN tude) upon change in temperature by a few tens of degrees; the volume viscosity TJ' is proportional to the shear viscosity and has the same order of magnitude. It has been shown that the acoustical characteristics of highly viscous liquids (sound velocity v and the damping coefficient o referred to the frequency) depend on the sound frequency w and the viscosity in virtually exactly the same way: the frequency and temperature changes of v and ojw for a given liquid depend on the product WTJ and not on the parameters separately. Conse- quently, these characteristics can be regarded as depending only on the nondimensional product w r, where r is some characteristic time interval proportional to the viscosity; the physical meaning of this time interval is made clear in Sec. 3. In connection with this circumstance, the acoustic characteristics of highly viscous liquids are usu- ally measured at a fixed frequency by varying only the temperature and thus at the same time changing the viscosity. It is more convenient to change the temperature than to carry out measure- ments at different frequencies, and the results that are obtained can be treated both as frequency de- pendences and viscosity (temperature) depend- ences. We note that a small departure from such a rule is still observed experimentally and consists in a slow growth of the limiting value of the sound velocity (at high and low frequencies) with de- crease in temperature. This departure is taken into account in what follows both in the theory and in the comparison between theory and experiment. In accord with the data found in the literature, the following characteristics of the dependence of the velocity and sound absorption on the value of wr in a number of highly viscous liquids are typical. a) For each temperature, there are limiting low-frequency and limiting high-frequency values of the sound velocity, v0 and v00 , in the given liq- uid. The difference between the high-frequency limiting value and the value v of the sound veloc- ity at a certain frequency w decreases asymptoti- cally like lNwr with increase of the quantity wr. Relaxation theory gives a fall-off like law 1/(wr)2 for this difference. b) At low frequencies the quantity ojw is pro- portional to wr, as in the relaxational case, but at high frequencies this quantity falls off like 1/..fWT, while relaxation theory leads to a fall off like l/wr. c) The maximum value of o lies close to the beginning of the dispersion curve, and not at its middle as follows for the relaxation case; a graph of the dependence of absorption on w r is much broader than for the relaxational case. d) In the entire dispersion region, the quantity (v2 - v~)/(v~ - v2) is directly proportional to wr, while for the relaxational case (for small disper- sion) this quantity is proportional to (wr) 2• Thus, in spite of a certain qualitative agree- ment, the relaxation mechanism actually cannot explain the frequency and temperature dependence of the sound velocity and absorption in highly vis- cous liquids. It has been found by Fabelinskil and Krivokhizha [ 41 that quantitative agreement with experiment for the dispersion of sound velocity could be materially improved if it was assumed that the relaxation time is not constant for a given material at a given temperature but is proportional to the square root of the frequency. It is clear that such an assumption is not compatible in principle with relaxation theory. This observation, which formulates the dependence of the acoustical char- acteristics of the medium on the square root of the frequency, is the starting point in the develop- ment of the theory set forth in the present re- search. In highly viscous liquids, in addition to the longitudinal elastic (sonic) waves, there can also be propagated transverse elastic (shear) waves, for which there is also observed an anomalous ab- sorption and dispersion. The same also holds for electromagnetic waves in strongly viscous liquids. In a given strongly viscous liquid, the regions of dispersion are identical for all these three types of waves; for a fixed frequency the temperatures at which the values of the absorption for sonic and electromagnetic waves reach a maximum are vir- tually identical (there are no data in the literature on the absorption of shear waves). Such an identi- cal behavior allows us to assume that some com- mon mechanism lies behind the observed behavior of viscous liquids. Relaxation theory, which has customarily been called upon up to the present time for an explanation of the anomalous absorp- tion and dispersion in liquids, disagrees with ex- periment for each of these types of waves. The theory given below is in excellent agreement with experiment for all types of waves. 2. HYPOTHESIS ON THE MICROINHOMO- GENEOUS STRUCTURE OF HIGHLY VISCOUS LIQUIDS In the search for a mechanism which could ex- plain the acoustic behavior of highly viscous liq- uids, we turned our attention to the well known characteristics of sound propagation in microinho- mogeneous media. A medium is called microinho- PROPAGATION OF SOUND IN STRONGLY VISCOUS LIQUIDS 895 mogeneous if it contains two or more components and can be divided into regions which are small in comparison with the wavelength but which still contain many heterogeneous parts. With respect to the passage of a wave, such regions can be re- garded as lying in a uniform pressure field. The features of the acoustic behavior of such media are determined by nonacoustic processes which take place between the components. The first microinhomogeneous medium (a sus- pension of heavy particles in liquid)