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SHORT INTRODUCTION TO ACOUSTIC THEORY

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Introduction

Acoustic theory for heterogeneous system should yield a relationship between some measured macroscopic acoustic properties, such as sound speed, attenuation, acoustic impedance, angular dependence of the scattered sound, etc., and some microscopic characteristics of the heterogeneous system, such as its composition, structure, electric surface properties, particle size distribution, etc. This relationship should be valid for a wide frequency range on the MHz scale and at a high-volume fraction of the dispersed phase.

ECAH theory

Despite one hundred years of almost continuous effort by many distinguished scientists, there is still no single theory that would meets all of these requirements. For example, the best-known theory, abbreviated as ECAH, following the names of its creators: Epstein, Carhart, Allegra and Hawley [1, 2], meets the first requirement. However, it completely fails the second. The ECAH theory is constructed in two stages. We call the first stage the “single particle theory”, since it attempts to account for all ultrasound disturbances surrounding just a single particle. This stage relates the microscopic properties of both the liquid and the particle to the system properties at a “single particle level”. The second stage, which we refer to as the “macroscopic theory”, then relates this “single particle level” to the macroscopic level at which we actually obtain our experimental raw data.

Extension of ECAH theory

Both parts of the ECAH theory (“single particle” and “macroscopic”) neglect any particle-particle interactions. Therefore, they are valid only for a dilute system. Evidently, in the ECAH “macroscopic theory” the total attenuation is regarded as simply a superposition of the contributions from the individual particle.  Consequently, only reflected compression wave determines attenuation coefficient.  Clearly, this part, derived by Epstein and Carhart [1], is similar to the well-known optical theorem that declares that the extinction cross section depends only on the scattering amplitude in the forward direction.

There have been several attempts to extend this general two-stage acoustic theory to concentrated systems by incorporating particle-particle interactions. Obviously, these extensions increase the complexity of the ECAH theory. However, the original ECAH theory, even without any modifications, is already very complex. Consequently, modifications to implement the particles interactions, such as outlined briefly above, are practically impossible

If we combine scattering and absorption in a single mathematical model, such as the ECAH or modification to it, we are forced to use at all times an extended set of input parameters. Many of these parameters may be unknown in a given case. Even in the case when a given mechanism may be unimportant, the relevant physical properties may still be required because of the complicated perhaps very nonlinear characteristics of a “unified” approach. Separating scattering and absorption opens the way to minimize the number of required input parameters.

Possible simplifications

One might ask, is it necessary to develop such a general theory? Is it possible to introduce some simplifications, while at the same time providing room for more readily implementing these particle interactions? Such simplifications are possible, indeed. It turns out that there are some quite general peculiarities of ultrasound propagation through complex liquids that allow us to simplify the theoretical process. Historically these peculiarities prompted the introduction of six different mechanisms of sound interaction with colloids. Here is a short heuristic description of them all.

Viscous attenuation

The “viscous” mechanism is hydrodynamic in nature. Accordingly, it is related to the shear waves generated by the particle oscillating in the acoustic pressure field. These shear waves appear because of the difference in the densities of the particles and the medium. The density contrast causes particle motion with respect to the medium. Consequently, the liquid layers in the particle vicinity slide relative to each other. The non-stationary sliding motion of the liquid near the particle is referred to as the “shear wave”. This mechanism causes losses of acoustic energy due to shear friction converting it into the heat. Viscous losses are dominant for small rigid particles with sizes less than 3 microns, such as oxides, pigments, paints, ceramics, cement, and graphite. The viscous mechanism is closely related to the electrokinetic mechanism which is also associated with the shear waves.

Thermal attenuation

The “thermal” mechanism is thermodynamic in nature. It is related to the temperature gradients generated near the particle surface. These temperature gradients appear due to the thermodynamic coupling between pressure and temperature. Dissipation of the acoustic energy in these temperature gradients dominates attenuation effect for soft particles, including emulsion droplets and latex droplets.

Scattering attenuation

The “scattering” mechanism is essentially the same as in the case of light scattering. Acoustic scattering does not produce dissipation of the acoustic energy. Particles simply redirect a part of the acoustic energy flow. Therefore, this portion of the sound does not reach the receiving sound transducer. The scattering mechanism contributes to the overall attenuation, and this contribution is significant for larger particles with a diameter larger roughly 3 microns.

Intrinsic attenuation

The “intrinsic” mechanism refers to the losses of the acoustic energy due to the interaction of the sound wave with the material of the particles and the medium. It is unrelated to the state of the division of the colloidal dispersion. Thus, there is no dependence on particle size. It must be taken into account when the overall attenuation is low, when particles are small or low volume fractions.

Structural attenuation

The “structural” mechanism links acoustics with rheology. It occurs when particles build some network. Oscillation of these inter-particle bonds can cause additional energy dissipation.

Electrokinetic mechanism

The “electrokinetic” mechanism describes the interaction of ultrasound with the double layer of the particles. Oscillation of the charged particles in the acoustic field leads to generation of an alternating electrical field, and consequently to an alternating electric current. This mechanism is the basis for electro-acoustic measurements. However, it turns out its contribution to acoustic attenuation is negligible. This makes acoustic measurements completely independent of the electrical properties of the dispersion, including the properties of the double layer.

Adsorption and scattering of ultrasound

We can divide all of the mechanisms of ultrasound attenuation into two groups, depending on the way of the acoustic energy transformation. The ultrasound attenuation in the heterogeneous system arises either from (1) absorption (the conversion of the acoustic energy into the thermal energy) or (2) scattering (the re-direction of the incident acoustic energy from the incident beam). The term “extinction cross section” reflects combined effects of the scattering and absorption. In this respect, ultrasound is similar to the light. There is a well-known formula for the light, “extinction = absorption + scattering” which is also applicable for the ultrasound. This formula is the basis for the acoustic theory that serves as a basis for our software that calculates particle size distribution from the measured attenuation spectra.

Adsorption of energy is often neglected in light scattering. In acoustics the situation is dramatically different. The absorption of ultrasound is easy to calculate.  Set of input parameters is simple and accessible. The absorption of ultrasound by solid rigid particles depends only on their density, which is readily available or can be easily measured. In the case of ultrasound, absorption is not a complicating factor, like for the light scattering. Instead, it is very important source of information about the particles, especially sub-micron particles and nanoparticles. Ignoring this term means ignoring the major advantage of ultrasound over light as the characterization technique.

Peculiarities of Acoustics in complex liquids

Finally, we offer several arguments in support of our view of retaining the historical viewpoint of Rayleigh and others.

  • Sub-micron particles

For acoustics, sub-micron particles do not scatter ultrasound at all in the frequency range under 100 MHz. They only absorb ultrasound, converting it to the heat. Consequently, there is no need to develop a general complex scattering-absorption theory for such sub-micron particles.

  • Multiple scattering elimination

For acoustics there is a very simple way to eliminate the nonlinear effects of multiple scattering. (We define scattering here in the classical sense, as the interaction of the compression waves scattered by particles with other particles). The effects of multiple scattering are completely eliminated by following Morse’s suggestion [3] to measure the intensity of the incident ultrasound pulse after transmission. The intensity of this ultrasound is independent on multiple scattering, because it occurs after pulse propagation forward. However, particle-particle interactions through viscous or thermal absorption mechanisms contributes to the overall attenuation effect.

  • Acoustics and rheology

The ultrasound attenuation in pure liquids and gels is due to absorption only. There is no scattering. Accordingly, this allows interpreting acoustic spectra in rheological terms and to use an acoustic spectrometer as a high frequency rheometer.

  • Separation of scattering and adsorption of ultrasound.

Separation of scattering and absorption phenomena provides more insight as to the nature of the attenuation phenomena. The unified approach is like a “black box”. We input information and get an answer without any understanding of the processes going on. In the case of light, it is practically impossible to separate absorption and scattering in measurement, whereas in the case of ultrasound it easily achievable.

  • Ill-defined problem of calculation PSD.

Calculation of the Particle Size Distribution (PSD) from the attenuation spectra is a classical ill-defined problem. Careful selection of a’priori independent information minimizes the likelihood of multiple solutions. Such information can be more readily employed in helping to solve the inverse problem when the mechanisms concerning the sound attenuation can be linked to all available a’prior independent information. Obviously, the unified or “black box” approach does not easily provide format for this purpose.

Particle-particle interaction

The importance of the particle-particle interactions is different for the different mechanisms. For example, the attenuation of rigid heavy particles becomes a nonlinear function of the volume fraction above 10%vl, reaching a maximum attenuation at 15%vl. Interestingly, this non-linear behavior is independent of the particle shape. Moreover, it occurs at the same critical concentration even for profoundly non-spherical particles. Perhaps this happens because, for long wavelengths, the particles behave essentially as point sources. Therefore, the shape effects are less important. Evidently, this same phenomenon exists in light scattering.

The thermal loss mechanism, which is thermodynamic in nature, is less dependent on the particle-particle interactions. This was shown to be true for several different polymer latices and emulsion, see ref. [4].

The importance of particle-particle interactions, as it relates to the scattering mechanism, depends very much on the method of the measurement. Generally, we can measure the sound scattered at some angle to the incident beam. In contrary, we can consider only the decrease in the intensity of the incident wave as it travels through the colloid. It is not widely understood that this choice of the measurement technique plays a very important role in the defining the necessary theory. In fact, the effect of “multiple scattering” can be minimized by the measuring the attenuation of the incident wave as was clearly pointed out by Morse [3]. Hence, by choosing to measure the attenuation of the incident beam, the scattering mechanism becomes much less dependent on particle-particle interactions.

Summary

Evidently, particle-particle interactions become an important consideration for viscous losses at fairly low volume fraction. Such interactions are relatively unimportant for the scattering losses, even at much higher volume fraction. Thus, we can conclude that ultrasound absorption is the most important mechanism to address when developing a more general theory that takes into account hydrodynamic, thermodynamic and specific particle interactions. Development of an extended scattering theory to account for such particle interactions, the so-called “multiple scattering” problem, is of much less importance for acoustics. However, multiple scattering is indeed a major concern when attempting to analyze similar concentrated systems using optical systems

We conclude that there is a strategic approach for deriving acoustic theory which is an alternative to the ECAH theory. We give up the idea of considering simultaneously all the mechanisms of ultrasound attenuation for all ka. In return we gain the ability to incorporate particle-particle interactions into the theory of absorption. Consequently, acoustics becomes more suitable for characterizing real concentrated dispersions and emulsions.

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