acoustic theory...

ACOUSTIC THEORY FOR PARTICULATES (extraction from Chapter 4, [56]).

Any acoustic theory for particulates 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.  It is also desirable that this theory satisfies a set of requirements, summarized as follows: 

1.    The theory must be valid for a wide particle size range, from 10 nanometer to 1 millimeter, and for ultrasonic frequencies from 1 to 100 MHz. This is roughly equivalent in magnitude to a ka range of 0.00001 to 100. (k is the compression wavenumber)

2.    The theory should be valid for concentrated suspensions, and therefore must take into account particle-particle interactions.

3.    The theory should reflect the multiple mechanisms of ultrasound interaction with colloids, namely: viscous, thermal, scattering, intrinsic, structural, and electrokinetic. The heuristic description of these mechanisms is given below.   

Despite one hundred years of almost continuous effort by many distinguished scientists, there is still no single theory that 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, completely fails the second, and takes into account only four of the six mechanisms mentioned in the third.

The ECAH theory is constructed in two stages. We call the first stage the “single particle theory”, since it attempts to account for all of the ultrasound disturbances surrounding just a single particle. This stage relates the microscopic properties of both the fluid and 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.

Both parts of the ECAH theory (“single particle” and “macroscopic”) neglect any particle-particle interactions and are therefore valid only for dilute systems. For instance, in the ECAH “macroscopic theory” the total attenuation is regarded as simply a superposition of the contributions from each particle, and is determined only by the reflected compression wave. 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 [3]. 

There have been several attempts to extend this general two-stage acoustic theory to concentrated systems by incorporating particle-particle interactions. The following are examples of three different approaches for each stage:

For the Single Particle theory:

1.    Isolated particle in ECAH theory [1, 2].

2.    Cell model [14].

3.    Core-shell model [15, 16].

For the Macroscopic theory:

1.    Scattering and multiple-scattering theories [1, 4-8].

2.    Coupled phase model [9-11].

3.    Phenomenological theory by Temkin [12, 13].

Obviously these extensions increase the complexity of the ECAH theory. However, the original ECAH theory, even without any modifications, is already very complex, as a result of the authors’ attempt to construct a universal theory that is valid not only for any ka value, but also for all interaction mechanisms between the sound and the colloidal system.  Modifications to implement these particle interactions, such as outlined briefly above, are therefore practically impossible.

 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 indeed possible. It turns out that there are some quite general peculiarities of ultrasound propagation through colloids that allow us to simplify the theoretical process. Historically these peculiarities prompted the introduction of six different mechanisms of sound interaction with colloids. We give here a short heuristic description of them all.

1). The “viscous” mechanism is hydrodynamic in nature. 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. As a result, 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 is important for acoustics. It causes losses of acoustic energy due to shear friction. Viscous dissipative 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. 

2). The “thermal” mechanism is thermodynamic in nature and is related to the temperature gradients generated near the particle surface. These temperature gradients are due to the thermodynamic coupling between pressure and temperature. Dissipation of acoustic energy caused by thermal losses is the dominant attenuation effect for soft particles, including emulsion droplets and latex droplets.

3). The “scattering” mechanism is essentially the same as in the case of light scattering. Acoustic scattering does not produce dissipation of acoustic energy. Particles simply redirect a part of the acoustic energy flow, and as a result 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 exceeding roughly 3 microns.

4). The “intrinsic” mechanism refers to losses of  acoustic energy due to the interaction of the sound wave with the material of the particles and the medium, considered as homogeneous phases on a molecular level, and unrelated to the state of division of the colloidal dispersion.. It must be taken into account when the overall attenuation is low, which might happened for small particles or low volume fractions.

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

6). 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 electroacoustic measurements. However, it turns out its contribution to acoustic attenuation is negligible [55], which makes acoustic measurements completely independent of the electrical properties of the dispersion, including the properties of the double layer.

We can divide all of the mechanisms of ultrasound attenuation into two groups, depending on the way the acoustic energy is transformed in the colloid. The ultrasound attenuation in a colloid arises either from (1) absorption (the conversion of acoustic energy into thermal energy) or (2) scattering (the re-direction of incident acoustic energy from the incident beam). The combined effects of scattering and absorption can be described as the extinction cross section, Σ, which is an effective area whose product, with the incident intensity, is equal to the power lost from the sound beam [17-19]. In this respect, ultrasound is similar to light. There is a well known formula for light, “extinction = absorption + scattering”  which is also applicable to sound. This formula is the basis for the acoustic theory and therefore we consider it in some details in the following section. 

Extinction = absorption + scattering. Superposition approach.

There is a recent trend to snub this extinction equation, to consider viscous and thermal dissipations as additional forms of “scattering”, and to picture all particle-particle interactions at high concentration as simply additional aspects of “multiple scattering”. According to this nouveau view, the very term absorption is absorbed into a “Unified Scattering Theory “and the term extinction becomes extinct.

We think that crossing this Rubicon is misleading at best, hides a general understanding of the underlying mechanisms, and complicates rather than simplifies any extensions of the theory.

Adepts of this new philosophy offer several arguments for this new perspective.

First they claim that it is simply a semantic change. They argue that within the framework of the ECAH formulation it is impossible, in any case, to clearly separate the various mechanisms. The various components become hidden behind a veil of mathematical formalism, and this complexity defies intuitive understanding in terms of the underlying physical phenomena.  This lack of clarity is then used as a reason to discard terminology and philosophical underpinnings that have a long and honorable history.

In rebuttal to this first claim, we take strong exception to this semantic argument.  It not only completely alters the historical perspective, but the very language is a contradiction in terms. According to Webster, we find the following definitions:

Scatter: to separate and go in several direction; to reflect and refract in an irregular, diffuse matter.

Absorption: a taking in and not reflecting; partial loss of power of light or radio waves passing through a medium.

It is obvious that Webster’s definition stresses “absorption” as being quite different from “scattering”, emphasizing that it is “not  reflecting”.

This difference was obvious, and seemed important to the founder of the scattering theory, Lord Rayleigh. He wrote more than one hundred years ago in his book “The Theory of Sound. Vol.2” the following:

“…When we inquire into the mechanical question, it is evident that sound is not destroyed by obstacles as such. In the absence of dissipative forces, what is not transmitted must be reflected. Destruction depends upon viscosity and upon conduction of heat; but the influence of these agencies is enormously augmented by the contact of solid matter exposing a large surfaces. At such a surface the tangential as well as the normal motion is hindered, and a passage of heat to and fro takes place, as the neighboring air is heated and cooled during its condensations and rarefactions. With such rapidity of alternation as we are concerned with in the case of audible sounds, these influences extend to only a very thin layer of the air and of the solid, and are thus greatly favored by a fine state of division…   “

It is clear that Lord Rayleigh considered sound absorption as especially peculiar for in colloids, systems with the finite state of division and extended surfaces.

Secondly, the adepts appeal to the light propagation through the colloids. The general formula extinction=scattering + absorption, since it expresses only a basic conservation of energy, must be valid for light and sound. However, in the case of light the “scattering” term certainly dominates discussion in  scientific talks and publications. The terms “extinction” and “absorption” are rarely mentioned, and only then in special handbooks [3].

Figure 1  Attenuation spectra calculated for monodisperse colloids with three different sizes showing superposition of viscous absorption and rigid particle scattering. The arrows show the point corresponding to a frequency of 100 MHz.

This neglect of absorption phenomena in optics might be often justified, but it hardly justifies carrying over this bias in case of sound. The role of absorption is very different in acoustics as compared to optics. In optics, absorption is an electrodynamic effect, whereas in acoustics it is either hydrodynamic or thermodynamic. For optics, absorption presents a difficult problem because it requires information about the imaginary component of the refractive index and this information is generally not available. For optics, this normally leaves only one choice - to ignore the absorption of light. In this case, the description of light propagation thorough a colloid is reduced to a consideration of   scattering alone. In many real colloids, this simplification works well enough, because the absorption is often negligible.

In acoustics the situation is dramatically different. The absorption of ultrasound is easy to calculate. No special properties of the particles are required. As we will see later, 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. Rather it is very important source of information about the particles, especially sub-micron particles. Ignoring this term means ignoring the major advantage of ultrasound over light as the characterization technique. 

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

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

2.     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 to measure the intensity of the incident ultrasound after transmission. The intensity of this ultrasound is not affected by multiple scattering, but it may be affected by particle-particle interactions through viscous or thermal absorption mechanisms. We prove this statement experimentally later.

3.      The ultrasound attenuation in pure liquids and gels is via absorption only. There is no scattering there. This allows us to interpret acoustic spectra in rheological terms and to use n acoustic spectrometer as a high frequency rheometer.     

4.     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 which may be unknown in a given case. Even in the case where 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. This issue is discussed in details at the end of this Chapter.

5.      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.

Figure 2  Measured attenuation spectra for particle size standard silica BCR-70.

6. Calculation of the PSD from attenuation spectra is a classical ill-defined problem. The likelihood of multiple solutions can be minimized by carefully using all a’priori and independent information. Such information can be more readily employed in helping to solve the inverse problem if the mechanisms concerning the sound attenuation can be linked to all available a’prior independent information. The unified or “black box” approach does not easily provide format for this purpose

Taking into account all these argument, we have decided to proceed in this book with the classical meaning of the terms “scattering” and “absorption”, following Rayleigh, Morse, Sewell, and other distinguished scientists.

There is one more important argument supporting our decision. This last argument requires more detail consideration and justification.

7.     In the case of light it is practically impossible to separate absorption and scattering [3] in measurement, whereas in the case of ultrasound it easily achievable.

For ultrasound, absorption and scattering are distinctly separated in the frequency domain. As a general rule, for a given particle size, the absorption of ultrasound is dominant  at low frequencies, whereas scattering is dominant at high frequencies, with an overlap region existing within a narrow intermediate frequency range.

This is illustrated in Figure 1 (where the sum of the viscous absorption and scattering is plotted as a function of ka) for particles of three different sizes. The low frequency peaks correspond to the viscous absorption for different size particles, the high frequency peak corresponds to the scattering losses.

Figure 2 shows the measured attenuation spectrum for silica BCR-70. This material is a certified standard with a median size of 2.9 microns [52]. This attenuation spectrum demonstrates very clearly the transition from viscous absorption losses at low frequency, to scattering losses at high frequency. In this case, the transition range occurs in the vicinity of 20 MHz.

As with light, ultrasound scattering decreases rapidly with decreasing particle size. For ultrasound frequencies below 100 MHz, scattering becomes unimportant for particles below 1 micron. Thus for samples containing such small particles it makes no sense to employ a complicated theory that combines both absorption and scattering; an absorption theory would be sufficient. In the opposite case of large particles with ka >1, ultrasound absorption is negligible and any theory need take into account only scattering. 

Although Figure 1 considered only viscous type absorption, this separation between absorption and scattering is also observed for the thermal loss mechanism. Anson and Chivers [20] studied seventy two liquids and showed that thermal losses are important up to ka=0.5. Other mechanisms dominate for higher ka, except in those emulsions where only the thermal properties differ substantially between the two phases.

This separation of the loss mechanisms in the frequency domain allows us to express the total attenuation, aT , as the sum of the attenuation due to the absorption, aab , and the attenuation due to the scattering, asc. In addition, we should add a background intrinsic attenuation ain  because the dispersion medium attenuates ultrasound as does any other liquid. Thus:

                                                    (1)

A further simplification can be considered from the fact that ultrasound absorption is important for small ka, that is for wavelengths larger than the particle size. It is the so-called Rayleigh region [21, 22], or long wavelength limit. At this limit the two general components of ultrasound absorption (viscous and thermal) are additive. This then allows us to consider these two effects separately, and to neglect their possible coupling. The total attenuation now becomes:

                                             (2)

The potential contribution of any structural losses in a structured colloid would lead to the modification of the viscous losses, and to additional energy dissipation as a result of oscillation of the inter-particle bonds. This last contribution can be accounted for by adding a further term to the total attenuation:

                                 (3)

The simple superposition expression for the measured attenuation is not universally applicable. There are some exceptions [23]. However, Equation 3 is valid for a very large number of practical concentrated colloids.

Figure 3. Measured attenuation spectra for 0.3 micron diameter rutile in water.

The most important advantage of this superposition approach is that, compared to the ECAH theory, it allows us to take particle–particle interactions into account more easily.  Several independent groups have determined the range of volume fractions for which such particle-particle interactions become important. The conclusion from these studies is that the viscous loss mechanism is the most sensitive to such interactions.

Figure 4  Measured attenuation spectra for 0.1 micron neoprene latex in water.

Figure 5  Measured attenuation spectra for 120 micron quartz particles in water.

For example, Uric [24, 25] showed that the attenuation of rigid heavy particles becomes a nonlinear function of volume fraction above 10%vl, reaching a maximum attenuation at 15%vl. This was confirmed later by Hampton [26], Blue and McLeroy [27], Marlow et al [53], and Dukhin and Goetz [14]. It is interesting that we find this non-linear behavior to be independent of particle shape, and that it occurs at the same critical concentration, even for profoundly non-spherical particles [27]. Perhaps this happens because, for long wavelengths, the particles behave essentially as point sources; shape effects are therefore not pronounced. This same phenomenon is found in light scattering [3].  

The thermal loss mechanism, which is thermodynamic in nature, is less dependent on particle-particle interactions. This was shown to be true for several different polymer latices by Dukhin and Goetz [28, 29], and also for several emulsions by McClements, Hemar et al [15, 16, 30]. 

The importance of particle-particle interactions, as it relates to the scattering mechanism, depends very much on the method of measurement. In principle, we can measure the sound scattered at some angle to the incident beam, or, instead, 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 measurement technique plays a very important role in defining the necessary theory. In fact, the effect of “multiple scattering” can be minimized by measuring the attenuation of the incident wave as was clearly pointed out by Morse [17]:

 “…whether multiple scattering is important or not, the attenuation of the incident wave in traversing a distance z of region R is given by factor Exp(-Np Πz), where Π is the total scattering cross-section, Np is the number concentration of particles. When multiple scattering is important, some energy is absorbed by the scattered wave before it emerges from R, and some of the scattered wave is rescattered, but the formulas for the Σ and attenuation still tells us what happens to the incident wave, the part of the wave which has not been affected by scattering...”

Hence, by choosing to measure the attenuation of the incident beam, the scattering mechanism becomes much less dependent on particle-particle interactions. The result is that the variation of the transmitted wave intensity remains a linear function, of the volume fraction, up to much higher volume fractions than would be possible if directly monitoring the off-axis scattered sound. For example, Busby and Richardson [31] showed that the attenuation for large 95 micron glass spheres remains a linear function of the ultrasound path length up to 18%vl. Atkinson and Kytomaa [32, 33] concluded that the attenuation of 1 mm particles remains linear to 30%vl.

We have also collected data that confirms these observations by using a variety of disperse systems, each of which exhibits only one of the several possible mechanisms of ultrasound attenuation. Accordingly, Figure 4.3 shows the attenuation spectra for a dispersion of 0.3 micron rutile particles, for which we would expect mainly viscous losses.  For comparison, Figure 4.4 presents the attenuation spectra for a dispersion of 0.16 micron latex particles, for which the thermal losses are dominant. Finally, Figure 5 illustrates the attenuation for a dispersion of 120 micron quartz particles, for which scattering would be dominant.

The viscous attenuation in Figure 3 is strongly volume fraction dependent, leading also to a shift in the critical frequency from about 25 MHz to 60 MHz.  This frequency shift is much less pronounced for the thermal attenuation shown in Figure 4. Furthermore, we see that the critical frequency for the scattering attenuation in Figure 5 is virtually unchanged with volume fraction.

Figure 6  Measured attenuation at one particular frequency as a function of volume fraction for viscous, thermal and scattering losses.