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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.
To better illustrate
the difference in volume fraction dependence, we can extract the viscous,
thermal and scattering attenuation at a single frequency and plot this versus
volume fraction (Figure 6). The viscous attenuation is a linear function of
volume fraction up to only about 10%vl. In contrast, the thermal attenuation
remains linear up to about 30%vl. Finally, the scattering attenuation remains
linear up to 46%vl. Linear dependence
with volume fraction is a strong indication of the absence of particle-particle
interactions. Conversely, deviation from linearity implies the presence of such
interactions. The boundary, between the linear and nonlinear regions,
determines the volume fraction limit for any dilute case theory.
In summary, particle-particle interactions become an
important consideration for viscous losses at fairly low volume fraction,
whereas such interactions are relatively unimportant for scattering losses,
even at very high 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
The general structure of this expanded ultrasound absorption
theory remains the same as in the ECAH theory. Again there are two stages:
“single particle” and “macroscopic”. There are currently several versions of
the “single particle” theory that addresses particle-particle interactions.
They are based on the “cell model”, or the “effective media model”, or their
combination in the “core-shell model”. For both absorption mechanisms (viscous
and thermal) the “coupled phase model” can be used as the “macroscopic” theory
in place of the Epstein version of the optical theorem.
There is a substantial difference between the ways
absorption and scattering are measured. Absorption irreversibly converts
acoustic energy to heat. The measured intensity of the ultrasound after
propagation through colloid, It , is what remains of the
incident power, Ii , after subtraction of the energy absorbed
by the particles and liquid. For scattering, the situation is more complicated
because the energy disturbed by the particles is simply redirected, although
remaining as acoustic energy. This difference provides many options when
performing a scattering type measurement.
For instance, it is possible to measure the angular
dependence of scattered sound, a possibility that simply does not exist for
absorbed sound. Such angular dependence
measurements were made by Kol’tsova and Mikhailov [34] for very large graphite
particles in gelatin media, and by Faran [35] for metal cylinders. However, the
measurement of such angular dependence of the scattered sound is not common.
The more typical way
to measure the scattered sound is to determine the attenuation and/or sound
speed. Interpretation of a scattering measurement made in this mode is somewhat
more complicated than characterizing simple absorption, because part of the
scattered power may still reach the detector together with the remainder of the
incident beam. For this reason we speak of three components of scattered sound:
coherent, incoherent and multiple scattered. There are different ways to
separate these components, either in theory or in the experimental design of an
instrument. What we want to stress here, however, is that absorption and scattering
are very different, not only in their nature and frequency dependencies, but
also in the ways the measurement can be performed.
In summary, 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, but are rewarded with the ability to
incorporate particle-particle interactions into the theory of absorption. As a
result, acoustics can be more easily applied to the characterization of real
concentrated colloidal systems. Consequently, in the Chapter 4 of the book [56]
we describe this new theory, which we call the “superposition theory”. The term
“superposition” reflects superposition of the various mechanisms of the
ultrasound extinction on an additive basis.
As mentioned previously, the “single particle theory” might
be derived on the basis of either a “cell model” or a “core-shell model”. We
prefer to use a “cell model” for the viscous losses, since this mechanism is
hydrodynamic in nature and the use of a cell model is well established for such
effects. At present there is no corresponding “cell model” theory for thermal
losses, but in principle the “core-shell model” theory can be used. It is not yet clear which approach is the
most advantageous for characterizing thermal losses.
The macroscopic part of the “superposition” acoustic theory
is a combination of the “scattering theory” and the “coupled phase model”. We
suggest that the “scattering theory” be used only for characterizing ultrasound
scattering, whereas the “coupled phase model” provides a better framework to
describe ultrasound absorption.
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