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
particle size distribution, etc. This relationship should be valid
for wide frequency range on MHz scale, high volume fraction of dispersed phase.
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. 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 , 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, 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
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
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 electro-acoustic measurements. However, it turns out
its contribution to acoustic attenuation is negligible , 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 heterogeneous system 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
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 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. 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 and nanoparticles. 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.
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
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
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 an acoustic spectrometer as a high frequency rheometer.
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
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
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.
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.
The importance of particle-particle
interactions is different for different mechanisms. For example, 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 by several
authors, see Ref. . 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 . 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
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 , and
also for several emulsions by McClements, Hemar et al [see Ref. ].
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 .
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.
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
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.
Epstein, P.S. and Carhart R.R., “The Absorption of Sound in
Suspensions and Emulsions”, J. Acoust. Soc. Amer., 25, 3, 553-565 (1953)
Allegra, J.R. and Hawley, S.A. “Attenuation of Sound in
Suspensions and Emulsions: Theory and Experiments”, J. Acoust. Soc. Amer., 51, 1545-1564 (1972)
Morse, P.M. and Uno
Ingard, K., “Theoretical Acoustics”, 1968
McGraw-Hill, NY, Princeton University Press, NJ, 925 p, (1986)
A.S. and Goetz, P.J. “Ultrasound for characterizing
colloids”, Elsevier, 2002
Hemar, Y. and Herrmann, N. “Incorporation of
thermal overlap effects into multiple scattering theory”, J. Acous. Soc. Am.,
105, 2, 915-918 (1999)http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000105000002000915000001&idtype=cvips&gifs=yes&ref=no