ISO 20998-2:2013

INTERNATIONAL ISO
STANDARD 20998-2
First edition
2013-08-15
Measurement and characterization of
particles by acoustic methods —
Part 2:
Guidelines for linear theory
Caractérisation des particules par des méthodes acoustiques —
Partie 2: Théorie linéaire
Reference number
©
ISO 2013
© ISO 2013
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ii © ISO 2013 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviated terms . 2
5 Mechanism of attenuation (dilute case) . 4
5.1 Introduction . 4
5.2 Excess attenuation coefficient . 4
5.3 Specific attenuation mechanisms . 5
5.4 Linear models . 5
6 Determination of particle size. 7
6.1 Introduction . 7
6.2 Inversion approaches used to determine PSD . 8
6.3 Limits of application. 9
7 Instrument qualification . 9
7.1 Calibration . 9
7.2 Precision . 9
7.3 Accuracy .10
8 Reporting of results .11
Annex A (informative) Viscoinertial loss model .12
Annex B (informative) ECAH theory and limitations .13
Annex C (informative) Example of a semi-empirical model .16
Annex D (informative) Iterative fitting .19
Annex E (informative) Physical parameter values for selected materials .21
Annex F (informative) Practical example of PSD measurement .22
Bibliography .30
Foreword
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The committee responsible for this document is ISO/TC 24, Particle characterization including sieving,
Subcommittee SC 4, Particle characterization.
ISO 20998 consists of the following parts, under the general title Measurement and characterization of
particles by acoustic methods:
— Part 1: Concepts and procedures in ultrasonic attenuation spectroscopy
— Part 2: Guidelines for linear theory
iv © ISO 2013 – All rights reserved

Introduction
It is well known that ultrasonic spectroscopy can be used to measure particle size distribution (PSD)
in colloids, dispersions, and emulsions (References [1][2][3][4]). The basic concept is to measure the
frequency-dependent attenuation or velocity of the ultrasound as it passes through the sample. The
attenuation spectrum is affected by scattering or absorption of ultrasound by particles in the sample,
and it is a function of the size distribution and concentration of particles (References [5][6][7]). Once
this relationship is established by empirical observation or by theoretical calculations, one can estimate
the PSD from the ultrasonic data. Ultrasonic techniques are useful for dynamic online measurements in
concentrated slurries and emulsions.
Traditionally, such measurements have been made off-line in a quality control lab, and constraints
imposed by the instrumentation have required the use of diluted samples. By making in-process
ultrasonic measurements at full concentration, one does not risk altering the dispersion state of the
sample. In addition, dynamic processes (such as flocculation, dispersion, and comminution) can be
observed directly in real time (Reference [8]). These data can be used in process control schemes to
improve both the manufacturing process and the product performance.
ISO 20998 consists of two parts:
— 20998-1 introduces the terminology, concepts, and procedures for measuring ultrasonic
attenuation spectra;
— 20998-2 provides guidelines for determining particle size information from the measured spectra
for cases where the spectrum is a linear function of the particle volume fraction.
A further part addressing the determination of particle size for cases where the spectrum is not a linear
function of volume fraction is planned.
INTERNATIONAL STANDARD ISO 20998-2:2013(E)
Measurement and characterization of particles by
acoustic methods —
Part 2:
Guidelines for linear theory
1 Scope
This part of ISO 20998 describes ultrasonic attenuation spectroscopy methods for determining the size
distributions of a particulate phase dispersed in a liquid at dilute concentrations, where the ultrasonic
attenuation spectrum is a linear function of the particle volume fraction. In this regime, particle–
particle interactions are negligible. Colloids, dilute dispersions, and emulsions are within the scope of
this part of ISO 20998. The typical particle size for such analysis ranges from 10 nm to 3 mm, although
particles outside this range have also been successfully measured. For solid particles in suspension, size
measurements can be made at concentrations typically ranging from 0,1 % volume fraction up to 5 %
volume fraction, depending on the density contrast between the solid and liquid phases, the particle
size, and the frequency range.
NOTE See References [9][10].
For emulsions, measurements may be made at much higher concentrations. These ultrasonic methods
can be used to monitor dynamic changes in the size distribution.
While it is possible to determine the particle size distribution from either the attenuation spectrum or
the phase velocity spectrum, the use of attenuation data alone is recommended. The relative variation in
phase velocity due to changing particle size is small compared to the mean velocity, so it is often difficult
to determine the phase velocity with a high degree of accuracy, particularly at ambient temperature.
Likewise, the combined use of attenuation and velocity spectra to determine the particle size is not
recommended. The presence of measurement errors (i.e. “noise”) in the magnitude and phase spectra
can increase the ill-posed nature of the problem and reduce the stability of the inversion.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. For dated references, only the edition cited applies. For undated
references, the latest edition of the referenced document (including any amendments) applies.
ISO 14488:2007, Particulate materials — Sampling and sample splitting for the determination of
particulate properties
ISO 20998-1:2006, Measurement and characterization of particles by acoustic methods — Part 1: Concepts
and procedures in ultrasonic attenuation spectroscopy
3 Terms and definitions
For the purposes of this document, the terms and definitions in ISO 20998-1 and the following apply.
3.1
coefficient of variation
ratio of the standard deviation to the mean value
3.2
dimensionless size parameter
representation of particle size as the product of wave number and particle radius
3.3
particle radius
one-half of the particle diameter
3.4
wave number
ratio of 2π to the wavelength
4 Symbols and abbreviated terms
A matrix representing the linear attenuation model
A coefficients of series expansion in ECAH theory
n
a particle radius
c speed of sound in liquid
C specific heat at constant pressure
p
C particle projection area divided by suspension volume
PF
CV coefficient of variability (ratio of the standard deviation to the mean value)
E extinction at a given frequency
ECAH Epstein-Carhart-Allegra-Hawley (theory)
f frequency
i
H identity matrix
h Hankel functions of the first kind
n
I transmitted intensity of ultrasound
I incident intensity of ultrasound
i the imaginary number
inv() matrix inverse operation
K extinction efficiency (extinction cross section divided by particle projection area)
K matrix representation of the kernel function (the ultrasonic model)
T
K transpose of matrix K
k(f, x) kernel function
k , k , k wave numbers of the compressional, thermal, and shear waves
c T s
ka dimensionless size parameter
P Legendre polynomials
n
PSD particle size distribution
2 © ISO 2013 – All rights reserved

q solution vector (representation of the PSD)
q (x) volume weighted density function of the PSD
Q (x) volume weighted cumulative PSD
s standard deviation
x particle diameter
th
x the 10 percentile of the cumulative PSD
th
x median size (50 percentile)
th
x the 90 percentile of the cumulative PSD
x minimum particle size in a sample
min
x maximum particle size in a sample
max
α total ultrasonic attenuation coefficient
α attenuation spectrum
α absolute attenuation coefficient divided by the frequency, α = (α/f)
α excess attenuation coefficient, α = α – α
exc exc L
α alternate definition of excess attenuation coefficient where α ’ = α – α
exc’ exc int
α measured attenuation spectrum
exp
a intrinsic absorption coefficient of the dispersion
int
α attenuation coefficient of the continuous (liquid) phase
L
α attenuation spectrum predicted by the model, given a trial PSD
mod
α attenuation coefficient of the discontinuous (particulate) phase
P
α elastic scattering component of the attenuation coefficient
sc
α thermal loss component of the attenuation coefficient
th
α viscoinertial loss component of the attenuation coefficient
vis
β volume thermal expansion coefficient
T
error in the fit, Δ= ααα−α
Δ
expmod
Δ Tikhonov regularization factor
Δl thickness of the suspension layer
ΔQ fraction of the total projection area containing a certain particle size class
η viscosity of the liquid
κ thermal conductivity
λ ultrasonic wavelength
μ shear modulus
'
density of the liquid and particle, respectively
ρ, ρ
ϕ volume concentration of the dispersed phase
χ chi-squared value
Ψ compression wave
c
Ψ shear wave
s
Ψ thermal wave
T
ω angular frequency (i.e. 2π times the frequency)
5 Mechanism of attenuation (dilute case)
5.1 Introduction
As ultrasound passes through a suspension, colloid, or emulsion, it is scattered and absorbed by the
discrete phase with the result that the intensity of the transmitted sound is diminished. The attenuation
coefficient is a function of ultrasonic frequency and depends on the composition and physical state of
the particulate system. The measurement of the attenuation spectrum is described in ISO 20998-1.
5.2 Excess attenuation coefficient
The total ultrasonic attenuation coefficient, α, is due to viscoinertial loss, thermal loss, elastic scattering,
and the intrinsic absorption coefficient, α , of the dispersion (References [1][10]):
int
αα=+αα++α (1)
visthscint
The intrinsic absorption is determined by the absorption of sound in each homogenous phase of the
dispersion. For pure phases, the attenuation coefficients, denoted α for the continuous (liquid) phase
L
and α for the discontinuous (particulate) phase, are physical constants of the materials. In a dispersed
P
system, intrinsic absorption occurs inside the particles and in the continuous phase, therefore,
αφ≈−()1 ⋅+αφ⋅α (2)
int LP
The excess attenuation coefficient is usually defined to be the difference between the total attenuation
and the intrinsic absorption in pure (particle-free) liquid phase (References [4][7]):
αα=−α (3)
excL
With this definition, the excess attenuation coefficient is shown to be the incremental attenuation caused by
the presence of particles in the continuous phase. Combining Formulae (1), (2), and (3), it can be seen that
αα=+αα++φα⋅−()α (4)
excvis th sc PL
4 © ISO 2013 – All rights reserved

The viscoinertial, thermal, and elastic scattering terms depend on particle size, but α and α do not.
L P
Thus, the excess attenuation coefficient contains a term that does not depend on size. When working
with aqueous dispersions and rigid particles, this term can often be neglected, so that
αα≈+αα+ (5)
excvis th sc
However, in some emulsions, the ultrasonic absorption in the oily phase can be significant. In that case,
the definition of the excess attenuation coefficient given in Formula (3) may be modified as
αα=−α (6)
exc' int
In this situation, Formula (5) is still valid. It should be noted that some authors express attenuation
coefficient as a reduced quantity αα=(/ f ) , dividing the absolute attenuation coefficient by the frequency.
5.3 Specific attenuation mechanisms
5.3.1 Scattering
Ultrasonic scattering is the redirection of acoustic energy away from the incident beam, so it is elastic
(no energy is absorbed). The scattering is a function of frequency and particle size.
5.3.2 Thermal losses
Thermal losses are due to temperature gradients generated near the surface of the particle as it is
compressed by the acoustic wave. The resulting thermal waves radiate a short distance into the liquid
and into the particle. Dissipation of acoustic energy caused by thermal losses is the dominant attenuation
effect for soft colloidal particles, including emulsion droplets and latex droplets.
5.3.3 Viscoinertial losses
Viscoinertial losses are due to relative motion between the particles and the surrounding fluid. The
particles oscillate with the acoustic pressure wave, but their inertia retards the phase of this motion.
This effect becomes more pronounced with increasing contrast in density between the particles and
the medium. As the liquid flows around the particle, the hydrodynamic drag introduces a frictional loss.
Viscoinertial losses dominate the total attenuation for small rigid particles, such as oxides, pigments,
and ceramics. An explicit calculation of the attenuation due to viscoinertial loss is given in Annex A for
the case of rigid particles that are much smaller than the wavelength of sound in the fluid.
5.4 Linear models
5.4.1 Review
The attenuation of ultrasound in a dispersed system is caused by a variety of mechanisms (see 5.3), the
significance of which depends on material properties, particle size, and sound frequency. Moreover, for
some material systems, a linear relationship between sound attenuation and particle concentration can
be observed up to concentrations of 20 % volume fraction or more, while for others, such a relationship
exists only at low concentrations. This situation has led to a variety of models; two principal approaches
may be distinguished.
The first is the scattering theory, which aims at the scattered sound field around a single particle.
Based on this, the propagation of sound through the dispersed system can be calculated. By assuming
independent scattering events and neglecting multiple scattering, the attenuation turns out to be
linearly dependent on the particle concentration.
The fundamentals of the scattering theory were already presented by Rayleigh, but his approach ignored
the energy dissipation by shear waves and thermal waves (viscoinertial and thermal losses). A well-
known scattering theory is the ECAH (Epstein-Carhart-Allegra-Hawley) theory, a short introduction to
which is given in Annex B. The ECAH theory includes sound scattering as well as the viscoinertial and
the thermal losses. It can be applied to homogenous, spherical particles with no limit regarding material
properties, particle size, or sound frequency.
The second principal approach in modelling is to consider only the attenuation by viscoinertial and
thermal losses, which is admissible in the long wavelength limit (where x ⪻ λ or, equivalently, ka ⪻
1) only. That restriction facilitates the inclusion of nonlinear concentration effects that are caused by
the interaction of shear waves and/or thermal waves. Consequently, most of these theories are beyond
the scope of this part of ISO 20998. However, linear solutions can be obtained in the limiting case of
vanishing particle concentration (ϕ → 0). In general, these theories then agree with the ECAH theory
(with regard to the modelled attenuation mechanism). Purely linear models are that of Reference [11]
for the viscoinertial loss mechanism and that of Reference [12] for the thermal loss mechanism, both of
which agree with ECAH results (Reference [7]).
The theoretical models may fail to accurately explain measured attenuation spectra since they hold true
only for homogenous, spherical particles and require the knowledge of several physical parameters of
the dispersed system. In such situations, semi-empirical approaches may be used that are based on the
observation that for spheres we get
α =f()xf ,
vis

α =f()xf ,
th
and

α =f()xf .
sc
The application and derivation of such a semi-empirical model is described in Annex C.
5.4.2 Physical parameters
A number of physical properties affect the propagation of ultrasound in suspensions and emulsions.
These properties (listed in Table 1) are included in the ECAH model described in Annex B. In most
practical applications, many of these parameters are not known, and it is therefore difficult to compare
theory with experimental observation directly. Fortunately, approximate models can be employed for
many situations (cf. 5.3.1), which reduces the number of influential parameters. Moreover, some of
these parameters only weakly affect the attenuation and, therefore, do not need to be known with high
accuracy. Typical material systems are listed in Table 2 together with the material properties that most
significantly affect the attenuation.
Table 1 — The complete set of properties for both particle and medium that affect the
ultrasound propagation through a colloidal suspension
Dispersion medium Dispersed particle Units
−3
Density Density kg ⋅ m
Shear viscosity (microscopic) Pa ⋅ s
Shear modulus Pa
−1
Sound speed Sound speed M ⋅ s
−1 −1
Absorption Absorption Np ⋅ m , dB ⋅ m
−1 −1
Heat capacity at constant pressure Heat capacity at constant pressure J ⋅ kg ⋅ K
6 © ISO 2013 – All rights reserved

Table 1 (continued)
Dispersion medium Dispersed particle Units
−1 −1
Thermal conductivity Thermal conductivity W ⋅ m ⋅ K
−1
Thermal expansion Thermal expansion K
NOTE The decibel (dB) is commonly used as a unit of attenuation, so absorption is often expressed in units of
−1 −1
dB ⋅ m or dB ⋅ cm .
Table 2 — Material properties that have the most significant effect on ultrasonic attenuation
System Properties of the particle Properties of the liquid
Rigid submicron particles Density Density, sound speed, shear viscosity
Soft submicron particles Thermal expansion Thermal expansion
Large soft particles Density, sound speed, elastic constants Density, sound speed
Large rigid particles Density, sound speed, shape Density, sound speed
6 Determination of particle size
6.1 Introduction
This section describes procedures for estimating the particle size distribution from an ultrasonic
attenuation spectrum.
In general, the observed ultrasonic attenuation spectrum, which forms the data function α, is dependent
on the particle size distribution and on the particle concentration. In dilute suspensions and emulsions,
the sound field interacts with each particle independently. That is, the attenuation of sound is formed
by the superposition of individual, uncorrelated events, and the spectrum is a linear function of
concentration. In this case, a linear theory such as the ECAH model described in Annex B can be applied
to determine the particle size distribution.
Within the linear theory, the attenuation of sound is related to a PSD by the following formula:
 
αφ()ff=⋅((αα)(−+fk)) φ⋅⋅(,fx)(qx)dx (7)
exci Pi Li i 3
∫
where ϕ is the volume concentration of the dispersed phase and q (x) is the volume weighted density
function of the PSD. The function k( f, x) is called the kernel function, and it models the physical
interactions between the ultrasound and the particles.
The inversion problem, i.e. the determination of the continuous function q (x) from a (discrete)
attenuation spectrum, is an ill-posed problem. Any measured discrete attenuation spectrum cannot
reveal all details of q (x). Moreover, signal noise further reduces the amount of accessible information
on q (x). For that reason, the inversion problem has to be modified by restricting the space of possible
solutions. Two principal approaches may be distinguished:
a) the approximation of q (x) by a given PSD function, where the parameters of this function are
determined by a nonlinear regression;
b) the discretization of the size axis x plus imposing additional constraints on the solution vector q
(regularization.)
These two approaches are described in 6.2.
NOTE The choice of inversion approach does not depend on the choice of theory used to calculate the
attenuation spectrum.
The performance of the algorithms depends on the material system, on the measurement instrument,
as well as on the size distribution. It is further related to the information content of the measured
attenuation spectrum, which is determined by the covered frequency range, by the signal noise, to a
lower extent by the number of frequencies, and primarily by the structure of the kernel functions k( f, x)
(Reference [13]).
6.2 Inversion approaches used to determine PSD
6.2.1 Optimization of a PSD function
In the case of colloidal dispersions, i.e. in the long wavelength regime, the spectra are very smooth so
that very little information appears to be contained in the data. In order to extract the PSD from the
attenuation data, a model function might be assumed, effectively reducing the number of free parameters
to be fit to the data (Reference [9]). A typically used model function is the log normal distribution (cf.
[33]
ISO 9276-2:—, Annexes A and B):
 
 
1 1 1 x
 
qx()=−expln (8)
 
 
2 s x
sx 2π
 50 
 
where x is the median size and s is the standard deviation of ln(x). The solution of the inversion problem
is found by minimizing the residual Δ:
Δ= ααα−α (9)
expmod
where α is measured and α is calculated by Formula (7) using, for example, viscoinertial loss
exp mod
(see Annex A), ECAH (see Annex B), or some other suitable model. The model parameters of the best
fitting function can be obtained from an optimization strategy. These are iterative algorithms, the
general principal of which is described in Annex D. Care must be taken to ensure that the optimization
strategy does not result in a local minimum of the residual Δ, which could cause a significant error in
the estimated PSD.
6.2.2 Regularization
Model functions restrict the solution q (x) with regard to the number of modes or the skewness,
which may obscure relevant details in the distribution function. As shown in Annex C, it is possible to
derive an inversion without model parameters for the estimated PSD (References [14][15][16]). If the
information content of α is sufficient, an alternative approach is to introduce size fractions and to re-
write Formula (7) in its discrete form:
ααα=⋅φφ()αα−+α ⋅⋅K q (10)
pL
where the matrix K is the discrete representation of the ultrasonic model, giving the attenuation as a
function of particle size. Solving this formula is an ill-conditioned problem, as the signal noise is extremely
magnified. Formal solution may yield results that are physically unacceptable, for example, negative size
fractions or a discontinuous PSD. To avoid such results, one can modify the problem by assuming certain
properties of the shape of the distribution function (or the solution vector q, respectively). The most
8 © ISO 2013 – All rights reserved

popular regularization is based on the smoothness of the distribution function, which can be quantified
T
via q · H · q leading to the modified objective function:
2 T
χδ=−αα KH⋅+qq⋅⋅ ⋅q (11)
exp
T
Here, H is the identity matrix, K is the transpose of matrix K, and δ is a suitable Tikhonov regularization
factor (Reference [14]). The solution is then obtained from:
TT2
q=+invK()KHδ ⋅ K αα (12)
exp
For small values of the regularization factor δ, the solution q is highly affected by the signal noise
showing strong oscillations with large negative values. In contrast, very large regularization factors
yield such smooth solutions that the characteristic features of the PSD are lost. In order to select an
optimal regularization factor, different strategies can be applied (References [17][18][19][20]).
6.3 Limits of application
The typical particle size for ultrasonic analysis ranges from 10 nm to 3 mm, although particles outside
this range have also been successfully measured. Measurements can be made with a linear model for
concentrations of the dispersed phase ranging from 0,1 % volume fraction up to 5 % volume fraction or
more, depending on the density contrast between the continuous and the dispersed phases. In the case of
emulsions, measurements may be made at much higher concentrations (approaching 50 % volume fraction).
The application of linear theoretical models requires the knowledge of the relevant model parameters.
Users should therefore be aware of possible changes in those parameters, e.g. variation of the particle
concentration. In particular, processes including a change of the phase (e.g. dissolution) or a change
in temperature may defy an analysis with theoretical models. In such a case, users are referred to the
world of chemometrics, i.e. to methods for data treatment and statistical modelling (e.g. with neural
networks, multiple regression, etc.).
7 Instrument qualification
7.1 Calibration
Ultrasonic spectroscopy systems are based on first principles. Thus, calibration in the strict sense is not
required; however, it is still necessary and desirable to confirm the accurate operation of the instrument
by a qualification procedure. See ISO 20998-1 for recommendations.
7.2 Precision
7.2.1 Reference samples
For testing precision, reference samples with an x /x ratio in the range of 1,5 to 10 should be used.
90 10
It is desirable that reference samples used to determine precision are non-sedimenting and comprising
spherical particles with diameters in the range of 0,1 μm to 1 μm. The concentration shall be in the range
of 1 % to 5 % volume fraction.
7.2.2 Repeatability
The requirements given in ISO 20998-1 shall be followed. The instrument should be clean, and the
liquid used for the background measurement should be virtually free of particles. Execute at least five
consecutive measurements with the same dispersed sample aliquot or dispersed single shot samples.
Calculate the mean and coefficient of variation (CV) for the x , x , and x . An instrument is considered
10 50 90
to meet this part of ISO 20998 for repeatability if the CV for each of the x , x , and x is smaller than
10 50 90
10 %. If a larger CV value is obtained, then all potential error sources shall be checked.
7.2.3 Reproducibility
Reproducibility tests shall follow the same protocol as that for repeatability. At least three distinct
samples of the same reference material shall be measured, and the mean and CV for the x , x , and
10 50
x shall be calculated. A CV larger than that of repeatability may be expected due to differences in
sampling or dispersion or between analysts or instruments. The certification for the reference material
will contain information about the acceptable error for that material.
7.3 Accuracy
7.3.1 Qualification procedure
In the qualification step, the accuracy of the total measurement procedure is examined. It is essential
that a written procedure is available that describes the sub-sampling, the sample dispersion, the
ultrasonic measurement, and the calculation of the PSD in full detail. This procedure shall be followed
in its entirety and the title and version number reported.
7.3.2 Reference samples
Certified reference materials (see ISO 35) are preferred in the measurement of accuracy. These materials
have a known size distribution of spherical particles with an x /x ratio in the range of 1,5to 10. It is
90 10
preferred that the median size of the certified reference material be chosen so that it lies within the size
range contemplated for the end-use application. For single shot analysis, the full contents of the container
shall be used. If sub-sampling is necessary, this shall be done with due care while using a method that
has been proven to yield adequate results (see ISO 14488). If a protocol for sampling, dispersion, or
measurement is not available, the procedure that is used shall be reported with the final results.
7.3.3 Instrument preparation
The advice given in ISO 20998-1 should be followed. The instrument should be clean, and the liquid used
for the background measurement should be free of particles.
7.3.4 Accuracy test
The written test protocol defined in 7.3.1 shall be followed for the accuracy test, which measures the
PSD of the selected reference material. Single shot analysis may be applied. Analysis of sub-samples
is permitted if the procedure for sub-sampling is also written and is documented to provide good
repeatability. Analysis shall be made on five consecutive sample aliquots, and the average value and CV
of the median size shall be calculated.
7.3.5 Qualification acceptance criteria
The 95 % tolerance limits stated for each certified size value of the standard reference material
specification form a set of maximum and minimum values that define the stated parameter. The
qualification test shall be accepted as passing the requirement of this part of ISO 20998 if the resulting
measured particles size distribution achieves both of the following criteria.
a) The reported average value of the median size measured in the qualification test is no smaller than
90 % of the minimum value and no larger than 110 % of the maximum value.
b) The reported CV of the median size does not exceed 10 %.
If a larger deviation is obtained, then all potential error sources should be checked. If it is not possible
to meet the qualification criteria of this section, then this failure shall be noted on the final PSD report.
If a higher standard of accuracy is required for any reason, then a reference material should be chosen
with a narrow confidence interval and a total protocol for sampling, dispersion, and measurement
should be used that guarantees minimum deviation.
10 © ISO 2013 – All rights reserved

8 Reporting of results
The particle size distribution results shall be reported according to the guidelines in ISO 20998-1, Clause 5.
Annex A
(informative)
Viscoinertial loss model
Viscoinertial loss (see 5.3.3) may be calculated in the long wavelength limit from Formula (A.1). The form
shown here is from Reference [5], but it is derived from the explicit analytical solutions of Reference [7]
and is mathematically equivalent to results obtained by many authors (References [11][21][22]).
∞
 
ρρ'−
φ C ()
 
d
α =   (A.1)
vis  
  22
2 cρ  
 
∞∞ −2
 
ρρ'+ CC+ ω
()id() 
 
where the dissipative and inertial drag coefficients (Reference [5]) are given by
9η
∞
C =+()1 Y (A.2)
d
2a
1 9
 
∞−1
CY=+1 (A.3)
i  
2 2
 
with the dimensionless parameter Y defined by
ωρ
Ya= (A.4)
η
In the formulae above,
α is the viscous attenuation coefficient
vis
a is the particle radius
c is the speed of sound in liquid
ω is the angular frequency of ultrasound (i.e. 2π times the frequency)
is the density of the liquid and particle, respectively
ρ, p '
η is the viscosity of the liquid
ϕ is the volume concentration of the particle
12 © ISO 2013 – All rights reserved

Annex B
(informative)
ECAH theory and limitations
B.1 Introduction
The Epstein-Carhart-Allegra-Hawley (ECAH) theory is derived from Reference [6] on sound attenuation
in liquid/liquid systems (emulsions). Reference [7] later generalized that theory to include elastic solid
particles as well as fluid particles in a liquid suspending medium. This theory is one of many linear
scattering theories, each of which has made assumptions about the particle system and how it reacts to
sound waves (see Figure B.1).
Figure B.1 — Linear models of ultrasonic scattering
B.2 Calculation of attenuation
ECAH theory considers the propagation of sound through a suspension or emulsion via three distinct
types of wave: a compression wave, ψ , a thermal wave, ψ , and a shear wave, ψ . Since the incident
c T s
sound beam is generally a compression wave, the other two types are generated at the boundary of the
discontinuous phase. These waves are solutions of the wave formulae:
∇+ k ψ =0 (B.1)
()
cc
∇+k ψ =0 (B.2)
()TT
∇+ k ψ =0 (B.3)
()
ss
Here, the wave numbers k , k , and k are defined by
c T s
ω
k =+iα (B.4)
c 0
c
ωρC
p
ki=+1 (B.5)
()
T
2κ
ωρ
k = (B.6)
s
μ
where ω is the angular frequency, c is the speed of sound, α is the absorption, ρ is the density, C is
0 p
the specific heat at constant pressure, κ is the thermal conductivity, μ is the shear modulus, and i is the
imaginary number. These waves appear in the interior of the particle and in the continuous phase (the
fluid). Within the fluid, the wave number corresponding to the shear wave [Formula (B.6)] is calculated
by replacing the shear modulus μ with the factor -iωη, where η is the viscosity.
In the case of a spherical particle, the general solution of this system of formulae can be represented as
an expansion of Legendre polynomials. The reflected compression wave, for example, is
∞
r n
Ψ =+in21 Ah ka P cosθ (B.7)
() () ()
c ∑ nn cn
n=0
where h are Hankel functions of the first kind and P are Legendre polynomials. The coefficients
n n
A specify the reflected compression wave completely. In total, there are six sets of coefficients that
n
describe the three waves in the medium and the three waves inside the particle. These coefficients
are related through the boundary conditions, which require continuity of these physical parameters at
the surface of the particle: radial velocity, tangential velocity, temperature, heat flux, radial stress, and
tangential stress. Explicit formulae for the boundary conditions for emulsions, are given in Reference
[2], pp. 114–115 and the boundary conditions for suspensions of solids are discussed in Reference [7].
14 © ISO 2013 – All rights reserved

The resulting system of formulae is solved to determine the coefficients A in Formula (B.7). The
n
ultrasonic attenuation, α, is then given by Reference [6]:
∞
3φ
α =− 21nA+ Re (B.8)
()
excn∑
2ka
c n=0
where a is the radius of the particle. In the long wavelength limit, where k a < < 1, coefficients of order
c
n > 1 can be ignored. This limiting case applies to many suspensions of microscopic particles when the
ultrasonic frequency is well below 100 MHz (Reference [10]). In this case,
3φ
α =− ReAA+3  (B.9)
exc  01
2ka
c
Formulae for determining A are given in Reference [28]. Explicit analytical solutions for A (thermal
n 0
loss) and A (viscous loss) are given in Reference [7] for the long wavelength limit in the case of rigid,
dense particles.
B.3 Limitations of ECAH theory
It should be noted that Formulae (B.8) and (B.9) are only valid for dilute suspensions, where particle-
particle interactions can be neglected.
Annex C
(informative)
Example of a semi-empirical model
There are many semi-empirical models; this annex provides one example.
The ultrasonic extinction, E, of a suspension or emulsion of mono-disperse particles with the diameter
x can be described by the Lambert-Beer’s law according to Reference [4].
 
I
E =−ln =⋅ΔΔlC ⋅Kf ,xl=⋅α (C.1)
()
 
PF i
I
 0 
f
i
The extinction, E, at a given frequency, f , is linearly dependent on the thickness of the suspension layer,
i
Δl, the projection area-concentration, C , and the extinction efficiency (normalized extinction cross
PF
section), K, where
particleprojectionarea
C = (C.2)
PF
suspensionvolume
extinction cross section
K = (C.3)
particle projection area
The extinction efficiency, K, is a function of frequency and particle size. In a polydisperse system, K is
integrated over all particle size fractions from x to x :
min max
x
max
Ef =⋅ΔlC ⋅ Kf ,xq⋅ xdx (C.4)
() () ()
iPFi 2
∫
x
min
The integral in Formula (C.4) can be approximated as sum:
Ef ≅⋅ΔΔlC ⋅ Kf ,xq⋅ xx⋅ (C.5)
() () ()
iPFi∑ jj2 j
j
If now extinction measurements are performed at various frequencies, this results in a linear system of
formulae:
KK
   
Ef() qx⋅Δ 
11,,1 j
1 21 1
   
 
 =⋅ΔlC ⋅   ⋅  (C.6)
PF
 
 
 
 
 
Ef() KK qx⋅
j ii,1 ,j 2ii
 
 
 
The system of formulae (C.6) is numerically unstable and must be solved by suitable algorithms.
To calculate the particle size distribution, one must know the extinction cross section K as a function of
the dimensionless size parameter ka defined in Formula (C.7):
 2π x 
ka= (C.7)
  
λ 2
  
The dimensionless size parameter is defined as the product of wave number and particle radius, and
it equals π times the ratio of particle diameter, x, to ultrasonic wavelength, λ,
...