ISO 20998-2:2013
(Main)Measurement and characterization of particles by acoustic methods — Part 2: Guidelines for linear theory
Measurement and characterization of particles by acoustic methods — Part 2: Guidelines for linear theory
ISO 20998-2:2013 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 ISO 20998-2:2013. 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. 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.
Caractérisation des particules par des méthodes acoustiques — Partie 2: Théorie linéaire
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Standards Content (Sample)
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 20998-2:2013(E)
©
ISO 2013
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ISO 20998-2:2013(E)
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ISO 20998-2:2013(E)
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
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ISO 20998-2:2013(E)
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
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ISO 20998-2:2013(E)
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.
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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
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ISO 20998-2:2013(E)
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
0
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
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ISO 20998-2:2013(E)
q solution vector (representation of the PSD)
q (x) volume weighted density function of the PSD
3
Q (x) volume weighted cumulative PSD
3
s standard deviation
x particle diameter
th
x the 10 percentile of the cumulative PSD
10
th
x median size (50 percentile)
50
th
x the 90 percentile of the cumulative PSD
90
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
2
η viscosity of the liquid
κ thermal conductivity
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ISO 20998-2:2013(E)
λ ultrasonic wavelength
μ shear modulus
'
density of the liquid and particle, respectively
ρ, ρ
ϕ volume concentration of the dispersed phase
2
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
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ISO 20998-2:2013(E)
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
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ISO 20998-2:2013(E)
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
2
α =f()xf ,
vis
2
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
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ISO 20998-2:2013(E)
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
3
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)
3
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
3
on q (x). For that reason, the inversion problem has to be modified by restricting the space of possible
3
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
3
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.
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ISO 20998-2:2013(E)
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):
2
1 1 1 x
qx()=−expln (8)
3
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
50
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,
3
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 (Referen
...
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