ISO 20998-2:2022
(Main)Measurement and characterization of particles by acoustic methods — Part 2: Linear theory
Measurement and characterization of particles by acoustic methods — Part 2: Linear theory
This document specifies requirements for 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 document. 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 % by volume up to 5 % by volume, depending on the density contrast between the solid and liquid phases, the particle size, and the frequency range[9],[10]. For emulsions, measurements can be made at much higher concentrations. These ultrasonic methods can be used to monitor dynamic changes in the size distribution.
Caractérisation des particules par des méthodes acoustiques — Partie 2: Théorie linéaire
General Information
Relations
Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 20998-2
Second edition
2022-08
Measurement and characterization of
particles by acoustic methods —
Part 2:
Linear theory
Caractérisation des particules par des méthodes acoustiques —
Partie 2: Théorie linéaire
Reference number
ISO 20998-2:2022(E)
© ISO 2022
---------------------- Page: 1 ----------------------
ISO 20998-2:2022(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2022
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting on
the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address below
or ISO’s member body in the country of the requester.
ISO copyright office
CP 401 • Ch. de Blandonnet 8
CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii
© ISO 2022 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 20998-2:2022(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviations .2
5 Mechanism of attenuation (dilute case) . 4
5.1 Introduction . 4
5.2 Excess attenuation coefficient . 4
5.3 Specific attenuation mechanisms . 5
5.3.1 Scattering . 5
5.3.2 Thermal losses . 5
5.3.3 Viscoinertial losses . . 5
5.3.4 Non-monotonic relaxation mechanisms . 5
5.4 Linear models . 5
5.4.1 Review . 5
5.4.2 Physical parameters . 6
6 Determination of particle size . 7
6.1 Introduction . 7
6.2 Inversion approaches used to determine PSD . 8
6.2.1 Optimization of a PSD function . 8
6.2.2 Regularization . 8
6.3 Limits of application . 9
7 Instrument qualification .9
7.1 Calibration . 9
7.2 Precision . . . 9
7.2.1 Reference samples . 9
7.2.2 Repeatability . 10
7.2.3 Reproducibility . 10
7.3 Accuracy . 10
7.3.1 Qualification procedure . 10
7.3.2 Reference samples . 10
7.3.3 Instrument preparation . 10
7.3.4 Accuracy test . 10
7.3.5 Qualification acceptance criteria. 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 .17
Annex D (informative) Iterative fitting .20
Annex E (informative) Physical parameter values for selected materials .22
Annex F (informative) Practical example of PSD measurement .23
Bibliography .32
iii
© ISO 2022 – All rights reserved
---------------------- Page: 3 ----------------------
ISO 20998-2:2022(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.
ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www.iso.org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www.iso.org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to
the World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see
www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 24, Particle characterization including
sieving, Subcommittee SC 4, Particle characterization.
This second edition cancels and replaces the first edition (ISO 20998-2:2013), which has been
technically revised.
The main changes are as follows:
— References to relaxation mechanisms that affect attenuation
— Additional explanatory notes for Table 1
— Clarification of notation used in Formula (9)
— Minor editorial changes
A list of all parts in the ISO 20998 series can be found on the ISO website.
Any feedback or questions on this document should be directed to the user’s national standards body. A
complete listing of these bodies can be found at www.iso.org/members.html.
iv
© ISO 2022 – All rights reserved
---------------------- Page: 4 ----------------------
ISO 20998-2:2022(E)
Introduction
It is well known that ultrasonic spectroscopy can be used to measure particle size distribution (PSD) in
[1],[2],[3],[4]
colloids, dispersions, and emulsions . 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
[5],[6],[7]
of the size distribution and concentration of particles . 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
[8]
observed directly in real time . This data can be used in process control schemes to improve both the
manufacturing process and the product performance.
While it is possible to determine the particle size distribution from either the attenuation spectrum or
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.
v
© ISO 2022 – All rights reserved
---------------------- Page: 5 ----------------------
INTERNATIONAL STANDARD ISO 20998-2:2022(E)
Measurement and characterization of particles by acoustic
methods —
Part 2:
Linear theory
1 Scope
This document specifies requirements for 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 document. 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 % by volume up to 5 % by
volume, depending on the density contrast between the solid and liquid phases, the particle size, and
[9],[10]
the frequency range . For emulsions, measurements can be made at much higher concentrations.
These ultrasonic methods can be used to monitor dynamic changes in the size distribution.
2 Normative references
The following documents are referred to in the text in such a way that some or all of their content
constitutes requirements of this document. For dated references, only the edition cited applies. For
undated references, the latest edition of the referenced document (including any amendments) applies.
ISO 14488, 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 given in ISO 20998-1 and the following
apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
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 wavenumber (3.4) and particle radius (3.3)
3.3
particle radius
half of the particle diameter
1
© ISO 2022 – All rights reserved
---------------------- Page: 6 ----------------------
ISO 20998-2:2022(E)
3.4
wavenumber
ratio of 2π to the wavelength
4 Symbols and abbreviations
For the purposes of this document, the following abbreviations and symbols apply.
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 variation (ratio of the standard deviation to the mean value)
E extinction at a given frequency
ECAH Epstein-Carhart-Allegra-Hawley (theory)
f frequency
i
g() an arbitrary function
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
k.a dimensionless size parameter
ln() natural logarithm
P Legendre polynomials
n
PSD particle size distribution
q solution vector (representation of the PSD)
2
© ISO 2022 – All rights reserved
---------------------- Page: 7 ----------------------
ISO 20998-2:2022(E)
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 , x minimum and maximum particle diameters in a sample
min max
α total ultrasonic attenuation coefficient
α attenuation spectrum
α
absolute attenuation coefficient divided by the frequency, αα = f
()
excess attenuation coefficient, αα=−α
α
exc
excL
α alternate definition of excess attenuation coefficient where αα=−α
′
exc’ exc int
α measured attenuation spectrum
exp
α 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
Τ
Δ error in the fit
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
λ ultrasonic wavelength
μ shear modulus
ρ, ρ′ density of the liquid and particle, respectively
3
© ISO 2022 – All rights reserved
---------------------- Page: 8 ----------------------
ISO 20998-2:2022(E)
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 given by the viscoinertial loss, thermal loss, elastic
[1],[10]
scattering, and the intrinsic absorption coefficient α of the dispersion which can also include
int
a variety of relaxation effects not alluded to in Reference [1] and [10], for example, solvent-ion and
macromolecule-solvent effects which need to be accounted for separately due to their differing non-
[11],[12],[13]
monotonic frequency dependencies .
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 absorption 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
αα≈−1 φφ⋅+ ⋅α (2)
()
intL P
Excess attenuation coefficient is usually defined to be the difference between the total attenuation and
[4],[7]
the intrinsic absorption in pure (particle-free) liquid phase :
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) to (3), it can be
seen that
αα=+αα++φ⋅−αα (4)
()
excvis th sc PL
4
© ISO 2022 – All rights reserved
---------------------- Page: 9 ----------------------
ISO 20998-2:2022(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 excess attenuation coefficient given in Formula (3) can be modified as in Formula (6)
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 quantityαα = f , dividing the absolute attenuation coefficient by the
()
frequency.
5.3 Specific attenuation mechanisms
5.3.1 Scattering
Ultrasonic scattering is the re-direction 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.
5.3.4 Non-monotonic relaxation mechanisms
Whereas according to Reference [14] the intrinsic absorption contains contributions that reflect
“translational” molecular motion and the relaxation of both “rotational” and “vibrational” degrees of
molecular freedom, non-monotonic relaxation mechanisms should be accounted for separately before
the required monotonic attenuation spectrum can be obtained. One approach to the identification and
quantification of relaxation effects is found in Reference [11].
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 % by volume or more, while for others such a relationship exists
5
© ISO 2022 – All rights reserved
---------------------- Page: 10 ----------------------
ISO 20998-2:2022(E)
only at low concentrations. This situation has led to a variety of models; two principal approaches can
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 k.a < < 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 document. 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
[15]
to the modelled attenuation mechanism). Purely linear models are that of Urick for the viscoinertial
[16]
loss mechanism and that of Isakovich for the thermal loss mechanism, both of which agree with
[7]
ECAH results .
The theoretical models sometimes 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 = gx f
()
vis
2
α = gx f
()
th
and
α = gx f ,
()
sc
where g is an arbitary function.
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. see 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.
6
© ISO 2022 – All rights reserved
---------------------- Page: 11 ----------------------
ISO 20998-2:2022(E)
Table 1 — 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) Shear viscosity (microscopic) Note 2 and 4 Pa• s
Shear Modulus Note 3 and 4 Pa
−1
Sound speed Sound speed m • s
−1 −1
Absorption Absorption Np•m , dB•m Note 1
−1 −1
Heat capacity at constant pressure Heat capacity at constant pressure J• kg • K
−1 −1
Thermal conductivity Thermal conductivity W• m • K
−1
Thermal expansion Thermal expansion K
NOTE 1 The decibel (dB) is commonly used as a unit of attenuation, so absorption is often expressed in units
of dB/m or dB/cm.
NOTE 2 Shear viscosity of dispersed particles applies to liquid particles i.e. droplets only.
NOTE 3 Shear Modulus applies only to solid particles.
NOTE 4 Shear modulus μ can be replaced by Shear viscosity η using the relationship μω=−i η .
x
s
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 clause describes procedures for estimating the particle size distribution from an ultr
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.