ISO 20998-2:2022

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

• Status: Published
• Publication Date: 22-Aug-2022
• ICS: 19.120 - Particle size analysis. Sieving
• Technical Committee: ISO/TC 24/SC 4 - Particle characterization
• Drafting Committee: ISO/TC 24/SC 4/WG 14 - Acoustic methods
• Current Stage: 6060 - International Standard published
• Start Date: 23-Aug-2022
• Due Date: 07-May-2022
• Completion Date: 23-Aug-2022

Relations

• Revises: ISO 20998-2:2013 - Measurement and characterization of particles by acoustic methods - Part 2: Guidelines for linear theory
• Effective Date: 23-Apr-2020

Overview

ISO 20998-2:2022 - "Measurement and characterization of particles by acoustic methods - Part 2: Linear theory" defines requirements for ultrasonic attenuation spectroscopy used to determine particle size distributions (PSD) in dilute particulate suspensions. The standard applies where the ultrasonic attenuation spectrum is a linear function of particle volume fraction, i.e., particle‑particle interactions are negligible. Typical measurable particle sizes are 10 nm to 3 mm, with solid-suspension concentrations commonly from 0.1 % to 5 % by volume (higher for emulsions). The document supports dynamic monitoring of PSD changes and complements Part 1 (concepts and procedures).

Key topics and technical requirements

• Linear attenuation regime - conditions where excess attenuation scales linearly with particle volume fraction; scope includes colloids, dilute dispersions and emulsions.
• Mechanisms of attenuation - treatment of scattering, thermal losses and viscoinertial losses, and discussion of non‑monotonic relaxation mechanisms.
• Analytical models - summary of linear models and theoretical frameworks (including references to ECAH - Epstein‑Carhart‑Allegra‑Hawley theory) and their limitations.
• PSD determination - inversion approaches for retrieving PSD from attenuation spectra:
  • Optimization of a PSD function
  • Regularization methods to stabilize ill‑posed inverse problems (e.g., Tikhonov)
• Limits of application - criteria based on particle size, concentration, density contrast and frequency range.
• Instrument qualification and metrology - calibration, precision (repeatability and reproducibility), accuracy testing, reference samples and acceptance criteria.
• Reporting requirements - standardized presentation of results and uncertainty considerations.
• Informative annexes - practical examples, semi‑empirical models, parameter tables and iterative fitting procedures.

Applications and users

ISO 20998-2:2022 is practical for:

• Laboratories and instrument manufacturers developing or validating ultrasonic particle‑sizing systems.
• Process engineers and quality control teams performing online or at‑line monitoring of dispersions, emulsions and slurries.
• R&D and materials characterization in industries such as pharmaceuticals, coatings, food, ceramics, mineral processing and oil & gas where PSD affects performance. Benefits include non‑destructive measurement, capability for dynamic monitoring, and suitability for a wide size range and various concentrations (within stated limits).

Related standards

• ISO 20998-1:2006 - Concepts and procedures in ultrasonic attenuation spectroscopy (normative reference).
• ISO 14488 - Sampling and sample splitting for particulate materials (normative reference for sampling practice).

Keywords: ISO 20998-2:2022, ultrasonic attenuation spectroscopy, particle size distribution, PSD, acoustic particle characterization, ECAH theory, viscoinertial losses, instrument qualification, inversion, regularization.

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 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: [phone]
Email: [e-mail]
Website: www.iso.org
Published in Switzerland
ii
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
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
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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
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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
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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
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
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
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
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)
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 , 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
η viscosity of the liquid
κ thermal conductivity
λ ultrasonic wavelength
μ shear modulus
ρ, ρ′ density of the liquid and particle, respectively
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 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
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
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:

α = gx f
()
vis
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.
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 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:
 
αα()ff=⋅φφ()()−α ()fk+⋅ ()fx, ⋅qx()dx (7)
exci Pi Li i 3
∫
where ϕ is the volume concentration of the dispersed phase, q (x) the volume weighted density function
of the PSD. The function K(f ,x) is called the kernel function, and it models the physical interactions
i
between ultrasound and the particles.
The inversion problem, i.e. 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).
3 3
For that reason, the inversion problem has to be modified by restricting the space of possible solutions.
Two principal approaches can 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 discretisation 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, 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)
i
[17]
.
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 can be assumed, effectively reducing the number of free parameters
[9]
to be fit to the data . A typically used model function is the log normal distribution (see Annex A of
Reference [18]):
 
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 is found by minimizing the error in the fit, Δ, which is defined to
be the Euclidean distance between two vectors:
12/
N
 
Δ= ααα−=α  αα−  (9)
()
expmod expmnnod
∑
 
n=1 
In Formula (9), α is the measured spectrum, and the spectrum predicted by the model, α , is
exp mod
calculated by Formula (7) using for example viscoinertial loss (see Annex A), ECAH (see Annex B), or
some other suitable model. Individual components of these vectors are represented by α and
exp
n
α , respectively, and N is the size of the vectors. The model parameters of the best fitting function
mod
n
can be obtained from an optimization strategy. These are iterative algorithms, the general principal of
which is described in Annex D. Care shall be taken to ensure that the optimization strategy does not
result in a local minimum of the residual Δ, which can cause a significant error in the estimated PSD.
Using the parameter values shown in Annex E, Annex F provides a detailed example of how to determine
the PSD by iterative optimization of Formula (9).
6.2.2 Regularization
Model functions restrict the solution q (x) with regard to the number of modes or the skewness, which
might obscure relevant details in the distribution function. As shown in Annex C, it is possible to derive
[19],[20],[21]
an inversion without model parameters for the estimated PSD . 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 +⋅Kq⋅ (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 can 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 popular regularisation is based on the smoothness of the distribution function,
T
which can be quantified via q ⋅H⋅q leading to the modified objective function:
2 T
χδ=−αα Kq⋅+ ⋅⋅qH⋅q (11)
exp
T
In Formula (11) and (12), H is the identity matrix, K is the transpose of matrix K, and δ is a suitable
[19]
Tikhonov regularization factor . The solution is then obtained from:
TT2
qi= nv KK+⋅δ HK αα (12)
()
exp
For small values of the regularisation factor δ the solution q is highly affected by the signal noise
showing strong oscillations with large negative values. In contrast, very large regularisation factors
yield such smooth solutions, that the characteristic features of the PSD are lost. In order to select an
[22],[23],[24],[25]
optimal regularisation factor, different strategies can be applied .
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 % by volume up to 5 % or more by volume,
depending on the density contrast between the continuous and the dispersed phases. In the case of
emulsions, measurements can be made at much higher concentrations (approaching 50 % by volume).
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, for example, variation of the
particle concentration. In particular, processes including a change of the phase (e.g. dissolution) or a
change in temperature sometimes 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).
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 % by volume.
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 document for repeatability if the CV for each of the x , x and x is smaller than 10 %. If a
10 50 90
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 repeatability. At least three distinct samples of
the same reference material shall be measured, and the mean and CV for the x , x , and x shall be
10 50 90
calculated. A CV larger than that of repeatability could 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 being examined. It is
essential that a written procedure is available that describes sub-sampling, 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 Reference [26]) are required in the measurement of accuracy. These
materials have a known size distribution with an x /x ratio in the range of 1,5 to 10. It is preferred
90 10
that the median size of the certified reference material will 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 according to ISO 14488 or
another method that has been proven to yield adequate results. 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 upper and the lower limit of the uncertainty range of the certified values (expanded uncertainty
on a 95 % confidence level) give the interval in which the true size values lie with a high probability.
The qualification test shall be accepted as passing the requirement of this document if the resulting
measured particle 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 subclause, 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.
8 Reporting of results
The particle size distribution results shall be reported according to ISO 20998-1:2006, Clause 5.
Annex A
(informative)
Viscoinertial loss model
Viscoinertial loss (see 5.3.3) can be calculated in the long wavelength limit from Formula (A.1). The
form shown in this Annex A is from Reference [5], but it is derived from the explicit analytical solutions
[15],[27],[28]
of Reference [7] and is mathematically equivalent to results obtained by many authors .
∞ 2
 
C ρρ′−
φ ()
  d
α = (A.1)
 

vis 
 
 2 cρ  
∞∞ −2
 
′
ρρ+ CC+ ω
() ()
id
 
 
[5]
where the dissipative and inertial drag coefficients are given by
9η
∞
C =+()1 Y (A.2)
d
2a
∞−1
1 9
CY=+1 (A.3)
()
i
2 2
with the dimensionless parameter Y defined by
ωρ
Ya= (A.4)
2η
In Formulae (A.1) to (A.4),
α = viscous attenuation coefficient
a = particle radius
c = speed of sound in liquid
ω = angular frequency of ultrasound (i.e. 2π times the frequency)
ρ, ρ′ = density of the liquid and particle, respectively
η = viscosity of the liquid
ϕ = volume concentration of the particle
Annex B
(informative)
ECAH theory and limitations
B.1 Introduction
The Epstein-Carhart-Allegra-Hawley (ECAH) theory is derived from the original work by Epstein
[6] [7]
and Carhart on sound attenuation in liquid/liquid systems (emulsions). Allegra and Hawley 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 equation as given in Formulae (B.1) to
(B.3):
2 2
(∇+k )ψ =0 (B.1)
c c
2 2
(∇+k )ψ =0 (B.2)
TT
2 2
()∇+k ψ =0 (B.3)
ss
The wave numbers k , k , and k are computed in Formulae (B.4), (B.5), and (B.6):
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
...