ISO 23484:2023
(Main)Determination of particle concentration by small-angle X-ray scattering (SAXS)
Determination of particle concentration by small-angle X-ray scattering (SAXS)
This document deals with the application of small-angle X-ray scattering (SAXS) for the measurement of the particle concentration in suspensions. In this document, only the concentration of sufficiently monodisperse spherical particles is treated, which means that the width of the size distribution is typically below about 50 % of the mean diameter. Here, the differential scattering cross section can be calculated based on the form factor, which depends only on the momentum transfer q and the particle radius r. Furthermore, this document is limited to dilute systems. A dilute system in the sense of SAXS means that particle interactions are absent. In case of long-range interactions (Coulomb forces between the particles), special care needs to be taken and a reduction of the concentration can be necessary.
Détermination de la concentration de particules par diffusion des rayons X aux petits angles (SAXS)
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INTERNATIONAL ISO
STANDARD 23484
First edition
2023-07
Determination of particle
concentration by small-angle X-ray
scattering (SAXS)
Détermination de la concentration de particules par diffusion des
rayons X aux petits angles (SAXS)
Reference number
ISO 23484:2023(E)
© ISO 2023
---------------------- Page: 1 ----------------------
ISO 23484:2023(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2023
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 2023 – All rights reserved
---------------------- Page: 2 ----------------------
ISO 23484:2023(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 Principle of the method . 3
5.1 Particle size detection limits . 4
5.2 Particle concentration detection limits . 4
5.3 Effects of polydispersity . 5
6 Apparatus . 6
7 Preliminary procedures and instrument set-up . 7
8 Sample preparation .7
9 Measurement and data correction procedures . 8
10 Determination of the particle concentration . 8
11 Repeatability .11
12 Documentation and test report .11
12.1 Test report . 11
12.2 Technical records .12
Annex A (informative) Inter-laboratory comparison .13
Bibliography .14
iii
© ISO 2023 – All rights reserved
---------------------- Page: 3 ----------------------
ISO 23484:2023(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing documents 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.
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 2023 – All rights reserved
---------------------- Page: 4 ----------------------
ISO 23484:2023(E)
Introduction
Small-angle X-ray scattering (SAXS) is a well-established method to obtain structural information on
inhomogeneities in materials at the nanoscale, typically between 1 nm and 100 nm, and is thus perfectly
suited for nanoparticulate systems. Under certain conditions, the upper limit can be extended to
200 nm and beyond. For sufficiently monodisperse spherical particles, the observed oscillations of the
scattered intensity as a function of the momentum transfer, which is directly related to the scattering
angle and the wavelength of the incident X-rays, enable the size determination of nanoparticles. In
order to determine their concentration in a liquid (also called suspending medium, solvent or matrix),
the absolute differential scattering cross section has to be determined, thus the ratio of the scattered
intensity to the incident intensity. Assumptions on the particle shape are required, which can be based
on microscopic techniques like electron microscopy. Furthermore, the electron density difference
between the particles and the liquid needs to be known.
The concentration of nanoparticles, thus particles in the size range between about 1 nm to 100 nm, is
one of the most important parameters for nanoparticle use in industry, medicine and research, and is
expected to become relevant as well for regulatory purposes, especially in the pharmaceutic sector.
The application of SAXS for the determination of the mean particle size and size distribution has been
described in ISO 17867. This document covers the extension to obtain the nanoparticle concentration
as well from SAXS measurements. User-friendly commercial SAXS instruments are available worldwide
from several manufacturers for both routine and more sophisticated analyses, and state-of-the-art
research instruments are available at synchrotron radiation facilities.
As in all particle size measurement techniques, care is required in all aspects of the use of the
instrument, collection of data, and further interpretation. Therefore, there is a need for a document
that allows users to obtain good inter-laboratory agreement on the accuracy and reproducibility of the
technique.
Since all illuminated particles present in the X-ray beam are measured simultaneously, SAXS results are
ensemble and time averaged across all the particle orientations which are present in the sample.
v
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INTERNATIONAL STANDARD ISO 23484:2023(E)
Determination of particle concentration by small-angle
X-ray scattering (SAXS)
1 Scope
This document deals with the application of small-angle X-ray scattering (SAXS) for the measurement
of the particle concentration in suspensions. In this document, only the concentration of sufficiently
monodisperse spherical particles is treated, which means that the width of the size distribution is
typically below about 50 % of the mean diameter. Here, the differential scattering cross section can be
calculated based on the form factor, which depends only on the momentum transfer q and the particle
radius r. Furthermore, this document is limited to dilute systems. A dilute system in the sense of SAXS
means that particle interactions are absent. In case of long-range interactions (Coulomb forces between
the particles), special care needs to be taken and a reduction of the concentration can be necessary.
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 17867, Particle size analysis — Small angle X-ray scattering (SAXS)
3 Terms and definitions
For the purposes of this document, the following terms and definitions 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
particle
minute piece of matter with defined physical boundaries
Note 1 to entry: A physical boundary can also be described as an interface.
Note 2 to entry: A particle can move as a unit.
Note 3 to entry: This definition applies to nano-objects.
[SOURCE: ISO/TS 80004-6:2021, 3.9]
3.2
particle size
linear dimension of a particle (3.1) determined by a specified measurement method and under specified
measurement conditions
Note 1 to entry: Different methods of analysis are based on the measurement of different physical properties.
Independent of the particle property actually measured, the particle size can be reported as a linear dimension,
e.g. as an equivalent spherical diameter.
[SOURCE: ISO/TS 80004-6:2021, 4.1.1]
1
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ISO 23484:2023(E)
3.3
particle size distribution
distribution of particles (3.1) as a function of particle size (3.2)
Note 1 to entry: Particle size distribution may be expressed as cumulative distribution or a distribution density
(distribution of the fraction of material in a size class, divided by the width of that class).
[SOURCE: ISO/TS 80004-6:2021, 4.1.2]
3.4
suspension
heterogeneous mixture of materials comprising a liquid and a finely dispersed solid material
[SOURCE: ISO/TS 80004-6:2021, 3.13]
3.5
concentration
amount-of-substance of a component divided by the volume of the system
[SOURCE: ISO 18113-1:2022, 3.2.12]
3.6
particle number concentration
number of particles per unit of volume of suspension
Note 1 to entry: The particle number concentration can also be given as number of particles per unit of mass of
suspension. Literature values for the density of the liquid can be used for the conversion as, in most cases, the
low content of particles for which this document is applicable will not affect the sample density significantly.
[SOURCE: ISO 29464:2017, 3.2.131]
4 Symbols and abbreviated terms
The symbols and abbreviated terms used in this document are listed in Table 1.
Table 1 — Symbols
Symbol Description Unit (with prefix)
-1
C Particle number concentration l
Median of lognormal size distribution nm
d
ln
Number-weighted mean particle diameter nm
d
num
f , f Atomic scattering factors
1 2
g (r) Number-weighted particle size distribution
num
I Primary beam intensity without sample
in
I(q) Scattered intensity (or scattering intensity)
M Molar mass g/mol
N Number of particles
-1
N Avogadro constant mol
A
P(q, r) Particle form factor as functions of q-value and particle radius, r
Momentum transfer or q-value, magnitude of the scattering vector given
-1
q nm
by q = (4π /λ) sin θ
r Particle radius nm
r Thomson radius fm
e
S(q,r) Structure factor as functions of q-value and particle radius, r
T Transmission
2
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ISO 23484:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit (with prefix)
t Optimum sample thickness mm
o
w Sample thickness mm
Z Number of protons
λ Wavelength of the incident X-rays in vacuum nm
-1
μ Linear absorption coefficient mm
3
ρ Mass density g/cm
-3
ρ Electron density nm
e
-3
ρ Electron density of particles nm
e_P
-3
ρ Electron density of the liquid nm
e_L
-3
Δρ Electron density difference nm
e
σ Standard deviation of Gaussian size distribution nm
dΣ
-1 --1
q Differential scattering cross section per volume cm sr
()
dΩ
σ Standard deviation of logarithm of particle size distribution
ln
2θ Scattering angle deg or rad
Ω Solid angle of a detector pixel sr
5 Principle of the method
When X-rays pass through matter, a small fraction of the radiation can be scattered due to electron
density differences in the matter. The scattered radiation intensity profile (as a function of the scattering
angle or momentum transfer, q), contains information that can be used to deduce morphological
characteristics of the material. In the small-angle regime (typically 2θ < 5°; wavelength dependent),
information on the particle dimensions within the material is available from the elastic scattering
arising from the electron density contrast between the particles and the medium in which they reside,
typically a liquid. For sufficiently monodisperse spherical nanoparticles, the scattering pattern consist
of concentric rings, corresponding to oscillations of the scattered intensity as function of the scattering
angle or momentum transfer, q. If the (electron) density of the particles and the surrounding liquid
are known, the nanoparticle concentration can be determined by comparing the calculated and the
measured differential scattering cross section. The method requires the calibration of the q-axis and
the intensity axis. The absolute scattering cross section can be obtained by using either:
— primary or secondary standards such as water, Lupolen or glassy carbon with calculable or known
scattering cross section;
— an area detector with very high dynamic range (such as hybrid-pixel detectors) so that the incident
radiation (direct beam) and the scattered radiation can be measured;
— an area detector with known quantum efficiency for the scattered radiation and an additional
detector (such as a calibrated photodiode) to determine the incident photon flux.
Calibration with reference materials consisting of nanoparticles with known concentration is not
required in these cases.
If an absolute intensity calibration is not possible, it is still possible to determine the mass concentration
from the extrapolated forward scattering intensity using a similar monodisperse reference material
with known mass concentration.
At increased concentrations, i.e. those higher than ten volume %, particle-particle interactions and
inter-particle interference can be relevant. Such interactions require sophisticated data modelling and
expert knowledge for data interpretation, which is beyond the scope of this document. In practice, a
concentration ladder may be explored to determine the dependence of reported size on concentration.
3
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ISO 23484:2023(E)
If available, each sample shall be measured twice: in its original concentration and diluted 1:1 to allow
identification of concentration artefacts.
5.1 Particle size detection limits
The determination of the particle size (mean particle diameter) and the size distribution shall be in
accordance with ISO 17867. The accessible size range strongly depends on the instrument. In order
to register at least one minimum in the scattered intensity as function of the momentum transfer, the
lower diameter limit is typically a few nm. The higher diameter limit is about hundred nm for most
laboratory instruments, but the range can be extended to several hundred nm at dedicated ultra small
angle X-ray scattering (USAXS) instruments which are available at some synchrotron radiation facilities
and some laboratory instruments.
5.2 Particle concentration detection limits
The particle number concentration limits vary as well with the instrument, but even more with the
size and the (electron) density of the particles. The scattered intensity of spherical particles scales
with the sixth power of the particle size, thus the accessible lower number concentration limit of large
particles is orders of magnitude lower. On the other hand, the scattered intensity scales with the square
of the electron density difference between particles and liquid. Therefore, much lower concentration
ranges are accessible for gold nanoparticles compared to polystyrene particles. Table 2 provides typical
orders of magnitude for the lower limit of detection (LLD) for nanoparticles of different diameters and
materials suspended in water. The accessible concentration ranges are schematically shown in Figure 1.
Table 2 — Lower limit of detection for nanoparticle number concentration
Material Density Diameter LLD number concentration
3 -1
g/cm nm l
20
Polystyrene 1,05 10 10
17
Polystyrene 1,05 100 10
17
Silica 2,65 10 10
14
Silica 2,65 100 10
15
Silver 10,49 10 10
12
Silver 10,49 100 10
14
Gold 19,28 10 10
11
Gold 19,28 100 10
4
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ISO 23484:2023(E)
Key
X diameter/nm
Y number of particles/litre
1 lower limit for polystyrene particles in water
2 lower limit for silica particles in water
3 lower limit for gold particles in water
4 lower limit for silver particles in water
5 upper limit for all particles corresponding to 10 % volume fraction
NOTE For low-density particles like polystyrene, the low concentration limit can be decreased, for example,
by using ethanol instead of water as liquid.
Figure 1 — Schematic representation of the accessible concentration ranges for spherical
nanoparticles of four different materials (gold, silver, silica and polystyrene) in aqueous
suspension as function of the particle diameter
5.3 Effects of polydispersity
The issue of polydispersity is extremely important for most real samples. As mentioned above, the
scattered intensity is proportional to the sixth power of the radius, thus the scatt
...
ISO/FDIS 23484:2022(E)
ISO /TC 24/SC 4/WG 10
Secretariat: BSI
Date: 2023-02-16
Determination of particle concentration by small-angle X-ray
scattering (SAXS)
2022-12-08
---------------------- Page: 1 ----------------------
Détermination de la concentration de particules par diffusion des rayons X aux petits angles (SAXS)
FDIS stage
---------------------- Page: 2 ----------------------
ISO/FDIS 23484:2023(E)
© ISO 2022, Published in Switzerland2023
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 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel.Phone: + 41 22 749 01 11
Fax + 41 22 749 09 47
E-mail: copyright@iso.org
www.iso.org
Website: www.iso.org
Published in Switzerland
© ISO 2022 – All rights reserved iii
© ISO 2023 – All rights reserved iii
---------------------- Page: 3 ----------------------
ISO/FDIS 23484:2023(E)
Contents
1
Foreword vi
Introduction vii
1 Scope 1
2 Normative references 1
3 Terms and definitions 1
4 Symbols and abbreviations 3
5 Principle of the method 4
5.1 Particle size detection limits 4
5.2 Particle concentration detection limits 4
5.3 Effects of polydispersity 6
6 Apparatus 7
7 Preliminary procedures and instrument set-up 8
8 Sample preparation 9
9 Measurement and data correction procedures 9
10 Determination of the particle concentration 9
11 Repeatability 12
12 Documentation and test report 12
12.1 Test report 12
12.2 Technical records 13
Annex A (informative) Inter-laboratory comparison 14
Bibliography 15
Foreword . vi
Introduction .vi i
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Symbols and abbreviations . 3
5 Principle of the method . 4
5.1 Particle size detection limits . 4
5.2 Particle concentration detection limits . 4
5.3 Effects of polydispersity . 6
6 Apparatus . 7
7 Preliminary procedures and instrument set-up . 8
8 Sample preparation . 9
9 Measurement and data correction procedures . 9
10 Determination of the particle concentration . 9
11 Repeatability . 12
12 Documentation and test report . 12
12.1 Test report . 12
iv © ISO 2022 – All rights reserved
iv © ISO 2023 – All rights reserved
---------------------- Page: 4 ----------------------
ISO/FDIS 23484:2023(E)
12.2 Technical records . 13
Annex A (informative) Inter-laboratory comparison . 14
Bibliography . 15
© ISO 2022 – All rights reserved v
© ISO 2023 – All rights reserved v
---------------------- Page: 5 ----------------------
ISO/FDIS 23484:2023(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing documents 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).
Field Code Changed
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).
Field Code Changed
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation onof 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 the following URL:
www.iso.org/iso/foreword.html.
Field Code Changed
This document was prepared by Technical Committee ISO/TC 24, Particle characterization including
sieving, Subcommittee SC 4, Particle characterization.
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.
vi © ISO 2022 – All rights reserved
vi © ISO 2023 – All rights reserved
---------------------- Page: 6 ----------------------
ISO/FDIS 23484:2023(E)
Introduction
Small-angle X-ray scattering (SAXS) is a well-established method to obtain structural information on
inhomogeneities in materials at the nanoscale, typically between 1 nm and 100 nm, and is thus perfectly
suited for nanoparticulate systems. Under certain conditions, the upper limit can be extended to 200 nm
and beyond. For sufficiently monodisperse spherical particles, the observed oscillations of the scattered
intensity as a function of the momentum transfer, which is directly related to the scattering angle and the
wavelength of the incident X-rays, enable the size determination of nanoparticles. In order to determine
their concentration in a liquid (also called suspending medium, solvent or matrix), the absolute
differential scattering cross section has to be determined, thus the ratio of the scattered intensity to the
incident intensity. Assumptions on the particle shape are required, which can be based on microscopic
techniques like electron microscopy. Furthermore, the electron density difference between the particles
and the liquid needs to be known.
The concentration of nanoparticles, thus particles in the size range between about 1 nm to 100 nm, is one
of the most important parameters for nanoparticle use in industry, medicine and research, and is
expected to become relevant as well for regulatory purposes, especially in the pharmaceutic sector. The
application of SAXS for the determination of the mean particle size and size distribution has been
described in ISO 17867:2020. This document covers the extension to obtain the nanoparticle
concentration as well from SAXS measurements. User-friendly commercial SAXS instruments are
available worldwide from several manufacturers for both routine and more sophisticated analyses, and
state-of-the-art research instruments are available at synchrotron radiation facilities.
As in all particle size measurement techniques, care is required in all aspects of the use of the instrument,
collection of data, and further interpretation. Therefore, there is a need for a document that allows users
to obtain good inter-laboratory agreement on the accuracy and reproducibility of the technique.
Since all illuminated particles present in the X-ray beam are measured simultaneously, SAXS results are
ensemble and time averaged across all the particle orientations which are present in the sample.
© ISO 2022 – All rights reserved vii
© ISO 2023 – All rights reserved vii
---------------------- Page: 7 ----------------------
DRAFT DOCUMENT ISO/FDIS 23484:2022(E)
Determination of the particle concentration by small-angle X-ray
scattering (SAXS)
21 Scope
This document deals with the application of small-angle X-ray scattering (SAXS) for the measurement of
the particle concentration in suspensions. In this document, only the concentration of sufficiently
monodisperse spherical particles is treated, which means that the width of the size distribution shouldis
typically be below about 50 % of the mean diameter. Here, the differential scattering cross section can
be calculated based on the form factor, which depends only on the momentum transfer q and the particle
radius r. Furthermore, this document is limited to dilute systems. A dilute system in the sense of SAXS
means that particle interactions are absent. In case of long-range interactions (Coulomb forces between
the particles), special care hasneeds to be taken and a reduction of the concentration mightcan be
necessary.
32 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 17867:2020, Particle size analysis — Small angle X-ray scattering (SAXS)
43 Terms and definitions
For the purposes of this document, the following terms and definitions 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/obphttps://www.iso.org/obp
— — IEC Electropedia: available at https://www.electropedia.org/https://www.electropedia.org/
3.1
particle
minute piece of matter with defined physical boundaries
Note 1 to entry: A physical boundary can also be described as an interface.
Note 2 to entry: A particle can move as a unit.
Note 3 to entry: This general particle definition applies to nano-objects.
[SOURCE: ISO/TS 80004 -6:2021(en),, 3.9]
3.2
particle size
linear dimension of a particle (3.1)(3.1) determined by a specified measurement method and under
specified measurement conditions
Note 1 to entry: Different methods of analysis are based on the measurement of different physical properties.
Independent of the particle property actually measured, the particle size can be reported as a linear dimension, e.
g. as an equivalent spherical diameter.
[SOURCE: ISO/TS 80004 -6:2021(en),, 4.1.1]
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---------------------- Page: 8 ----------------------
ISO/FDIS 23484:2023(E)
3.3
particle size distribution
distribution of particles (3.1)(3.1) as a function of particle size (3.2)(3.2)
Note 1 to entry: Particle size distribution may be expressed as cumulative distribution or a distribution density
(distribution of the fraction of material in a size class, divided by the width of that class).
[SOURCE: ISO/TS 80004 -6:2021(en),, 4.1.2]
3.4
suspension
heterogeneous mixture of materials comprising a liquid and a finely dispersed solid material
[SOURCE: ISO/TS 80004-6:2021(en),, 3.13]
3.5
concentration
amount-of-substance of a component divided by the volume of the system
[SOURCE: ISO 18113-1:2022(en),, 3.2.12]
3.6
particle number concentration
number of particles per unit of volume of suspension
[ADOPTED: ISO 29464:2017(en), 3.2.131]
Note 1 to entry: The particle number concentration can also be given as number of particles per unit of mass of
suspension. Literature values for the density of the liquid can be used for the conversion as, in most cases, the low
content of particles for which this document is applicable will not affect the sample density significantly.
2 © ISO 2022 – All rights reserved
2 © ISO 2023 – All rights reserved
---------------------- Page: 9 ----------------------
ISO/FDIS 23484:2023(E)
[SOURCE: ISO 29464:2017, 3.2.131]
54 Symbols and abbreviations
The symbols and abbreviations used in this document are listed in Table 1. Table 1.
Table 1 — Symbols
Symbol Description Unit (with prefix)
-1
C Particle number concentration l
̄ ¯
𝑑𝑑 𝑑𝑑 Median of lognormal size distribution nm
𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙
¯
𝑑𝑑 Number-weighted mean particle diameter nm
num
f1, f2 Atomic scattering factors
gnum(r) Number-weighted particle size distribution
I Primary beam intensity with sample
out
Iin Primary beam intensity without sample
I(q) Scattered intensity (or scattering intensity)
M Molar mass g/mol
N Number of particles
-1
𝑁𝑁 Avogadro constant mol
𝐴𝐴
P(q, r) Particle form factor as functions of q-value and particle radius, r
Momentum transfer or q-value, magnitude of the scattering vector given
-1
q nm
by q = (4π /λ) sin θ
-1
qmin Minimum accessible q-value nm
-1
q Maximum accessible q-value nm
max
r Particle radius nm
re Thomson radius fm
S(q,r) Structure factor as functions of q-value and particle radius, r
T Transmission
to Optimum sample thickness mm
3
V Volume of particle nm
w Sample thickness mm
Z Number of protons
λ Wavelength of the incident X-rays in vacuum nm
-1
μ Linear absorption coefficient mm
3
Mass density g/cm³cm
ρρ
-3
Electron density nm
ρeρe
-3
Electron density of particles nm
ρeρe_P
-3
Electron density of the liquid nm
ρeρe_L
-3
𝛥𝛥𝜌𝜌 Electron density difference nm
𝑒𝑒
σ Standard deviation of Gaussian size distribution nm
𝑑𝑑𝑑𝑑
-1 --1
(𝑞𝑞)(𝑞𝑞) Differential scattering cross section per volume cm sr
𝑑𝑑𝑑𝑑
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ISO/FDIS 23484:2023(E)
Symbol Description Unit (with prefix)
σ Standard deviation of logarithm of particle size distribution
ln
2θ Scattering angle deg or rad
85 Principle of the method
When X-rays pass through matter, a small fraction of the radiation maycan be scattered due to electron
density differences in the matter. The scattered radiation intensity profile (as a function of the scattering
angle or momentum transfer, q), contains information that can be used to deduce morphological
characteristics of the material. In the small-angle regime (typically 2θ < 5°; wavelength dependent),
information on the particle dimensions within the material is available from the elastic scattering arising
from the electron density contrast between the particles and the medium in which they reside, typically
a liquid. For sufficiently monodisperse spherical nanoparticles, the scattering pattern consist of
concentric rings, corresponding to oscillations of the scattered intensity as function of the scattering
angle or momentum transfer, q. If the (electron) density of the particles and the surrounding liquid are
known, the nanoparticle concentration can be determined by comparing the calculated and the measured
differential scattering cross section. The method requires the calibration of the q-axis and the intensity
axis. The absolute scattering cross section can be obtained by using either:
— — primary or secondary standards such as water, Lupolen or glassy carbon with calculable or
known scattering cross section;
— — an area detector with very high dynamic range (such as hybrid-pixel detectors) so that the
incident radiation (direct beam) and the scattered radiation can be measured;
— — an area detector with known quantum efficiency for the scattered radiation and an additional
detector (such as a calibrated photodiode) to determine the incident photon flux.
Calibration with reference materials consisting of nanoparticles with known concentration is not
required in these cases.
If an absolute intensity calibration is not possible, it is still possible to determine the mass concentration
from the extrapolated forward scattering intensity using a similar monodisperse reference material with
known mass concentration.
At increased concentrations, i.e. those higher than ten volume %, particle-particle interactions and inter-
particle interference can be relevant. Such interactions require sophisticated data modelling and expert
knowledge for data interpretation, which is beyond the scope of the present standard.this document. In
practice, a concentration ladder may be explored to determine the dependence of reported size on
concentration. If available, each sample shall be measured twice: in its original concentration and diluted
1:1 to allow identification of concentration artefacts.
8.15.1 Particle size detection limits
The determination of the particle size (mean particle diameter) and the size distribution is describedshall
be in detail inaccordance with ISO 17867:2020. The accessible size range strongly depend on the
instrument. In order to register at least one minimum in the scattered intensity as function of the
momentum transfer, the lower diameter limit is typically a few nm. The higher diameter limit is about
hundred nm for most laboratory instruments, but the range can be extended to several hundred nm at
dedicated USAXS (ultra small angle X-ray scattering (USAXS) instruments which are available at some
synchrotron radiation facilities and some laboratory instruments.
8.25.2 Particle concentration detection limits
The particle number concentration limits vary as well with the instrument, but even more with the size
and the (electron) density of the particles. The scattered intensity of spherical particles scales with the
sixth power of the particle size, thus the accessible lower number concentration limit of large particles is
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ISO/FDIS 23484:2023(E)
orders of magnitude lower. On the other hand, the scattered intensity scales with the square of the
electron density difference between particles and liquid. Therefore, much lower concentration ranges are
accessible for gold nanoparticles compared to polystyrene particles. Table 2Table 2 provides typical
orders of magnitude for the lower limit of detection (LLD) for nanoparticles of different diameters and
materials suspended in water. The accessible concentration ranges are schematically shown in
Figure 1.Figure 1.
Table 2 — Lower limit of detection for nanoparticle number concentration
Material densityDensity diameterDiameteLLD number concentration
3 -1
g/cm r l
nm
20
Polystyrene 1,05 10 10
17
Polystyrene 1,05 100 10
17
Silica 2,65 10 10
14
Silica 2,65 100 10
15
Silver 10,49 10 10
12
Silver 10,49 100 10
14
Gold 19,28 10 10
11
Gold 19,28 100 10
1x1024
1
2
1x1022
3
4
1x1020
5
1x1018
1x1016
1x1014
1x1012
1x1010
1 2 5 10 20 50 100 200
X
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Y
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ISO/FDIS 23484:2023(E)
Key
X diameter / /nm
Y number of particles / liter/litre
1 lower limit for polystyrene particles in water
2 lower limit for silica particles in water
3 lower limit for gold particles in water
4 lower limit for silver particles in water
5 upper limit for all particles corresponding to 10 % volume fraction
NOTE For low-density particles like polystyrene, the low concentration limit can be decreased, for example, by
using ethanol instead of water as liquid.
Figure 1 — Schematic representation of the accessible concentration ranges for spherical
nanoparticles of four different materials (gold, silver, silica and polystyrene) in aqueous
suspension as function of the particle diameter
8.35.3 Effects of polydispersity
The issue of polydispersity is extremely important for most real samples. As mentioned above, the
scattered intensity is proportional to the sixth power of the radius, thus the scattering from larger
particles can hamper the detection of smaller size fractions in poly-dispersed samples. Thus, the samples
have to be sufficiently monodisperse. If the size distribution is too broad, minima are no longer observed
in the scattering curve, and thus a unique model fitting is no longer possible. Also, a size distribution
asymmetry, thus a non-Gaussian or non-lognormal size distribution would lead to deviations for the
particle concentration.
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ISO/FDIS 23484:2023(E)
96 Apparatus
The general design of a SAXS instrument is shown in Figure 2.Figure 2. The SAXS set-up consists of X-ray
source, optics, collimation system, sample holder, beam stop, and detector. The greatest challenges in
SAXS are to separate the parasitic scattering from the collimation system and the unscattered,
transmitted beam (“direct beam”) from the scattered radiation at small angles (around 0,1°). The direct
beam is normally blocked by a beam stop and parasitic scattering should be eliminated. The need for
separation of primary and scattered beam makes collimation of the primary beam mandatory.
Key
X 2θ or q
Y scattered intensity
1 X-ray source
2 optics
3 collimation system
4 sample
a
2θ.
Figure 2 — Schematic design of a SAXS instrument, consisting of X-ray source, optics,
collimation system, sample holder, beam stop and X-ray detector
The above outlined principles strictly apply only for scattering patterns that are obtained with ideal
point-collimation optics, i.e. point shaped X-ray beam cross section, and monochromatic radiation.
However, a widely popular camera design (Kratky camera or slit collimators) uses line collimation, i.e.
the probing X-ray beam is a narrow ribbon with line-shaped cross section. This has the advantage of
producing higher intensities in the weak outer part of the scattering curve (towards large q), but
generally requires numerical corrections (desmearing). The two design principles are shown in
Figure 3.Figure 3.
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ISO/FDIS 23484:2023(E)
Key
1 X-ray source
2 collimation system
3 sample
Figure 3 — Schematic view of point- (left) and line-collimation (right) optics
107 Preliminary procedures and instrument set-up
The momentum transfer q is related to the scattering angle and the wavelength. Wavelength calibration
can be performed before conducting an experiment and thus would be classified as a preliminary
procedure, but this is only required for polychromatic sources. If characteristic X-ray emission lines (e.g.
copper Kα or molybdenum Kα lines) are used, a suitable absorber can be used to check that the right
emission line has been selected correctly (Nickelnickel for Cu Kα, Zirconiumzirconium for Mo Kα). The
scattering angle follows from the geometry of the experimental set-up and can be obtained from the pixel
size of the X-ray detector and the sample-to-detector distance. If the latter can be varied, the distance can
be determined by triangulation.
The utilization of a calibration material for the q-value, such as silver behenate, is a simple and convenient
alternative to calibrate the q-axis.
For the determination of the particle number concentration, the scattered intensity mustshall be scaled
to absolute units. For this purpose, a variety of auxiliary calibration materials are available, including
water and glassy carbon. Alternatively, many current SAXS instruments can determine this directly by
means of calibrated detectors, or measurement of the unattenuated primary beam intensity if an area
detector with very high linearity is used. Otherwise, or if the incident photon flux is very high (e.g. at
synchrotron radiation beamlines), it is also possible to determine the incident photon flux (e.g. with a
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ISO/FDIS 23484:2023(E)
calibrated ion chamber or photodiode) and to use the previously determined quantum efficiency of the
area detector. The use of semi-transparent beamstops to measure the beam intensity is not
recommended due to the radiation hardening effects of such, which can lead to inaccurate values.
All calibrations should be described in the analysis report.
118 Sample preparation
Sample preparation is simple and fast for SAXS measurements. The required sample volumes are small,
typically in a range of 5 μLμl to 50 μLμl for liquids, if copper radiation is used.
Liquid samples are usually measured inside a thin-walled sample cell, typically a capillary with a diameter
between 0,5 mm to 2 mm when the liquid primarily contains water or hydrocarbons. Particles in a liquid
that contains heavy atoms, for example, chlorine in chloroform, should be measured in smaller diameter
capillaries as the atoms strongly absorb the incident radiation, or higher energy radiation should be used.
The sample thickness has to be determined, which is easier for capillaries with rectangular cross-section,
but their wall thickness is usually larger.
It is strongly recommended to measure liquid samples in a re-fillable, well characterized or flow-
throughsamplethrough sample cell, as this greatly reduces the risks of errors encountered in the
background subtraction procedure (incorrigible effects may lead to inaccurate subtraction if non-
identical sample cells are used for the two measurements).
129 Measurement and data correction procedures
If possible, a SAXS particle-concentration experiment should consistsconsist of at least two
measurements using the same sample holder and preferably the same acquisition time:
a) a) a background measurement (containing signals from the liquid or matrix, the (same) sample cell
windows, the parasitic instrument radiation, background radiation and detector noise).);
b) b) a sample measurement (containing signals from the particles, the liquid or matrix, the sample
cell windows, the parasitic instrument radiation, background radiation and detector noise);).
Care shall be taken that the scattering of the window material of the sample cell, the parasitic scattering
of the SAXS instrument, and the dark count rate of the detector are removed. The transmission from the
sample and background/liquid and efficiency variation over the detector shall be taken into account.
The statistical quality of the scattering pattern improves with increasing intensity and complies with
standard statistics for signals obtained by the subtraction of two independent measurements.
As the data analysis methods are sensitive to the quality of the data, the scattering signal of the sample
shall be carefully extracted from the total measured signal. These correction steps are performed by the
software provided with commercial instruments, or through custom software. At best, they propagate
the data uncertainties through the co
...
FINAL
INTERNATIONAL ISO/FDIS
DRAFT
STANDARD 23484
ISO/TC 24/SC 4
Determination of particle
Secretariat: BSI
concentration by small-angle X-ray
Voting begins on:
2023-03-03 scattering (SAXS)
Voting terminates on:
Détermination de la concentration de particules par diffusion des
2023-04-28
rayons X aux petits angles (SAXS)
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BEING ACCEPTABLE FOR INDUSTRIAL, TECHNO-
ISO/FDIS 23484:2023(E)
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LIGHT OF THEIR POTENTIAL TO BECOME STAN-
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---------------------- Page: 1 ----------------------
ISO/FDIS 23484:2023(E)
FINAL
INTERNATIONAL ISO/FDIS
DRAFT
STANDARD 23484
ISO/TC 24/SC 4
Determination of particle
Secretariat: BSI
concentration by small-angle X-ray
Voting begins on:
scattering (SAXS)
Voting terminates on:
Détermination de la concentration de particules par diffusion des
rayons X aux petits angles (SAXS)
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ii
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NATIONAL REGULATIONS. © ISO 2023
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ISO/FDIS 23484:2023(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 Principle of the method . 3
5.1 Particle size detection limits . 4
5.2 Particle concentration detection limits . 4
5.3 Effects of polydispersity . 5
6 Apparatus . 6
7 Preliminary procedures and instrument set-up . 7
8 Sample preparation .7
9 Measurement and data correction procedures . 8
10 Determination of the particle concentration . 8
11 Repeatability .11
12 Documentation and test report .11
12.1 Test report . 11
12.2 Technical records . 11
Annex A (informative) Inter-laboratory comparison .13
Bibliography .14
iii
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ISO/FDIS 23484:2023(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing documents 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 nongovernmental, 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.
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
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ISO/FDIS 23484:2023(E)
Introduction
Small-angle X-ray scattering (SAXS) is a well-established method to obtain structural information on
inhomogeneities in materials at the nanoscale, typically between 1 nm and 100 nm, and is thus perfectly
suited for nanoparticulate systems. Under certain conditions, the upper limit can be extended to
200 nm and beyond. For sufficiently monodisperse spherical particles, the observed oscillations of the
scattered intensity as a function of the momentum transfer, which is directly related to the scattering
angle and the wavelength of the incident X-rays, enable the size determination of nanoparticles. In
order to determine their concentration in a liquid (also called suspending medium, solvent or matrix),
the absolute differential scattering cross section has to be determined, thus the ratio of the scattered
intensity to the incident intensity. Assumptions on the particle shape are required, which can be based
on microscopic techniques like electron microscopy. Furthermore, the electron density difference
between the particles and the liquid needs to be known.
The concentration of nanoparticles, thus particles in the size range between about 1 nm to 100 nm, is
one of the most important parameters for nanoparticle use in industry, medicine and research, and is
expected to become relevant as well for regulatory purposes, especially in the pharmaceutic sector.
The application of SAXS for the determination of the mean particle size and size distribution has been
described in ISO 17867. This document covers the extension to obtain the nanoparticle concentration
as well from SAXS measurements. User-friendly commercial SAXS instruments are available worldwide
from several manufacturers for both routine and more sophisticated analyses, and state-of-the-art
research instruments are available at synchrotron radiation facilities.
As in all particle size measurement techniques, care is required in all aspects of the use of the
instrument, collection of data, and further interpretation. Therefore, there is a need for a document
that allows users to obtain good inter-laboratory agreement on the accuracy and reproducibility of the
technique.
Since all illuminated particles present in the X-ray beam are measured simultaneously, SAXS results are
ensemble and time averaged across all the particle orientations which are present in the sample.
v
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FINAL DRAFT INTERNATIONAL STANDARD ISO/FDIS 23484:2023(E)
Determination of particle concentration by small-angle
X-ray scattering (SAXS)
1 Scope
This document deals with the application of small-angle X-ray scattering (SAXS) for the measurement
of the particle concentration in suspensions. In this document, only the concentration of sufficiently
monodisperse spherical particles is treated, which means that the width of the size distribution is
typically below about 50 % of the mean diameter. Here, the differential scattering cross section can be
calculated based on the form factor, which depends only on the momentum transfer q and the particle
radius r. Furthermore, this document is limited to dilute systems. A dilute system in the sense of SAXS
means that particle interactions are absent. In case of longrange interactions (Coulomb forces between
the particles), special care needs to be taken and a reduction of the concentration can be necessary.
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 17867, Particle size analysis — Small angle X-ray scattering (SAXS)
3 Terms and definitions
For the purposes of this document, the following terms and definitions 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
particle
minute piece of matter with defined physical boundaries
Note 1 to entry: A physical boundary can also be described as an interface.
Note 2 to entry: A particle can move as a unit.
Note 3 to entry: This definition applies to nano-objects.
[SOURCE: ISO/TS 800046:2021, 3.9]
3.2
particle size
linear dimension of a particle (3.1) determined by a specified measurement method and under specified
measurement conditions
Note 1 to entry: Different methods of analysis are based on the measurement of different physical properties.
Independent of the particle property actually measured, the particle size can be reported as a linear dimension,
e.g. as an equivalent spherical diameter.
[SOURCE: ISO/TS 800046:2021, 4.1.1]
1
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ISO/FDIS 23484:2023(E)
3.3
particle size distribution
distribution of particles (3.1) as a function of particle size (3.2)
Note 1 to entry: Particle size distribution may be expressed as cumulative distribution or a distribution density
(distribution of the fraction of material in a size class, divided by the width of that class).
[SOURCE: ISO/TS 800046:2021, 4.1.2]
3.4
suspension
heterogeneous mixture of materials comprising a liquid and a finely dispersed solid material
[SOURCE: ISO/TS 800046:2021, 3.13]
3.5
concentration
amount-of-substance of a component divided by the volume of the system
[SOURCE: ISO 181131:2022, 3.2.12]
3.6
particle number concentration
number of particles per unit of volume of suspension
Note 1 to entry: The particle number concentration can also be given as number of particles per unit of mass of
suspension. Literature values for the density of the liquid can be used for the conversion as, in most cases, the
low content of particles for which this document is applicable will not affect the sample density significantly.
[SOURCE: ISO 29464:2017, 3.2.131]
4 Symbols and abbreviations
The symbols and abbreviations used in this document are listed in Table 1.
Table 1 — Symbols
Symbol Description Unit (with prefix)
1
C Particle number concentration l
Median of lognormal size distribution nm
d
ln
Numberweighted mean particle diameter nm
d
num
f , f Atomic scattering factors
1 2
g (r) Numberweighted particle size distribution
num
I Primary beam intensity with sample
out
I Primary beam intensity without sample
in
I(q) Scattered intensity (or scattering intensity)
M Molar mass g/mol
N Number of particles
1
N Avogadro constant mol
A
P(q, r) Particle form factor as functions of qvalue and particle radius, r
Momentum transfer or qvalue, magnitude of the scattering vector given
1
q nm
by q = (4π /λ) sin θ
1
q Minimum accessible qvalue nm
min
1
q Maximum accessible qvalue nm
max
r Particle radius nm
2
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ISO/FDIS 23484:2023(E)
TTabablele 1 1 ((ccoonnttiinnueuedd))
Symbol Description Unit (with prefix)
r Thomson radius fm
e
S(q,r) Structure factor as functions of qvalue and particle radius, r
T Transmission
t Optimum sample thickness mm
o
3
V Volume of particle nm
w Sample thickness mm
Z Number of protons
λ Wavelength of the incident X-rays in vacuum nm
1
μ Linear absorption coefficient mm
3
ρ Mass density g/cm
3
ρ Electron density nm
e
3
ρ Electron density of particles nm
e_P
3
ρ Electron density of the liquid nm
e_L
3
Δρ Electron density difference nm
e
σ Standard deviation of Gaussian size distribution nm
dΣ
1 1
Differential scattering cross section per volume cm sr
()q
dΩ
σ Standard deviation of logarithm of particle size distribution
ln
2θ Scattering angle deg or rad
5 Principle of the method
When X-rays pass through matter, a small fraction of the radiation can be scattered due to electron
density differences in the matter. The scattered radiation intensity profile (as a function of the scattering
angle or momentum transfer, q), contains information that can be used to deduce morphological
characteristics of the material. In the small-angle regime (typically 2θ < 5°; wavelength dependent),
information on the particle dimensions within the material is available from the elastic scattering
arising from the electron density contrast between the particles and the medium in which they reside,
typically a liquid. For sufficiently monodisperse spherical nanoparticles, the scattering pattern consist
of concentric rings, corresponding to oscillations of the scattered intensity as function of the scattering
angle or momentum transfer, q. If the (electron) density of the particles and the surrounding liquid
are known, the nanoparticle concentration can be determined by comparing the calculated and the
measured differential scattering cross section. The method requires the calibration of the q-axis and
the intensity axis. The absolute scattering cross section can be obtained by using either:
— primary or secondary standards such as water, Lupolen or glassy carbon with calculable or known
scattering cross section;
— an area detector with very high dynamic range (such as hybrid-pixel detectors) so that the incident
radiation (direct beam) and the scattered radiation can be measured;
— an area detector with known quantum efficiency for the scattered radiation and an additional
detector (such as a calibrated photodiode) to determine the incident photon flux.
Calibration with reference materials consisting of nanoparticles with known concentration is not
required in these cases.
If an absolute intensity calibration is not possible, it is still possible to determine the mass concentration
from the extrapolated forward scattering intensity using a similar monodisperse reference material
with known mass concentration.
3
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ISO/FDIS 23484:2023(E)
At increased concentrations, i.e. those higher than ten volume %, particle-particle interactions and
inter-particle interference can be relevant. Such interactions require sophisticated data modelling and
expert knowledge for data interpretation, which is beyond the scope of this document. In practice, a
concentration ladder may be explored to determine the dependence of reported size on concentration.
If available, each sample shall be measured twice: in its original concentration and diluted 1:1 to allow
identification of concentration artefacts.
5.1 Particle size detection limits
The determination of the particle size (mean particle diameter) and the size distribution shall be in
accordance with ISO 17867. The accessible size range strongly depend on the instrument. In order to
register at least one minimum in the scattered intensity as function of the momentum transfer, the
lower diameter limit is typically a few nm. The higher diameter limit is about hundred nm for most
laboratory instruments, but the range can be extended to several hundred nm at dedicated ultra small
angle X-ray scattering (USAXS) instruments which are available at some synchrotron radiation facilities
and some laboratory instruments.
5.2 Particle concentration detection limits
The particle number concentration limits vary as well with the instrument, but even more with the
size and the (electron) density of the particles. The scattered intensity of spherical particles scales
with the sixth power of the particle size, thus the accessible lower number concentration limit of large
particles is orders of magnitude lower. On the other hand, the scattered intensity scales with the square
of the electron density difference between particles and liquid. Therefore, much lower concentration
ranges are accessible for gold nanoparticles compared to polystyrene particles. Table 2 provides typical
orders of magnitude for the lower limit of detection (LLD) for nanoparticles of different diameters and
materials suspended in water. The accessible concentration ranges are schematically shown in Figure 1.
Table 2 — Lower limit of detection for nanoparticle number concentration
Material Density Diameter LLD number concentration
3 1
g/cm nm l
20
Polystyrene 1,05 10 10
17
Polystyrene 1,05 100 10
17
Silica 2,65 10 10
14
Silica 2,65 100 10
15
Silver 10,49 10 10
12
Silver 10,49 100 10
14
Gold 19,28 10 10
11
Gold 19,28 100 10
4
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ISO/FDIS 23484:2023(E)
Key
...
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