ISO 23484:2023

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)

General Information

• Status: Published
• Publication Date: 04-Jul-2023
• ICS: 19.120 - Particle size analysis. Sieving
• Technical Committee: ISO/TC 24/SC 4 - Particle characterization
• Drafting Committee: ISO/TC 24/SC 4/WG 10 - Small angle X-ray scattering method
• Current Stage: 6060 - International Standard published
• Start Date: 05-Jul-2023
• Due Date: 29-Jun-2024
• Completion Date: 05-Jul-2023

Overview

ISO 23484:2023 - Determination of particle concentration by small-angle X‑ray scattering (SAXS) - specifies how to measure particle number and mass concentration in suspensions using SAXS. The standard is limited to sufficiently monodisperse, spherical nanoparticles (size-distribution width typically < ~50% of mean diameter) and to dilute systems where particle–particle interactions are negligible. It describes the principle of SAXS concentration determination, instrument and calibration options, measurement and data-correction procedures, and reporting requirements.

Key technical topics and requirements

• Scope and assumptions
  • Applies to monodisperse spherical particles in liquid suspensions; requires knowledge or assumption of particle shape and electron density contrast.
  • Limited to dilute systems; long-range interactions (e.g., Coulomb) require lower concentrations or special care.
• Principle
  • Concentration is derived from the absolute differential scattering cross section by comparing measured and calculated scattering using the particle form factor P(q,r).
  • Absolute intensity (ratio of scattered to incident beam) must be established for number-concentration determination.
• Calibration approaches
  • Use of primary/secondary standards (e.g., water, Lupolen, glassy carbon) with known scattering cross sections.
  • High-dynamic-range area detectors that can measure direct and scattered beams.
  • Detectors with known quantum efficiency combined with a calibrated photodiode to determine incident flux.
  • If absolute calibration is not possible, mass concentration can be estimated from extrapolated forward scattering using a monodisperse reference material.
• Instrument and measurement controls
  • q-axis and intensity-axis calibration, sample thickness optimization, and data-correction procedures are specified.
  • Each sample should be measured at its original concentration and at a 1:1 dilution to detect concentration artefacts.
• Limits and sensitivity
  • Size sensitivity typically covers ~1 nm to 100 nm (lab SAXS), extendable with USAXS; detection limits depend strongly on particle radius (scattered intensity ∝ radius^6) and electron-density contrast (∝ Δρ^2).
• Quality and reporting
  • Sections on repeatability, documentation, test report contents, and inter-laboratory comparison (informative annex).

Applications and users

• Practical for materials scientists, nanoparticle manufacturers, pharmaceutical analysts, environmental labs, and synchrotron users who need accurate ensemble-averaged nanoparticle concentrations in liquid suspensions.
• Useful in quality control, formulation development, regulatory submissions, and research where knowledge of particle concentration (number or mass) is critical.

Related standards

• ISO 17867 - Particle size analysis - Small angle X‑ray scattering (SAXS) (referenced for size and distribution determination)
• Other ISO terminology and particle-characterization references cited within the document.

Keywords: ISO 23484:2023, small-angle X‑ray scattering, SAXS, particle concentration, nanoparticles, suspensions, monodisperse, differential scattering cross section, calibration, detection limits.

Standards Content (Sample)

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 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
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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 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
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
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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
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expressions related to conformity assessment, as well as information about ISO's adherence to
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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
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
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]
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
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
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.
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
Polystyrene 1,05 10 10
Polystyrene 1,05 100 10
Silica 2,65 10 10
Silica 2,65 100 10
Silver 10,49 10 10
Silver 10,49 100 10
Gold 19,28 10 10
Gold 19,28 100 10
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
...