Particle size analysis -- Small angle X-ray scattering (SAXS)

This document specifies a method for the application of small-angle X-ray scattering (SAXS) to the estimation of mean particle sizes in the 1 nm to 100 nm size range. It is applicable in dilute dispersions where the interaction and scattering effects between the particles are negligible. This document describes several data evaluation methods: the Guinier approximation, model-based data fitting, Monte-Carlo?based data fitting, the indirect Fourier transform method and the expectation maximization method. The most appropriate evaluation method is intended to be selected by the analyst and stated clearly in the report. While the Guinier approximation only provides an estimate for the mean particle diameter, the other methods also give insight in the particle size distribution.

Analyse granulométrique -- Diffusion des rayons X aux petits angles (SAXS)

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INTERNATIONAL ISO
STANDARD 17867
Second edition
2020-10
Particle size analysis — Small angle
X-ray scattering (SAXS)
Analyse granulométrique — Diffusion des rayons X aux petits
angles (SAXS)
Reference number
ISO 17867:2020(E)
ISO 2020
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ISO 17867:2020(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2020

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

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Published in Switzerland
ii © ISO 2020 – All rights reserved
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ISO 17867:2020(E)
Contents Page

Foreword ........................................................................................................................................................................................................................................iv

Introduction ..................................................................................................................................................................................................................................v

1 Scope ................................................................................................................................................................................................................................. 1

2 Normative references ...................................................................................................................................................................................... 1

3 Terms and definitions ..................................................................................................................................................................................... 1

4 Symbols .......................................................................................................................................................................................................................... 2

5 Principle of the method ................................................................................................................................................................................. 2

6 Apparatus and procedure ........................................................................................................................................................................... 3

7 Preliminary procedures and instrument set-up ................................................................................................................. 6

8 Sample preparation ........................................................................................................................................................................................... 6

9 Measurement procedure .............................................................................................................................................................................. 7

10 Data correction procedures ...................................................................................................................................................................... 8

11 Calculation of the mean particle diameter ................................................................................................................................ 9

11.1 General ........................................................................................................................................................................................................... 9

11.2 Guinier approximation ..................................................................................................................................................................... 9

11.3 Model fitting ...........................................................................................................................................................................................10

12 Size distribution determination ........................................................................................................................................................10

12.1 Limitations of size distribution determination from SAXS data ...............................................................10

12.2 A brief overview of methods ....................................................................................................................................................11

12.3 Goodness of fit: evaluating fits ...............................................................................................................................................11

12.4 Model-based data fitting ..............................................................................................................................................................11

12.5 Monte Carlo-based data fitting ...............................................................................................................................................12

12.6 Indirect Fourier transform (IFT) method .....................................................................................................................13

12.7 Expectation maximization (EM) method ......................................................................................................................13

13 Repeatability .........................................................................................................................................................................................................14

14 Documentation and test report..........................................................................................................................................................14

14.1 Test report ................................................................................................................................................................................................14

14.2 Technical records ...............................................................................................................................................................................14

Annex A (informative) General principle ......................................................................................................................................................16

Annex B (informative) System qualification ..............................................................................................................................................24

Bibliography .............................................................................................................................................................................................................................25

© ISO 2020 – All rights reserved iii
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ISO 17867:2020(E)
Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT), see www .iso .org/

iso/ foreword .html.

This document was prepared by Technical Committee ISO/TC 24, Particle characterization including

sieving, Subcommittee SC 4, Particle characterization.

This second edition cancels and replaces the first edition (ISO 17867:2015), which has been technically

revised. The main changes compared to the previous edition are as follows:

— inclusion of various methods for the extraction of particle size distribution by using the SAXS method;

— correction of technical terms.

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 2020 – All rights reserved
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ISO 17867:2020(E)
Introduction

This document deals with small-angle X-ray scattering (SAXS), which is performed for particle

size analysis in the 1 nm to 100 nm size range. Under certain conditions (narrow size distributions,

appropriate instrumental configuration, and idealised shape) the limit of 100 nm can be significantly

extended. In ideal circumstances, SAXS can determine the mean particle diameter and particle size

distribution, surface area, and sometimes particle shape in a reasonably rapid measurement time. User-

friendly commercial instruments are available worldwide from a number of 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 an International Standard

that allows users to obtain good interlaboratory agreement on the accuracy and reproducibility of the

technique.

SAXS can be applied to any hetero-phase system, in which the two or more phases have a different

electron density. In most cases, the electron density corresponds reasonably well to the mass density.

SAXS is sensitive to the squared electron density difference. For fixed volume fractions, it does not

matter whether the particles constitute the denser phase and the solvent (or matrix) is the less-dense

phase or vice versa. Thus, pore size distributions can be measured with SAXS in the same way as size

distributions of oil droplets in emulsions or solid particles in suspensions. Core-shell-nanoparticles can

also be investigated, but low density (e.g. organic) shells are not detected if the core has a significantly

higher density. To obtain the outer particle diameter including the shell, other methods should be used.

Although SAXS allows the determination of particle size, size distribution, surface area, and sometimes

particle shape in concentrated solutions, in powders and in bulk materials, this document is limited

to the description of particle sizes in 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 or the addition of salt can

be necessary.

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.

The shape of the particles can be assigned to a basic geometry: spheroid, disk, or cylinder. This does not

exclude more detailed information about the shape of the particle being obtained. However, the method

of calculation for more detailed shape analysis is very complex to be included in an International

Standard at this time. The sizes of irregularly shaped nanoparticles can be assessed by the radius of

gyration (R ) as obtained by classic Guinier analysis.

The size and size distribution of particles with basic shapes (sphere, disk, cylinder, core-shell, etc.) can

be determined from curve fitting for relatively narrow size distributions. The reliability of the method

of calculation for broader distributions depends on prior knowledge of the distribution.

This document assumes isotropically oriented nanoparticles of any shape in a test procedure. No

dimension of the nanoparticle shall be larger than defined by the scattering accessible to the specific

SAXS instrument. This generally limits the largest measurable particle size of the conventional

technique to 100 nm, although this limit can be significantly extended in samples with a very narrow

size distribution.

Small-angle neutron scattering is not described in this document but can be used without restriction

because the theory and application are similar.
A list of suitable references for further reading is given in the Bibliography.
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INTERNATIONAL STANDARD ISO 17867:2020(E)
Particle size analysis — Small angle X-ray scattering (SAXS)
1 Scope

This document specifies a method for the application of small-angle X-ray scattering (SAXS) to the

estimation of mean particle sizes in the 1 nm to 100 nm size range. It is applicable in dilute dispersions

where the interaction and scattering effects between the particles are negligible. This document

describes several data evaluation methods: the Guinier approximation, model-based data fitting, Monte-

Carlo–based data fitting, the indirect Fourier transform method and the expectation maximization

method. The most appropriate evaluation method is intended to be selected by the analyst and stated

clearly in the report. While the Guinier approximation only provides an estimate for the mean particle

diameter, the other methods also give insight in the particle 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 26824, Particle characterization of particulate systems — Vocabulary
ISO/TS 80004-2, Nanotechnologies — Vocabulary — Part 2: Nano-objects
3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 26824, ISO/TS 80004-2 and

the following apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at http:// 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 26824:2013, 1.1]
3.2
particle size

linear dimension of a particle 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 is reported as a linear dimension, e.g. as

the equivalent spherical diameter.
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ISO 17867:2020(E)

Note 2 to entry: Examples of size descriptors are those based at the opening of a sieve or a statistical diameter,

e.g. the Feret diameter, measured by image analysis.

Note 3 to entry: In ISO 9276-1:1998, the symbol x is used to denote the particle size. However, it is recognized that

the symbol d is also widely used to designate these values. Therefore the symbol x may be replaced by d.

3.3
radius of gyration
square root of the ratio of the moment of inertia to the particle mass

Note 1 to entry: Guinier radius (i.e. radius of gyration) is expressed in nanometres. Typical average radii are in

the range of 1 nm to 50 nm.
[SOURCE: ISO 26824:2013, 10.2]
4 Symbols
Symbol Description Unit
d Volume-squared-weighted mean particle diameter nm
d Number-weighted mean particle diameter nm
num
g (r) Intensity-weighted particle size distribution
int
g (r) Number-weighted particle size distribution
num
g (r) Volume-weighted particle size distribution
vol
I Primary beam intensity with sample
out
I Primary beam intensity without sample
I(q) Scattered intensity (or scattering intensity)
M Number of degrees of freedom in fitting
N Number of particles
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 by
q nm
q = 4π/λ sin(θ)
q Small angle resolution, minimum accessible q-value nm
min
q Maximum accessible q-value nm
max
r Particle radius nm
R Radius of gyration (Guinier radius, see A.4) nm
t Optimum sample thickness mm
T Transmission
V Volume of particle nm
2θ Scattering angle deg or rad
λ Wavelength of the incident X-rays in vacuum nm
μ Linear absorption coefficient mm
ρ(r) Electron density distribution nm
σ Standard deviation of size distribution nm
5 Principle of the method

When electromagnetic radiation passes through matter, a small fraction of the radiation may 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. When X-rays are used to probe a geometrically

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ISO 17867:2020(E)

ordered group of particles or molecules (“crystals”), the well-known X-ray diffraction pattern is obtained

at wide scattering angles, which can be used to characterize the unit cell and lattice constants of such

crystalline material. In the small-angle regime (typically 2θ < 5 °; wavelength dependent), information

on the particle or pore 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. This

is analogous to static light scattering and small-angle neutron scattering. A diagrammatic form of the

angular dependence of the X-ray scattered intensity of a titanium dioxide mixture (rutile and anatase)

is shown in Figure 1.
Key
X scattering angle 2θ (in degrees)
Y intensity
1 SAXS range
2 XRD range

Figure 1 — X-ray scattering diagram illustrating the small-angle SAXS region (left hand side) and

the wide-angle XRD (X-ray diffraction) region (right hand side) of a titanium dioxide powder

At low concentration, the small-angle scattering region contains information about the particle

morphology, which may be evaluated to extract either the particle size (distribution), or particle shape.

Only in very few cases is it possible to obtain both size and shape information. Different regions in

q are dominated by signals from a particular length range, and so can contain distinct (exploitable)

information aspects. In the low-q range, the Guinier approximation can be applied as an indicative

method to get an intensity weighted mean size, provided the particles are smaller than 2π/q . Model

min

fitting can be applied in the full range of q to compute a traceable particle size and size distribution

with associated uncertainties. Both methods can fail depending on data quality and particle properties.

At increased concentrations, i.e. those higher than typically one 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 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. The result of both measurements shall be

arithmetically averaged and the uncertainty enhanced by the variation. If dilution is not possible for

technical reasons, this shall be stated in the report. In particular, the Guinier approximation is highly

sensitive to concentration-induced scattering effects.
6 Apparatus and procedure
A diagrammatic form of a SAXS instrument is shown in Figure 2.
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ISO 17867:2020(E)
Key
X 2θ or q
Y scattered intensity
1 X-ray source
2 optics
3 collimation system
4 sample
2θ.

Figure 2 — Diagrammatic form of a SAXS instrument, consisting of X-ray source, optics,

collimation system, sample holder, beam stop, and X-ray detector

The SAXS set-up consists of X-ray source, optics, collimation system, sample holder, beam stop,

and detector. In order to extract meaningful information from the measurement, the following key

parameters define the capability of the system:

— q-range: q and q ; number of sampled points in the Guinier region for Guinier approximation;

min max
— detector sensitivity and system background noise.

Most available X-ray sources produce divergent beams which shall be collimated for SAXS measurements.

With laboratory X-ray sources, multilayer optics are commonly used but basic SAXS measurements can

also be achieved with slit collimation. The X-ray flux on the sample is generally higher when optics is

used. Furthermore, multilayer coated optics can be used to generate a monochromatic X-ray beam.

The greatest challenge in SAXS is to separate 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.
There are two main options to collimate an X-ray beam (see Figure 3).

— Point collimation systems have multiple pinholes or crossed slits that limit the shape of the X-ray

beam to a low divergence and a small dimension (typically, the beam spot on the sample is less

than 0.8 mm in diameter). The scattering is normally centro-symmetrically distributed around

the primary X-ray beam. For isotropic samples, the scattering pattern in the detection plane

perpendicular to the X-ray beam exhibits circular contour lines around the point of incidence of the

primary beam. The illuminated sample volume is smaller than in line-collimation. Point collimation

allows the study of isotropic and anisotropic systems.

— Line-collimation instruments confine the beam in one dimension so that the beam profile is a long

and narrow line. The beam dimensions can be adapted to accommodate a given sample geometry.

Typical dimensions are 20 mm × 0,3 mm. The illuminated sample volume is larger compared to

point-collimation and the scattered intensity at the same flux density is proportionally larger. If

the system is isotropic, the resulting smearing can be reverted using a deconvolution procedure,

albeit at the cost of magnified uncertainties of the observed intensities. The investigation of

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ISO 17867:2020(E)

anisotropic nanostructure with such line-collimated instruments is not as straightforward as for

point collimation.

In addition, both the point and line collimation systems can use either a parallel or focused beam (see

Figure 4).

The scattered radiation (containing the morphological information, as described in Clause 5) forms a

pattern that contains the information on the size and structure of the sample. This pattern is detected

typically by a 1-dimensional or 2-dimensional flat X-ray detector situated behind the sample and

perpendicular to the direction of the primary beam. Some multipurpose diffractometers that combine

SAXS and diffraction use a scanning point (0-dimensional) detector. Detector classes commonly used

include photon-counting and integration type detectors, although in practice, the least problematic/

distorted measurements come from high-dynamic range, direct-detection, photon-counting detectors.

Key
1 X-ray source
2 collimation system
3 sample
Figure 3 — Point and line collimation types used in SAXS
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ISO 17867:2020(E)
a) Parallel beam
b) Point (2D) or line (1D) focused beam
Key
1 X-ray source
2 mirror
3 sample
Figure 4 — Focused and parallel beam set-up
7 Preliminary procedures and instrument set-up

Wavelength calibration (see Annex B) 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 (Nickel for Cu Kα,

Zirconium for Mo Kα). Utilization of a calibration material, such as silver behenate, should form part of

a full system qualification and fit-for-purpose specification as noted in Annex B.

Only if, in addition to the mean particle diameter and the particle size distribution, the (absolute)

concentration or volume fraction of scatterers is to be determined, the intensity shall be scaled to

absolute units. For this purpose, a variety of auxiliary calibration materials are available, including

water and glassy carbon. Alternatively, some instruments can determine this directly by means

of calibrated detectors, or measurement of the unattenuated primary beam intensity on the SAXS

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.
8 Sample preparation

Sample preparation is simple and fast for SAXS measurements. The required sample volumes are small,

typically in a range of 5 μl to 50 μl for liquids and pastes, if copper radiation is used. Solid samples

2 2

require an area of (1 × 1) mm to (1 × 20) mm . The sample thickness is typically smaller than 1 mm and

can be tuned to optimize the scattering and limit the X-ray absorption, depending on the composition of

the sample.

Liquid samples are usually measured inside a thin-walled capillary, the diameter of which is typically

between 0,5 mm to 2 mm when the liquid primarily contains water or hydrocarbons. Solvents that

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ISO 17867:2020(E)

contain 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. Viscous samples can be measured better in a paste cell. It is strongly recommended to measure

liquid samples in a re-fillable or flow-through container, 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 containers are used for the two measurements).

Pastes, powders, and vacuum sensitive materials can be mounted into a sample holder with windows,

which shall be transparent to X-rays and exhibit little scattering themselves. Frequently used window

materials include polyimide films, mica or silicon nitride (see Reference [18]). Care should be taken

that the scattering from the window material does not affect the result of the measurement and can

be appropriately subtracted. For example, polyimide films exhibit a broad small-angle diffraction peak

in the vicinity of q approximately 0,7 nm , which shall be correctly subtracted in the background

subtraction procedure.

Solids can be clamped onto frames with or without additional window foils for protection against the

vacuum. The sample thickness shall be chosen in line with the respective absorption of the material

(see Reference [17]). The optimum thickness, t , is given by
t = (1)

where μ is the linear absorption coefficient of the material. The optimum specimen thickness

corresponds to a ratio of the primary beam intensity with an
...

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