ISO 17867:2015
(Main)Particle size analysis — Small-angle X-ray scattering
Particle size analysis — Small-angle X-ray scattering
Small-angle X-ray scattering (SAXS) is a well-established technique that allows structural information to be obtained about inhomogeneities in materials with a characteristic length from 1 nm to 100 nm. Under certain conditions (narrow size distributions, appropriate instrumental configuration, and idealised shape) the limit of 100 nm can be significantly extended. ISO 17687:2015 specifies a method for the application of SAXS to the estimation of mean particle sizes in dilute dispersions where the interaction between the particles is negligible. This International Standard allows two complementary data evaluation methods to be performed, model fitting and Guinier approximation. The most appropriate evaluation method shall be selected by the analyst and stated clearly in the report. SAXS is sensitive to electron density fluctuations. Therefore, particles in solution and pores in a matrix can be studied in same way.
Analyse granulométrique — Diffusion des rayons X aux petits angles
General Information
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Standards Content (Sample)
INTERNATIONAL ISO
STANDARD 17867
First edition
2015-05-01
Particle size analysis — Small-angle
X-ray scattering
Analyse granulométrique — Diffusion des rayons X aux petits angles
Reference number
ISO 17867:2015(E)
©
ISO 2015
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ISO 17867:2015(E)
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ISO 17867:2015(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Symbols and abbreviations . 1
4 Principle of the method . 2
5 Apparatus and procedure . 3
6 Preliminary procedures and instrument set-up . 5
7 Sample preparation . 5
8 Measurement procedure . 6
9 Calculation of the mean particle diameter . 7
9.1 General . 7
9.2 Guinier approximation . 8
9.3 Model fitting . 8
10 Repeatability . 9
11 Documentation and test report. 9
11.1 Test report . 9
11.2 Technical records .10
Annex A (informative) General principles .11
Annex B (informative) Working size range and resolution .20
Annex C (informative) System qualification .21
Bibliography .22
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ISO 17867:2015(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
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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 on the meaning of ISO specific terms and expressions related to conformity
assessment, as well as information about ISO’s adherence to the WTO principles in the Technical Barriers
to Trade (TBT), see the following URL: Foreword — Supplementary information.
The committee responsible for this document is ISO/TC 24, Particle characterization including sieving,
Subcommittee SC 4, Particle characterization.
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ISO 17867:2015(E)
Introduction
This International Standard deals with Small-Angle X-ray Scattering (SAXS), which is performed
for particle size analysis in the 1 nm to 100 nm size range. In ideal circumstances, it can provide an
estimate of particle size, average size and its 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 inter-laboratory 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.
The so-called ‘particle’ is always the phase with the smaller volume fraction. Because SAXS is sensitive
to the squared electron density difference, 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.
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 International Standard
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 has to be taken and a reduction of the concentration or the addition of salt
might 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.
g
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 International Standard 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 measureable 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 International Standard, 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 17867:2015(E)
Particle size analysis — Small-angle X-ray scattering
1 Scope
Small-angle X-ray scattering (SAXS) is a well-established technique that allows structural information
to be obtained about inhomogeneities in materials with a characteristic length from 1 nm to 100 nm.
Under certain conditions (narrow size distributions, appropriate instrumental configuration, and
idealised shape) the limit of 100 nm can be significantly extended. This International Standard specifies
a method for the application of SAXS to the estimation of mean particle sizes in dilute dispersions
where the interaction between the particles is negligible. This International Standard allows two
complementary data evaluation methods to be performed, model fitting and Guinier approximation.
The most appropriate evaluation method shall be selected by the analyst and stated clearly in the report.
SAXS is sensitive to electron density fluctuations. Therefore, particles in solution and pores in a matrix
can be studied in same way.
2 Normative references
The following documents, in whole or in part, are normatively referenced in this document and are
indispensable for its application. 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 9276-1, Representation of results of particle size analysis — Part 1: Graphical representation
ISO 9276-2, Representation of results of particle size analysis — Part 2: Calculation of average particle
sizes/diameters and moments from particle size distributions
ISO/TS 27687, Nanotechnologies — Terminology and definitions for nano-objects — Nanoparticle, nanofibre
and nanoplate
3 Symbols and abbreviations
Table 1 — Symbols
Symbol Name Unit
d
Volume-squared-weighted mean particle diameter nm
vs
d
Number-weighted mean particle diameter nm
num
I Primary beam intensity with sample
out
I Primary beam intensity without sample
in
I(q) Scattered intensity (or scattering intensity)
Momentum transfer or q-value, magnitude of the scattering vector given by
−1
q nm
q = (4π /λ)sinθ
r Particle radius nm
R Radius of gyration (Guinier radius, see A.4) nm
g
t Optimum sample thickness mm
o
T Transmission
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ISO 17867:2015(E)
Table 1 (continued)
Symbol Name Unit
3
V Volume of particle nm
λ Wavelength of the incident X-rays in vacuum nm
2θ Scattering angle deg or rad
−1
μ Linear absorption coefficient mm
σ Standard deviation of size distribution
4 Principle of the method
When electromagnetic radiation impinges on matter, a small fraction of the radiation is scattered. As a
function of the scattering angle or momentum transfer, q-value, the scattered radiation intensity profile
contains information that can be used to obtain various characteristics of the material. In particular,
when X-rays impinge on a geometrically ordered group of particles or molecules, this gives rise to
the well-known X-ray diffraction pattern at wide scattering angles which is used to characterize the
unit cell and lattice constants of the material. In the small-angle regime (typically 2θ < 5°; wavelength
dependent), information on the size of particles or pores within the material is available from the elastic
(no change in wavelength) scattering arising from the electron density contrast between the particles
and the medium in which they reside. This is analogous to static light 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.
Y
10
10
1
8
10
2
6
10
4
10
0,1 110 100
X
Key
1 SAXS range
2 XRD range
X scattering angle 2θ/deg
Y intensity
Figure 1 — X-ray scattering diagram illustrating the small-angle SAXS region (left hand side)
and the wide-angle XRD region (right hand side) of a titanium dioxide powder
At low concentration, the small-angle scattering region contains information about particle size, size
distribution, and particle shape, for which different ranges of q are evaluated. The Guinier approximation
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ISO 17867:2015(E)
can be applied in the low-q range to get an intensity weighted mean size, when the particles are smaller
than 2π/q . Model fitting can be applied in the full range of q to compute traceable particle size and
min
size distribution with associated uncertainties. Porod’s law can be applied to the high-q range to get an
indication of the particle shape. This last method does not provide particle size and is therefore outside
the scope of this International Standard. Note that all three 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 standard. In
practice, a concentration ladder may be explored in order 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 exclude 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 and the uncertainty shall be marked. Note that the radius of gyration is
more affected by concentration than model fitting.
5 Apparatus and procedure
A diagrammatic form of a SAXS instrument is shown in Figure 2.
Y
4
23
a
1
X
Key
1 X-ray source
2 optics
3 collimation system
4 sample
a 2θ
X 2θ or q
Y scattered intensity
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.
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ISO 17867:2015(E)
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 incoming primary beam from the scattered radiation
at small angles (around 0.1°). The direct beam should be 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 pinholes or crossed slits that shape the X-ray beam to a small
dimension (typically, the beam spot on the sample is less than 0,8 mm in diameter) that illuminates
the sample. The scattering is centro-symmetrically distributed around the primary X-ray beam.
For isotropic samples, the scattering pattern in the detection plane consists of circles around 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 dimension can be adjusted according to the type of sample for studies.
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 removed using deconvolution. The investigation
of anisotropic systems is not as straightforward as for point collimation.
In addition, the point and line collimation systems can use either a parallel or focused beam (see Figure 4).
The majority of the generated X-rays will simply transmit through the diluted sample without
interacting with the particles. The X-rays scattered by the particles form a scattering 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 detector. There are a number of types of detector routinely employed, for instance,
photon-counting and integration type detectors. The scattering pattern contains the information on the
structure of the sample.
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 17867:2015(E)
1
3
2
a) Point (2D) or line (1D) focused beam
1
3
2
b) Parallel beam
Key
1 X-ray source
2 mirror
3 sample
Figure 4 — Focused and parallel beam set-up
6 Preliminary procedures and instrument set-up
Wavelength calibration (see Annex C) can be performed before conducting an experiment and thus would
be classified as a preliminary procedure, but this is not routinely done in the laboratory. If characteristic
X-rays of copper are used, a nickel absorber can be used to check that Cu Kα radiation has been selected
correctly. Utilization of calibration materials, for example, silver behenate, should form part of a full
system qualification and fit-for-purpose specification as noted in Annex C.
7 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.
Liquid samples are usually measured inside a thin-walled capillary, the diameter of which is around 1
mm to 2 mm when the liquid primarily contains water or hydrocarbons. Solvents that contain heavy
atoms, for example, chlorine in chloroform, should be measured in smaller diameter capillaries as the
atoms strongly absorb the incident radiation. Viscous samples can be measured better in a paste cell.
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 foils. Care should be taken that the scattering from the window material
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ISO 17867:2015(E)
does not affect the result of the measurement. Polyimide films exhibit a broad small-angle diffraction
−1
peak in the vicinity of q approximately 0,7 nm , which has to be taken into account in data interpretation.
Solids can be clamped onto frames with or without additional window foils for protection against the
[14]
vacuum. The sample thickness shall be chosen in line with the respective absorption of the material.
The optimum thickness, t , is given by
o
t =1/μ (1)
o
where μ is the linear absorption coefficient of the material. The optimum specimen thickness corresponds
to a ratio of the primary beam intensity with and without sample, I and I , of:
out in
−−μt 1
II/%==ee ~37 (2)
outin
Thus, the ideal specimen will transmit about 37 % of the incident radiation, and the specimen thickness
can be adjusted accordingly to optimize transmission. Any sample treatment (for example, dilution,
sonication, or centrifugation) may affect the particle size distribution and should be described in the
analysis report.
8 Measurement procedure
Every SAXS particle-sizing experiment consists of at least two measurements using the same sample
holder and the same acquisition time:
a) sample scattering;
b) solvent/matrix scattering, the so-called blank or background experiment.
This is the minimum requirement for determining the scattering of the particles, which is the difference
between the two scattering measurements. A typical example for this procedure is given in Figure 5. Care
has to 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/matrix material and efficiency variation over the detector shall be taken into account.
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ISO 17867:2015(E)
Key
1 solvent
2 particle scattering
3 particle dispersion
−1
X q/nm
Y scattered intensity/(q)
Figure 5 — Typical SAXS profiles of a particle dispersion,
the solvent and the difference (the corrected signal only due to particle scattering)
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.
9 Calculation of the mean particle diameter
9.1 General
After background subtraction and desmearing (if required, see Annex A), the mean particle diameter
can be calculated according to two different approaches.
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ISO 17867:2015(E)
9.2 Guinier approximation
2
For the Guinier approximation (explained in detail in A.3), ln(I) is plotted as a function of q (Guinier
plot). As the scattered intensity at very small angles is approximated by a Gaussian function
1
22
Iq =−IRexp q (3)
()
0 g
3
which can be transformed to
1
22
lnIq =ln IR− q (4)
()
0 g
3
A straight line can fit the data in the Guinier region which is typically up to qR around 1. The slope is
g
1
2
then equal to − R
g
3
For monodisperse homogeneous spherical particles, the volume-squared-weighted mean particle
diameter can be calculated from R according to:
g
5
dR=2 (5)
vs g
3
According to ISO 9276-2 and Reference [3], d corresponds to D and x .
vs 8,6 26,
9.3 Model fitting
For model fitting, the full range of q can be fitted by a model function for a polydisperse ensemble of
particles according to:
∞
Iq =NP qr, gr dr+c (6)
() () ()
∫
0
where
N is a scaling factor (including the number of particles, the electron density difference, the
intensity of the primary beam etc.);
P(q,r) is the form factor;
g(r) represents the distribution and c a constant background.
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For homogenous spheres, the form factor is given by:
2
4π
Pq(),sr =− ()inqr qrcosqr (7)
3
q
The most common distributions are lognormal and Gaussian: A Gaussian size distribution is described by:
2 2
rd− /2 ρ −d /2
() ∞ ()
numnum
gr =exp /exp dρ (8)
()
∫
2 2
0
2σ 2σ
and a lognormal distribution can be written as:
2
1
lnrd−ln
ln
2
1
gr =−exp (9)
()
2
πσr
2 2σ
ln ln
where the mean diameter d can be transformed to the number-weighted mean particle diameter of a
ln
Gaussian distribution d according to:
num
2
dd= exp/σ 2 (10)
num ln ()ln
N, c, the standard deviation of the size distribution (σ or σ ) and the mean particle diameter ( d or
ln
num
d ) are the fit parameters. From the determined size distribution, also the volume-weighted and
ln
intensity-weighted mean particle diameters can be calculated.
According to ISO 9276-2, d corresponds to D and x .
num 1,0 1,0
Information on the particle size can also be obtained from other evaluation methods in real space or
Fourier space as explained in A.5.
10 Repeatability
Repeated measurements of the same sample can indicate if the material is changing during the duration
of the experiment and therefore can be an indicator of degradation under the X-ray beam. Additionally,
sample-to-sample measurements will indicate homogeneity or heterogeneity of the material. Sample-
to-sample heterogeneity and instability of a sample/material over time can only be detected if the
heterogeneity and instability create effects that can be distinguished beyond the method repeatability.
The method repeatability shall be established for each individual instrument in a validation study on a
suitable, calibration standard. A frequently used material is silver behenate.
11 Documentation and test report
11.1 Test report
Test reports should be prepared in line with ISO 9276-1 and ISO 9276-2. The sections “identification
of the method used”, “test results”, and “description of the test item” in the scope of this International
Standard contain the following information:
a) reference to this International Standard;
b) the mean particle diameter d and its uncertainty, including a clear statement whether this represents
a number, volume, or intensity weighted mean. In the absence of a full uncertainty evaluation, the
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ISO 17867:2015(E)
standard deviation from several repeated measurements should be provided as estimate of the
repeatability. ISO/IEC Guide 98-3 can assist here, but expert judgment may have to be employed.
The standard error of the slope, u , can be obtained as a linear regression analysis output from ma
...
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