ISO/TS 14411-1:2017

Preparation of particulate reference materials - Part 1: Polydisperse material based on picket fence of monodisperse spherical particles

ISO/TR 14411-1:2017 describes the preparation of polydisperse spherical particles based on a picket fence of quasi-monodisperse reference materials, the characterization of its monodisperse components with acceptable uncertainty and the estimation of the uncertainty of the mixture of these particles. This type of material is normally suitable for all particle characterization methods within the appropriate limits of the techniques. An example of using these reference materials in a reliability calculation for a mass-based cumulative size distribution is provided. ISO/TR 14411-1:2017 itself to the technical specificities of preparation beyond the general requirements for certified and non-certified reference materials as described in ISO Guide 30, ISO Guide 31, ISO Guide 35 and ISO 17034.

Préparation des matériaux de référence à l'état particulaire — Partie 1: Matériaux polydispersés composés d'un ensemble de particules sphériques monodispersées

General Information

• Status: Published
• Publication Date: 09-May-2017
• ICS: 19.120 - Particle size analysis. Sieving
• Technical Committee: ISO/TC 24/SC 4 - Particle characterization
• Drafting Committee: ISO/TC 24/SC 4/WG 11 - Sample preparation and reference materials
• Current Stage: 9093 - International Standard confirmed
• Start Date: 21-Nov-2024
• Completion Date: 13-Dec-2025

Overview

ISO/TS 14411-1:2017 specifies how to prepare polydisperse particulate reference materials built from a “picket fence” of quasi‑monodisperse spherical particles. The Technical Specification sets out the technical specifics for selecting, characterizing and combining monodisperse fractions and for estimating the uncertainty of the resulting mixtures. These picket‑fence materials are intended to support comparability across particle sizing methods within the applicable limits of each technique.

Key topics and requirements

• Material requirements for individual monodisperse fractions (sphericity, homogeneity and general properties).
• Characterization of monodisperse components:
  • Particle size distribution (number- and mass-/volume-based)
  • Aspect ratio (shape considerations)
  • Density (pycnometry, hydrostatic balance references)
  • Refractive index
• Preparation procedures for picket‑fence distributions:
  • Preparation from dry powders and from suspensions
  • Preparation of individual “pickets” and controlled mixing to form polydisperse samples
  • Determination of particle mass fractions and sample handling considerations
• Uncertainty estimation for mixtures:
  • Uncertainty contributions for count-based and mass-based size distributions
  • Effects of limited particle counts, mixture fractions and sampling
  • Uncertainty in microscopic measurements (including neighbouring particles)
  • Combining multiple uncertainty components into a combined uncertainty
• Informative annexes providing worked examples such as reliability calculations for mass-based cumulative size distribution and multiple‑picket cases.

Applications and who uses it

ISO/TS 14411-1:2017 is directly relevant for:

• Reference material producers preparing certified and non‑certified particulate RMs
• Metrology institutes and calibration laboratories establishing traceability for particle size
• Analytical laboratories validating and verifying particle sizing methods (laser diffraction, microscopy, particle counters, sedimentation techniques)
• Instrument manufacturers seeking benchmark materials to demonstrate method performance and comparability
• Regulatory and quality assurance teams that require robust uncertainty budgets and inter‑method comparability for particle characterization.

Practical uses include method validation, interlaboratory comparisons, instrument performance verification and developing measurement uncertainty estimates for particle size distributions.

Related standards

This TS complements general guidance for reference materials and certification such as ISO Guide 30, ISO Guide 31, ISO Guide 35 and ISO 17034 and references particle‑specific standards (e.g., density and aspect ratio definitions). For full preparation protocols, characterization methods and worked uncertainty examples refer to the full ISO/TS 14411-1:2017 document.

Standards Content (Sample)

TECHNICAL ISO/TS
SPECIFICATION 14411-1
First edition
2017-05
Preparation of particulate reference
materials —
Part 1:
Polydisperse material based on picket
fence of monodisperse spherical
particles
Préparation des matériaux de référence à l’état particulaire —
Partie 1: Matériaux polydispersés composés d’un ensemble de
particules sphériques monodispersées
Reference number
©
ISO 2017
© ISO 2017, Published in Switzerland
All rights reserved. Unless otherwise specified, 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
Ch. de Blandonnet 8 • CP 401
CH-1214 Vernier, Geneva, Switzerland
Tel. [phone]
Fax [phone]
[e-mail]
www.iso.org
ii © ISO 2017 – All rights reserved

Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions and symbols . 1
3.1 Terms and definitions . 1
3.2 Symbols . 2
4 Material requirements for preparing the individual monodisperse fractions .3
4.1 General description . 3
4.2 Requirements on the general properties of the material for individual pickets . 3
5 Characterization of the individual monodisperse fractions . 4
5.1 Particle size distribution . 4
5.2 Aspect ratio . 5
5.3 Density . 5
5.4 Refractive index . 5
6 Preparation of picket-fence distributions . 5
6.1 General . 5
6.2 Preparation of individual pickets . 6
6.2.1 General. 6
6.2.2 Preparation of suspensions from dry powders . 6
6.2.3 Determination of the particle mass fraction of suspensions . 6
6.3 Preparation of a picket fence distribution. 6
6.3.1 General. 6
6.3.2 Preparation from dry powders . 6
6.3.3 Preparation from suspensions . 7
7 Estimation of uncertainties . 7
7.1 General . 7
7.2 Uncertainty of a volume-based size distribution due to limited number of
particles counted . 7
7.3 Uncertainty of a number-based size distribution . 8
7.4 Picket-fence distributions composed of more than two kinds of quasi-
monodisperse particles . 9
7.5 Uncertainty of a count base size distribution due to various number fraction .10
7.6 Uncertainty of a mass base size distribution due to various mass fractions .10
7.7 Uncertainty estimation based on the data before or after the mixing process .10
7.8 Uncertainty due to microscopic scale measurement .12
7.9 Uncertainty due to surrounding particles in microscopic measurement .12
7.10 Other uncertainty contributions .12
7.11 Combined uncertainty .12
Annex A (informative) Picket-fence distributions composed of more than two kinds of
quasi-monodisperse particles .13
Annex B (informative) Example of reliability calculation for a mass-based cumulative size
distribution transformed from the number-based size distribution .16
Annex C (informative) Example of uncertainty estimation due to mixture fraction and
sample size .19
Annex D (informative) Uncertainty estimation of various cases .21
Bibliography .28
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 on 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: w w w . i s o .org/ iso/ foreword .html.
This document was prepared by Technical Committee TC 24, Particle characterization including sieving,
Subcommittee SC 4, Particle characterization.
iv © ISO 2017 – All rights reserved

Introduction
The measurement of the particle size distribution can be accomplished by a number of techniques
which measure some 1-D characteristic of the particle and usually equate this to an equivalent size
assuming ideal shapes (usually spherical). Thus, these techniques usually require or assume knowledge
of some other constant in order to calculate the particle size distribution. Each of these techniques
measures different properties which makes the equivalent particle size a method-defined measurand.
Comparability of results therefore requires application of the same methods, which in turn requires
standardization.
This unsatisfactory situation of fundamental lack of comparability could be improved by a better
understanding of the effects influencing the various methods. Since the sample material represents the
link between the different methods, it is of central importance that it should meet as many physical
assumptions of the considered methods as possible. A feasible approach is mixing known amounts of
spherical, monodisperse particle fractions to create a polydisperse mixture (“picket fence distribution”).
The individual particles should be spherical, as many sizing methods assume the particles to be
spherical. Using particles that are in fact spherical fulfils this assumption, so the results of the various
methods should be the same as far as the particle shape is concerned. A further advantage of spherical
particles is that their size can be described by a single parameter only, the particle diameter.
The individual fractions of the mixture need to be monodisperse, as only then it is possible to trace the
particle diameter back to the standard meter with an acceptable uncertainty and to get mixtures of
theoretically known particle size distributions in the end.
These materials should be used as follows.
The monodisperse particle fractions can be used to demonstrate equivalence of results with
these ideal particles. If a method gives deviating results, the method is not yet fully understood
and further investigation of the deviation is needed. The polydisperse mixtures can be used to
challenge measurement methods to see what the output is. Final outcome should be a comprehensive
understanding of the methods including particle dispersion, particle transport, physical principle and
evaluation leading to better comparability of results. The approach described in this document is based
on Reference [22] and Reference [23].
A second approach is developing a theoretical framework for more accurate measurement of particle
size distributions. Also, this approach is fundamentally limited to spherical particles of equal density,
to be applicable to different methods.
This document describes preparation protocols of picket fence distributions of spherical, quasi-
monodisperse particulate reference materials.
TECHNICAL SPECIFICATION ISO/TS 14411-1:2017(E)
Preparation of particulate reference materials —
Part 1:
Polydisperse material based on picket fence of
monodisperse spherical particles
1 Scope
This document describes the preparation of polydisperse spherical particles based on a picket fence
of quasi-monodisperse reference materials, the characterization of its monodisperse components with
acceptable uncertainty and the estimation of the uncertainty of the mixture of these particles. This
type of material is normally suitable for all particle characterization methods within the appropriate
limits of the techniques. An example of using these reference materials in a reliability calculation for a
mass-based cumulative size distribution is provided.
This document limits itself to the technical specificities of preparation beyond the general requirements
for certified and non-certified reference materials as described in ISO Guide 30, ISO Guide 31,
ISO Guide 35 and ISO 17034.
2 Normative references
There are no normative references in this document.
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— IEC Electropedia: available at http:// www .electropedia .org/
— ISO Online browsing platform: available at http:// www .iso .org/ obp
3.1.1
aspect ratio
ratio of minimum Feret diameter to the maximum Feret diameter of a particle
[SOURCE: ISO 26824:2013, 4.5, modified]
3.1.2
pycnometry
method wherein particle density is obtained from the measured mass of sample with a given
calibrated volume
[SOURCE: ISO 26824:2013, 2.4]
3.1.3
apparent particle density
particle mass in the dry status divided by the volume it would occupy including all pores, closed or
open, and surface fissures
[SOURCE: ISO 13317-4:2014, 3.1]
3.1.4
hydrostatic balance
method to measure particle density based on particle dynamic sedimentation velocity with known
fluid density and viscosity condition
3.1.5
reference material
material, sufficiently homogeneous and stable with respect to one or more specified properties, which
has been established to be fit for its intended use in a measurement process
[SOURCE: ISO Guide 30:2015]
3.1.6
certified reference material
reference material (3.1.5) characterized by a metrologically valid procedure for one or more specified
properties, accompanied by a certificate that provides the value of the specified property, its associated
uncertainty, and a statement of metrological traceability
[SOURCE: ISO Guide 30:2015]
3.1.7
nominal value
designated diameter in terms of a target value in a given specification
Note 1 to entry: The nominal value is the target diameter for an individual picket as calculated from the upper
and lower size of the picket fence distribution (3.1.8), the number of pickets and the requirement of equal spacing
of pickets on a lognormal scale. Actual values may differ from the nominal ones due to the availability of suitable
material
3.1.8
picket fence distribution
mixture of several monodisperse particle fractions (pickets)
3.2 Symbols
Derived
Symbol Quantity Unit
unit
α Particle mass fraction for the suspension of picket i kg/kg mg/kg
i
(0)
α Particle number fraction for the suspension of picket i — —
i
Δx Particle size interval in size range i m μm
i
Uncertainty of the parameter given in the index
δ — —
NOTE: In other fields of measurement science and ISO/
IEC Guide 98–3 (GUM), the symbol u is used instead.
δx Uncertainty of the size x m μm
i i
g Parameter defined by Formula (A.1) — —
M Parameter used in Formula (A.5) — —
m Mass of the vessel for the dry mass determination kg mg
b
m Dry mass (vessel and particles) in the dry mass determination kg mg
d
Mass of suspension i used for the preparation of the pick-
m kg mg
i
et-fence distribution
2 © ISO 2017 – All rights reserved

Derived
Symbol Quantity Unit
unit
m Mass of solvent of picket i kg mg
l,i
m Mass of particles of picket i in suspension kg mg
p,i
m Mass of suspension used for the dry mass determination kg mg
s
Mass of the particles of picket i in the final picket fence dis-
m , kg mg
x i
tribution
N, N Total number of particles and particle number of picket i — —
i
n Number of particles in size range i — —
i
n Total number of pickets — —
picket
p Total number of uncertainty factors — —
−1 −1
q (x) Density distribution by number m μm
−1 −1
q (x) Density distribution by volume or mass m μm
Q (x) Cumulative distribution by number — —
True cumulative distribution by number and mass with
Q *,Q * — —
0,i 3,i
logarithmic abscissa
Q (x) Cumulative distribution by volume or mass — —
−3 −3
ρ Particle density of picket i kg m g cm
i
s Standard deviation of the particle size distribution m μm
s Geometric standard deviation — —
g
Parameter used in Formula (6) to give confidence level,
u = 1,96 for 95 % probability reliability
u — —
NOTE  This corresponds to the coverage factor k in ISO/
IEC Guide 98–3 (GUM).
x x Particle diameter and particle diameter in size range i m μm
, i
x Diameter of the smallest picket m μm
l
x Diameter of the largest picket m μm
u
x
50,,0 i
Median diameter of particle i based on number and mass m μm
x
50,,3 i
Median particle size of cumulative volume or mass distribu-
x m μm
50,3
tion
Most reliable median particle size of a cumulative volume or
x* m μm
50,3
mass distribution with logarithmic abscissa
4 Material requirements for preparing the individual monodisperse fractions
4.1 General description
The material of the individual pickets shall be suitable for particle size measurement using image
analysis methods within dry or aqueous environment.
4.2 Requirements on the general properties of the material for individual pickets
The material of the individual pickets shall meet the following requirements.
a) The particles shall be spherical without significant macroscopic concavities, outgrowths or pores.
The aspect ratio of all particles shall exceed a value of 0,95. A typical mean aspect ratio should be
0,97. Alternatively, the ellipse ratio shall exceed 0,95, a typical value should be 0,97 or above.
b) When dispersed in pure water, no colour bleeding is allowed. The optical homogeneity of the
material is very important to be as uniform as possible. This applies for the particles within one
monodisperse fraction, as well as for a comparison of the particles of two different monodisperse
fractions.
c) The particle surface should be smooth without any contaminations or adhesions.
d) The apparent density of the material has to exceed the density of the dispersing liquid for the
particles not to float in wet applications. Furthermore, the apparent density should not be too
high for avoiding sedimentation effects. Therefore, a value within the range above 1 000 kg/m
and smaller than 2 500 kg/m seems to be optimal for aqueous applications. Particles of higher
densities can be used if a liquid with higher density or viscosity is used.
The apparent density of the material is important to be as constant as possible for different particle
sizes. Variations of ±0,5 % with respect to the mean value of the apparent density may be accepted.
e) The material should not contain any kind of fragmented particles or coarse outliers, e.g.
agglomerates. Any other material coming in contact with the particles should not be dyed by
adhering dust or abrasion.
f) The particles shall have a high chemical stability and be non-soluble in dispersant media.
g) The material should be easily dispersible in the chosen liquid. No particle agglomerates or
flocculation should be detectable after dispersion. It is allowed to support the particle dispersion
using dispersing agents or ultrasound.
h) The particles should not be disrupted by ultrasound pressure in dispersant media. The mechanical
strength should be as high as possible since the material should be able to withstand a typical dry
dispersion procedure without getting crushed. Nevertheless, it is not possible to define a concrete
value since there are several different dry dispersion procedures not allowing for a reliable
theoretical calculation of stress parameters.
i) The particles should not agglomerate under normal environmental conditions. Their electrostatic
behaviour should allow for using them, e.g. on a vibratory chute without adhering to the chute itself.
j) The material should provide a shelf life of at least two years after production without appreciably
changing its physical properties. All important storage conditions have to be known, e.g. necessary
UV-protection/light-protection.
k) The swelling of the material suspended in pure dispersant media should be as low as possible. In
any case, it should not exceed a value of 0,8 % referred to the particle diameter in dry condition.
The swelling behaviour shall be specified in the sample preparation procedure
l) The size of the particle-liquid interface in dispersion should be negligible compared to the particle
diameter.
5 Characterization of the individual monodisperse fractions
5.1 Particle size distribution
Particle size should be determined by a method that provides traceable results. The requirements for
the individual methods are given below.
— Results shall be traceable to the International System of Units (SI) either by using CRMs with
traceable reference values or by being calibrationless.
— The methods shall be validated in a way that allows estimation of a measurement uncertainty.
— Uncertainty estimates for the various size fractions are available.
4 © ISO 2017 – All rights reserved

— All results and characteristic values have to be given in terms of a volume-based particle size
distribution, Q (x).
It should be ensured that the measured particle size distributions do not overlap. This is achieved if
each distribution meets the following requirements:
a) The distribution width given in terms of the ratio x /x should be 1,12 or smaller.
90 10
b) The actual mass median particle diameter of each mono-disperse fraction should not deviate by
more than 4 % from the nominal diameter calculated from the lognormal distribution.
c) The uncertainty of the actual mass median particle diameter, calculated from traceable results
from a suitable optical method, x , should be in the range of 0,99 x * to 1,01 x * with 95 %
50,3 50,3 50,3
reliability where x * is the most reliable mass median diameter. Larger uncertainties will result
50,3
in larger uncertainty for the final distribution.
It is possible to compensate for not meeting one of the above criteria by setting stricter limits on others.
For example, a higher ratio x /x is permissible if the actual mean particle diameters deviate less
90 10
than 4 % from the nominal ones. Regardless of failing to meet an individual requirement, the basic
requirement of non-overlapping distributions shall be met.
5.2 Aspect ratio
The aspect ratio should be measured by a suitable optical method measuring at least 10 000 particles
by random sampling. Fewer particles would not allow demonstrating fulfilment of the criteria set for
the aspect ratio.
5.3 Density
The particle density should be measured by pycnometry or hydrostatic balance.
5.4 Refractive index
The refractive index shall be measured by any suitable method, e.g. by the liquid immersion method.
6 Preparation of picket-fence distributions
6.1 General
A picket-fence distribution should include at least one complete decade in particle size.
A picket-fence distribution should contain an uneven number of mono-disperse fractions and not less
than 7 different mono-disperse particle fractions within a range of one decade of particle size. All
fractions should be equally spaced on a logarithmic scale.
The nominal diameter x of picket i is calculated from the lower diameter, x , the upper diameter, x , and
i l u
the total number of pickets, n , using Formula (1):
picket
loglxx− og
() ()
ul
i−1
()
n −1
picket
x =⋅x 10 (1)
i l
All pickets should consist of the same material to minimize differences in density or optical properties.
6.2 Preparation of individual pickets
6.2.1 General
In most cases, it will be impossible to weigh the particles as dry powders due to the intrinsic uncertainty
of weighing. In addition, many particles are distributed as suspensions, so drying and weighing
increases the risk of agglomeration.
NOTE The uncertainty of the balance alone for weighing 1 mg with an analytical balance (display 0,000 00 g)
is about 15 %. Using a microbalance, this uncertainty is typically reduced to less than 0,5 %, but static influences
(e.g. incomplete transfer from the weighing boat to the final vessel) increase this uncertainty.
If direct weighing is possible, 6.2.2 and 6.2.3 can be skipped. For particles available as suspensions,
6.2.2 can be skipped.
6.2.2 Preparation of suspensions from dry powders
Weigh an appropriate amount of the dry powder (m ) into a known amount (m ) of the chosen solvent
p,i l,i
and homogenize according to the instruction of the particle producer. The particle mass fraction of
picket i is calculated as Formula (2):
m
pi,
α = (2)
i
mm+
pi,,li
6.2.3 Determination of the particle mass fraction of suspensions
For particles available as suspensions, the particle mass fraction shall be determined. Follow the
following steps.
a) Clean and dry a vessel and determine its mass, m .
b
b) Weigh an appropriate amount of the suspension (m ) into the vessel. The amount taken should not
s
contain less than 500 mg of solids.
c) Slowly evaporate the solvent to achieve constant mass (m ). The final drying temperature should
d
be high enough to evaporate all solvent, but should not cause shrinkage of particles. If in doubt,
contact the provider of the suspension for appropriate drying conditions. Constant mass is achieved
when subsequent weighings differ by less than 1 mg.
d) Determine the particle mass fraction of picket i in the suspension as given in Formula (3):
mm−
db
α = (3)
i
mm−
sb
6.3 Preparation of a picket fence distribution
6.3.1 General
Picket-fence distributions shall be prepared as “one-shot” materials directly from gravimetric weighing
of the individual pickets to avoid errors from subsampling of a homogenized sample. The uncertainties
of the weighing should be less than 0,1 % of the weighed mass.
6.3.2 Preparation from dry powders
Weigh equal masses of the dry powders for each picket, m , into a vessel and homogenize.
x,i
NOTE This is the simplest approach for the preparation of a picket fence distribution, but the accuracy of
weighing sets a limit to the use of this approach.
6 © ISO 2017 – All rights reserved

6.3.3 Preparation from suspensions
Prepare a picket fence distribution from i individual suspension as follows.
a) Determine the required mass, m , of each suspension so that α · m is constant for all pickets.
i i i
b) Homogenize each suspension and weigh the amount calculated in a) into a vessel. Avoid weighing
less than 0,5 g to minimize weighing errors.
c) The mass of the particles of picket i (m ) is calculated as given in Formula (4):
x,i
mm=⋅α (4)
xi, ii
The combined influence of the subsampling process can be assessed by preparing several suspensions
from the same set of pickets and subject them to measurement.
7 Estimation of uncertainties
7.1 General
In order to represent particle size distribution of the reference material, the uncertainty of the picket-
fence distribution shall be indicated. The following uncertainty contributions shall be included in the
combined uncertainty of the particle size distribution.
7.2 Uncertainty of a volume-based size distribution due to limited number of
particles counted
The detailed derivation of the formulae below is given in D.1.
For the general case, the maximum uncertainty of the cumulative distribution for the size range i is as
given in Formula (5) to Formula (9):
∂Q ∂Q ∂Q
3,ii3, 3,i
δδQ ≤ Q + δδQ ++ QQ=δ (5)
3,i 01, 02, 0,n 33,iN,
∂Q ∂Q ∂Q
01, 02, 0,n
3 3
xx− gg−
()
∂Q ()jj+1 ni
3,i
i>j   = (6)
∂Q
g
0,j
n
33 3
xg −−xx g
()
∂Q jn jj+1 i
3,i
i = j   = (7)
∂Q
0,j g
n
3 3
0−−xx g
∂Q ()jj+1 i
3,i
i < j   = (8)
∂Q
g
0,j
n
0−xg
∂Q
ji
3,i
i < j, j = n   = (9)
∂Q
0,j g
n
By use of Tschebyscheff theory, the uncertainty at 95 % confidence (= 5 % uncertainty) for the general
distribution is determined by References [24] and [25]. The relation confidence reliability level and
parameter u for general and normal distributions is shown in Figure 1. The parameter value of u is 4,47
for general distribution. The derivation of the 4,47 for the general distribution is indicated in Annex D.
QQ1−
()
0,ii0,
δQ =4,47 (10)
0,,iN
N
The uncertainty δ x for the cumulative mass distribution in size range i is calculated by Formula (11):
i
δ Q
3,,iN
δ x = (11)
iN,
q
3,i
By use of Formulae (D.9), (D.10), (D.18) and (D.19), the uncertainty of the particle diameter for
cumulative mass distribution is fully determined.
Key
1 normal distribution
2 general distribution
P reliability level, expressed in per cent
u parameter, dimensionless
Figure 1 — Relation between reliability level and parameter u
7.3 Uncertainty of a number-based size distribution
The uncertainty δ x for the cumulative number distribution in size range i is calculated by
i
Formula (12):
δ Q
0,,iN
δ x = (12)
iN,
q
0,i
where δQ is calculated by Formula (D.18).
0,i,N
8 © ISO 2017 – All rights reserved

7.4 Picket-fence distributions composed of more than two kinds of quasi-
monodisperse particles
A picket-fence distribution composed of three-kinds of quasi-monodisperse particles is indicated in
Annex A.
The cumulative distribution composed of m kinds of quasi-monodisperse particles is shown in Figure 2.
Key
x particle diameter, expressed in micrometers
Q (x) mass base cumulative distribution, dimensionless
q mass base distribution density, expressed in reciprocal of micrometers
3,k
α mass fraction of particles included in picket k, dimensionless
k
Figure 2 — Particle size distribution composed of more than two kinds of quasi-monodisperse
particles
The density distribution by mass is given as Formula (13):
m
qx()= α qx() (13)
33∑ jj,
j=1
where α is the mass fraction of the sample j.
j
For the cumulative distribution by mass, the size x is in the size range of sample k, Formula (14) is
satisfied.
x
Qx()=+αα ++ αα+ qx()dx (14)
31 21kk− 3,k
∫
x
k,min
7.5 Uncertainty of a count base size distribution due to various number fraction
The number fraction of i kind of particles is calculated as given in Formula (15):
N
()
i
α = (15)
i
NN++
()
1 m
The symbol N indicates particle number of picket i.
i
The uncertainty due to m kinds of number fraction is calculated by Formula (16):
2 22
   
∂QQ∂
0 ()0 0 ()0
δ Q =  δα  ++  δα  (16)
0,,i α 1 m
()0 ()0
   
∂α ∂α
1 m
   
7.6 Uncertainty of a mass base size distribution due to various mass fractions
The mass fraction of i kinds of particles is calculated as Formula (17):
m
i
α = (17)
i
mm++
()
1 m
The symbol m indicates particle mass of picket i.
i
The uncertainty due to m kinds of mass fraction is calculated by Formula (18):
2 2
 ∂QQ  ∂ 
3 3
δ Q = δα ++ δα (18)
   
3,,i α 1 m
∂α ∂α
 1   m 
The calculation method of the terms in Formula (16) and Formula (17) is shown in D.2 and D.3.
Examples of uncertainty estimation to prepare sample preparation are shown in Annex C.
7.7 Uncertainty estimation based on the data before or after the mixing process
7.7.1 General
The uncertainty of m kinds of quasi-monodisperse particles can be estimated by use of the data before or
after the mixing process. Both methods are acceptable in calculating the uncertainty. For the uncertainty
estimation based on the data after mixing process, it is not necessary to include the uncertainty due to
the mixture fraction, but the total sample size should be large enough. For the uncertainty estimation
based on the data before mixing process, it is necessary to include the uncertainty contributions due
to the mixture and the sample size of each size distribution. The calculation method of the uncertainty
based on the data before mixing process is shown in D.2 and D.3.
7.7.2 Uncertainty of count and mass base cumulative distribution based on the data after the
mixing process
Formula (19) and Formula (20) are used to calculate the uncertainty. In this case, the particle size
distribution based on the mixed particles is examined.
δδQQ= + δ Q (19)
()
()
00,,iN ad
10 © ISO 2017 – All rights reserved

δδQQ= + δ Q (20)
() ()
33,,iN ad
The values of δQ due to total sample size N is calculated by Formula (D.18) and δQ is an additional
0,i,N ad
uncertainty contribution. The value of δQ is calculated by Formula (21):
3,i,N
∂Q ∂Q ∂Q
3,ii3, 3,i
δδQ = Q + δδQ ++ Q (21)
3,i,N 01, 02, 0,n
∂Q ∂Q ∂Q
01, 02, 0,n
The uncertainty of particle size is calculated by Formula (22) and Formula (23):
δ Q
δx = (22)
q
0,i
δ Q
δx = (23)
q
3,i
7.7.3 Uncertainty of count and mass base cumulative distribution based on the data before the
mixing process
In this case, the data of particle size distribution before the mixing process is examined. The uncertainty
due to mixture fraction should be considered.
Formula (24) and Formula (25) are used to calculate the uncertainty:
2 2
δδQQ= + δδQQ+ (24)
()
() ()
00,,Ni0 ,aα d
2 2
δδQQ= + δδQQ+() (25)
() ()
33,,Ni3 ,aα d
The term δQ indicates the uncertainty of count base cumulative distribution and the value is
0,N
calculated by the size distribution and number fraction of each picket. The term δQ indicates
0,i,α
uncertainty due to number fraction of each picket.
2 22
   
∂QQ∂
0 ()0 0 ()0
   
δQ = δα ++ δα (26)
0,,i α 1 m
()0 ()0
   
∂α ∂α
1 m
   
2 2
()0 ()0
   
∂α ∂α
()0 ii
δα =  δN  ++  δN  (27)
i 1 m
   
∂N ∂N
1 m
   
The uncertainty of the number of particles of picket i depends on the uncertainty of the particle diameter
of picket i, the uncertainty of mass of picket i and the uncertainty of the density of the particles of picket i.
91 1
2 2 2
δδNN=⋅ x ++δm δρ (28)
ii i i i
2 2 2
x m ρ
i i i
The detailed derivation in Formula (28) is shown in D.4.
The each term of δQ is determined by use of Formula (27) and Formula (28). The term δQ calculated
0,i,α 3,i,α
by Formula (18) is uncertainty due to mass fraction. For the uncertainty of mass base cumulative
distribution, the data of mass base size distribution and mass fraction of each picket are used.
The detailed calculation method is shown in D.3.
In this case, the uncertainty of particle size for count and mass base is calculated by Formula (29) and
Formula (30):
δ Q
δx = (30)
q
0,i
δ Q
δx = (31)
q
3,i
7.8 Uncertainty due to microscopic scale measurement
In order to obtain particle size by microscopic method, a calibrated scale with values traceable
to the International System of Units shall be used. The uncertainty due to the microscopic scale
measurement changes with magnification factor and position in a screen. The corresponding
deviation is indicated by δ .
7.9 Uncertainty due to surrounding particles in microscopic measurement
It is important to know the uncertainty due to surrounding particles in a screen in the microscopic
measurement. This factor depends on magnification factor and particle diameter. The corresponding
deviation is indicated by δ .
7.10 Other uncertainty contributions
When other uncertainty contributions (e.g. microscope resolution, threshold setting in digital image
analysis, etc.) are known, they shall be included to calculate total uncertainty, δ .
7.11 Combined uncertainty
The combined uncertainty for a cumulative count and mass distribution is given as Formula (32):
p
δδQQ= (32)
()
00,,ti∑
i=1
p
δδQQ= (33)
()
33,,ti∑
i=1
where δQ andδQ are the individual uncertainty contribution i, 1 ≤ i ≤ p, and p is the total number
0,i 3,i
of uncertainties. The combined uncertainty of particle size for count and mass base is calculated by
Formula (34) and Formula (35):
δ Q
0,t
δ x = (34)
0,t
q
0,i
δ Q
3,t
δ x = (35)
3,t
q
0,i
The example calculation of the uncertainty based on limited number of sample size is shown in Annex B.
12 © ISO 2017 – All rights reserved

Annex A
(informative)
Picket-fence distributions composed of more than two kinds of
quasi-monodisperse particles
For the sake of illustrative clarity, a simplified example using only three rather than the recommended
seven pickets is shown.
In order to calculate the uncertainty of picket-fence distributions composed of three kinds of nearly
mono-disperse particles, a modified size distribution with maximum and minimum particle size
based on log-normal distribution is considered. Figure A.1 shows the relationship between cumulative
distribution and a parameter g defined by Formula (A.1):
lnxx−ln
g= (A.1)
lns
g
Key
g parameter defined Formula (A.1), dimensionless
Q undersize distribution, dimensionless
g , g parameters corresponding to the minimum and the maximum particle diameters, dimensionless
0 1
M , M fractions of particles between minimum particle and median particle, and median particle and maximum
0 1
particle, respectively, dimensionless
U , U undersize distributions of the log-normal distribution corresponding to the minimum and the maximum
0 1
particle diameters, dimensionless
U fraction of particles having parameters from g to infinity, dimensionless
2 1
Figure A.1 — Corrected distribution with known maximum and minimum particle sizes
The parameters g and g in Figure A.1 correspond to the minimum and the maximum particle
0 1
diameters, respectively. The solid line shows the perfect log-normal distribution and the dotted line
indicates the cumulative distribution truncated by the minimum and the maximum particle sizes. The
undersize in this case is represented as Formulae (A.2), (A.3), and (A.4):
g < g Qg = 0 (A.2)
0 ()
 
g
11 ς
g ≤ g ≤ g  Qg =−exp dς (A.3)
()
0 1
∫
 
g
M 2
2π 0
 
g > g     Qg = 1 (A.4)
1 ()
where
M = M +M = 1 − U − U (A.5)
0 1 0 2
As the particle size distribution is truncated at the minimum size and the maximum size, M, is smaller
than unity. In order to calculate the uncertainty of picket-fence distribution composed of three kinds of
nearly mono-disperse particles, the true particle size distribution is represented by Formula (A.6) to
Formula (A.11). When the three kinds of nearly mono-disperse particles that follows Formula (A.2) to
Formula (A.5) are uniformly mixed with N , N and N particles, the mass fraction of the particle i is α .
1 2 3 i
The parameter g and minimum and maximum particle diameters are as follows:
lnxx−ln
50,,31
g = (A.6)
lns
g,1
lnxx−ln
50,,32
g = (A.7)
lns
g,2
lnxx−ln
50,,33
g = (A.8)
lns
g,3
xx= explgsn , xx= explgsn (A.9)
() ()
15,max 03,,11,max g,1 15,min 03,,11,min g,1
xx= explgsn , xx= explgsn (A.10)
() ()
25,max 03,,22,max g,2 25,min 03,,22,min g,2
xx= explgsn , xx= explgsn (A.11)
() ()
35,max 03,,23,max g,3 35,min 03,,33,min g,3
where g is the parameter g of particle i. In Formula (A.6) to Formula (A.11), the mass median diameter
i
of particle i is represented by x . The maximum particle diameter of particle i is smaller than the
50,,3 i
minimum particle diameter of particle i + 1.
xx≤ (A.12)
ii,max +1,min
Then, the cumulative distribution is represented by Formula (A.13) to Formula (A.19):
x < x    Qg =0 (A.13)
()
1,min
 
g
α
1 ς
x ≤ x ≤ x     Qg =−exp dς (A.14)
()
1,min 1,max
∫
 
M g 2
2π 1,min
 
x < x < x       Qg =α (A.15)
()
1,max 2,min
14 © ISO 2017 – All rights reserved

 
g
α
1 ς
x ≤ x ≤ x    Qg =+α exp− dς (A.16)
2,min 2,max ()
∫
 
g
M 2
2π 2,min
 
x < x < x     Qg =+αα (A.17)
()
2,max 3,min
31 2
 
g
α
1 ς
x ≤ x ≤ x    Qg =+αα +−exp dς (A.18)
3,min 3,max ()
31 2
∫
 
g
M 2
2π 3,min
 
x < x       Qg =1 (A.19)
()
3,max
A typical distribution composed of three kinds of nearly mono-disperse particles is shown in Figure A.2.
Key
1 count base distribution
2 mass base distribution
x particle diameter, expressed in micrometers
Q(x) cumulative distribution, dimensionless
α mass fraction of particles included in picket k, dimensionless
k
Figure A.2 — Particle size distribution composed of three kinds of quasi- monodisperse
particles
Annex B
(informative)
Example of reliability calculation for a mass-based cumulative size
distribution transformed from the number-based size distribution
In this example case, the particles are composed of three kinds of quasi-monodisperse particles and the
particle properties before mixing are as follows.
The mass fractions of particles 1,2, and 3 are 0,35, 0,35 and 0,30, respectively.
x =1 μm s =10, 5 (B.1)
50,,01 g,1
x =3 μm s =10, 5 (B.2)
5002,, g,2
x =6 μm s =10, 5 (B.3)
50,,03 g,3
Total sample size:  N = 90 000.
In this example case, the mass fraction of α , α , α is selecte
...