ASTM E2578-07(2022)

Standard Practice for Calculation of Mean Sizes/Diameters and Standard Deviations of Particle Size Distributions

SIGNIFICANCE AND USE
5.1 Mean particle diameters defined according to the Moment-Ratio (M-R) system are derived from ratios between two moments of a particle size distribution.
SCOPE
1.1 The purpose of this practice is to present procedures for calculating mean sizes and standard deviations of size distributions given as histogram data (see Practice E1617). The particle size is assumed to be the diameter of an equivalent sphere, for example, equivalent (area/surface/volume/perimeter) diameter.  
1.2 The mean sizes/diameters are defined according to the Moment-Ratio (M-R) definition system.2,3,4  
1.3 The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.  
1.4 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.5 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 31-Aug-2022
• ICS: 19.120 - Particle size analysis. Sieving
• Technical Committee: E56 - Nanotechnology
• Drafting Committee: E56.02 - Physical and Chemical Characterization

Relations

• Refers: ASTM E1617-09(2024) - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 01-Feb-2024
• Refers: ASTM E1617-09(2019) - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 01-Apr-2019
• Refers: ASTM E1617-09(2014)e1 - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 01-Apr-2014
• Refers: ASTM E1617-09 - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 01-Mar-2009
• Refers: ASTM E1617-97(2007) - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 01-Apr-2007
• Refers: ASTM E1617-97 - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 10-Apr-1997
• Refers: ASTM E1617-97(2002) - Standard Practice for Reporting Particle Size Characterization Data
• Effective Date: 10-Apr-1997

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: E2578 − 07 (Reapproved 2022)
Standard Practice for
Calculation of Mean Sizes/Diameters and Standard
Deviations of Particle Size Distributions
This standard is issued under the fixed designation E2578; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
2.1 ASTM Standards:
1.1 The purpose of this practice is to present procedures for
E1617Practice for Reporting Particle Size Characterization
calculating mean sizes and standard deviations of size distri-
Data
butions given as histogram data (see Practice E1617). The
particle size is assumed to be the diameter of an equivalent
3. Terminology
sphere, for example, equivalent (area/surface/volume/
3.1 Definitions of Terms Specific to This Standard:
perimeter) diameter.
3.1.1 diameter distribution, n—the distribution by diameter
of particles as a function of their size.
1.2 The mean sizes/diameters are defined according to the
2,3,4
Moment-Ratio (M-R) definition system.
3.1.2 equivalent diameter, n—diameter of a circle or sphere
whichbehavesliketheobservedparticlerelativetoordeduced
1.3 The values stated in SI units are to be regarded as
from a chosen property.
standard. No other units of measurement are included in this
3.1.3 geometric standard deviation, n—exponential of the
standard.
standard deviation of the distribution of log-transformed par-
1.4 This standard does not purport to address all of the
ticle sizes.
safety concerns, if any, associated with its use. It is the
3.1.4 histogram, n—a diagram of rectangular bars propor-
responsibility of the user of this standard to establish appro-
tional in area to the frequency of particles within the particle
priate safety, health, and environmental practices and deter-
size intervals of the bars.
mine the applicability of regulatory limitations prior to use.
3.1.5 lognormal distribution, n—a distribution of particle
1.5 This international standard was developed in accor-
size, whose logarithm has a normal distribution; the left tail of
dance with internationally recognized principles on standard-
alognormaldistributionhasasteepslopeonalinearsizescale,
ization established in the Decision on Principles for the
whereas the right tail decreases gradually.
Development of International Standards, Guides and Recom-
3.1.6 mean particle size/diameter, n—size or diameter of a
mendations issued by the World Trade Organization Technical
hypothetical particle such that a population of particles having
Barriers to Trade (TBT) Committee.
thatsize/diameterhas,forapurposeinvolved,propertieswhich
are equal to those of a population of particles with different
sizes/diameters and having that size/diameter as a mean
size/diameter.
3.1.7 moment of a distribution, n—a moment is the mean
value of a power of the particle sizes (the 3rd moment is
This practice is under the jurisdiction ofASTM Committee E56 on Nanotech-
proportional to the mean volume of the particles).
nology and is the direct responsibility of Subcommittee E56.02 on Physical and
3.1.8 normal distribution, n—a distribution which is also
Chemical Characterization.
Current edition approved Sept. 1, 2022. Published October 2022. Originally
known as Gaussian distribution and as bell-shaped curve
approved in 2007. Last previous edition approved in 2018 as E2578– 07 (2018).
because the graph of its probability density resembles a bell.
DOI: 10.1520/E2578-07R22.
Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of Definition 3.1.9 number distribution, n—the distribution by number of
Systems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp.
particles as a function of their size.
233–241.
Alderliesten, M., “Mean Particle Diameters. From Statistical Definition to
PhysicalUnderstanding,” Journal of Biopharmaceutical Statistics,Vol15,2005,pp. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
295–325. contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
Mugele, R.A., and Evans, H.D., “Droplet Size Distribution in Sprays,” Journal Standards volume information, refer to the standard’s Document Summary page on
of Industrial and Engineering Chemistry, Vol 43, 1951, pp. 1317–1324. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2578 − 07 (2022)
¯
3.1.10 order of mean diameter, n—the sum of the subscripts
size D is mostly represented by D. The r-th sample moment
¯
¯
p and q of the mean diameter D .
about the mean D, denoted by M , is defined by:
p,q
r
3.1.11 particle, n—a discrete piece of matter.
r
21 ¯
M : 5 N n ~D 2 D! (2)
r ( i i
i
3.1.12 particle diameter/size, n—some consistent measure
of the spatial extent of a particle (see equivalent diameter).
6.1.2 The best-known example is the sample variance M .
3.1.13 particle size distribution, n—a description of the size
This M always underestimates the population variance
and frequency of particles in a population.
σ (squared standard deviation). Instead, M multiplied by
D 2
N/(N–1) is used, which yields an unbiased estimator, s , for
3.1.14 population, n—a set of particles concerning which D
the population variance. Thus, the sample variance s has to
statistical inferences are to be drawn, based on a representative
D
be calculated from the equation:
sample taken from the population.
¯
3.1.15 sample, n—a part of a population of particles.
n ~D 2 D!
( i i
N
i
s 5 M 5 (3)
3.1.16 standard deviation, n—most widely used measure of
D 2
N 2 1 N 2 1
the width of a frequency distribution.
6.1.3 Its square root is the standard deviation s of the
3.1.17 surface distribution, n—the distribution by surface D
sample (see also 6.3). If the particle sizes D are lognormally
area of particles as a function of their size.
distributed, then the logarithm of D,lnD, follows a normal
3.1.18 variance, n—a measure of spread around the mean;
¯
distribution (Gaussian distribution).The geometric mean D of
g
square of the standard deviation.
the particle sizes D equals the exponential of the (arithmetic)
3.1.19 volume distribution, n—thedistributionbyvolumeof
mean of the (lnD)-values:
particles as a function of their size.
N
4. Summary of Practice
¯ 21 n
Π i
D 5 exp N n ~lnD ! 5 D (4)
g ( i i i
4.1 Samples of particles to be measured should be repre- @ #
i
!
i
sentative for the population of particles.
4.2 The‘frequency’ofaparticularvalueofaparticlesize D
6.1.4 The standard deviation s of the (lnD)-values can be
lnD
can be measured (or expressed) in terms of the number of
expressed as:
particles, the cumulated diameters, surfaces or volumes of the
¯
particles. The corresponding frequency distributions are called n $ln~D /D !%
( i i g
i
Œ
Number, Diameter, Surface, or Volume distributions. s 5 (5)
lnD
N 2 1
4.3 As class mid points D of the histogram intervals the
i
¯
6.2 Definition of Mean Diameters D :
p,q
arithmetic mean values of the class boundaries are used.
¯
6.2.1 Themeandiameter D ofasampleofparticlesizesis
p,q
4.4 Particle shape factors are not taken into account, al-
defined as 1/(p – q)-th power of the ratio of the p-th and the
though their importance in particle size analysis is beyond
q-th moment of the Number distribution of the particle sizes:
doubt.
’ 1/ p2q
~ !
M
4.5 A coherent nomenclature system is presented which p
¯
D 5 if pfiq (6)
F G
p,q ’
M
q
conveys the physical meanings of mean particle diameters.
6.2.2 Using Eq 1, Eq 6 can be rewritten as:
5. Significance and Use
1/~p2q!
p
n D
( i i
5.1 Mean particle diameters defined according to the
i
¯
D 5 if pfiq (7)
Moment-Ratio (M-R) system are derived from ratios between p,q
q
3 n D 4
( i i
i
two moments of a particle size distribution.
6.2.3 The powers p and q may have any real value. For
6. Mean Particle Sizes/Diameters
equal values of p and q it is possible to derive from Eq 7 that:
6.1 Moments of Distributions:
q
n D lnD
6.1.1 Moments are the basis for defining mean sizes and ( i i i
i
¯
D 5 exp if p 5 q (8)
q,q
standard deviations. A random sample, containing N elements
q
3 n D 4
( i i
i
fromapopulationofparticlesizes D,enablesestimationofthe
i
moments of the size distribution of the population of particle
6.2.4 If q = 0, then:
’
sizes. The r-th sample moment, denoted by M , is defined to
r
N
be:
n lnD
( i i
’ 21 r
i
M : 5 N n D (1)
¯ n
r i i i
( Π
D 5 exp 5 D (9)
0,0 i
i
3 n 4
i !
(
i
i
(
where N5 n , D is the midpoint of the i-th interval and n
i i i
i
¯
6.2.5 D is the well-known geometric mean diameter. The
is the number of particles in the i-th size class (that is, class
0,0
’
¯
frequency). The (arithmetic) sample mean M of the particle physical dimension of any D is equal to that of D itself.
1 p,q
E2578 − 07 (2022)
¯
6.2.6 Mean diameters D of a sample can be estimated
2 ¯2
p,q
n D 2 ND
i i 1,0
(
i
from any size distribution f (D) according to equations similar
r
Œ
s 5 (12)
D
N 2 1
to Eq 7 and 8:
m 1/p2q
which can be rewritten as:
p2r
f ~D !D
( r i i
i
¯
2 2
¯ ¯
D 5 if pfiq (10)
s 5 c=D 2 D (13)
p,q m
2,0 1,0
3 q2r4
f ~D !D
( r i i
i
with:
and:
c 5 =N/ N 2 1 (14)
~ !
m
p2r
f D D lnD 6.3.2 In practice, N >> 100, so that c ≈ 1. Hence:
~ !
( r i i i
i
¯
D 5 exp if p 5 q (11)
m
p,p
¯2 ¯2
p2r =
3 4 s' D 2 D (15)
2,0 1,0
f ~D !D
( r i i
i
6.3.3 The standard deviation s of a lognormal Number
lnD
where:
distributionofparticlesizes Dcanbeestimatedby(seeEq12):
f (D) = particle quantity in the i-th class,
r i
¯
D = midpoint of the i-th class interval,
$ ~ !%
i n ln D /D
( i i 0,0
i
r = 0, 1, 2, or 3 represents the type of quantity, viz. Œ
s 5 (16)
lnD
N 2 1
number, diameter, surface, volume (or mass)
respectively, and
6.3.4 In particle-size analysis, the quantity s is referred to
g
m = number of classes. 2
as the geometric standard deviation althou
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