ASTM E2578-07(2012)
(Practice)Standard Practice for Calculation of Mean Sizes/Diameters and Standard Deviations of Particle Size Distributions
Standard Practice for Calculation of Mean Sizes/Diameters and Standard Deviations of Particle Size Distributions
SIGNIFICANCE AND USE
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. , ,
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 and health practices and determine the applicability of regulatory limitations prior to use.
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Designation: E2578 − 07 (Reapproved 2012)
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 3.1.1 diameter distribution, n—the distribution by diameter
of particles as a function of their size.
1.1 The purpose of this practice is to present procedures for
3.1.2 equivalent diameter, n—diameter of a circle or sphere
calculating mean sizes and standard deviations of size distri-
butions given as histogram data (see Practice E1617). The whichbehavesliketheobservedparticlerelativetoordeduced
from a chosen property.
particle size is assumed to be the diameter of an equivalent
sphere, for example, equivalent (area/surface/volume/
3.1.3 geometric standard deviation, n—exponential of the
perimeter) diameter.
standard deviation of the distribution of log-transformed par-
ticle sizes.
1.2 The mean sizes/diameters are defined according to the
2,3,4
Moment-Ratio (M-R) definition system.
3.1.4 histogram, n—a diagram of rectangular bars propor-
tional in area to the frequency of particles within the particle
1.3 The values stated in SI units are to be regarded as
size intervals of the bars.
standard. No other units of measurement are included in this
standard.
3.1.5 lognormal distribution, n—a distribution of particle
size, whose logarithm has a normal distribution; the left tail of
1.4 This standard does not purport to address all of the
alognormaldistributionhasasteepslopeonalinearsizescale,
safety concerns, if any, associated with its use. It is the
whereas the right tail decreases gradually.
responsibility of the user of this standard to establish appro-
priate safety and health practices and determine the applica-
3.1.6 mean particle size/diameter, n—size or diameter of a
bility of regulatory limitations prior to use.
hypothetical particle such that a population of particles having
thatsize/diameterhas,forapurposeinvolved,propertieswhich
2. Referenced Documents
are equal to those of a population of particles with different
2.1 ASTM Standards: sizes/diameters and having that size/diameter as a mean
size/diameter.
E1617Practice for Reporting Particle Size Characterization
Data
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
3. Terminology
proportional to the mean volume of the particles).
3.1 Definitions of Terms Specific to This Standard:
3.1.8 normal distribution, n—a distribution which is also
known as Gaussian distribution and as bell-shaped curve
because the graph of its probability density resembles a bell.
This practice is under the jurisdiction ofASTM Committee E56 on Nanotech-
3.1.9 number distribution, n—the distribution by number of
nology and is the direct responsibility of Subcommittee E56.02 on Physical and
Chemical Characterization.
particles as a function of their size.
Current edition approved May 1, 2012. Published May 2012. Originally
3.1.10 order of mean diameter, n—the sum of the subscripts
approved in 2007. Last previous edition approved in 2007 as E2578– 07. DOI:
10.1520/E2578-07R12. ¯
p and q of the mean diameter D .
p,q
Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of Definition
Systems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp. 3.1.11 particle, n—a discrete piece of matter.
233–241.
3.1.12 particle diameter/size, n—some consistent measure
Alderliesten, M., “Mean Particle Diameters. From Statistical Definition to
of the spatial extent of a particle (see equivalent diameter).
PhysicalUnderstanding,” Journal of Biopharmaceutical Statistics,Vol15,2005,pp.
295–325.
3.1.13 particle size distribution, n—a description of the size
Mugele, R.A., and Evans, H.D., “Droplet Size Distribution in Sprays,” Journal
and frequency of particles in a population.
of Industrial and Engineering Chemistry, Vol 43, 1951, pp. 1317–1324.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
3.1.14 population, n—a set of particles concerning which
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
statistical inferences are to be drawn, based on a representative
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. sample taken from the population.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2578 − 07 (2012)
3.1.15 sample, n—a part of a population of particles. ¯
n ~D 2 D!
( i i
N
i
3.1.16 standard deviation, n—most widely used measure of s 5 M 5 (3)
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
square of the standard deviation.
g
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)
4.1 Samples of particles to be measured should be repre- ~ !
g i i i
(
@ #
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 2
¯
$ ~ !%
n ln D /D
( i i g
Number, Diameter, Surface, or Volume distributions. i
Œ
s 5 (5)
lnD
N 2 1
4.3 As class mid points D of the histogram intervals the
i
¯
arithmetic mean values of the class boundaries are used. 6.2 Definition of Mean Diameters D :
p,q
¯
4.4 Particle shape factors are not taken into account, al-
6.2.1 Themeandiameter D ofasampleofparticlesizesis
p,q
though their importance in particle size analysis is beyond
defined as 1/(p – q)-th power of the ratio of the p-th and the
doubt.
q-th moment of the Number distribution of the particle sizes:
4.5 A coherent nomenclature system is presented which
’ 1/~p2q!
M
p
¯
conveys the physical meanings of mean particle diameters.
D 5 if pfiq (6)
F G
p,q ’
M
q
5. Significance and Use
6.2.2 Using Eq 1, Eq 6 can be rewritten as:
5.1 Mean particle diameters defined according to the 1/~p2q!
p
n D
( i i
i
Moment-Ratio (M-R) system are derived from ratios between
¯
D 5 if pfiq (7)
p,q
q
two moments of a particle size distribution. 3 n D 4
i i
(
i
6. Mean Particle Sizes/Diameters
6.2.3 The powers p and q may have any real value. For
equal values of p and q it is possible to derive from Eq 7 that:
6.1 Moments of Distributions:
6.1.1 Moments are the basis for defining mean sizes and
q
n D lnD
( i i i
standard deviations. A random sample, containing N elements i
¯
D 5 exp if p 5 q (8)
q,q
q
fromapopulationofparticlesizes D,enablesestimationofthe
3 n D 4
i ( i i
i
moments of the size distribution of the population of particle
’
sizes. The r-th sample moment, denoted by M , is defined to
r 6.2.4 If q = 0, then:
be:
N
’ 21 r
M : 5 N n D (1) n lnD
r ( i i i i
(
i
i
¯ n
Π i
D 5 exp 5 D (9)
0,0 i
3 n 4
!
( i
(
i
where N5 n , D is the midpoint of the i-th interval and n
i
i i
i
i
is the number of particles in the i-th size class (that is, class
¯
’
6.2.5 D is the well-known geometric mean diameter. The
frequency). The (arithmetic) sample mean M of the particle 0,0
¯
¯ physical dimension of any D is equal to that of D itself.
size D is mostly represented by D . The r-th sample moment p,q
¯
about the mean D, denoted by M , is defined by: ¯
6.2.6 Mean diameters D of a sample can be estimated
r
p,q
from any size distribution f (D) according to equations similar
r
21 r
¯
M : 5 N n ~D 2 D! (2)
r ( i i
to Eq 7 and 8:
i
m 1/p2q
6.1.2 The best-known example is the sample variance M .
p2r
f D D
~ !
r i i
This M always underestimates the population variance (
i
¯
D 5 if pfiq (10)
p,q m
σ (squared standard deviation). Instead, M multiplied by
D 2
q2r
2 3 4
f D D
~ !
N/(N–1) is used, which yields an unbiased estimator, s , for
( r i i
D
i
the population variance. Thus, the sample variance s has to
D
be calculated from the equation: and:
E2578 − 07 (2012)
m
6.3.2 In practice, N >> 100, so that c ≈ 1. Hence:
p2r
f D D lnD
~ !
( r i i i
i
¯
D 5 exp if p 5 q (11) ¯2 ¯2
p,p m =
s' D 2 D (15)
2,0 1,0
p2r
3 4
f D D
~ !
( r i i
i
6.3.3 The standard deviation s of a lognormal Number
lnD
distributionofparticlesizes Dcanbeestimatedby(seeEq12):
where:
f (D) = particle quantity in the i-th class,
r i ¯
n $ln~D /D !%
( i i 0,0
D = midpoint of the i-th class interval, i
i
Œ
s 5 (16)
lnD
N 2 1
r = 0, 1, 2, or 3 represents the type of quantity, viz.
number, diameter, surface, volume (or mass)
6.3.4 In particle-size analysis, the quantity s is referred to
g
respectively, and
as the geometric standard deviation although it is not a
m = number of classes.
standard deviation in its true sense:
6.2.7 If r = 0 and we put n = f (D), then Eq 10 reduces to
i 0 i
s 5 exp@s # (17)
g lnD
the familiar form Eq 7.
¯
6.4 Relationships Between Mean Diameters D :
p,q
6.3 Standard Deviation:
6.4.1 It can be shown that:
6.3.1 According to Eq 3, the standard deviation of the
Number distribution of a sample of particle sizes can be
¯ ¯
D # D if
...
This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: E2578 − 07 (Reapproved 2012) E2578 − 07 (Reapproved 2012)
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. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. 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.
2,3,4
1.2 The mean sizes/diameters are defined according to the Moment-Ratio (M-R) definition system.
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 and health practices and determine the applicability of regulatory
limitations prior to use.
2. Referenced Documents
2.1 ASTM Standards:
E1617 Practice for Reporting Particle Size Characterization Data
3. Terminology
3.1 Definitions of Terms Specific to This Standard:
3.1.1 diameter distribution, n—the distribution by diameter of particles as a function of their size.
3.1.2 equivalent diameter, n—diameter of a circle or sphere which behaves like the observed particle relative to or deduced from
a chosen property.
3.1.3 geometric standard deviation, n—exponential of the standard deviation of the distribution of log-transformed particle
sizes.
3.1.4 histogram, n—a diagram of rectangular bars proportional in area to the frequency of particles within the particle size
intervals of the bars.
3.1.5 lognormal distribution, n—a distribution of particle size, whose logarithm has a normal distribution; the left tail of a
lognormal distribution has a steep slope on a linear size scale, whereas the right tail decreases gradually.
3.1.6 mean particle size/diameter, n—size or diameter of a hypothetical particle such that a population of particles having that
size/diameter has, for a purpose involved, properties which 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 proportional
to the mean volume of the particles).
3.1.8 normal distribution, n—a distribution which is also known as Gaussian distribution and as bell-shaped curve because the
graph of its probability density resembles a bell.
This practice is under the jurisdiction of ASTM Committee E56 on Nanotechnology and is the direct responsibility of Subcommittee E56.02 on Physical and Chemical
Characterization.
Current edition approved May 1, 2012. Published May 2012. Originally approved in 2007. Last previous edition approved in 2007 as E2578 –07. – 07. DOI:
10.1520/E2578-07R12.
Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of Definition Systems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp. 233–241.
Alderliesten, M., “Mean Particle Diameters. From Statistical Definition to Physical Understanding,” Journal of Biopharmaceutical Statistics, Vol 15, 2005, pp. 295–325.
Mugele, R.A., and Evans, H.D., “Droplet Size Distribution in Sprays,” Journal of Industrial and Engineering Chemistry, Vol 43, 1951, pp. 1317–1324.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E2578 − 07 (2012)
3.1.9 number distribution, n—the distribution by number of particles as a function of their size.
¯
3.1.10 order of mean diameter, n—the sum of the subscripts p and q of the mean diameter D .
p,q
3.1.11 particle, n—a discrete piece of matter.
3.1.12 particle diameter/size, n—some consistent measure of the spatial extent of a particle (see equivalent diameter).
3.1.13 particle size distribution, n—a description of the size and frequency of particles in a population.
3.1.14 population, n—a set of particles concerning which statistical inferences are to be drawn, based on a representative sample
taken from the population.
3.1.15 sample, n—a part of a population of particles.
3.1.16 standard deviation, n—most widely used measure of the width of a frequency distribution.
3.1.17 surface distribution, n—the distribution by surface area of particles as a function of their size.
3.1.18 variance, n—a measure of spread around the mean; square of the standard deviation.
3.1.19 volume distribution, n—the distribution by volume of particles as a function of their size.
4. Summary of Practice
4.1 Samples of particles to be measured should be representative for the population of particles.
4.2 The ‘frequency’ of a particular value of a particle size D can be measured (or expressed) in terms of the number of particles,
the cumulated diameters, surfaces or volumes of the particles. The corresponding frequency distributions are called Number,
Diameter, Surface, or Volume distributions.
4.3 As class mid points D of the histogram intervals the arithmetic mean values of the class boundaries are used.
i
4.4 Particle shape factors are not taken into account, although their importance in particle size analysis is beyond doubt.
4.5 A coherent nomenclature system is presented which conveys the physical meanings of mean particle diameters.
5. 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.
6. Mean Particle Sizes/Diameters
6.1 Moments of Distributions:
6.1.1 Moments are the basis for defining mean sizes and standard deviations. A random sample, containing N elements from
a population of particle sizes D , enables estimation of the moments of the size distribution of the population of particle sizes. The
i
’
r-th sample moment, denoted by M , is defined to be:
r
’ 21 r
M :5 N n D (1)
r ( i i
i
(
where N5 n , D is the midpoint of the i-th interval and n is the number of particles in the i-th size class (that is, class frequency).
i i i
i
’
¯ ¯
The (arithmetic) sample mean M of the particle size D is mostly represented by D . The r-th sample moment about the mean D,
denoted by M , is defined by:
r
r
¯
M :5 N n ~D 2 D! (2)
r ( i i
i
6.1.2 The best-known example is the sample variance M . This M always underestimates the population variance σ (squared
2 2 D
standard deviation). Instead, M multiplied by N/(N–1) is used, which yields an unbiased estimator, s , for the population
2 D
variance. Thus, the sample variance s has to be calculated from the equation:
D
¯
n ~D 2 D!
( i i
N
i
s 5 M 5 (3)
D 2
N 2 1 N 2 1
6.1.3 Its square root is the standard deviation s of the sample (see also 6.3). If the particle sizes D are lognormally distributed,
D
¯
then the logarithm of D, lnD, follows a normal distribution (Gaussian distribution). The geometric mean D of the particle sizes
g
D equals the exponential of the (arithmetic) mean of the (lnD)-values:
N
¯ 21 n
Π i
D 5 exp N n ~lnD ! 5 D (4)
g ( i i i
@ #
i
!
i
E2578 − 07 (2012)
6.1.4 The standard deviation s of the (lnD)-values can be expressed as:
lnD
¯
n $ln~D /D !%
( i i g
i
Œ
s 5 (5)
lnD
N 2 1
¯
6.2 Definition of Mean Diameters D :
p,q
¯
6.2.1 The mean diameter D of a sample of particle sizes is defined as 1/(p – q)-th power of the ratio of the p-th and the q-th
p,q
moment of the Number distribution of the particle sizes:
’ 1/~p2q!
M
p
¯
D 5 if pfiq (6)
F G
’
p,q
M
q
6.2.2 Using Eq 1, Eq 6 can be rewritten as:
1/~p2q!
p
n D
( i i
i
¯
D 5 if pfiq (7)
p,q
q
3 n D 4
( i i
i
6.2.3 The powers p and q may have any real value. For equal values of p and q it is possible to derive from Eq 7 that:
q
n D lnD
i i i
(
i
¯
D 5 exp if p 5 q (8)
q,q
q
3 n D 4
( i i
i
6.2.4 If q = 0, then:
N
n lnD
i i
(
i
¯ n
Π i
D 5 exp 5 D (9)
0,0 i
3 n 4
!
( i
i
i
¯ ¯
6.2.5 D is the well-known geometric mean diameter. The physical dimension of any D is equal to that of D itself.
0,0 p,q
¯
6.2.6 Mean diameters D of a sample can be estimated from any size distribution f (D) according to equations similar to Eq
p,q r
7 and 8:
m 1/p2q
p2r
f D D
~ !
r i i
(
i
¯
D 5 if pfiq (10)
p,q m
q2r
3 4
f D D
~ !
( r i i
i
and:
m
p2r
f D D lnD
~ !
r i i i
(
i
¯
D 5 exp if p 5 q (11)
p,p m
p2r
3 4
f D D
~ !
r i i
(
i
where:
f (D ) = particle quantity in the i-th class,
r i
D = midpoint of the i-th class interval,
i
r = 0, 1, 2, or 3 represents the type of quantity, viz. number, diameter, surface, volume (or mass) respectively, and
m = number of classes.
6.2.7 If r = 0 and we put n = f (D ), then Eq 10 reduces to the familiar form Eq 7.
i 0 i
6.3 Standard Deviation:
6.3.1 According to Eq 3, the standard deviation of the Number distribution of a sample of particle sizes can be estimated from:
2 ¯ 2
n D 2 ND
( i i 1,0
i
Œ
s 5 (12)
D
N 2 1
which can be rewritten as:
¯ 2 ¯ 2
=
s 5 c D 2 D (13)
2,0 1,0
with:
E2578 − 07 (2012)
¯
TABLE 1 Nomenclature for Mean Particle Diameters D
p,q
Systematic
Nomenclature
Code
¯
harmonic mean volume diameter
D
23.0
¯
diameter-weighted harmonic mean volume
D
22.1
diameter
¯
surface-weighted harmonic mean volume di-
D
21.2
ameter
¯ harmonic mean surface diameter
D
22.0
¯
diameter-weighted harmonic mean surface
D
21.1
diameter
¯
harmonic mean
...
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