ASTM E2578-07
(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
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 E 1617). The particle size is assumed to be the diameter of an equivalent sphere, e.g., 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 This practice uses SI (Systeme International) units as 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.
General Information
Relations
Standards Content (Sample)
NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation:E2578–07
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 3.1.2 equivalent diameter, n—diameter of a circle or sphere
whichbehavesliketheobservedparticlerelativetoordeduced
1.1 The purpose of this practice is to present procedures for
from a chosen property.
calculating mean sizes and standard deviations of size distri-
3.1.3 geometric standard deviation, n—exponential of the
butions given as histogram data (see Practice E1617). The
standard deviation of the distribution of log-transformed par-
particle size is assumed to be the diameter of an equivalent
ticle sizes.
sphere, e.g., equivalent (area/surface/volume/perimeter) diam-
3.1.4 histogram, n—a diagram of rectangular bars propor-
eter.
tional in area to the frequency of particles within the particle
1.2 The mean sizes/diameters are defined according to the
2,3,4
size intervals of the bars.
Moment-Ratio (M-R) definition system.
3.1.5 lognormal distribution, n—a distribution of particle
1.3 This practice uses SI (Système International) units as
size, whose logarithm has a normal distribution; the left tail of
standard.
alognormaldistributionhasasteepslopeonalinearsizescale,
1.4 This standard does not purport to address all of the
whereas the right tail decreases gradually.
safety concerns, if any, associated with its use. It is the
3.1.6 mean particle size/diameter, n—size or diameter of a
responsibility of the user of this standard to establish appro-
hypothetical particle such that a population of particles having
priate safety and health practices and determine the applica-
thatsize/diameterhas,forapurposeinvolved,propertieswhich
bility of regulatory limitations prior to use.
are equal to those of a population of particles with different
2. Referenced Documents sizes/diameters and having that size/diameter as a mean
size/diameter.
2.1 ASTM Standards:
3.1.7 moment of a distribution, n—a moment is the mean
E1617 Practice for Reporting Particle Size Characterization
value of a power of the particle sizes (the 3rd moment is
Data
proportional to the mean volume of the particles).
3. Terminology
3.1.8 normal distribution, n—a distribution which is also
known as Gaussian distribution and as bell-shaped curve
3.1 Definitions of Terms Specific to This Standard:
because the graph of its probability density resembles a bell.
3.1.1 diameter distribution, n—the distribution by diameter
3.1.9 number distribution, n—the distribution by number of
of particles as a function of their size.
particles as a function of their size.
3.1.10 order of mean diameter, n—thesumofthesubscripts
p and q of the mean diameter D .
This practice is under the jurisdiction ofASTM Committee E56 on Nanotech-
p,q
nologyandisthedirectresponsibilityofSubcommitteeE56.02onCharacterization:
3.1.11 particle, n—a discrete piece of matter.
Physical, Chemical, and Toxicological Properties.
3.1.12 particle diameter/size, n—some consistent measure
Current edition approved Nov. 1, 2007. Published November 2007. DOI:
of the spatial extent of a particle (see equivalent diameter).
10.1520/E2578-07.
Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of Definition 3.1.13 particle size distribution, n—adescriptionofthesize
Systems,” Part. Part. Syst. Charact., 7, 1990, pp. 233-241.
and frequency of particles in a population.
Alderliesten, M., “Mean Particle Diameters. From Statistical Definition to
3.1.14 population, n—a set of particles concerning which
Physical Understanding,” J. Biopharm.Statist., 15, 2005, pp. 295-325.
statistical inferences are to be drawn, based on a representative
Mugele, R. A., Evans, H. D., “Droplet Size Distribution in Sprays,” Ind. Eng.
Chem., 43, 1951, pp. 1317-1324.
sample taken from the population.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
3.1.15 sample, n—a part of a population of particles.
contact ASTM Customer Service at service@astm.org. For Annual Book of ASTM
3.1.16 standard deviation, n—most widely used measure of
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website. the width of a frequency distribution.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
E2578–07
3.1.17 surface distribution, n—the distribution by surface the particle sizes D equals the exponential of the (arithmetic)
area of particles as a function of their size. mean of the (lnD)-values:
3.1.18 variance, n—a measure of spread around the mean;
N
square of the standard deviation.
21 n
i
P
3.1.19 volume distribution, n—the distribution by volume
D 5exp@N n ~ln D !# 5 D (4)
(
g i i Πi
i
i
of particles as a function of their size.
6.1.4 The standard deviation s of the (lnD)-values can be
lnD
4. Summary of Practice
expressed as:
4.1 Samples of particles to be measured should be repre-
n $ln~D/D !%
(
i i g
sentative for the population of particles.
i
Œ
s 5 (5)
ln D
N–1
4.2 The‘frequency’ofaparticularvalueofaparticlesize D
can be measured (or expressed) in terms of the number of
6.2 Definition of Mean Diameters D :
p,q
particles, the cumulated diameters, surfaces or volumes of the
6.2.1 Themeandiameter D ofasampleofparticlesizesis
p,q
particles. The corresponding frequency distributions are called
defined as 1/(p – q)-th power of the ratio of the p-th and the
Number, Diameter, Surface, or Volume distributions.
q-th moment of the Number distribution of the particle sizes:
4.3 As class mid points D of the histogram intervals the
i
’ 1/~p2q!
M
p
arithmetic mean values of the class boundaries are used.
D 5 if p fi q (6)
p,q F G
’
M
4.4 Particle shape factors are not taken into account, al- q
though their importance in particle size analysis is beyond
6.2.2 Using Eq 1, Eq 6 can be rewritten as:
doubt.
p 1/~p2q!
nD
(
i i
4.5 A coherent nomenclature system is presented which
i
D 5 if p fi q (7)
p,q
q
conveys the physical meanings of mean particle diameters. F G
nD
( i i
i
5. Significance and Use
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:
5.1 Mean particle diameters defined according to the
Moment-Ratio (M-R) system are derived from ratios between q
nD lnD
(
i i i
i
two moments of a particle size distribution.
D 5exp if p 5 q (8)
q,q
q
F G
nD
( i i
i
6. Mean Particle Sizes/Diameters
6.2.4 If q = 0, then:
6.1 Moments of Distributions:
N
6.1.1 Moments are the basis for defining mean sizes and
nlnD
( i i
standard deviations. A random sample, containing N elements
i
n
i
P
D 5exp 5 D (9)
0,0 Πi
fromapopulationofparticlesizes D,enablesestimationofthe F G
i n
( i
i
i
moments of the size distribution of the population of particle
’
6.2.5 D is the well-known geometric mean diameter.The
sizes. The r-th sample moment, denoted by M , is defined to
0,0
r
be: physical dimension of any D is equal to that of D itself.
p,q
6.2.6 Mean diameters D of a sample can be estimated
’ 21 r
p,q
M :5 N nD (1)
(
r i i
i from any size distribution f (D) according to equations similar
r
to Eq 7 and 8:
(
where N 5 n, D is the midpoint of the i-th interval and
i i
i
m 1/p2q
n is the number of particles in the i-th size class (i.e., class
i p2r
f ~D !D
’ (
r i i
frequency). The (arithmetic) sample mean M of the particle
i
D 5 if p fi q (10)
p,q
m
size D is mostly represented by D . The r-th sample moment
q2r
3 4
f ~D !D
( r i i
about the mean D, denoted by M , is defined by:
i
r
21 r
M :5 N n ~D – D! (2) and:
r ( i i
i
m
p2r
6.1.2 The best-known example is the sample variance M .
2 f ~D !D lnD
(
r i i i
i
This M always underestimates the population variance
D 5exp if p 5 q (11)
p,p
m
p2r
s (squared standard deviation). Instead, M multiplied by 3 4
D 2 f D !D
~
( r i i
i
N/(N–1) is used, which yields an unbiased estimator, s , for
D
the population variance. Thus, the sample variance s has to
D
where:
be calculated from the equation:
f (D) = particle quantity in the i-th class,
r i
D = midpoint of the i-th class interval,
i
n ~D – D!
(
i i
N
i
r = 0, 1, 2 or 3 represents the type of quantity, viz.
s 5 M 5 (3)
D 2
N–1 N–1
number, diameter, surface, volume (or mass) re-
spectively, and
6.1.3 Its square root is the standard deviation s of the
D
m = number of classes.
sample (see also 6.3). If the particle sizes D are lognormally
distributed, then the logarithm of D,lnD, follows a normal 6.2.7 If r = 0 and we put n = f (D), then Eq 10 reduces to
i 0 i
distribution (Gaussian distribution).The geometric mean D of the familiar form Eq 7.
g
E2578–07
TABLE 1 Nomenclature for Mean Particle Diameters D
p,q D # D (19)
p21, q21 p,q
Systematic
Nomenclature 6.4.2 Differences between mean diameters decrease accord-
Code
ing as the uniformity of the particle sizes D increases. The
D harmonic mean volume diameter
23.0
equalsignapplieswhenallparticlesareofthesamesize.Thus,
D diameter-weighted harmonic mean volume diameter
22.1
D surface-weighted harmonic mean volume diameter the differences between the values of the mean diameters
21.2
provide already an indication of the dispersion of the particle
D harmonic mean surface diameter
22.0
sizes.
D diameter-weighted harmonic mean surface diameter
21.1
6.4.3 Another relations
...
Questions, Comments and Discussion
Ask us and Technical Secretary will try to provide an answer. You can facilitate discussion about the standard in here.