# ASTM E2578-07(2018) 14.02

Designation: E2578 − 07 (Reapproved 2018)Standard Practice forCalculation of Mean Sizes/Diameters and StandardDeviations of Particle Size Distributions1This standard is issued under the fixed designation E2578; the number immediately following the designation indicates the year oforiginal adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. Asuperscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 The purpose of this practice is to present procedures forcalculating mean sizes and standard deviations of size distri-butions given as histogram data (see Practice E1617). Theparticle size is assumed to be the diameter of an equivalentsphere, for example, equivalent (area/surface/volume/perimeter) diameter.1.2 The mean sizes/diameters are defined according to theMoment-Ratio (M-R) definition system.2,3,41.3 The values stated in SI units are to be regarded asstandard. No other units of measurement are included in thisstandard.1.4 This standard does not purport to address all of thesafety concerns, if any, associated with its use. It is theresponsibility of the user of this standard to establish appro-priate safety, health, and environmental practices and deter-mine the applicability of regulatory limitations prior to use.1.5 This international standard was developed in accor-dance with internationally recognized principles on standard-ization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recom-mendations issued by the World Trade Organization TechnicalBarriers to Trade (TBT) Committee.2. Referenced Documents2.1 ASTM Standards:5E1617 Practice for Reporting Particle Size CharacterizationData3. Terminology3.1 Definitions of Terms Specific to This Standard:3.1.1 diameter distribution, n—the distribution by diameterof particles as a function of their size.3.1.2 equivalent diameter, n—diameter of a circle or spherewhich behaves like the observed particle relative to or deducedfrom a chosen property.3.1.3 geometric standard deviation, n—exponential of thestandard deviation of the distribution of log-transformed par-ticle sizes.3.1.4 histogram, n—a diagram of rectangular bars propor-tional in area to the frequency of particles within the particlesize intervals of the bars.3.1.5 lognormal distribution, n—a distribution of particlesize, whose logarithm has a normal distribution; the left tail ofa 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 ahypothetical particle such that a population of particles havingthat size/diameter has, for a purpose involved, properties whichare equal to those of a population of particles with differentsizes/diameters and having that size/diameter as a meansize/diameter.3.1.7 moment of a distribution, n—a moment is the meanvalue of a power of the particle sizes (the 3rd moment isproportional to the mean volume of the particles).3.1.8 normal distribution, n—a distribution which is alsoknown as Gaussian distribution and as bell-shaped curvebecause the graph of its probability density resembles a bell.3.1.9 number distribution, n—the distribution by number ofparticles as a function of their size.1This practice is under the jurisdiction of ASTM Committee E56 on Nanotech-nology and is the direct responsibility of Subcommittee E56.02 on Physical andChemical Characterization.Current edition approved Jan. 1, 2018. Published January 2018. Originallyapproved in 2007. Last previous edition approved in 2012 as E2578 – 07 (2012).DOI: 10.1520/E2578-07R18.2Alderliesten, M., “Mean Particle Diameters. Part I: Evaluation of DefinitionSystems,” Particle and Particle Systems Characterization, Vol 7, 1990, pp.233–241.3Alderliesten, M., “Mean Particle Diameters. From Statistical Definition toPhysical Understanding,” Journal of Biopharmaceutical Statistics, Vol 15, 2005, pp.295–325.4Mugele, R.A., and Evans, H.D., “Droplet Size Distribution in Sprays,” Journalof Industrial and Engineering Chemistry, Vol 43, 1951, pp. 1317–1324.5For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at service@astm.org. For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United StatesThis international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for theDevelopment of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.13.1.10 order of mean diameter, n—the sum of the subscriptsp 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 measureof the spatial extent of a particle (see equivalent diameter).3.1.13 particle size distribution, n—a description of the sizeand frequency of particles in a population.3.1.14 population, n—a set of particles concerning whichstatistical inferences are to be drawn, based on a representativesample 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 ofthe width of a frequency distribution.3.1.17 surface distribution, n—the distribution by surfacearea 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 ofparticles as a function of their size.4. Summary of Practice4.1 Samples of particles to be measured should be repre-sentative for the population of particles.4.2 The ‘frequency’of a particular value of a particle size Dcan be measured (or expressed) in terms of the number ofparticles, the cumulated diameters, surfaces or volumes of theparticles. The corresponding frequency distributions are calledNumber, Diameter, Surface, or Volume distributions.4.3 As class mid points Diof the histogram intervals thearithmetic mean values of the class boundaries are used.4.4 Particle shape factors are not taken into account, al-though their importance in particle size analysis is beyonddoubt.4.5 A coherent nomenclature system is presented whichconveys the physical meanings of mean particle diameters.5. Significance and Use5.1 Mean particle diameters defined according to theMoment-Ratio (M-R) system are derived from ratios betweentwo moments of a particle size distribution.6. Mean Particle Sizes/Diameters6.1 Moments of Distributions:6.1.1 Moments are the basis for defining mean sizes andstandard deviations. A random sample, containing N elementsfrom a population of particle sizes Di, enables estimation of themoments of the size distribution of the population of particlesizes. The r-th sample moment, denoted by Mr’, is defined tobe:Mr’: 5 N21(iniDir(1)where N5(ini, Diis the midpoint of the i-th interval and niis the number of particles in the i-th size class (that is, classfrequency). The (arithmetic) sample mean M1’of the particlesize D is mostly represented by D¯. The r-th sample momentabout the mean D¯, denoted by Mr, is defined by:Mr: 5 N21(ini~Di2 D¯!r(2)6.1.2 The best-known example is the sample variance M2.This M2always underestimates the population varianceσD2(squared standard deviation). Instead, M2multiplied byN/(N–1) is used, which yields an unbiased estimator, sD2, forthe population variance. Thus, the sample variance sD2has tobe calculated from the equation:sD25NN 2 1M25(ini~Di2 D¯!2N 2 1(3)6.1.3 Its square root is the standard deviation sDof thesample (see also 6.3). If the particle sizes D are lognormallydistributed, then the logarithm of D,lnD, follows a normaldistribution (Gaussian distribution). The geometric mean D¯gofthe particle sizes D equals the exponential of the (arithmetic)mean of the (lnD)-values:D¯g5 exp@N21(ini~lnDi!#5!NΠiDini(4)6.1.4 The standard deviation slnDof the (lnD)-values can beexpressed as:slnD5Œ(ini$ln~Di/D¯g!%2N 2 1(5)6.2 Definition of Mean Diameters D¯p,q:6.2.1 The mean diameter D¯p,qof a sample of particle sizes isdefined as 1/(p – q)-th power of the ratio of the p-th and theq-th moment of the Number distribution of the particle sizes:D¯p,q5FMp’Mq’ G1/~p2q!if pfiq (6)6.2.2 Using Eq 1, Eq 6 can be rewritten as:D¯p,q53(iniDip(iniDiq41/~p2q!if pfiq (7)6.2.3 The powers p and q may have any real value. Forequal values of p and q it is possible to derive from Eq 7 that:D¯q,q5 exp3(iniDiqlnDi(iniDiq4if p 5 q (8)6.2.4 If q = 0, then:D¯0,05 exp3(inilnDi(ini45!NΠiDini(9)6.2.5 D¯0,0is the well-known geometric mean diameter. Thephysical dimension of any D¯p,qis equal to that of D itself.E2578 − 07 (2018)26.2.6 Mean diameters D¯p,qof a sample can be estimatedfrom any size distribution fr(D) according to equations similarto Eq 7 and 8:D¯p,q53(imfr~Di!Dip2r(imfr~Di!Diq2r41/p2qif pfiq (10)and:D¯p,p5 exp3(imfr~Di!Dip2rlnDi(imfr~Di!Dip2r 4if p 5 q (11)where:fr(Di) = particle quantity in the i-th class,Di= midpoint of the i-th class interval,r = 0, 1, 2, or 3 represents the type of quantity, viz.number, diameter, surface, volume (or mass)respectively, andm = number of classes.6.2.7 If r = 0 and we put ni= f0(Di), then Eq 10 reduces tothe familiar form Eq 7.6.3 Standard Deviation:6.3.1 According to Eq 3, the standard deviation of theNumber distribution of a sample of particle sizes can beestimated from:sD5Œ(iniDi22 ND¯1,02N 2 1(12)which can be rewritten as:s 5 c=D¯2,022 D¯1,02(13)with:c 5 =N/~N 2 1! (14)6.3.2 In practice, N >> 100, so that c ≈ 1. Hence:s =D¯2,022 D¯1,02(15)6.3.3 The standard deviation slnDof a lognormal Numberdistribution of particle sizes D can be estimated by (see Eq 12):slnD5Œ(ini$ln~Di/D¯0,0!%2N 2 1(16)6.3.4 In particle-size analysis, the quantity sgis referred toas the geometric standard deviation2although it is not astandard deviation in its true sense:sg5 exp@slnD# (17)6.4 Relationships Between Mean Diameters D¯p,q:6.4.1 It can be shown that:D¯p,0# D¯m,0if p # m (18)and that:D¯p21, q21# D¯p,q(19)6.4.2 Differences between mean diameters decrease accord-ing as the uniformity of the particle sizes D increases. Theequal sign applies when all particles are of the same size. Thus,the differences between the values of the mean diametersprovide already an indication of the dispersion of the particlesizes.6.4.3 Another relationship very useful for relating severalmean particle diameters has the form:@D¯p,q#p2q5 D¯p,0p/D¯q,0q(20)6.4.4 For example, for p = 3 and q =2:D¯3,25D¯3,03/D¯2,02.6.4.5 Eq 20 is particularly useful when a specific meandiameter cannot be measured directly. Its value may becalculated from two other, but measurable mean diameters.6.4.6 Eq 7 also shows that:D¯p,q5 D¯q,p(21)6.4.7 This simple symmetry relationship plays an importantrole in the use of D¯p,q.6.4.8 The sum O of the subscripts p and q is called the orderof the mean diameter D¯p,q:O 5 p1q (22)6.4.9 For lognormal particle-size distributions, there exists avery important relationship between mean diameters:TABLE 1 Nomenclature for Mean Particle Diameters D¯p,qSystematicCodeNomenclatureD¯23.0harmonic mean volume diameterD¯22.1diameter-weighted harmonic mean volumediameterD¯21.2surface-weighted harmonic mean volume di-ameterD¯22.0harmonic mean surface diameterD¯21.1diameter-weighted harmonic mean surfacediameterD¯21.0harmonic mean diameterD¯0.0geometric mean diameterD¯1.1diameter-weighted geometric mean diameterD¯2.2surface-weighted geometric mean diameterD¯3.3volume-weighted geometric mean diameterD¯1.0arithmetic mean diameterD¯2.1diameter-weighted mean diameterD¯3.2surface-weighted mean diameterD¯4.3volume-weighted mean diameterD¯2.0mean surface diameterD¯3.1diameter-weighted mean surface diameterD¯4.2surface-weighted mean surface diameterD¯5.3volume-weighted mean surface diameterD¯3.0mean volume diameterD¯4.1diameter-weighted mean volume diameterD¯5.2surface-weighted mean volume diameterD¯6.3volume-weighted mean volume diameterE2578 − 07 (2018)3D¯p,q5 D¯0,0exp@~p1q!slnD2/2# (23)6.4.10 Eq 23 is a good approximation for a sample if thenumber of particles in the sample is large (N > 500), thestandard deviation σlnD< 0.7 and the order O of D¯p,qnot largerthan 10. Erroneous results will be obtained if these require-ments are not fulfilled. For lognormal particle-sizedistributions, the values of the mean diameters of the sameorder are equal. Conversely, an equality between the values ofthese mean diameters points to lognormality of a particle-sizedistribution. For this type of distribution a mean diameter D¯p,qcan be rewritten as D¯j,j, where j =(p + q)/2 = O/2, if O is even.6.4.11 Sample calculations of mean particle diameters and(geometric) standard deviation are presented in Appendix X1.7. Nomenclature of Mean Particle Sizes/Diameters67.1 Table 1 presents the M-R nomenclature of meandiameters, an unambiguous list without redundancy. Thisnomenclature conveys the physical meanings of mean particlediameters.7.2 The mean diameter D¯3.2(also called: Sauter-diameter) isinversely proportional to the volume specific surface area.8. Keywords8.1 distribution; equivalent size; mass distribution; meanparticle size; mean particle diameter; moment; particle size;size distribution; surface distribution; volume distributionAPPENDIX(Nonmandatory Information)X1. SAMPLE CALCULATIONS OF MEAN PARTICLE DIAMETERSX1.1 Estimation of mean particle diameters and standarddeviations can be demonstrated by using an example from theliterature citing the results of a microscopic measurement of asample of fine quartz (Table X1.1).3The notation of the classboundaries in Table X1.1 was chosen to remove any doubts asto the classification of a particular particle size. A histogram ofthese data is shown in Fig. X1.1. The standard deviation of thissize distribution, according to Eq 12, equals 2.08 µm. Thegeometric standard deviation, according to Eq 16 and 17,equals 1.494.X1.1.1 Values of some mean particle diameters D¯p,qof thissize distribution, calculated according to Eq 7 and 8, are:D¯0,054.75 µm, D¯1,055.14 µm, D¯2,055.55 µm, D¯3,055.95 µm,andD¯3,256.84 µm, D¯3,357.26 µm, D¯4,357.64 µmX1.1.2 Fig. X1.2 shows that the distribution indeed is fairlylognormal, because the data points on lognormal probabilitypaper fit a straight line.X1.1.3 This lognormal probability plot allows for a graphi-cal estimation of the geometric mean diameter D¯0,0and thegeometric standard deviation sg:X1.1.3.1 For lognormal distributions, the value of D¯0,0equals the median value, the 50 % point of the distribution,being about 4.8 µm.X1.1.3.2 The values of the particle sizes at the 2.3 % and97.7 % points are about 2.15 µm and 10.8 µm, respectively.This range covers four standard deviations. Therefore, thestand