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Pharmaceutical Physics

n the July 2001 Pharmaceutical Technology article about particle-size determination, the question of what constitutesa “correct” method for the determination of particle-size dis-tributions was addressed (1). A correct method was defined as

one whose sample was obtained by an appropriate sampling pro-cedure, in which the sample was prepared properly and intro-duced into the instrument, and in which all instrumental para-meters were used correctly for the analysis. It also was pointedout that all of the correct (but differing) particle-size results ob-tained through various methodologies are equally accurate, buteach method simply might be expressing its correct results indifferent terms. When viewed in this light, the decision aboutwhich particle-size methodology is most appropriate for a givensituation can be seen as a simple matter of accuracy versus pre-cision. If absolute accuracy is most important, then one mustconduct rigorous research to verify that the method finallyadopted does indeed yield particle-size results that are absolutelyindicative of the characteristics of the bulk material. If, however,one is more interested in developing profiles of lot-to-lot vari-ability, then the use of any of the available methods that yieldscorrect results is appropriate.

The next several articles in the “Pharmaceutical Physics” col-umn series will examine a variety of correct methodologies thatcan be used to deduce information about the shape and sizedistribution of particles. However, one cannot begin to addressthose topics without a prior exposition of what is meant by theshape and size of the particles that consitute a powdered solid.

Particle shapeIt is not possible to discuss rationally the size of a particle orany distribution associated with the sizes of an ensemble of par-ticles without first considering the three-dimensional charac-teristics of the particle itself. This is because the size of a parti-cle is expressed either in terms of linear dimension characteristicsderived from its shape or in terms of its projected surface orvolume. As will be shown, some methods of expressing parti-cle size discard any concept of particle shape and instead ex-press the size in terms of some type of equivalent spherical size.

An appropriate starting place for a discussion of particle shapecan be found in USP General Test �776� (see Figure 1) (2). In theshape performance aspect of this particular test procedure, USP

Particle-Size Distribution, Part I

Representations of ParticleShape, Size, and DistributionHarry G. Brittain

Harry G. Brittain, PhD, is the director ofthe Center for Pharmaceutical Physics, 10Charles Road, Milford, NJ 08848, tel. 908.996.3509, fax 908.996.3560, hbrittain@earthlink.net. He is a member of Pharma-ceutical Technology’s Editorial AdvisoryBoard.

I

Particle-size determinations are undertakento obtain information about the sizecharacteristics of an ensemble of particles.Because the particles being studied are notusually the exact same size, information isrequired regarding the average particle sizeand the distribution of sizes about thisaverage. However, the concept of particlesize is irrevocably derived from aspects ofparticle shape and morphology because theidea of a particle diameter proceeds frompreconceived shape factors.

www.pharmtech.com

requires that “for irregularly shaped particles, characterizationof particle size must also include information on particle shape.”The general method defines several descriptors of particle shape(see Figure 2). The USP definitions of these shape parameters are● acicular: slender, needle-like particle of similar width and

thickness● columnar: long, thin particle with a width and thickness that

are greater than those of an acicular particle● flake: thin, flat particle of similar length and width

● plate: flat particle of similar length and width but with greaterthickness than flakes

● lath: long, thin, blade-like particle● equant: particles of similar length, width, and thickness; both

cubical and spherical particles are included.In ordinary practice, one rarely observes discrete particles but

typically is confronted with particles that have aggregated or ag-glomerated into more-complex structures. USP provides sev-eral terms that describe any degree of association:● lamellar: stacked plates● aggregate: mass of adhered particles● agglomerate: fused or cemented particles● conglomerate: mixture of two or more types of particles● spherulite: radial cluster● drusy: particle covered with tiny particles (2).

The particle condition also can be described by another se-ries of terms:● edges: angular, rounded, smooth, sharp, fractured● optical: color, transparent, translucent, opaque● defects: occlusions, inclusions.

Furthermore, surface characteristics can be described as● cracked: partial split, break, or fissure● smooth: free of irregularities, roughness, or projections● porous: having openings or passageways● rough: bumpy, uneven, not smooth● pitted: small indentations.

The pharmaceutical descriptors of particle shape are derivedfrom the general concept of crystallographic habit. The exactshape acquired by a crystal will depend on various factors suchas the temperature, pressure, and composition of the crystal-lizing solution. Nevertheless, precipitation of a given compoundgenerally creates a characteristic shape or outline. Because thefaces of a crystal must reflect the internal structure of the solid,

Plate Tabular

Equant

Blade

Acicular

Columnar

Figure 1: Description of particle shape as defined by USP.

Figure 2: Growth along certain crystal directions can profoundly alter the characteristic habit of various crystals.

TabularLamellar Equant Columnar Acicular

Isom

etricTetrag

on

alH

exago

nal

1011 0111

0001

the angles between any two faces of a crystal will remain thesame even if the crystal growth is accelerated or retarded in onedirection or another (see Figure 2). Optical crystallographersusually will catalogue the various crystal faces and documentthe angles between them as they identify the crystal system towhich the given particle belongs. When the particle is particu-larly well formed, a description of symmetry elements also iscompiled.

For many individuals, however, the concept of qualitativeshape descriptors has proven inadequate, and this deficiencyhas necessitated the definition of more quantitatively definedshape coefficients (3). For instance, Heywood describes the elon-gation ratio, n, as

[1]

and the flakiness ratio, m, as

[2]

in which T is the particle thickness (the minimum distance be-tween two parallel planes that are tangential to opposite surfacesof the particle), B is the breadth of the particle (the minimumdistance between two parallel planes that are perpendicular tothe planes defining the thickness), and L is the particle length(the distance between two parallel planes that are perpendicu-lar to the planes defining thickness and breadth) (4).

Particle sizeIt really is not possible to continue a discussion of particle shapeor size without first developing definitions of particle diame-ter. This step is, of course, rather trivial for a spherical particlebecause its size is uniquely determined by its diameter. For ir-regular particles, however, the concept of size requires defini-tion by one or more parameters. It often is most convenient todiscuss particle size in terms of derived diameters such as aspherical diameter that is in some way equivalent to some sizeproperty of the particle. These latter properties are calculatedby measuring a size-dependent property of the particle and re-lating it to a linear dimension.

Certainly the most commonly used measurements of particlesizes are the length (the longest dimension from edge to edge of aparticle oriented parallel to the ocular scale) and the width (thelongest dimension of the particle measured at right angles to thelength). Intuitive as these properties may be, their definition stillis best shown in Figure 3a. Closely related to these properties aretwo other descriptors of particle size: Feret’s diameter, which is the

m � BT

n � LB

distance between imaginary parallel lines tangent to a ran-domly oriented particle and perpendicular to the ocularscale, and Martin’s diameter, which is the diameter of theparticle at the point that divides a randomly oriented par-ticle into two equal projected areas (see Figure 3b).

The coordinate system associated with the measurementis implicit in the definitions of length, width, Feret’s di-ameter, and Martin’s diameter because the magnitude ofthese quantities requires some reference point. As such,these descriptors are most useful when discussing particle

size as measured by microscopy because the particles are im-mobile. Defining spatial descriptors for freely tumbling parti-cles is considerably more difficult and hence requires the defi-nition of a series of derived particle descriptors. However, giventhe popularity of techniques such as electrozone sensing or laserlight scattering, derived statements of particle diameter are ex-tremely useful.

All of the derived descriptors for particle size begin with thehomogenization of the length and width descriptors into eithera circular or spherical equivalent and make use of the ordinarygeometrical equations associated with the derived equivalent.For instance, the perimeter diameter is defined as the diameterof a circle having the same perimeter as the projected outline ofthe particle. The surface diameter is the diameter of a sphere hav-ing the same surface area as the particle, and the volume dia-meter is defined as the diameter of a sphere having the same vol-ume as the particle. One of the most widely used deriveddescriptors is the projected area diameter, which is the diameterof a circle having the same area as the projected area of the par-ticle resting in a stable position. The concept of projected areadiameter is illustrated in Figure 3c.

Several other derived descriptors of particle diameter havebeen used for various applications. For instance, the sieve dia-meter is the width of the minimum square aperture throughwhich the particle will pass. Other descriptors that have beenused are the drag diameter, which is the diameter of a sphere hav-ing the same resistance to motion as the particle in a fluid of thesame viscosity and at the same velocity; the free-falling diameter,which is the diameter of a sphere having the same density andthe same free-falling speed as the particle in a fluid of the samedensity and viscosity; and the Stokes diameter, which is the free-falling diameter of a particle in the laminar-flow region.

Distribution of particle sizesAll analysts know that the particles that constitute real samples ofpowdered substances do not consist of any single type but insteadwill generally exhibit a range of shapes and sizes. Particle-size de-terminations therefore are undertaken to obtain information aboutthe size characteristics of an ensemble of particles. Furthermore,because the particles being studied are not the exact same size, in-formation is required about the average particle size and the dis-tribution of sizes about that average.

One could imagine the situation in which a bell-shaped curveis found to describe the distribution of particle sizes in a hy-pothetical sample; this type of system is known as the normaldistribution. Samples that conform to the characteristics of anormal distribution are described fully by a mean particle size

Feret's diameter

(a) (b) (c)

Length

Width

Martin'sdiameter

Projected area

diameter

Figure 3: Some commonly used descriptors of particle size.

and the standard deviation. Table I shows an example of a sam-ple exhibiting a normal distribution in which 3000 particleshave been sorted according to an undefined determiner of theirsize. In the usual data representation, the number of particlesin each size fraction is identified, and then one calculates thepercentage of particles in each size fraction. This calculation

yields the particle size histogram (seeFigure 4a). The number frequency ordi-narily is used to construct a cumulativedistribution, which can be ascending ordescending depending on the nature ofthe study and what information is re-quired (see Figure 4b).

The arithmetic mean of the ensembleof particle diameters is calculated usingthe relation

[3]

in which n is the number of particleshaving a diameter equal to di. The stan-dard deviation in the distribution thenis calculated using

[4]

In the example shown in Table I, one cal-culates that dav � 30.2 �m and that � =1.1.

The most commonly occurring valuein the distribution is the mode, which isthe value at which the frequency repre-sentation is a maximum value. The me-dian divides the frequency curve into twoequal parts and equals the particle sizeat which the cumulative representation

equals 50%. In a rigorous normal distribution, the mean, mode,and median have the same value. For a slightly skewed distrib-ution, however, the following approximate relationship holds:

[5]

It would be highly advantageous if powder distributions couldbe described by the normal distribution function because all ofthe statistical procedures developed for Gaussian distributionscould be used to describe the properties of the sample. How-ever, unless the range of particle sizes is extremely narrow, mostpowder samples cannot be described adequately by the normaldistribution function. The size distribution of the majority ofreal powder samples usually is skewed toward the larger end ofthe particle-size scale. Such powders are better described by thelog-normal distribution type. This terminology arose becausewhen the particle distribution is plotted by means of the loga-rithm of the particle size, the skewed curve is transformed intoone closely resembling a normal distribution (see Figure 5).

The distribution in a log-normal representation can be com-pletely specified by two parameters: the geometric median par-ticle size (dg) and the standard deviation in the geometric mean(�g). The geometric median is the particle size pertaining to the50% value in the cumulative distribution and is calculated using

mean�mode�3 mean�median

� ��d av � d i �

2

n

12

d av �ndi

n

Table I: Particle composition of a hypothetical sampleexhibiting a normal distribution.Size Number Number Percent Percent(�m) in Band Frequency Less Than Greater Than5 50 1.67 1.67 98.33

10 90 3.00 4.67 95.3315 110 3.67 8.33 91.6720 280 9.33 17.67 82.3325 580 19.33 37.00 63.0030 600 20.00 57.00 43.0035 540 18.00 75.00 25.0040 360 12.00 87.00 13.0045 170 5.67 92.67 7.3350 120 4.00 96.67 3.3355 60 2.00 98.67 1.3360 40 1.33 100.00 0.00Total 3000 100

20

15

10

5

0

Particle size (�m)

Num

ber

freq

uenc

y 100

80

60

40

20

0

(a) (b)

0 10 20 30 40 50 60

Particle size (�m)

Cum

ulat

ive

dist

ribut

ion

Figure 4: Particle-size representations for a hypothetical normal distribution. Shown are (a) thefrequency distribution and (b) the cumulative distribution.

20

15

10

5

0

20

15

10

5

0

Particle size (�m)

Num

ber

freq

uenc

y

(a) (b)

10 1000 10 20 30 40 50 60 70

Particle size (�m)

Num

ber

freq

uenc

y

Figure 5: Particle-size representations for a hypothetical log-normal distribution, plotted on a (a)linear scale and on a (b) logarithmic scale.

[6]

in which n is the number of particles having particle size equal

d g � antilogn log�d i �

n

to di. Two samples having identical dg and�g values can be said to have been drawnfrom the same total population and ex-hibit properties of characteristics of thetotal population.

In many applications, particle-size re-sults are processed by plotting the cu-mulative frequency data on a logarith-mic scale. If a straight line is obtained,the particle-size distribution is said toobey the log-normal function. The valueof dg is equal to the 50% value of the cu-mulative distribution. The value of �g

is obtained by dividing the 84.1% valueof the distribution by the 50% value.

Although the distribution in the log-normal representation is specified com-

pletely by the geometric median particle size and the geometricmean standard deviation, a number of other average values havebeen derived to define useful properties. These values are espe-cially useful when the physical significance of the geometric me-dian particle size is not clear. The arithmetic mean (dav) particlesize is defined as the sum of all particle diameters divided by thetotal number of particles and is calculated using Equation 3. Thesurface mean (ds) particle size is defined as the diameter of a hy-pothetical particle having an average surface area and is calcu-lated using

[7]

The volume mean (dv) particle size is the diameter of a hypo-thetical particle having an average volume and is obtained from

[8]

The volume-surface mean (dvs) particle size is the average sizebased on the specific surface per unit volume and is calculatedusing

[9]

For the distribution plotted in Figure 5, one can calculate thatdg � 32.91 �m, dav � 34.42 �m, ds � 35.93 �m, dv � 37.43 �m,and dvs � 40.62 �m.

Various types of physical significance have been attached tothe various expressions of particle size. For chemical reactions,the surface mean is important, although for pigments the vol-ume mean value is the appropriate parameter. Deposition ofparticles in the respiratory tract is related to the weight meandiameter, and the dissolution of particulate matter is related tothe volume-surface mean.

Particle-size distributions can be sorted according to the mass(or volume) of the particles contained within a given size bandor to the number of particles contained in the same size band.

d vs �nd i

3

nd i2

d v �nd i

3

n

13

d s �nd i

2

n

12

30

20

10

0

% mass frequency% number frequency Cumulative

% massCumulative % number

100

80

60

40

20

0

Size (�m)

% in

bon

d(a) (b)

0 15 30 45 60 75

Size (�m)

% fi

ner

Figure 6: Particle-size representations for a hypothetical log-normal distribution. Shown are (a)the frequency distribution and (b) the cumulative distribution. Each contains the differenceobtained when processing the data in terms of either particle number or particle mass.

1012

1620

30

40

50

70

100

140

200

270325

2,0001,500

1,000

500400300

200150

100

50

0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.5

Cumulative percentage of undersize particles

Mes

h (U

S s

tand

ard

siev

e se

ries)

USstandard

MM M��

1012

1620

30

40

50

70

100

140

200

270325

2,0001,500

1,000

500400300

200150

100

50

0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.5

Cumulative percentage of undersize particles

Mes

h (U

S s

tand

ard

siev

e se

ries)

USstandard

(b)

(a)

MM M��

Figure 7: Particle-size representations plotted in a log-probabilityformat for (a) a single hypothetical log-normal distribution and for(b) a hypothetical sample containing two log-normal distributionswhose average particle size differs by 50%.

With substances having real density values, the distrib-ution of the same ensemble of particles can look quitedifferent depending on how the data are plotted. Figure6 shows the frequency and cumulative distribution plotsfor the same sample, but the data have been separatelyprocessed in terms of the mass and particle numbers.

Unfortunately not every powdered sample is charac-terized by the existence of a single distribution, and thecharacter of real samples can be quite complicated. Rec-ognizing the existence of multimodal distributions is not alwaysa straightforward process, but their existence often can be de-tected by plotting the data on log-probability paper. The exis-tence of more than one particle population is indicated by achange in the slope of the line. Figure 7 shows a single log-nor-mal distribution and a multimodal sample consisting of twopopulations whose mean differed by approximately 50%. Thebreak in the log plot is clearly evident, but if one were to simplyplot the latter sample in either a frequency or cumulative view,one would not have been able to detect the existence of two par-ticle-size populations in the sample.

SummaryThis rather simplified discussion of particle shape, size, and dis-tribution represents only an introduction to the topic. Inter-ested readers should consult the primary sources in the list of

recommended references to additional information (see “Rec-ommended reading” sidebar). Most highly recommended arethe various editions of Particle Size Measurement by Allen be-cause they contain some of the most detailed and informativeexpositions available about these topics. However, the scope ofthe discussion in this opening article provides a sufficient basisfor the expositions of the various methodologies that will fol-low in subsequent installments of this column.

References1. H.G. Brittain, “What is the ‘Correct’ Method to Use for Particle-Size

Determination?” Pharm. Technol. 25 (7), 96–98 (2001).2. “Optical Microscopy,” General Test �776�, USP 24 (The United States

Pharmacopoeial Convention, Rockville, MD, 2000), pp. 1965–1967.3. T. Allen, Particle Size Measurement (Chapman and Hall, London, 3rd

ed.,1981) pp. 107–120.4. H. Heywood, J. Pharm. Pharmacol. (S15) 56T, (1963). PT

● R.R. Irani and C.F. Callis, Particle Size: Measurement, Interpretation, and Application (JohnWiley & Sons, New York, 1963).

● Z.K. Jelinek, Particle Size Analysis (Ellis Horwood Ltd., Chichester, 1970).● J.D. Stockham and E.G. Fochtman, Particle Size Analysis (Ann Arbor Science Publishers,

Ann Arbor, MI, 1977).● B.H. Kaye, Direct Characterization of Fine Particles (John Wiley & Sons, New York, 1981).● H.G. Barth, Modern Methods of Particle Size Analysis (John Wiley & Sons, New York, 1984).● T. Allen, Particle Size Measurement, 5th ed. (Chapman and Hall, London, 1997).

Recommended reading

©Reprinted from

December 2001AN ADVANSTAR ★ PUBLICATION

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