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Particle-SizeDistrib ribution, Part I
Representationsof Particle Shape, Size, andDistribution Harry G.Brittain G. Brittain
I Particle-size determinations are undertaken to obtain informa i nformatio tion n about the size size characteristi chara cteristics cs of an ensembl ensemble e of particles. particles. Because Bec ause the particles particles being studied studied are not usually the exact exact same same size, size,inf informatio ormation n is required requir ed regarding the average average particle particle size size and the distrib distribution ution of sizes about this average.However, average. However,the the concept concept of particle size is irrevoca irrevocabl bly y derived from from aspects aspects of particle shape and morphology because the idea id ea of a particle parti cle diameter diameter proce proceeds eds from preconceived preconce ived shape factors.
n the July 2001 Pharma Phar mace ceuti cal cal Technology Technology article about pa part rt icle-si icle-si ze de determin termina ation, ti on, the ques questi on of what what consti onstitut tute es a“ corre orr ect” method method for f or the t he de determinati termination on of pa part rticle-s icle-size ize distr i buti but i ons was was address ddressed (1). ( 1). A correct method was defined as one whose whose sample wasobta obt ained by an appr appropr opriiate sampli mpl ing proprocedure dur e, i n which whi ch the sa sample mpl e was was prepared prepared properly and int i ntroroduce duced into i nto the t he ins in strume tr ument, nt, and in i n which all all ins in strume tr umental ntal paraparameters were were use used corre corr ectly ctl y for t he anal analysi ysi s. I t also was was point poi nte ed out tha t hatt all of the correc correctt (but diffe dif feri ring ng)) pa part rticle-s icle-size ize res results ult s obtai ned through thr ough var vari ous methodol methodologiesare equally quall y acc accur urate ate,, but each method simply might be expressing its correct results in diff di ffe erent terms. terms. When When viewed viewed in thi t his s li ght, the th e de decis cision ion about which particle-size methodology is most appropriate for a given si tuation uati on can can be se seen as asi mple mpl e mat mat ter of accur accura acy ver versus precisi cisi on. I f absolut bsolute e accurac ccuracy y is most most import i mport ant, nt , then one must must conduct rigorous research to verify that the method finally adopted does indeed yield particle-size results that are absolutely indic indi cative ti veof the cha charac racte teri ris stics ti csof the bulk material. material. If, If , however, ver, one is more intere in teres sted in deve developing lopi ng profiles profi lesof lot-t lot -to-l o-lot ot variabili bil i ty, then the use use of any of the ava availil able methods methods that that yields correct correct result ul t s is appropr appropr i ate. The next next seve severral ar ar ti clesi n the t he “ Phar Pharmace maceuti ut i cal cal Physi Physi cs” column seri es will wil l exami xamine ne a vari vari et y of correct correct methodol methodologie ogies s that that can be used to deduce information about the shape and size distri bution buti on of pa part rt icles. Howeve However, r, one cannot cannot be begin gin to t o addre addres ss those those topics topi cswit hout a prior pri or expos exposititii on of what what is mea meant by the shape and size of t he par par t i clest hat consi consi tute tu te a powdered powdered soli soli d.
Parti Particle shape
directo r of H a r r y G . B r i t t a in in , P h D , is the directo the Center for Pharmaceutical Physics, 10 Cha rles rles Roa d, Milford, Milford, NJ 08848, 08848, tel. 908. 996.3509, fax 908.996.3560, hbrittain@ ea rthli rthlink.net. He is a membe r of PharmaAdviso ry ceutical Technolog Technolog y’s Editorial Adviso Board.
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Pharmaceutic al Technology DECEMBER 2001
I t i s not poss possi ble to dis di scuss cuss rati onally the t he si ze of a pa part rt i cle or any distri distr i buti but ion associated with wit h the si zes of an ens ense emble mbl e of parti cles cles without with out fi rst consi consi de deri ri ng the three t hree-di mens mensii onal onal charac charac-teris teri sti csof the part part icle icl e i tself tself.. This Thi s is beca becaus use e the si si zeof a part part i cle is expres expressed eithe eit herr in terms of line li nea ar dime dimens nsiion characteri racteris stics de deri ri ved ved from fr om its i ts shape shape or i n te t erms rm s of i t s proj ected cted sur surfac face e or volume. volume. As wil l be shown, some methods methods of expressing in g part part i cle si si ze dis di scard card any conce concept pt of parti part i cle shape shape and i nstead nstead expre press t he si zei n te t erms of some type of equivale qui valent nt spherica pheri call size. size. An appropri appropr i ate start startiing place place for a dis di scuss cussion of par particle ti cle shape hape can be found in USP Gener General al Test 776 (see Figure 1) (2). (2) . I n the t he shapeper performance for manceaspect pect of t his hi s parti part i cular te t est proce procedur dure e, USP USP www.pharmtech.com
requires that “for irregularly shaped parti cles, characteri zation plate: flat parti cle of simi lar length and width but with greater of particle size must also include information on particle shape.” thickness than flakes The general method definesseveral descriptors of particle shape lath: long, thin, blade-li ke particle (see Figure 2). The USP definitions of these shapeparameters are equant: particlesof similar length, width, and thickness; both acicular: slender, needle-like part icle of simi lar width and cubical and spherical particlesare included. thickness In ordinary practice, one rarely observes discrete particlesbut columnar: long, thi n part icle with a width and thi ckness that typically is confronted with particles that have aggregated or agare greater than those of an acicular particle glomerated into more-complex structures. USP provides sev flake: thin, flat particle of similar length and width eral terms that describe any degree of association: lamellar: stacked plates aggregate: mass of adhered particles agglomerate: fused or cemented part icles conglomerate: mixture of two or more types of particles spherul ite: radial cluster Plate Tabular drusy: part icle covered with tiny part icles (2). The particle conditi on also can be described by another series of terms: edges: angular, rounded, smooth, sharp, fractured optical: color, transparent, translucent, opaque defects: occlusions, inclusions. Furthermore, surface characteristics can be descri bed as cracked: partial spli t, break, or fissure smooth: free of irregularities, roughness, or projecti ons porous: having openings or passageways Equant Columnar rough: bumpy, uneven, not smooth pitted: small indentations. The pharmaceutical descriptors of particle shape are derived Blade from the general concept of crystallographic habit. The exact shape acquired by a crystal will depend on various factors such as the temperature, pressure, and compositi on of the crystallizing solution. Nevertheless, precipitati on of a given compound Acicular generally creates a characteristic shape or outline. Because the Figure 1: Description of partic le shape as def ined by USP. faces of a crystal must reflect the internal structure of the solid, Lamellar
Tabular
Equant
Columnar
Acicular I s o m e t r i c
T e t r a g o n a l
0001 H e x a g o n a l
1011
0111
Figure 2: Growth along certain crystal directions can profoundly alter the characteristic habit of various crystals. 40
Pharmaceutic al Technology DECEMBER 2001
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(a)
(b) h t n g L e
W i d t h
(c) Feret's diameter
Martin's diameter
Projected area diameter
Figure 3: Some commonly used descriptors of particle size.
the angles between any two faces of a crystal will remain the sameeven if the crystal growt h is accelerated or retarded in one directi on or another ( see Figure 2). Opti cal crystallographers usually will catalogue the various crystal faces and document the angles between them as they identify the crystal system to which the given part icle belongs. When the part icle is part icularly well formed, a description of symmetr y elements also is compiled. For many individuals, however, the concept of quali tative shape descri ptors has proven inadequate, and this deficiency has necessitated the definiti on of more quantitatively defi ned shape coefficients (3). For instance, Heywood describesthe elongation ratio, n, as [1] and the flakiness ratio, m, as [2] in which T is the particle thickness (the minimum di stance between two parallel planesthat are tangential to opposite surfaces of the parti cle), B is the breadth of the particle (the minimum distance between two parallel planes that are perpendicular to the planes defi ning the thickness), and L is the part icle length (the distance between two parallel planes that are perpendicular to the planes defining thickness and breadth) (4).
Particle size It really is not possible to conti nue a discussion of particle shape or size wit hout first developing definit ions of particle diameter. This step is, of course, rather trivial for a spheri cal particle because its size is uni quely determi ned by its diameter. For irregular parti cles, however, the concept of size requires definiti on by one or more parameters. It often is most convenient to discuss particle size in terms of derived di ameters such as a spherical diameter that i s in some way equivalent to some size property of the parti cle. These latter propert ies are calculated by measuring a size-dependent property of the particle and relating it to a linear dimension. Certainly the most commonly used measurements of particle sizes are the length (the longest dimension from edgeto edge of a particle ori ented parallel to the ocular scale) and thewidth (t he longest dimension of the particle measured at r ight angles to the length). Intuiti ve as these properties may be, their defi nition still is best shown in Figure 3a. Closely related to these properti es are two other descriptors of particle size: Feret’s diameter, 42
Pharmaceutic al Technology DECEMBER 2001
which is the distance between imaginary parallel lines tangent to a randomly oriented parti cle and perpendicular to the ocular scale, and Martin’s diameter, which is the diameter of the particle at the point that dividesa randomly oriented particle into two equal projected areas (see Figure 3b). The coordinate system associated with the measurement is implicit i n the definitions of length, width, Feret’s diameter, and Mart in’s diameter because the magnitude of these quantities requires some reference point. As such, these descriptors are most useful when discussing particle size as measured by microscopy because the particles are immobile. Defining spati al descriptors for freely tumbling parti cles is considerably more diff icult and hence requires the definiti on of a seri es of derived particle descriptors.However, given the popularity of techniques such as electrozone sensing or l aser light scattering, derived statements of particle diameter are extremely useful. All of the derived descriptors for particle size begin with the homogenization of the length and width descriptors into either a circular or spherical equivalent and make use of the ordinary geometrical equations associated with the derived equivalent. For instance, the perimeter diameter is defined as the diameter of a circle having the same peri meter asthe projected outline of the particle.The surfacediameter is the diameter of a sphere having the same surface area as the particle, and the volume diameter is defined as the diameter of a sphere having the samevolume as the part icle. One of the most wi dely used derived descriptors is the projected area diameter, which is the diameter of a circle having the same area asthe projected area of the parti cle resti ng in a stable position. The concept of projected area diameter is illustrated in Figure 3c. Several other deri ved descriptors of parti cle diameter have been used for various applicati ons. For instance, the sieve diameter is the widt h of the mini mum square apert ure through which the part icle will pass. Other descriptors that have been used are the drag diameter, which is the diameter of a sphere having the same resistanceto motion asthe parti cle in a fluid of the same viscosity and at the same velocity; the free-falling diameter, which is the diameter of a sphere having the same density and the same free-falling speed as the particle in a fluid of the same density and vi scosity; and the Stokes diameter, which is the freefalling diameter of a particle in the laminar-flow region.
Distributionof particlesizes All analysts know that the particles that constitute real samples of powdered substances do not consist of any single type but instead will generally exhibit a range of shapes and sizes. Parti cle-size determi nations therefore are undertaken to obtain information about the sizecharacteristi csof an ensemble of part icles. Furthermore, because the parti cles being studied are not the exact same size, information i s required about the average particle size and the distribution of sizes about that average. One could imagine the situation in which a bell-shaped curve is found to describe the distr ibution of part icle sizes in a hypothetical sample; thi s type of system is known as thenormal www.pharmtech.com
(a)
(b) n 100 o i t u 80 b i r t s i d 60 e v 40 i t a l u 20 m u C 0
20
y c n e15 u q e r 10 f r e b m 5 u N
0
0
Particle size ( m)
10 20
30
40
50 60
Particle size ( m)
Figure 4: Particle-size representations for a hypothetical normal distribution. Shown are (a) the frequency distribution and (b) the cum ulative distribution.
in each size fr action is identifi ed, and then one calculatesthe percentageof particlesin each sizefr action. Thi s calculati on yi elds the parti cle size histogram (see Figure 4a). The number frequency ordinari ly i s used to construct a cumulative distribut ion, which can be ascending or descending depending on the nature of the study and what information is required (see Figure 4b). The ari thmeti c mean of the ensemble of part icle diameters is calculated using the relation [3]
(a)
(b)
y 20
c n e u15 q e r f r 10 e b m u 5 N
in which n is the number of part icles having a diameter equal to d i . The standard deviation in the distribution t hen is calculated using
y 20
c n e u 15 q e r f r 10 e b m u 5 N
0
0 10
[4]
In the example shown in Table I, one calculates that d av 30.2 m and that = Particle size (m) Particle size ( m) 1.1. The most commonly occurring value Figure 5: Particle-size representations for a hypothetical log-normal distribution, plotted on a (a) in the distribution is themode, which is linear scale and on a (b) logarithmic scale. the value at which the frequency representation is a maximum value. The median divides the frequency curve into two equal parts and equals Table I: Particle composition of a hypothetical sample the particle size at which the cumulati ve representation equals exhibiting a norm al distr ibution. 50%. In a rigorous normal distribution, the mean, mode, and median have the same value. For a slightly skewed distribution, Size Num ber Num ber Percent Percent however, the following approximate relationship holds: ( m ) in Band Frequency Less Than Greater Than 0 10 20 30 40 50 60 70
5
50
1.67
1.67
98.33
10
90
3.00
4.67
95.33
15
110
3.67
8.33
100
[5]
91.67
It would be highly advantageous if powder distributions could be described by the normal distri bution functi on because all of 25 580 19.33 37.00 63.00 the stati stical procedures developed for Gaussian distributions 30 600 20.00 57.00 43.00 could be used to describe the properti es of the sample. How35 540 18.00 75.00 25.00 ever, unless the range of particle sizes is extremely narrow,most 40 360 12.00 87.00 13.00 powder samples cannot be descri bed adequately by the normal 45 170 5.67 92.67 7.33 distribution function. The size distribution of the majority of 50 120 4.00 96.67 3.33 real powder samples usually is skewed toward the larger end of 55 60 2.00 98.67 1.33 the particle-size scale. Such powders are better described by the 60 40 1.33 100.00 0.00 log-normal distribut ion type. This terminology arose because Total 3000 100 when the part icle distr ibuti on is plotted by means of the logadistribution. Samples that conform to t he characteri sti cs of a ri thm of the part icle size, the skewed curve is tr ansformed into normal distribut ion are described full y by a mean particle size one closely resembling a normal distribution (see Figure 5). and the standard deviation. Table I shows an example of a samThe distr ibuti on in a log-normal representati on can be comple exhibiting a normal distribution in which 3000 particles pletely specified by two parameters: the geometri c median parhave been sorted according to an undefi ned determiner of their ticle size(d g) and the standard deviation in the geometric mean size. In the usual data representati on, the number of part icles 20
280
9.33
17.67
82.33
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in which n is the number of particles having particle size equal to d i . Two samples (b) (a) 30 100 having identical d % mass g and g values can be frequency 80 d said to have been drawn from the same r n20 % number e o 60 n total populati on and exhibit properties b frequency i Cumulative f n i % mass of characteristicsof the total population. % 40 %10 Cumulative In many appli cati ons, parti cle-sizere20 % number sults are processed by plotting the cu0 0 mulati ve frequency data on a logari th0 15 30 45 60 75 mic scale. If a str aight li ne is obtained, Size (m) Size (m) the particle-size distribution is said to obey the log-normal function. The value of d Figure 6: Particle-size representations for a hypothetical log-norm al distribution. Shown are (a) g is equal to the 50% value of the cumulative distr ibuti on. The value of g the frequency distribution and (b) the cumulative distribution. Each contains the difference is obtained by dividing the 84.1% value obtained when processing the data in term s of either particle number or particle m ass. of the distr ibution by the 50% value. Although the distribution in t he lognormal representation is specified completely by the geometric median particle size and the geometric mean standard de(a) US M standard M M viation, a number of other averagevalues have been derived to 2,000 ) 10 s 12 define useful properties. These values are especially useful when 1,500 e i r 16 e the physical signifi cance of the geometric median particle size 1,000 s 20 e is not clear. The ari thmetic mean (d av ) particle size is defined v e 30 i 500 s as the sum of all parti cle diameters divi ded by the total num40 d 400 r a ber of part icles and i s calculated using Equati on 3. The surface 300 d 50 n 70 mean ( d a 200 t s) parti cle size is defi ned as the diameter of a hypo s 100 150 thetical particle having an average surface area and is calculated S 140 U 100 ( using h 200 s e270 M325 0.5 1 2
50 5 10 20 30 40 506070 80 90 95 98 99 99.5
[ 7]
Cumulative percentage of undersize particles
(b) US standard
) 10 s 12 e i r 16 e s 20 e v e i 30 s 40 d r a d 50 n a 70 t s 100 S 140 U ( h 200 s e270 M325 0.5 1 2
M
M
M 2,000 1,500 1,000 500 400 300 200 150 100
50 5 10 20 30 40 506070 80 90 95 98 99 99.5
The volume mean ( d v ) parti cle size is the diameter of a hypothetical particle having an average volume and is obtained from [ 8]
The volume-surface mean ( d vs) particle size is the average size based on the specific surface per unit volume and is calculated using [9]
Cumulative percentage of undersize particles
For the distr ibuti on plotted in Figure 5, one can calculate that 32.91 m, d av 34.42 m, d 35.93 m, d v 37.43 m, d g s and d vs 40.62 m. Various types of physical significance have been attached to the vari ous expressions of particle size. For chemical reactions, the surface mean is important, although for pigments the vol( g). The geometric median is the part icle size pertaining to the ume mean value is the appropr iate parameter. Deposition of 50% value in the cumulativedistribution and is calculated using particles in the respiratory tract is related to the weight mean diameter,and the dissolution of parti culate matter i s related to [6] the volume-surface mean. Particle-sizedistributions can be sorted according to the mass (or volume) of the particlescontained within a given sizeband Figure 7: Particle-size representations plotted in a log-probability format f or (a) a single hypothetical log-normal distribution and for (b) a hypothetical sample containing tw o log-normal distributions whose average particle size differs by 50%.
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Recommended r eading or to the number of particlescontained in the same size R.R.IraniandC.F.Callis,Particle Size: Measurement, Interpretation, and Application (John band. With substances having real density values, the disWiley&Sons,NewYork,1963). tribution of the same ensemble of particlescan look quite Z.K.Jelinek,Particle Size Analysis (EllisHorwoodLtd.,Chichester,1970). different depending on how the data are plotted.Figure J.D.StockhamandE.G.Fochtman,Particle Size Analysis (AnnArborSciencePublishers, 6 shows the frequency and cumulati ve distri bution plots AnnArbor,MI,1977). for the same sample, but the data have been separately B.H.Kaye,Direct Character ization o f Fine Part icles (JohnWiley&Sons,NewYork,1981). processed in terms of the mass and particle numbers. H.G.Barth,Modern Methods of Particle Size Analysis (JohnWiley&Sons,NewYork,1984). Unfortunately not every powdered sample is charac T.Allen,Particle Size Measuremen t, 5thed.(ChapmanandHall,London,1997). terized by the existence of a single distri bution, and the character of real samples can be quite complicated. Recogniz- recommended references to additional i nformation ( see“Recing the existence of mult imodal distri buti ons is not always a ommended reading” sidebar). Most highly recommended are straightforward process, but their existence often can be detected the various editions of Particle Size Measurement by Allen beby plotting the data on log-probability paper. The existence of cause they contain some of the most detailed and informative more than one particle population is indicated by a change in expositions available about these topics. However, the scope of the slope of the li ne.Figure 7 shows a single log-normal distri- the discussion in this opening article provides a suffi cient basis bution and a multimodal sample consisting of two populations for t he expositi ons of the vari ous methodologies that will folwhose mean differed by approximately 50%. The break in the low in subsequent installments of thi s column. log plot i s clearly evident, but if one were to simply plot the latter sample in either a frequency or cumulati ve view,one would References not have been able to detect the existence of two particle-size 1. H.G. Brittain, “What is the‘Correct’ Method to Use for Particle-Size Determination?” Pharm. Technol. 25(7), 96–98 (2001). populations in the sample.
Summary This rather simpli fied discussion of particle shape, size, and distr ibution represents only an int roduction to the topic. Interested readers should consult the primary sources in the list of
2. “Optical Mi croscopy,” General Test 776, USP 24 (The United States Pharmacopoeial Convention, Rockville, MD, 2000), pp. 1965–1967. 3. T.Allen, Parti cle Size Measurement (Chapman and Hall, London,3rd ed.,1981) pp. 107–120. 4. H. Heywood, J. Pharm. Pharmacol. (S15) 56T, (1963). PT
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