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Probability foundations of particle statistics
Probability Foundations of Particle Statistics H. -I-. McADAMS EncironnzenfalSection. if-eaponr Research Depr.. Cornell Aeronaurica~Lnborarory. Inc.. Buffalo, N-Y. (U.S.A.) (Received Nowmber 12, 1968)
Attempts to characterize systems of particles of carying geometry by a scalar quantity called “particle size“ hare given rise to a profusion of characteristic lengths
often
referred
to
as particle
“diameters”.
TypicalLy. these diameters are a- consequence of the measurement system, and J’urther proliferation of terms results when characteristic lengths are statistically reduced to some cersion of mean particle sire. Finally7 shape factors are introduced in an attempt to reconcile rhe tzrious quantities and to serre as conuersion factors from one system of measurement and statistical treatment to another. In the paper, it is shown that the literature
of particle
diameters,
particle statistics and shape factors can be unified through the notfon of an induced probability space. Applicable concepts of probability theory and the theor_r- of stochastic processes relecance to particle systems
are ra-iewed and their demonstrated.
A7ey Words-probability, probability space, particle size, particle shape, particle orientation, size distribution, mean particle ske, expectation, dimensional analysis_
IKi-RODUCTiON
Particle size statistics are expressions of the physicogeometric properties of particulate systems: dis-
tributions
of these
statistics
depend
on
both
particle geometry and the physical principles employed to sense that geometry. The intimate relation between the physical and geometric properties of particles has long been tacitly recognized in the realization that each method of particle size analysis has its own peculiar size regime of applicability. Conventional sieve analysis is useful down to some minimum size at which interparticle forces seriously interfere with sieving action ; sedimentation processes are Iimited by hydrodynamic and aerodynamic laws ; and Powder Techdogy
optical methods such as light or electron microscopy are subject to resolution limitations imposed by the wave mechanics of photons or electrons. Whereas the physical constrainis on particle size analysis can be ordered in terms of established physical laws. however, no organizing principle has evolved to place the geometric constraints on particle size analysis in proper perspective. It is the purpose of this paper to undertake such a unification and to show that concepts such as Martin-s diameter, Feret’s diameter, and the various statistical diameters fall within its purviewAs is well known, particle size statistics depend on the conventions used in defining the size parameter as well as on the physical principle employed in the measurement process. The very notion of “particle size” implies the existence of a single scalar quantity or characteristic length in terms of which particle geometry can be expressed_ In general, however, particles are irregular in shape and their geometry can not be reduced to a single dimension without appreciable sacrifice of information_ Even if shape is specified, some convention must be introduced whereby one or more characteristic lengths can be specified in terms of one or more measurement vectors in a reference fi-ame attached to the particle. Thus, orientation of the measurement vector, as well as particle shape, is an essential concern in prescribing what is meant by the “size” of a particle. TeIevision screens, though square in shape, are typically characterized (advantageously to the vendor) by a diagonal dimension, and the effect of orientation of the measurement vector can affect the characterization of particle size in powder systems in a similar manner- Indeed_ the effect may be even more spurious as a consequence of the fact that the orientation of the measurement vector often can not be controlled except in a statistical sense. The result is a statistical distribution of measurements determined in part by the geometry of the particles and in part by the ensemble of measurement vectors employed to probe the particulate systexx~
- Elsevk Sequoia !%A_, Lawam e - Printed in the Netherlands
PROBABILITY
FDUN~ATION.5
The particle size measurements and their distribution constitute a probability space induced from the space of measurement vectors by the powder system. The process by which such an induced space arises will be investigated. and its consequences for particle size analysis examined. Even though the details of the process may not be invoked in routine application of particle statistics, the probability foundations of these statistics determine ultimately their proper use.
DEFIXITIOSS
Ah?>
PRELDIIXARY
XOTIONS
For the purposes of this discussion, a particle size measurement will be considered to be the magnitude of a line vector whose initial and terminal points are determined by two distinct points on the particle surface_ Often the initial and terminal points of the measurement vector will actually be points lying on the particle surface, but it will not always be possible to define the measurement rector in this way. For example, it often occurs that only particle projections are available for measurement. In these instances, the points determining the measurement vector are not two points on the particle surface but rather their projection images_ Thus, in general. it is convenient to admit a transformation of the
particle surface and to define measurement
vectors
in terms of the image of the transformation rather than in terms of the pre-image. In many cases, the transformation will be the identity transformation, and the measurement vectors will be vectors originating and terminating on the particle surface. The proposed definition constitutes a considerable generalization of prevailing notions of a particle size parameter but will be seen to be advantageous to the statistical development which follows. Moreover, the definition will be seen to reduce, under appropriate restrictions, to the usual precepts of particle size analysis. For example, consider a spherical particle and an identity transformation of its surface. If no further constraints are invoked, any chord would qualify as a legitimate size parameter under the generalized definition_ If these chords are further restricted to those which pass through the geometric center of the particle, however, the set of allowed descriptors reduces to the set of sphere diameters, in consonance with prevailing notions. In general, the set of conventional descriptors will be seen to be subsets of the set of descriptors allowed under the proposed
OF PARTICLE
261
STATKl-ICS
definition. Indeed, it will often be advantageous to constrain the measurement vectors to pass throuz+ the geometric center of the particle, just as it w;lll
often be convenient to assume an -identity transformation of the particle surface_ But, in general. many instances will arise in which it will be impractical to impose these constraints, e.g., when only particle projected areas or plane crosssections are accessible for measurement. To define a statistical distribution of particle size measurements one must resolve such questions as the following : (I) What set or collection of measurement vectors are to be considered? (2) What relative frequency or probability is to be attached to subsets of these vectors? (3) What system of particles is to be considered? In the previous discussion, it was more or less assumed that the system of particles was confined to a single object In reality, however, it is some collection or population of particles that is of interest. In the sequel, it will be shown that a distribution of size measurements can arise within a single particle as a consequence of orientation of measurement vectors, and that this distribution is modified when a number of particles are considered_ Thus the total variation in particIe ,Sze measurements consists of intraparticle variation and interparticle variation.
Intraparticle variation arises as a consequence of particle shape and the orientation of measurement vectors relative to the particle reference frame. Interparticle variation arises as a consequence of variation in geometry from one particle to another.
PROBABILITY
CDSCEP-IS
In the interests of rigor in dealing with the statistical distribution of particie size measurements, it is essential to develop a few basic concepts of probability-in particular. the definition of a probability space and the manner in which a given probabiIity space can induce or determine another probability space A probability space consists of a collection of elementary events, a family of events which are subcollections of the elementary events, and a function which assigns to every event a probability. In the framework of particle size analysis, it is convenient to regard the set of allowable measurement vectors as the collection of elementary events. These vectors comprise a probability space if some way PonderTechnoi..2
(1968/59) x0-x8
262
H. T. MCADAM!j
can be found to group the vectors into subsets and assign to each subset a probability_ For example, we might admit into our collection only vectors passing through the geometric center of a sphere circumscribing the particle. If the collection of vectors sweeps out 47~ solid angle, then we might invoke a concept of “uniform probability density” and assign to any solid angle of magnitude 0 the probability 8/4x_ The result is a well-delined probability space which induces on the selected particle a probability space of characteristic lengths. Consider, for example, a cube of edge s circumscribed by a sphere of diameter s,/3, as shown in Fig. 1. Imagine a vector u passing through the center of the sphere and having initial point I and terminal point 7’ on the cube. Let the vector u be extended to intersect the surface of the circumscribing sphere at points S, and S,. As the points I and T move over the cube surface, the points Sr and S, move over the surface of the sphere- To any closed trajectory on the sphere surface there corresponds a definite area. This area, expressed as a fraction of the area of the hemisphere, is taken as the probability associated with a cone of measurement vectors, as shown in the figure. Thus, a means is provided whereby a distribution of random orientations induces a distribution of real numbers, the lengths of the measurement vectors originating and terminating on the surface of the cube. Let V denote the random variable associated with these lengths, and consider the case for V = q where F is a
specific value assumed by random variable V_ All vectors of length Y < Dare contained in a cone which intersects the cube face in a circle C, and the circumscribed sphere in a circle Cz_ Circle C, forms the base of a spherica segment, and the cube face, if projected onto the surface of the sphere, defines a four-sided spherical polygon of area (&)(4x)&/Z - ,/3)* = rz/Z Because of the symmetry of the cube, the probability Pr(V< v) is given as the ratio of the area of the circle C2 to the area of the spherical polygon_ Zvaluation of the areas involved gives, for the cumulative distribution of V. F(v) = 3 [I - (s/v)]
s< v< sJ2
(I)
and, upon differentiation, the probability function
density
j-(u) = (35)/f?
s,c v
(2)
Notice, however, that eqns. (1) and (2) define the distribution only over the interval s< v< s&Z, whereas the complete range of the random variable requires definition over the interval s< v< sJ3. With a bit more algebra and trigonometry’, one obtains (39/L?
s< v
f(v) = ( [(35)/u’] [ 1 - (4/x) cos- ‘(s/Jz=7)] sJ2,( vf sJ3 Integration produces the cumulative distribution function F(v) as shown in Fig. 2 The example illustrates the general procedure by which a distribution of measurement vector orientations, together with’ a particle shape, induces a distribution of characteristic lengths. The mean mx of a random variable X is its expectation. For a continuous random variable X with
probability density function defined as
fx(x), expectation is
m, = ECXJ = j;=xfxtxlh The expected value of the random variable V for a given particle shape, therefore, yields an average dimension my-_
Fig 1_ Derivationof cumulativedistributionof meas:,‘ement vectorltngthr
-
E[V]
= j;_
vfvivjdv
which might be called the irmaparticle mean for that shape, given the assumed distribution of measurement vector orientations Suppose, now, that the size of the cube, as Ponder
Tech?wL. 2 (1968169) 260-268
PROBABILITY
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FOUX~ATIONS
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1. . . .__ . .
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.-.-
_.:.I ::.:_/zT
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-.-._..
-- I‘ . ..
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: __ _ . . _ _ ..._ _.-_ iI/: . _---._ . _.
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measured by its edge length. is a random variable S-that is, we are dealing not with a single cube of edge s but with a collection of cubes having a probability density function Is(s)_ There exists an expected inter-particle value for the random variable S, namely E[S] = nz,.Then, if this collection of spheres is probed by measurement vectors in random orientation, the combined expected mean size nrc can be computed as a function of f\-(c) and fs(s)_ In particular. if all particles in the system are geometrically
similar
(in the case
assumed
above,
they are all cubes), the intraparticle mean mV can be expressed as a ratio nr&, which is constant regardless of the magnitude ofs. Then, if orientation of the measurement vectors is assumed to be independent of cube edge !engths, one has J-m
as the combined mean size. In general, of course, particle size, particle shape, and measurement vector orientation would be jointly distributed random variables. Consider now a case in which there is a certain degree of preferred orientation in the measurement vectors. For example, suppose that the cube is probed only by vectors lying in a plane passing through the center of the cube and parallel to a cube face. Let the measurement vectors pass through the
OF PARTICLE
263
STATISTICS
center of the cube and originate and terminate on the cube surface Further_ Irt these vectors be uniformly distributed in azimuth. Then a probability space of measurement vectors is defined, and it might be anticipaied that this distribution would induce a quite different distribution ofsize measurements than in the original example. In fact. it can be shown that the cumulatire distribution function ior the induced random variable V is
(4;5i)cos-1(s,!r)s,cr
ifr=s otherwise
Then E[V] = m,=s and my/s= 1. If the length of the cube edge is a random variable S, then the combined mean size is just the mean of the interparticle distribution_ It is seen. therefore. that particle size. shape and orientation are factors which must be considered simultaneously in any measurement of particle statistics. Even when shape is known and a particular geometric feature is chosen as characteristic dimension, one is unobtrusively being influenced by particle shape and orientation_ Indeed. the subtle interplay of particle size, particle shape and orientation is often insidious in determining the distribution of a particle statistic_ For example, the size distribution of spheres causes reIatively little diffrculty and it is fcr this reason, of course, that the concept of “equivalent sphere” is often invoked in an effort to simplify the description of particulate Pox-tierTt-chnoI_. 2
(196S;69)
X50-268
2&l
H. I-. MCADAMS
systems_ Note, however, that this simplification rests upon two important properties of spheres: (1) A sphere (like a cube) can be characterized by a single parameter, its radius, and (2) A sphere (unlike a cube) is, in a certain sense, a nonorientable body_ Particle shape ax4 c rientation, apparently eliminated from cansideration, are in fact only disguised through a nonchalant artifice. Once spherical geon etry is postulated, particle shape is specified and c rientation is also seemingly legislated out of rxister :e. A sphere, when rotated about its center. does not change its aspect to the viewer and orientation seems, therefore, of little concern. In our more general view of measurement vectors and their distribution, however, the sphere need not be probed by vectors passing through its center. Thus chord lengths are legitimate size measurements and may, indeed, be the only measurements generally available. Such a case is often encountered, not only for spheres but for other particles as well, when particles are embedded in a matrix and are accessible only by sections taken through the matrix. The diameters of the spheres must then be inferred from the mpzured diameters of spherical segments revealed by sectioning’. Each of the above assumed probability spaces for the measurement vectors, as well as many others which might be postulated, finds application, and each is appropriate within a given context_ For example, if we hare complete control over orientation-Le., we can probe the particle at will-we might prefer to measure cubes along an axis and quote their edge lengths_ In microscopic analysis, however, we might find it convenient to measure particles only in a fixed direction and, by assuming random azimuthal orientation of the particles, obtain a size parameter such as Martin’s diameter or Feret’s diameter. Finally, if the particles are suspended in a matrix in random orientation, as they might be XI turbidimetric measurement, observations may reflect random orientation in the sense of equal probability for all directions in space.
PARTICLE
SIZE
AVERAGES
AND
OTHER
STATISTICS
Average particle size, as is well known, assumes various meanings, and certainly the various familiar types of averages have been well discussed in the literature of particle size analysis Because of biases
often inherent in particle size measurement, shape factors are often introduced, and their formulation and use are also well documented. It is a thesis of this paper, however, that even with the existing backlog of theory pertaining to mean diameters and shape factors, subtleties and nuances can arise which may not fall within the framework of existing concepts. These subtleties become apparent, however, if the probability space underlying the statistical analysis is fully appreciated. As an example of the way in which bias can arise in the characterization of particle size, consider a cube with measurements taken through its center parallel to a face but in random azimuth. The measurements constitute realizations of a random variable V with probability density function s,c a< sJ2
fv (c) = (4!7c)[(4/(r ,/?=3] and expectation E[V]
= (4/z)
I’“’ C(rsdr)/(z*\/m)]
.s
= (45/n) log [s(l i- ,/2)-J/s = 1.12 s. Thus, if the random process is intended to yield an average value approximating the cube edge length. one must multiply the average value by the factor l/1.12=0.88 in order to eliminate the bias introduced by the method of measurement. As has been previously shown, if a population of cubes of various edge lengths were measured in the manner indicated, each cube would bc similarly affected, as would also be the expected value taken with respect to the random variable S denoting the distribution of cube edge lengths. It is, ofcourse, in dealing with the random variable S that the need arises again for rigor in definition of a probability space and for being sensitive to the way in which one probability space may give rise to another. Suppose, for example, that one is dealing with a collection of cubes having edge lengths varying between 1 cm and 2 cm_ Further, let us suppose that the number of cubes contained in any subinterval--say between 1.1 and 12 cm-is proportional to the length of the interval. Then, the probability density function of S is
fs(4 =
(01
l
Pon-drrTechnoL. Z (1%8/69) 260-268
PROBABILITY
FOUNDATIONS
Then
OF PARTICLE
265
Si-ATISTICS
distributed in the interval 1 ,( S’,( 4and construct a cumulative distribution of S according to F,(s) = j:
x’dx,j~x’dx =(2 -
I)/7
2 =
1
sds The corresponding
= 1.5 where 1.5 cm is a legitimate average particle size based on a distribution induced by the number density of particles. An equally legitimate average, in the statistical sense at least, is obtained ifone considers an induced probability density function for S in which the probability density associated with any interval between 1 and 2 cm is proportional to the volume of particles exhibiting edge lengths in this interval. Having assumed that the edge lengths are uniformly distributed between 1 and 2 cm, so that
f,(s) ={
1
l
0
otherwise
we note that particle voIumes are distributed in the interval I< S3 < 8. The cumulative distribution of S is given by I=&)=
_~~x’dr/S:r’d*-=(s~-l),15 ICC<2 .-.
and the probability density function is fs(s) ( ;h)zhenJs;
ss 2
Then
where 1.653 cm is the average size usually attributed the name t-olume mean diameter (or m mean diumetm if particle density is assumed constant) A similar analysis yields an average diameter which might be called the surface mean diameter*Here we note that the surface of one face of a cube is
l
This diamcta
is oftencallul
probability density function is
and
= (+) i;?ds
= 1.607
where 1.607 cm is the surface average diameter_ Quite clearly other mean diameters can arise ifthe distribution of measurements is defined in terms of other functicns of characteristic lengths of the particles Should appropriate occasion arise. for example there is no reason to reject a distributicn induced by an arbitrary function of particle dimensions, say the $ power of cube edge dimensions. In all instances, of course, the induced distribution will be affected by the set of allowed orientations constituting the probability space from which the intraparticle measurement distribution was induced. The compounding of complexities might well terminate here, were it not for the fact that the notion of expectation of a random variable can be generalized. In a manner quite analogous to the nap in which E [X] is defined for a random variable X, one can define the expectation E[g(X)] of some function of the random variable Then
and it is seen that the equation reduces to the usual expectation of the random variable X if g(X) = X. Through this generalized definition of expectation. quantities such as mean-volume diameter and mean-surface diameter arise Consider, for example, the case in which cubes have a uniform distribution of edge lengths in the interval l
E CsWI = E PSI the
mean vo!umcsulface
diameter
andis usedin connation withconsiderations of spaific surface per unitvolume; see for example Dalia Valle’_ Powder
Technol_. -7 (1968j69) X0-268
266
H. -I-_ MCADAMS
-
What
has been determined here is the average of a particle wherein expectation is performed with respect to a sample space based on numbers of particIes If one takes the cube root of E [g(S)], one obtains (3.75)3 = 1.55, the so-called mean-volume diameter_ This is an “equivalent” diameter, in the sense that it represents the size a cube would have to be in order to exhibit the mean volume 3.75 cm’_ Note, however, that instead of taking g(S)=S3. we might have taken g(S)=6S’. Then ~oizme
E [g(S)]
= E [6S=] 2
=
I 6szds=
15.
This number denotes the average surface arca of a cube, and ,/v= 1.58 is the “equivalent” size which a cube would have to assume in order to exhibit a mean area of 15 cm” in the abol:e sense.
SHAPE
FACTORS
In view of the multiplicity of charactlristic lengths which can be assigned to a particle or I o a collection of particles, it is of interest to disco\ :r, if possible, means whereby these- various le.&& can be reconciled To achieve such an sun requires that we be able to fmd dimensionless factors capable of transforming one characteristic dimension into another. The resuh is a shape factor, on which there is already a considerable body of literature. As noted by Herdanq shape factors can arise in at least three ways : (1) As factors of proportionality to reconciie discordant values obtained by different methods of measurement ; (2) As conversion factors for expressing measurements in terms of an equivalent sphere (or some other equivalent shape); (3) As non-dimensional ratios which relate such quantities as particle surface to the second power of a characteristic length, or particle vohune to the third power of a characteristic length. The first implies consideration of physicogeometric laws of similitude and requires that both the geometry of the particle as well as certain physical quantities be related so that scaling laws can be established. The second, by invoking a principle of “equivalence”. postulates an invariance which obtains ifa particle is transformed from one geometric
shape into another. The third considers particle shape fmed and seeks non-dimensional quantities which persist as factors describing the invariance of shape as size undergoes a scaIing transformation_ Evidently various combinations of these considerations can arise, with the result that a multiplicity of shape factors are produced if one considers a11 the contexts wherein scaling cuts across change in particle size, particle shape, and particle response to physical factors. Perhaps the simplest, or at least most straightforward examples of shape factors arising from measurement technique are factors which occur as a consequence of orientation of measurement vectors For example, consider the previously discussed case of a cube measured in two sets of orientations : (1) Along a principal axis througb its center; (2) Along random directions through its center. in a plane paralIe1 to a cube face, all azimuths being equally likelyIf the cube has side s, then nz, = s = mean characteristic length for the first case. and m,=s[(4/X) Iog(l+_J2)] = 1.12s = mean characteristic length for the second case. Then, for the cube, we can define a shape factor a2, = mJml
=
1.12
which might be called an “orientation shape factor” for cubes. It is quite evident that an unlimited number of such orientation shape factors might be constructed when it is realized that an unlimited number of ways exist for constraining or controlling the set of orientations allowed the measurement vectors. Often the set of orientations allowed is tied to or determined by the physical constraints of the measurement process. For example, if irregularly shaped particles are allowed to settle through a viscous medium, these particles often tend to orient themselves so as to experience maximum drag. (Falling leaves present an example.) Therefore, if measurement relies on Stokes’ law, the particles appear to settle at rates characteristic of spheres considerably smaller than spheres of volume and mass equivalent to those of the particles. The shape factor, in this instance. is the result of orientation induced by the measurement process itself. Another Powder Texhnd.
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PROBABILITY
FOUXXX4TIOSS
example. of course, is found in the microscopic measurement of particles in which preferred orientation is introduced by the specimen-preparation process Finally, even in the sieving process, lamellar or acicular particles seek preferred orientations for finding their way through the sieve openings, and this interaction influences the resultant sieve assessment of particle size. In fact, if one considers commonly employed methods cf size analysis, there are probably few in which the orientation of measurement vectors does not interact with the physical factors of the measurement process, and it is for this reason that the term physicogeometric is especially appropriate to particle size measurement processes. Shape factors arise as conversion factors for expressing measurements in terms of an equivalent shape as a straightforward consequence ofgeometry-. Somehow one must find an invariant meriting the term “size”, an invariant which does not depend on the ensemble of measurement vectors nor the physicogeometric laws by which the lengths of these vectors are sensed. That invariant is what is called “content” in the general geometry of N-dimensional space*. In N-dimensional space_ an orthotope is a closed N-dimensional figure bounded by a number of (N- I)-dimensional figures and having all edges perpendicular_ A regular orthotope is one having all edges of equal length. Thus, a line-se_gment is a (degenerate) regular orthotope in one-dimensional space, a square is a regular orthotope in twodimensional space, a cube is a regular orthotope in three-dimensional space. and so on_ The length of a line-segment is measured as a multiple of the unit length, the area of a rectangle is measured as a multiple of the unit square, and the volume of a
cube is measured as multiple of the unit cube. Collectively, length, area, volume, hypervolume are examples of content, and it is this content which is invariant under changes in orientation and measurement process In a mathematical sense we can regard each particle as plastic and continuously transformable into any required shape without either loss or gain of material or without change in material density_ For example, a sphere hypothetically can be transformed into a cube or tetrahedron by the proper application of force. Regardless of the multiplicity of forms assumed by the particle under this group of transformations, the volume of the particle is not changed. Thus it seems logical to identify the term
OF PARTICLE
STATISTICS
267
“size” with “content”-namely, volume if one is dealing with two-dimensional objects, area ifdealing with two-dimensional objects, and length if dealing with one-dimensional objects Though the postulating of an equivalent sphere is often invoked for standardizing the “content” of an irregular particle, a geometrically more logical choice of reference would be an equivalent cube, the regular orthotope in three-dimensional space. Indeed, mensuration formulas for the volume of a sphere, such as Volume = ($) jr? or Volume = (i)rc# are expressions of such a standardization. In the first formula, the factor_4x/3 can be considered as a shape factor for expressing the volume of the sphere in terms of a cube of side r, and in the second formula the factor xj6 can be considered as a shape factor for referring the sphere to a cube of side d. If the sphere is chosen as reference, then of course the applicable shape factors are simply the reciprocals 3147~ and 6/a respectivelyThe third class of shape factors arises in response to considerations of dimensional homogeneity and the formulation of non-dimensional constants in accordance with the Buckingham Pi Theorem_ If we admit only geometric quantities to be considered then only quantities involving length to some power need be involved In an N-dimensional polytope, the “contents.’ of geometrical figures of 1, 2, ___, N dimensions are recognized_ Thus a solid, or three-dimensional figure, exhibits volume, area (of its surface), and length (cf- previous discussion of characteristic lengths)_ One can therefore form such non-dimensional ratios as area/(length)’ volume/(leagth)3 and (volume)2i(area)3 Inasmuch as the characteristic lengths used in the denominator may have been derived subject to orientation and physical constraints, each of the above ratios is capable of proliferating into many, so that we would have, for example, a volume/ (length)3 based on Martin% diameter, Stokes’ diameter, and so on. The factors apply both directly and inversely. because in certain situations, such as adsorption measurements or pycnometxy,
268
H. T. MCADAhS
it may be area or volume which is sensed rather than some characteristic length. By inverse application of the shape factor. if it is known, one can deduce the corresponding characteristic length for the particle. An example is found in turbidimetric anaiysis, in which one senses light extinction by the presented area of particles in random orientation. By a theorem of Cauchy6, the mean presented area is one-fourth the area of the particle, and this relation is invariant regardless of particle shape, provided the particle is everywhere convex (that is, has no reentrant angles).
The orientations of vectors along which measurements are taken on particles, and the probability or relative frequency attached to these orientations, are seen to have far-reaching consequences in the evolving of particle statistics. The drstribution of intraparticle statistics is induced by the characteristic geometry of the particle acting in concert with an assumed or specified distribution of measurement vectors_ Because of physicogeometric properties of particles, orientation often interacts with or is determined by t&e physical measuring process. Moreover, it is shown that the various categories of shape factors are conditioned by orientation considerationsThe notion of an induced probability space provides a means whereby the multiplicity of charac:eristic lengths, statistical diameters and shape factors can be unified. By virtue of the fact that shape factors often reflect the consequence of physical as well as geometric controls, the use of two
or more
to The situation becomes extremely complex, hcwever, if particle size and shape vary simultaneously and are possibly correlated_ In using any particular set of particle statistics or method of measurement of particle size or shape, it is recommended that full cognizance be taken of the assumptions underlying the probability space applicable to the statistics The space should be chosen to be appropriate to the particular application at hand and need not adhere to recognized or long-used approaches_ It is believed that the concept of induced probability space, appropriately applied, provides a logic or rationale for this purpose. The application of this rationale, requiring as it does rigorous examination of assumptions, should make for improved understanding and interpretation of particle measurements obtain
physical
shape
principles
information
can be invoked
regarding
particles
REFERENCES 1 H. T. MCADASS, A statistical polyhedron model of abrasive &I-I, Z-mm_ ASME, MB, (4) (1963) 388-93. Z W. P. REID, Distribution of sizes of spheres in a solid from a study of dices of the solid, Tech. itfern. 996. U.S. i%cd Orhce l-es; starion, China me* Gzlz~. 1953. 3 J. M. DALLA VALL.E, Micromtitics, Pitman. New York. 1943.
p_ 39.
Statistics, Elsevier. Amsterdam, 1953, p_ 230. 5 D. M. Y. SOSWER~ An Introduction to the Geometry_ of_ N Dimmrrionr.Dover Publication New York. 1958. 6 M. A. CAUCHY, Note sur divers thCor&mesrelative5 la rectification da courbes, et B la quadrature dcs surfaces, Compt.
4 G. HERDAX, Small Parxicle
Rend_ I3 (1841) 106045.
Powder TecJmoL, 2 (1968/69) 260-268
Attempts to characterize systems of particles of carying geometry by a scalar quantity called “particle size“ hare given rise to a profusion of characteristic lengths
often
referred
to
as particle
“diameters”.
TypicalLy. these diameters are a- consequence of the measurement system, and J’urther proliferation of terms results when characteristic lengths are statistically reduced to some cersion of mean particle sire. Finally7 shape factors are introduced in an attempt to reconcile rhe tzrious quantities and to serre as conuersion factors from one system of measurement and statistical treatment to another. In the paper, it is shown that the literature
of particle
diameters,
particle statistics and shape factors can be unified through the notfon of an induced probability space. Applicable concepts of probability theory and the theor_r- of stochastic processes relecance to particle systems
are ra-iewed and their demonstrated.
A7ey Words-probability, probability space, particle size, particle shape, particle orientation, size distribution, mean particle ske, expectation, dimensional analysis_
IKi-RODUCTiON
Particle size statistics are expressions of the physicogeometric properties of particulate systems: dis-
tributions
of these
statistics
depend
on
both
particle geometry and the physical principles employed to sense that geometry. The intimate relation between the physical and geometric properties of particles has long been tacitly recognized in the realization that each method of particle size analysis has its own peculiar size regime of applicability. Conventional sieve analysis is useful down to some minimum size at which interparticle forces seriously interfere with sieving action ; sedimentation processes are Iimited by hydrodynamic and aerodynamic laws ; and Powder Techdogy
optical methods such as light or electron microscopy are subject to resolution limitations imposed by the wave mechanics of photons or electrons. Whereas the physical constrainis on particle size analysis can be ordered in terms of established physical laws. however, no organizing principle has evolved to place the geometric constraints on particle size analysis in proper perspective. It is the purpose of this paper to undertake such a unification and to show that concepts such as Martin-s diameter, Feret’s diameter, and the various statistical diameters fall within its purviewAs is well known, particle size statistics depend on the conventions used in defining the size parameter as well as on the physical principle employed in the measurement process. The very notion of “particle size” implies the existence of a single scalar quantity or characteristic length in terms of which particle geometry can be expressed_ In general, however, particles are irregular in shape and their geometry can not be reduced to a single dimension without appreciable sacrifice of information_ Even if shape is specified, some convention must be introduced whereby one or more characteristic lengths can be specified in terms of one or more measurement vectors in a reference fi-ame attached to the particle. Thus, orientation of the measurement vector, as well as particle shape, is an essential concern in prescribing what is meant by the “size” of a particle. TeIevision screens, though square in shape, are typically characterized (advantageously to the vendor) by a diagonal dimension, and the effect of orientation of the measurement vector can affect the characterization of particle size in powder systems in a similar manner- Indeed_ the effect may be even more spurious as a consequence of the fact that the orientation of the measurement vector often can not be controlled except in a statistical sense. The result is a statistical distribution of measurements determined in part by the geometry of the particles and in part by the ensemble of measurement vectors employed to probe the particulate systexx~
- Elsevk Sequoia !%A_, Lawam e - Printed in the Netherlands
PROBABILITY
FDUN~ATION.5
The particle size measurements and their distribution constitute a probability space induced from the space of measurement vectors by the powder system. The process by which such an induced space arises will be investigated. and its consequences for particle size analysis examined. Even though the details of the process may not be invoked in routine application of particle statistics, the probability foundations of these statistics determine ultimately their proper use.
DEFIXITIOSS
Ah?>
PRELDIIXARY
XOTIONS
For the purposes of this discussion, a particle size measurement will be considered to be the magnitude of a line vector whose initial and terminal points are determined by two distinct points on the particle surface_ Often the initial and terminal points of the measurement vector will actually be points lying on the particle surface, but it will not always be possible to define the measurement rector in this way. For example, it often occurs that only particle projections are available for measurement. In these instances, the points determining the measurement vector are not two points on the particle surface but rather their projection images_ Thus, in general. it is convenient to admit a transformation of the
particle surface and to define measurement
vectors
in terms of the image of the transformation rather than in terms of the pre-image. In many cases, the transformation will be the identity transformation, and the measurement vectors will be vectors originating and terminating on the particle surface. The proposed definition constitutes a considerable generalization of prevailing notions of a particle size parameter but will be seen to be advantageous to the statistical development which follows. Moreover, the definition will be seen to reduce, under appropriate restrictions, to the usual precepts of particle size analysis. For example, consider a spherical particle and an identity transformation of its surface. If no further constraints are invoked, any chord would qualify as a legitimate size parameter under the generalized definition_ If these chords are further restricted to those which pass through the geometric center of the particle, however, the set of allowed descriptors reduces to the set of sphere diameters, in consonance with prevailing notions. In general, the set of conventional descriptors will be seen to be subsets of the set of descriptors allowed under the proposed
OF PARTICLE
261
STATKl-ICS
definition. Indeed, it will often be advantageous to constrain the measurement vectors to pass throuz+ the geometric center of the particle, just as it w;lll
often be convenient to assume an -identity transformation of the particle surface_ But, in general. many instances will arise in which it will be impractical to impose these constraints, e.g., when only particle projected areas or plane crosssections are accessible for measurement. To define a statistical distribution of particle size measurements one must resolve such questions as the following : (I) What set or collection of measurement vectors are to be considered? (2) What relative frequency or probability is to be attached to subsets of these vectors? (3) What system of particles is to be considered? In the previous discussion, it was more or less assumed that the system of particles was confined to a single object In reality, however, it is some collection or population of particles that is of interest. In the sequel, it will be shown that a distribution of size measurements can arise within a single particle as a consequence of orientation of measurement vectors, and that this distribution is modified when a number of particles are considered_ Thus the total variation in particIe ,Sze measurements consists of intraparticle variation and interparticle variation.
Intraparticle variation arises as a consequence of particle shape and the orientation of measurement vectors relative to the particle reference frame. Interparticle variation arises as a consequence of variation in geometry from one particle to another.
PROBABILITY
CDSCEP-IS
In the interests of rigor in dealing with the statistical distribution of particie size measurements, it is essential to develop a few basic concepts of probability-in particular. the definition of a probability space and the manner in which a given probabiIity space can induce or determine another probability space A probability space consists of a collection of elementary events, a family of events which are subcollections of the elementary events, and a function which assigns to every event a probability. In the framework of particle size analysis, it is convenient to regard the set of allowable measurement vectors as the collection of elementary events. These vectors comprise a probability space if some way PonderTechnoi..2
(1968/59) x0-x8
262
H. T. MCADAM!j
can be found to group the vectors into subsets and assign to each subset a probability_ For example, we might admit into our collection only vectors passing through the geometric center of a sphere circumscribing the particle. If the collection of vectors sweeps out 47~ solid angle, then we might invoke a concept of “uniform probability density” and assign to any solid angle of magnitude 0 the probability 8/4x_ The result is a well-delined probability space which induces on the selected particle a probability space of characteristic lengths. Consider, for example, a cube of edge s circumscribed by a sphere of diameter s,/3, as shown in Fig. 1. Imagine a vector u passing through the center of the sphere and having initial point I and terminal point 7’ on the cube. Let the vector u be extended to intersect the surface of the circumscribing sphere at points S, and S,. As the points I and T move over the cube surface, the points Sr and S, move over the surface of the sphere- To any closed trajectory on the sphere surface there corresponds a definite area. This area, expressed as a fraction of the area of the hemisphere, is taken as the probability associated with a cone of measurement vectors, as shown in the figure. Thus, a means is provided whereby a distribution of random orientations induces a distribution of real numbers, the lengths of the measurement vectors originating and terminating on the surface of the cube. Let V denote the random variable associated with these lengths, and consider the case for V = q where F is a
specific value assumed by random variable V_ All vectors of length Y < Dare contained in a cone which intersects the cube face in a circle C, and the circumscribed sphere in a circle Cz_ Circle C, forms the base of a spherica segment, and the cube face, if projected onto the surface of the sphere, defines a four-sided spherical polygon of area (&)(4x)&/Z - ,/3)* = rz/Z Because of the symmetry of the cube, the probability Pr(V< v) is given as the ratio of the area of the circle C2 to the area of the spherical polygon_ Zvaluation of the areas involved gives, for the cumulative distribution of V. F(v) = 3 [I - (s/v)]
s< v< sJ2
(I)
and, upon differentiation, the probability function
density
j-(u) = (35)/f?
s,c v
(2)
Notice, however, that eqns. (1) and (2) define the distribution only over the interval s< v< s&Z, whereas the complete range of the random variable requires definition over the interval s< v< sJ3. With a bit more algebra and trigonometry’, one obtains (39/L?
s< v
f(v) = ( [(35)/u’] [ 1 - (4/x) cos- ‘(s/Jz=7)] sJ2,( vf sJ3 Integration produces the cumulative distribution function F(v) as shown in Fig. 2 The example illustrates the general procedure by which a distribution of measurement vector orientations, together with’ a particle shape, induces a distribution of characteristic lengths. The mean mx of a random variable X is its expectation. For a continuous random variable X with
probability density function defined as
fx(x), expectation is
m, = ECXJ = j;=xfxtxlh The expected value of the random variable V for a given particle shape, therefore, yields an average dimension my-_
Fig 1_ Derivationof cumulativedistributionof meas:,‘ement vectorltngthr
-
E[V]
= j;_
vfvivjdv
which might be called the irmaparticle mean for that shape, given the assumed distribution of measurement vector orientations Suppose, now, that the size of the cube, as Ponder
Tech?wL. 2 (1968169) 260-268
PROBABILITY
___.-- _.__. -. i
___~‘.__~‘__
__________
i
.
.
.
-
.
-
.
--
i
i
-_
.
FOUX~ATIONS
i
-
;
.
.
.
.
.
;
_
.._^_......L._
.
.
.
.
i
.
-
-
_
1. . . .__ . .
. ..+.
.-.-
_.:.I ::.:_/zT
__..
. . ___.___. -__ . . - _ _ __.. .. ^. . - . - --i __--____--_._.__.___--... . . - .___._. . . ___________ .__ _.......__ - _._._.. .._.
.;..I _ / . __.. _-_ .__ .^. -.-.. ^ ..-. --. _. ._...._--_ I
:
-.-._..
-- I‘ . ..
-
_
: __ _ . . _ _ ..._ _.-_ iI/: . _---._ . _.
.
/
,.T
.---
--
---
measured by its edge length. is a random variable S-that is, we are dealing not with a single cube of edge s but with a collection of cubes having a probability density function Is(s)_ There exists an expected inter-particle value for the random variable S, namely E[S] = nz,.Then, if this collection of spheres is probed by measurement vectors in random orientation, the combined expected mean size nrc can be computed as a function of f\-(c) and fs(s)_ In particular. if all particles in the system are geometrically
similar
(in the case
assumed
above,
they are all cubes), the intraparticle mean mV can be expressed as a ratio nr&, which is constant regardless of the magnitude ofs. Then, if orientation of the measurement vectors is assumed to be independent of cube edge !engths, one has J-m
as the combined mean size. In general, of course, particle size, particle shape, and measurement vector orientation would be jointly distributed random variables. Consider now a case in which there is a certain degree of preferred orientation in the measurement vectors. For example, suppose that the cube is probed only by vectors lying in a plane passing through the center of the cube and parallel to a cube face. Let the measurement vectors pass through the
OF PARTICLE
263
STATISTICS
center of the cube and originate and terminate on the cube surface Further_ Irt these vectors be uniformly distributed in azimuth. Then a probability space of measurement vectors is defined, and it might be anticipaied that this distribution would induce a quite different distribution ofsize measurements than in the original example. In fact. it can be shown that the cumulatire distribution function ior the induced random variable V is
(4;5i)cos-1(s,!r)s,cr
ifr=s otherwise
Then E[V] = m,=s and my/s= 1. If the length of the cube edge is a random variable S, then the combined mean size is just the mean of the interparticle distribution_ It is seen. therefore. that particle size. shape and orientation are factors which must be considered simultaneously in any measurement of particle statistics. Even when shape is known and a particular geometric feature is chosen as characteristic dimension, one is unobtrusively being influenced by particle shape and orientation_ Indeed. the subtle interplay of particle size, particle shape and orientation is often insidious in determining the distribution of a particle statistic_ For example, the size distribution of spheres causes reIatively little diffrculty and it is fcr this reason, of course, that the concept of “equivalent sphere” is often invoked in an effort to simplify the description of particulate Pox-tierTt-chnoI_. 2
(196S;69)
X50-268
2&l
H. I-. MCADAMS
systems_ Note, however, that this simplification rests upon two important properties of spheres: (1) A sphere (like a cube) can be characterized by a single parameter, its radius, and (2) A sphere (unlike a cube) is, in a certain sense, a nonorientable body_ Particle shape ax4 c rientation, apparently eliminated from cansideration, are in fact only disguised through a nonchalant artifice. Once spherical geon etry is postulated, particle shape is specified and c rientation is also seemingly legislated out of rxister :e. A sphere, when rotated about its center. does not change its aspect to the viewer and orientation seems, therefore, of little concern. In our more general view of measurement vectors and their distribution, however, the sphere need not be probed by vectors passing through its center. Thus chord lengths are legitimate size measurements and may, indeed, be the only measurements generally available. Such a case is often encountered, not only for spheres but for other particles as well, when particles are embedded in a matrix and are accessible only by sections taken through the matrix. The diameters of the spheres must then be inferred from the mpzured diameters of spherical segments revealed by sectioning’. Each of the above assumed probability spaces for the measurement vectors, as well as many others which might be postulated, finds application, and each is appropriate within a given context_ For example, if we hare complete control over orientation-Le., we can probe the particle at will-we might prefer to measure cubes along an axis and quote their edge lengths_ In microscopic analysis, however, we might find it convenient to measure particles only in a fixed direction and, by assuming random azimuthal orientation of the particles, obtain a size parameter such as Martin’s diameter or Feret’s diameter. Finally, if the particles are suspended in a matrix in random orientation, as they might be XI turbidimetric measurement, observations may reflect random orientation in the sense of equal probability for all directions in space.
PARTICLE
SIZE
AVERAGES
AND
OTHER
STATISTICS
Average particle size, as is well known, assumes various meanings, and certainly the various familiar types of averages have been well discussed in the literature of particle size analysis Because of biases
often inherent in particle size measurement, shape factors are often introduced, and their formulation and use are also well documented. It is a thesis of this paper, however, that even with the existing backlog of theory pertaining to mean diameters and shape factors, subtleties and nuances can arise which may not fall within the framework of existing concepts. These subtleties become apparent, however, if the probability space underlying the statistical analysis is fully appreciated. As an example of the way in which bias can arise in the characterization of particle size, consider a cube with measurements taken through its center parallel to a face but in random azimuth. The measurements constitute realizations of a random variable V with probability density function s,c a< sJ2
fv (c) = (4!7c)[(4/(r ,/?=3] and expectation E[V]
= (4/z)
I’“’ C(rsdr)/(z*\/m)]
.s
= (45/n) log [s(l i- ,/2)-J/s = 1.12 s. Thus, if the random process is intended to yield an average value approximating the cube edge length. one must multiply the average value by the factor l/1.12=0.88 in order to eliminate the bias introduced by the method of measurement. As has been previously shown, if a population of cubes of various edge lengths were measured in the manner indicated, each cube would bc similarly affected, as would also be the expected value taken with respect to the random variable S denoting the distribution of cube edge lengths. It is, ofcourse, in dealing with the random variable S that the need arises again for rigor in definition of a probability space and for being sensitive to the way in which one probability space may give rise to another. Suppose, for example, that one is dealing with a collection of cubes having edge lengths varying between 1 cm and 2 cm_ Further, let us suppose that the number of cubes contained in any subinterval--say between 1.1 and 12 cm-is proportional to the length of the interval. Then, the probability density function of S is
fs(4 =
(01
l
Pon-drrTechnoL. Z (1%8/69) 260-268
PROBABILITY
FOUNDATIONS
Then
OF PARTICLE
265
Si-ATISTICS
distributed in the interval 1 ,( S’,( 4and construct a cumulative distribution of S according to F,(s) = j:
x’dx,j~x’dx =(2 -
I)/7
2 =
1
sds The corresponding
= 1.5 where 1.5 cm is a legitimate average particle size based on a distribution induced by the number density of particles. An equally legitimate average, in the statistical sense at least, is obtained ifone considers an induced probability density function for S in which the probability density associated with any interval between 1 and 2 cm is proportional to the volume of particles exhibiting edge lengths in this interval. Having assumed that the edge lengths are uniformly distributed between 1 and 2 cm, so that
f,(s) ={
1
l
0
otherwise
we note that particle voIumes are distributed in the interval I< S3 < 8. The cumulative distribution of S is given by I=&)=
_~~x’dr/S:r’d*-=(s~-l),15 ICC<2 .-.
and the probability density function is fs(s) ( ;h)zhenJs;
ss 2
Then
where 1.653 cm is the average size usually attributed the name t-olume mean diameter (or m mean diumetm if particle density is assumed constant) A similar analysis yields an average diameter which might be called the surface mean diameter*Here we note that the surface of one face of a cube is
l
This diamcta
is oftencallul
probability density function is
and
= (+) i;?ds
= 1.607
where 1.607 cm is the surface average diameter_ Quite clearly other mean diameters can arise ifthe distribution of measurements is defined in terms of other functicns of characteristic lengths of the particles Should appropriate occasion arise. for example there is no reason to reject a distributicn induced by an arbitrary function of particle dimensions, say the $ power of cube edge dimensions. In all instances, of course, the induced distribution will be affected by the set of allowed orientations constituting the probability space from which the intraparticle measurement distribution was induced. The compounding of complexities might well terminate here, were it not for the fact that the notion of expectation of a random variable can be generalized. In a manner quite analogous to the nap in which E [X] is defined for a random variable X, one can define the expectation E[g(X)] of some function of the random variable Then
and it is seen that the equation reduces to the usual expectation of the random variable X if g(X) = X. Through this generalized definition of expectation. quantities such as mean-volume diameter and mean-surface diameter arise Consider, for example, the case in which cubes have a uniform distribution of edge lengths in the interval l
E CsWI = E PSI the
mean vo!umcsulface
diameter
andis usedin connation withconsiderations of spaific surface per unitvolume; see for example Dalia Valle’_ Powder
Technol_. -7 (1968j69) X0-268
266
H. -I-_ MCADAMS
-
What
has been determined here is the average of a particle wherein expectation is performed with respect to a sample space based on numbers of particIes If one takes the cube root of E [g(S)], one obtains (3.75)3 = 1.55, the so-called mean-volume diameter_ This is an “equivalent” diameter, in the sense that it represents the size a cube would have to be in order to exhibit the mean volume 3.75 cm’_ Note, however, that instead of taking g(S)=S3. we might have taken g(S)=6S’. Then ~oizme
E [g(S)]
= E [6S=] 2
=
I 6szds=
15.
This number denotes the average surface arca of a cube, and ,/v= 1.58 is the “equivalent” size which a cube would have to assume in order to exhibit a mean area of 15 cm” in the abol:e sense.
SHAPE
FACTORS
In view of the multiplicity of charactlristic lengths which can be assigned to a particle or I o a collection of particles, it is of interest to disco\ :r, if possible, means whereby these- various le.&& can be reconciled To achieve such an sun requires that we be able to fmd dimensionless factors capable of transforming one characteristic dimension into another. The resuh is a shape factor, on which there is already a considerable body of literature. As noted by Herdanq shape factors can arise in at least three ways : (1) As factors of proportionality to reconciie discordant values obtained by different methods of measurement ; (2) As conversion factors for expressing measurements in terms of an equivalent sphere (or some other equivalent shape); (3) As non-dimensional ratios which relate such quantities as particle surface to the second power of a characteristic length, or particle vohune to the third power of a characteristic length. The first implies consideration of physicogeometric laws of similitude and requires that both the geometry of the particle as well as certain physical quantities be related so that scaling laws can be established. The second, by invoking a principle of “equivalence”. postulates an invariance which obtains ifa particle is transformed from one geometric
shape into another. The third considers particle shape fmed and seeks non-dimensional quantities which persist as factors describing the invariance of shape as size undergoes a scaIing transformation_ Evidently various combinations of these considerations can arise, with the result that a multiplicity of shape factors are produced if one considers a11 the contexts wherein scaling cuts across change in particle size, particle shape, and particle response to physical factors. Perhaps the simplest, or at least most straightforward examples of shape factors arising from measurement technique are factors which occur as a consequence of orientation of measurement vectors For example, consider the previously discussed case of a cube measured in two sets of orientations : (1) Along a principal axis througb its center; (2) Along random directions through its center. in a plane paralIe1 to a cube face, all azimuths being equally likelyIf the cube has side s, then nz, = s = mean characteristic length for the first case. and m,=s[(4/X) Iog(l+_J2)] = 1.12s = mean characteristic length for the second case. Then, for the cube, we can define a shape factor a2, = mJml
=
1.12
which might be called an “orientation shape factor” for cubes. It is quite evident that an unlimited number of such orientation shape factors might be constructed when it is realized that an unlimited number of ways exist for constraining or controlling the set of orientations allowed the measurement vectors. Often the set of orientations allowed is tied to or determined by the physical constraints of the measurement process. For example, if irregularly shaped particles are allowed to settle through a viscous medium, these particles often tend to orient themselves so as to experience maximum drag. (Falling leaves present an example.) Therefore, if measurement relies on Stokes’ law, the particles appear to settle at rates characteristic of spheres considerably smaller than spheres of volume and mass equivalent to those of the particles. The shape factor, in this instance. is the result of orientation induced by the measurement process itself. Another Powder Texhnd.
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XC-268
PROBABILITY
FOUXXX4TIOSS
example. of course, is found in the microscopic measurement of particles in which preferred orientation is introduced by the specimen-preparation process Finally, even in the sieving process, lamellar or acicular particles seek preferred orientations for finding their way through the sieve openings, and this interaction influences the resultant sieve assessment of particle size. In fact, if one considers commonly employed methods cf size analysis, there are probably few in which the orientation of measurement vectors does not interact with the physical factors of the measurement process, and it is for this reason that the term physicogeometric is especially appropriate to particle size measurement processes. Shape factors arise as conversion factors for expressing measurements in terms of an equivalent shape as a straightforward consequence ofgeometry-. Somehow one must find an invariant meriting the term “size”, an invariant which does not depend on the ensemble of measurement vectors nor the physicogeometric laws by which the lengths of these vectors are sensed. That invariant is what is called “content” in the general geometry of N-dimensional space*. In N-dimensional space_ an orthotope is a closed N-dimensional figure bounded by a number of (N- I)-dimensional figures and having all edges perpendicular_ A regular orthotope is one having all edges of equal length. Thus, a line-se_gment is a (degenerate) regular orthotope in one-dimensional space, a square is a regular orthotope in twodimensional space, a cube is a regular orthotope in three-dimensional space. and so on_ The length of a line-segment is measured as a multiple of the unit length, the area of a rectangle is measured as a multiple of the unit square, and the volume of a
cube is measured as multiple of the unit cube. Collectively, length, area, volume, hypervolume are examples of content, and it is this content which is invariant under changes in orientation and measurement process In a mathematical sense we can regard each particle as plastic and continuously transformable into any required shape without either loss or gain of material or without change in material density_ For example, a sphere hypothetically can be transformed into a cube or tetrahedron by the proper application of force. Regardless of the multiplicity of forms assumed by the particle under this group of transformations, the volume of the particle is not changed. Thus it seems logical to identify the term
OF PARTICLE
STATISTICS
267
“size” with “content”-namely, volume if one is dealing with two-dimensional objects, area ifdealing with two-dimensional objects, and length if dealing with one-dimensional objects Though the postulating of an equivalent sphere is often invoked for standardizing the “content” of an irregular particle, a geometrically more logical choice of reference would be an equivalent cube, the regular orthotope in three-dimensional space. Indeed, mensuration formulas for the volume of a sphere, such as Volume = ($) jr? or Volume = (i)rc# are expressions of such a standardization. In the first formula, the factor_4x/3 can be considered as a shape factor for expressing the volume of the sphere in terms of a cube of side r, and in the second formula the factor xj6 can be considered as a shape factor for referring the sphere to a cube of side d. If the sphere is chosen as reference, then of course the applicable shape factors are simply the reciprocals 3147~ and 6/a respectivelyThe third class of shape factors arises in response to considerations of dimensional homogeneity and the formulation of non-dimensional constants in accordance with the Buckingham Pi Theorem_ If we admit only geometric quantities to be considered then only quantities involving length to some power need be involved In an N-dimensional polytope, the “contents.’ of geometrical figures of 1, 2, ___, N dimensions are recognized_ Thus a solid, or three-dimensional figure, exhibits volume, area (of its surface), and length (cf- previous discussion of characteristic lengths)_ One can therefore form such non-dimensional ratios as area/(length)’ volume/(leagth)3 and (volume)2i(area)3 Inasmuch as the characteristic lengths used in the denominator may have been derived subject to orientation and physical constraints, each of the above ratios is capable of proliferating into many, so that we would have, for example, a volume/ (length)3 based on Martin% diameter, Stokes’ diameter, and so on. The factors apply both directly and inversely. because in certain situations, such as adsorption measurements or pycnometxy,
268
H. T. MCADAhS
it may be area or volume which is sensed rather than some characteristic length. By inverse application of the shape factor. if it is known, one can deduce the corresponding characteristic length for the particle. An example is found in turbidimetric anaiysis, in which one senses light extinction by the presented area of particles in random orientation. By a theorem of Cauchy6, the mean presented area is one-fourth the area of the particle, and this relation is invariant regardless of particle shape, provided the particle is everywhere convex (that is, has no reentrant angles).
The orientations of vectors along which measurements are taken on particles, and the probability or relative frequency attached to these orientations, are seen to have far-reaching consequences in the evolving of particle statistics. The drstribution of intraparticle statistics is induced by the characteristic geometry of the particle acting in concert with an assumed or specified distribution of measurement vectors_ Because of physicogeometric properties of particles, orientation often interacts with or is determined by t&e physical measuring process. Moreover, it is shown that the various categories of shape factors are conditioned by orientation considerationsThe notion of an induced probability space provides a means whereby the multiplicity of charac:eristic lengths, statistical diameters and shape factors can be unified. By virtue of the fact that shape factors often reflect the consequence of physical as well as geometric controls, the use of two
or more
to The situation becomes extremely complex, hcwever, if particle size and shape vary simultaneously and are possibly correlated_ In using any particular set of particle statistics or method of measurement of particle size or shape, it is recommended that full cognizance be taken of the assumptions underlying the probability space applicable to the statistics The space should be chosen to be appropriate to the particular application at hand and need not adhere to recognized or long-used approaches_ It is believed that the concept of induced probability space, appropriately applied, provides a logic or rationale for this purpose. The application of this rationale, requiring as it does rigorous examination of assumptions, should make for improved understanding and interpretation of particle measurements obtain
physical
shape
principles
information
can be invoked
regarding
particles
REFERENCES 1 H. T. MCADASS, A statistical polyhedron model of abrasive &I-I, Z-mm_ ASME, MB, (4) (1963) 388-93. Z W. P. REID, Distribution of sizes of spheres in a solid from a study of dices of the solid, Tech. itfern. 996. U.S. i%cd Orhce l-es; starion, China me* Gzlz~. 1953. 3 J. M. DALLA VALL.E, Micromtitics, Pitman. New York. 1943.
p_ 39.
Statistics, Elsevier. Amsterdam, 1953, p_ 230. 5 D. M. Y. SOSWER~ An Introduction to the Geometry_ of_ N Dimmrrionr.Dover Publication New York. 1958. 6 M. A. CAUCHY, Note sur divers thCor&mesrelative5 la rectification da courbes, et B la quadrature dcs surfaces, Compt.
4 G. HERDAX, Small Parxicle
Rend_ I3 (1841) 106045.
Powder TecJmoL, 2 (1968/69) 260-268