Average nearest-neighbor distances between uniformly distributed finite particles

METALLOGRAPH Y 5, 97-111 (1972)

97

Average Nearest-Neighbor Distances Between Uniformly Distributed Finite Particles

P. P. BANSALa AND A. J. A R D E L L b

School of Engineering and Applied Science, University of California, Los Angeles, California

The average center-to-center distance (and hence the average free distance) between a finite particle and its nearest neighbor is calculated for three problems involving monodisperse particles uniformly distributed in space. These are: (1) the nearest-neighbor distance in a three-dimensional distribution of finite spheres; (2) the nearest-neighbor distance in a plane section normal to the axes of finite parallel cylinders; (3) the nearestneighbor distance in a plane section through the three-dimensional array of spheres. Our calculations are compared with their well-known point-particle counterparts; they show that the average free distances obtained from the point-particle approximations are reasonable only at small volume fractions (less than 0.05 in the more favorable two-dimensional problems). For each case considered, the range of volume fractions over which our calculations should be valid is discussed. The applicability of our results to various problems involving dispersions of a second phase is also discussed.

Introduction T h e average c e n t e r - t o - c e n t e r distance, ( R ) , b e t w e e n a particle and its nearest n e i g h b o r is an i m p o r t a n t spacing p a r a m e t e r e n c o u n t e r e d in a variety of m e t a l l u r gical problems. Specific examples can be found in various papers dealing w i t h the continuity or contiguity of particles of a single phase in a t w o - p h a s e mixture, x-3 and papers on dispersion s t r e n g t h e n i n g 4,5 in t w o - p h a s e alloys. I n a t h r e e - d i m e n s i o n a l r a n d o m array of " p o i n t " particles (that is, particles w i t h vanishingly small radii), the distribution of n e a r e s t - n e i g h b o r distances, f~(R), can be calculated exactly, e as d e m o n s t r a t e d first by H e r t z ~ and later by Chandrasekhar. 7 T h e distribution function, f~(R), is defined in such a m a n n e r that f~(R)dR is the probability that a point particle is located at a distance a Engineering Systems Department. b Materials Department. e We shall use the subscript v on quantities that pertain to the three-dimensional aggregate, and the subscript a or s on quantities that refer to a two-dimensional section through the aggregate. Copyright © 1972 by American Elsevier Publishing Company, Inc.

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between R and R + dR from its nearest neighbor. The value of (Rv> can then be calculated exactly from the relationship

= which yields the result 7

f:

Rf~(R) dR

(Rv> = 0.554Nv -1/a

(1)

(2)

where N , is the number of particles per unit volume. If, instead of point particles, we assume that the particles are monodisperse with finite radius r and volume fraction ¢, equation 2 can be expressed in the alternative form: (R~,> : 0.554(4~) t/3 r

(3)

which makes use of the relationship 4rrr3 3 N~ = ¢

(4)

The average nearest-neighbor separation, (Ra>, in a random two-dimensional (planar) section through the three-dimensional array can also be calculated exactly. The result is 1,2

(R.> -- N:1/2 2

(5)

where Na is the number of particles per unit area in the two-dimensional section. For finite monodisperse spherical particles of radius r and volume fraction ¢, equation 5 is usually expressed as

['rr'~'l 2 = \~-~,] r

(6)

which makes use of the fundamental relationship s

N~ = 2rN~

(7)

Equations 3 and 6 must be applied with caution when dealing with real systems in which the particles are finite and polydisperse because: (1) randomness, in the strict sense of the word, cannot exist, owing to the fact that the space occupied by a given particle is unavailable to other particles; (2) it is not quite correct to replace r by (r>, though this is frequently done ;4.5 (3) the mean free distance, (/>, between nearest neighbors is often the parameter of interest, and for relatively small values of ¢ equation 3, in particular, yields impossible results. To illustrate the latter point, consider the relationship l~etween (/> and : = -- 2~

(8)

Finite Particle Distances

99

which is also valid for polydisperse systems if we replace r by (r). If we assume that the particles are impenetrable, it is clear that (l) can never be less than zero. However, negative values of (1~) arc predicted by equation 3 when ~ > 0.0884, which can be easily verified by substitution into equation 8. We ean thus conclude that, if equation 2 is to be applied to systems containing finite particles in the form of equation 3, the volume fraction of the particles must be very small. The three limitations of equations 3 and 6 referred to above have generally been recognized by the various investigators who have had occasion to use them. However, there has apparently been no effort to determine whether better approximations for ( R ) (and (l)) exist for the physically realistic case of finite polydisperse particles. This paper is concerned specifically with this problem.

Theory To make the problem mathematically tractable, we consider only monodisperse assemblies of particles. The approach used here is the same as that used by Chandrasekhar, except for appropriate modifications which explicitly account for the finite radii of the particles. The particles are assumed to be impenetrable and are randomly (or, perhaps more appropriately, uniformly) distributed throughout the space available to them--that is, the space which is not occupied by other particles.

The Three-Dimensional Problem Let f~(R)dR denote the probability that the center-to-center distance, R, between a particle and its nearest neighbor lies between R and R + dR from the given particle. Following Chandrasekhar, fv(R) dR can be expressed as the product of two probabilities: (1) the probability that no other particle center exists within the spherical volume of radius R; (2) the probability that a particle center is contained within the spherical shell of radius R and thickness dR. The first probability can be found exactly. It is 1 -

L(R) aR r

where the lower limit of integration explicitly excludes interpenetration of the particles (that is, the center of one particle cannot be closer than 2r from the center of another). The second probability can be expressed as 4-rrR2 dRN v which is simply the volume of the spherical shell multiplied by the number of particles per unit volume. This expression must be an approximation, because a particle center cannot lie anywhere within the spherical shell. Nevertheless, it

P. P. Bansal and A. J. Ardell

100

should be a good approximation when the volume fraction is not too large, and exact when R >~ 2r. Combining the above two expressions and using equation 4, we obtain

17 R dR 3¢R~dR ,~(R) d R = [ X - - f f 2 / v ( ) ] ~

(9)

Dividing both sides of equation 9 by R ~ and differentiating with respect to R yields, after rearrangement of the remaining terms,

aft(R) _ 2dR f,~(R) R

3~R~ dR

(10)

On integrating equation 10 we find

fv(R) = eRie -~R3/r3.

(11)

The constant of integration, C, is readily found by normalizing fv(R) dR such that

f2

°f~(R) dR = 1

(12)

T

On substituting equation 11 into equation 12 and carrying out the integration, we obtain

f~(R) = 3¢-eSC'R2e-4'1~3/'3 rS

(2r < n < c~)

(13)

To find (R~), we use equation 1 (changing the lower limit of integration from 0 to 2r) and let x = ¢(R/r) 3, which yields

(R&

e8~ I °°xl/ae-Z dx ¢1/3 ds¢,

r _

eS~

~3

(14)

1--4

(~, 8¢)

where /'(~, 8¢) denotes the incomplete gamma function. Chandrasekhar's original result, equation 2, is readily recovered from equation 14 by recalling that, as r approaches zero, ¢ also approaches zero (the "volume fraction" of point particles is zero) in such a manner that r/qg/s-+(3/4rrN~) I/3, whereupon

(Rv) = (3/4"trN~)l/sF(~) which is equivalent to equation 2. If equation 14 is integrated once by parts, we obtain the result (Rv) --

eS4, /-oo -- 2 q - 3 ~ J s~ - x-elSe -x dx es'b -1 = 2 q- 3--~F(~, 8¢)

(15)

Finite Particle Distances

101

which illustrates more clearly the fact that (R~)/r must always be greater than 2. From equation 15 we also obtain the simple expression for {lv)/r (using equation

8): (Iv) r

--

es4,

, F '1

3¢1/~ ta,

8¢)

(16)

The dependence of (lv}/r on ¢ from equation 16 is shown in Fig. 1. For comparison, the dependence predicted by equation 3 (via equation 8) is also shown.

The Two-Dimensional Problems As was mentioned in the Introduction, equation 5 is valid for a plane section through a three-dimensional array of randomly distributed points. However, equation 5 applies equally well to a plane section perpendicular to an array of randomly spaced parallel lines. There are, accordingly, two cases of interest in our treatment of finite particles. We shall first consider the simpler case of parallel cylinders of equal radius, and then the more complicated case of the random plane section through the three-dimensional array of finite spheres. The Nearest-Neighbor Spacing of Finite Parallel Cylinders. We consider a plane section normal to the axes of parallel cylinders of circular cross section and radii rs. Once again, we assume that the rods are uniformly distributed throughout the unoccupied space available in the matrix. Let f,(R) dR denote the probability that the center of a cylinder lies between R and R + dR from its nearest neighbor. Using the same reasoning applied to the case of finite spherical particles, we find f~(R) = [ 1 -

.] 2r,['Rf s(R) dR12~rR

.

(17)

where the factor 27rR dRNs represents the probability of finding a cylinder center in the annulus of area 27rR dR when N~ is the number of cylinders per unit area in the plane normal to the cylinder axes. With ¢ = rrr~Ns, and algebra analogous to that used in equations 10 to 12, we find

fi(R) = r2~C2Re4*e-OR~M

(2r~ < R < oo)

(18)

Letting x = ¢(R/rs)9; the expression for (R~) is readily found to be

(R~) __ e4~' foo x t/2 e-~ dx rs

¢iTzJ 4¢ e4~F(~, 4¢)

(19)

102

P. P. Bansal and A. J. Ardell 7

4_-

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0

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I 0.006

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2

--/-

%%%%

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0 0.6

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0.02

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0.04

~

0.06

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0.10

0.4

0.2

0

I 0.2

I

I 0.4

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I 0.6

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FIG. 1. The dependence of /r on volume fraction over three decades of 4, (0.001 to 0.01, 0.01 to 0.1, and 0.1 to 1) for monodisperse spheres of radius r. The dashed curve shows the dependence of
Finite Particle Distances

103

After integrating equation 19 once by parts, and using equation 8, the mean free distance between nearest neighbors is found to be (l~) r~

e4*

1

-- 2-¢5~r(~, 44)

(20)

The ratio (ls)/rs is shown as a function of ¢ in Fig. 2. For comparison the dependence of (ls)/r~ in the parallel line approximation is also shown. In this case the expression for (l~)/r~ is rs = \~-¢]

--2

(21)

which follows from equation 5 with N, = N . and ¢ ~ 7rr]Ns.

The Nearest-Neighbor Spacing in a Plane Section Through a Three-Dimensional Array of Finite Spheres. This problem is complicated by the fact that a random plane through the three-dimensional array of particles will intersect various particles at different distances from their centers. Thus, in the plane section the particles will appear as circles of varying radii. Let r, represent the radius of one such circle. It is easy to show that the probability, g(r,) dra, that the radius of this circle lies between r, and r, + dra is r¢~ d r a

g(r,~) d r a - r(r2 _ r~)1/2

(0 < r, < r)

(22)

Suppose we start with a particle of sectioned radius, r~, and seek the probability that its nearest neighbor on the plane section lies between R and R + dR. This probability is clearly a conditional probability because it depends on the value of ra, and we shall denote it by ¢(Rlr~) dR. The dependence of ~(Rlra) o n ra arises because we can never observe a particle center closer than 2ra from the particle at the origin. This becomes clear when we consider the possible plane sections through two particles which just contact each other. Such particles will appear to be in contact on a plane section only when the plane contains their point of tangency, in which case their sectioned radii will be identical. The appropriate expression for ¢(RJr~) can therefore be developed in exactly the same manner as the distribution functions f~(R) and f~(R) were developed in the preceding sections. Accordingly, we have

¢(Rlra) dR=[1--f~2Rr?(R[r~)dR127rRN~dR

(23)

where the bracketed term represents the probability that no other particle center lies within the circle of radius R, and the other term represents the probability that a particle center is contained within the annulus of area 27rR dR. In equation 23, N~ is the number of particle centers per unit area in the plane section, which is related to N~ by equation 7. On substituting equation 7 into equation 23, also

P. P. Bansal and A. J. Ardell

104

20

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0.002


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0.004

1

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0.006

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0.008

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b t' I

4 3 2

~',,m

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FIG. 2. T h e d e p e n d e n c e of /r~in t h e pointparticle approximation.

Finite Particle Distances

105

making use of equation 4, and following the same mathematical procedures used previously, we find ¢(Rlra) = 36Re-('~/zr')(n~-4rD r2

(2ra < R < 0o)

(24)

where ¢(RIr.) has been normalized to unity. From standard procedures in probability theory, 9 it is easy to show that the joint probability density, f,,(R, r.), is simply L ( R , r~) = ¢(Rlr.) g(r.) (0 < r. < r; 2r. < n < ~ )

(25)

and that the average value of R--that is, (R.)--is given by

=

f o;

RL(R, r~) aR dro

(26)

ra

On substituting equations 22, 24, and 25 into equation 26, we find, after some algebraic manipulation, __ 1 f6d ez dx ~ ul/2e -u du r 6-¢j 0 ( 6 ¢ - x)~/2jx 1 f64/-(~, x)e ~ dx - - 6-¢j 0

(6¢ -

x)Tff

(27)

where x : 6¢(r./r) 2 and u = 3¢(R/r)2/2. If the expression for F(~, x) in equation 27 is integrated once by parts, the expression for ( R . ) / r becomes

7r

1 f6d/-,(½,x)e x dx

(28)

Now, the average planar radius,
f0

=

r~g(ra) dr~

and using the expression for g(r.) in equation 22. We thus find, from equation 28,



2+

1 -f6¢F(½, x)e*dx

3 q0

(29)

Therefore, in two dimensions, the average center-to-center distance between a particle and its nearest neighbor always exceeds twice the average planar radius. Using equation 8, we find, for the mean free distance between a particle and its nearest neighbor, (1(,) _ (r.>

1 .f6~F(½, x)e ~ dx

3"~j 0 (6¢ -- x) a/2

(30)

P. P. Bansal and A..7. Ardell

106

The dependence of (lo)l(ra) upon q~is shown in Fig. 3, where the prediction of the point-particle approximation is also shown for comparison. In this case, equation 6 was used to calculate (la)/(r.) after replacing r by (4/Tr)(r.), so that in the point-particle approximation (l.)/(r.) is given by ~--

\37r4,J

-- 2

(31)

Discussion The results of the previous section clearly demonstrate that care must be exercised when the point-particle calculations are employed for calculating the nearest-neighbor distances between finite particles. As can be seen in Fig. 1, the point-particle approximation is particularly bad for calculating the three-dimensional average nearest-neighbor distance, since (l,~)/r falls to zero at a volume fraction of only about 0.09 (as pointed out in the Introduction). At volume fractions as low as 0.05 the point-particle approximation results in an error of a factor of 2 in (l~)/r. This situation is not nearly so bad when using the point-particle approximation in either of the two-dimensional problems. The results of the finite particle calculations are nearly identical for both two-dimensional cases, as can be seen on comparing Figs. 2 and 3. From these figures it is also clear that the pointparticle approximations remain reasonable to values of 4, up to about 0.05. The errors in (ls)/rs and (la)/(r~) reach a factor of 2 when 4' --- 0.11. We may conclude from Figs. 1 through 3 that the present calculations are to be preferred over the point-particle approximations whenever 4, is sufficiently large. For computational purposes we have prepared tables of (l)/r for the three cases dealt with herein, which we will be pleased to supply on request. The average values of the center-to-center distances between nearest neighbors can, of course, be readily obtained simply by adding two to the tabulated values of the average free distances. It should be mentioned that all our calculations involved numerical integration on an IBM 360-91 computer. At this point it is legitimate to ask two important questions: (1) What is the maximum value of 4, at which the present calculations are likely to be valid and meaningful? (2) To what kinds of problems are they likely to apply, and under what conditions? The first question is difficult to answer. The present calculations certainly constitute a better level of approximation than their point-particle counterparts However, they cannot hold at volume fractions approaching unity, because neither spheres nor circular cylinders are space-filling. Even in structures represented by the densest possible packing of spheres (f.c.c. and h.c.p, lattices, for example), only 74% of space is filled. Indeed, in the random dense packing of rigid spheres, which is often used as a model of the liquid state, the fraction of occupied space is roughly 0.64. l° At this value of 4,, (lv)/r must equal zero in the

107

Finite Particle Distances

20

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4, FIG. 3. T h e dependence of (l,)/(r~) on volume fraction over three decades of ~ (0.001 to 0.01, 0.01 to 0.I, and 0.1 to 1) for a plane section through the three-dimensional array of monodisperse spheres of radius r and average section radius
P. P. Bansal and A. J. Ardell

108

absence of an embedding medium. When ¢ = 0.65, the value of (lv)/r predicted by equation 16 is 0.115. Therefore, it is reasonable to conclude that our calculations slightly overestimate the true values of (lv)/r in the range of ¢ from 0.4 to 0.65. Beyond ¢ z 0.74 the calculated spacings are meaningless. These conclusions should hold as well for the two-dimensional planar spacings through the three-dimensional section, given by equations 29 and 30. On the other hand, equations 19 and 20 are probably good approximations at still larger values of ¢, since cylinders of equal diameter can pack space to a maximum volume fraction of approximately 0.91. With respect to the problems to which our calculations should apply, this clearly depends on the physical aspects of a given situation. One of us 11 has recently used the results of the three-dimensional calculation in an attempt to evaluate the effect of ¢ on the coarsening of spherical precipitates in solid matrices. Corten, TM in his review of the mechanics and fracture behavior of composite materials, cites several examples where the distance between a cylindrical particle and its nearest neighbor influences the state of stress within the material. In these examples the cylinder centers are arranged on lattices, but any calculations involving random distributions of parallel fibers would necessarily involve the use of equation 19 or 20. Nearest-neighbor spacings in a plane section through a three-dimensional array of spheres have been frequently employed in problems involving precipitate dislocation interactions. Whenever random spatial distributions of particles are assumed, the point-particle approximation is used. This leads to the result that the yield stress, au, is proportional to ¢1/2 in some theories of precipitation hardening and dispersion strengthening. 4'5 This is a direct consequence of equation 31, since ~u is proportional to 1/(la). If we express equation 30 in the alternative, but equivalent, form,

(I')--~8~1/2--(ra)3-~¢~¢(6¢x)'/2F(~,x)e'dx-\3~'¢]

2 -- 6¢ -I-

(32)

we see that our calculation also leads to a ¢1/2 dependence at very small values of ¢. However, this only occurs for ¢ ,< 0.001, which are impractically small volume fractions. If we plot the values of (la)/(ra) versus ¢ in Fig. 3 on a log-log basis, we find slopes of approximately --0.61 (0.001 < ¢ < 0.01) and --0.83 (0.1 < ¢ < 0.4). Our calculations thus suggest a yield stress dependence upon volume fraction of the type au oc Cn(¢), with n(¢) varying from 0.61 to 0.83 for the volume fractions typical of most precipitation-hardened and dispersion-strengthened alloys. Along these same lines, we should mention that Kocks 1~ has pointed out that the nearest-neighbor spacing is not necessarily the appropriate spacing for estimating the stress necessary for dislocations to bypass pinning points. He has calculated a number of other point-particle spacings, based on a variety of differ-

Finite Particle Distances

109

ent criteria, and presented them in tabular form as ratios of 2(Ra) (as defined by equation 6). We do not know whether these ratios would remain unaffected if Kocks' spacings were calculated for finite particles and compared with the values of (la) obtained from equation 30. However, they would probably provide reasonable corrections, as first approximations, in problems involving other spacing criteria. In the examples above, we have implicitly taken for granted that the physical disposition of the particles is consistent with the geometrical requirements of our calculations, which we shall reiterate: the particles should be uniformly distributed throughout the medium in which they are embedded; they should be impenetrable. The requirement of a uniform spatial distribution implies that a particle is equally likely to be found anywhere in the space not occupied by another particle. This clearly implies that the particles should be non-interacting and, in the case of precipitates at least, should have nucleated randomly throughout the matrix. An example of interacting precipitates is provided by the 7'(Ni3A1) precipitates in aged Ni-A1 alloys, which become aligned along (100) crystallographic directions in the matrix because of elastic interactions. 14 The precipitates evidently nucleate randomly, since the alignment becomes increasingly pronounced as they grow. An excellent example of nonrandom "nucleation" is provided by precipitates which form as a result of spinodal decomposition. 15 An estimate of the extent to which the spatial distribution of particles approaches randomness can be obtained by use of a procedure described by Ashby and Ebeling. 16 Particle impenetrability excludes the application of our calculations to sintering processes. "Sintering" is used here in a general sense, and it includes the coalescence of precipitates during growth. For example, if two particles coalesce, their centers gradually move together until the individual particles are no longer distinguishable, and during this process their centers will be closer than 2r. Whether or not such processes are occurring can of course be ascertained only by examination of the microstructure. Finally, we come to the question of whether our calculations can be applied, without modification, to systems in which the particles are polydisperse (which is the usual case). All the spacing problems become considerably more complicated, if not impossible, to solve when the particle sizes are assumed to be distributed at the outset of the development. We are then forced to make simplifying assumptions regarding the significance of the parameter r. Strictly speaking, it is not legitimate to replace r by (r), even in the point-particle approximations. Instead, r should be replaced according to the following substitution scheme (which is exact for the point-particle approximations): 3-d spheres, equation 3 r--~(r3)1/3 2-d cylinders, equation 21 rs-+(r])l/2 2-d spheres, equation 6 r-+((r3)/(r))J/2

110

P. P. Bansal and A. J. Ardell

Since we cannot prove that the same substitutions should be valid for our finite particle calculations, it is impossible to say whether they are preferable to the simple substitution of ( r ) for r. We can state with reasonable confidence, however, that unless the particle size distribution is very broad, the error incurred by using ( r ) in place of r in our equations will be unlikely to exceed 10~o. In most cases this error will be small compared with that incurred by using the pointparticle approximations.

Summary The average free distance between a particle and its nearest neighbor has been calculated for three cases in which the particles are finite, monodisperse, and uniformly distributed in space. These are the nearest neighbor distances in: (1) a three-dimensional distribution of spheres; (2) a plane section normal to the axes of parallel circular cylinders; (3) a plane section through the three dimensional array of spheres. The spacings derived from our calculations are clearly preferable to those derived from the corresponding point particle approximations, except at small volume fractions where both methods converge to yield nearly identical results. Since the particle shapes dealt with herein cannot pack to fill space, our calculations slightly overestimate the true nearest-neighbor spacings for volume fractions approaching those which represent the densest possible packing of spheres (0.74) and circular cylinders (0.91). The applicability of our calculations to several problems is briefly discussed. It is emphasized that if the particles have nucleated, grown, or have been introduced into their matrix in such a manner that they are not randomly distributed throughout the unoccupied space available to them, then our calculations cannot be expected to apply. The same conclusion holds for particles which have evolved by a sintering or coalescence process (i.e., our calculations apply only when the particles are impenetrable). We believe that our calculations can be applied, without serious error, to polydisperse systems of particles by substituting ( r ) for r in the final equations.

Acknowledgment We are grateful to the U.S. Atomic Energy Commission for their support of this research.

References 1. J. Gurland, in Quantitative Microscopy (R. T. deHoff and F. N. Rhines, eds.), McGraw-Hill, New York, 1968, Chapter 9, p. 279. 2. E. E. Underwood, Quantitative Stereology, Addison-Wesley, Reading, Massachusetts, 1970, Chapter 4, p. 80.

Finite Particle Distances 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

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F. Forscher, J. Franklin Inst., 259, (1955) 107. A. Kelly and R. B. Nicholson, Progr. Mater. Sci., 10, (1963) 149. K. H. Westmacott, C. W. Fountain, and R. J. Stirton, Acta Met., 14, (1966) 1628. P. Hertz, Math. Ann., 67, (1909) 387. S. Chandrasekhar, Rev. Mod. Phys., 15, (1943) 1. R. L. Fullman, Trans. Met. Soc., AIME, 197, (1953) 447. G. P. Wadsworth and J. G. Bryan, Introduction to Probability and Random Variables, McGraw-Hill, New York, 1959, Chapter 5, p. 163. G. D. Scott, Nature, 188, (1960) 908. A. J. Ardell, Acta Met., to be published. H. T. Corten, in Modern Composite Materials (L. J. Broutman and R. H. Krock, eds.), Addison-Wesley, Reading, Massachusetts, 1967, Chapter 2, p. 27. U. F. Kocks, Aeta Met., 14, (1966) 1629. A. J. Ardell and R. B. Nicholson, Acta Met., 14, (1966) 1295. J. W. Cahn, Trans. Met. Soc. AIME, 242, (1968) 166. M. F. Ashby and R. Ebeling, Trans. Met. Soc. AIME, 236, (1966) 1396.

Accepted August 4, 1971