JOURNAL OF CHEMICAL PHYSICS

VOLUME 110, NUMBER 6

8 FEBRUARY 1999

Local volume fraction fluctuations in periodic heterogeneous media J. Quintanillaa) Department of Mathematics, University of North Texas, Denton, Texas 76203

S. Torquatob) Princeton Materials Institute and Department of Civil Engineering & Operations Research, Princeton University, Princeton, New Jersey 08544

~Received 17 June 1998; accepted 5 November 1998! Although the volume fraction is a constant for statistically homogeneous media, on a spatially local level it fluctuates and depends on the observation window size. In this article, we develop exact analytical expressions for the full local-volume fraction distribution function of periodic arrangements of rods, rectangles, and cubes in a matrix. These formulas depend on the inclusion density and window size. © 1999 American Institute of Physics. @S0021-9606~99!71306-1#

random variable. As shown in Fig. 1, such random placement allows variation in the proportion of the window covered by particles. In this article, we obtain exact analytical expressions for the full distribution function of j for periodic rods, rectangles, and cubes. We then calculate the standard deviation of j for these systems. We do this by working directly with the statistical description of these periodic systems. By contrast, the moments of j for random media were studied by us through integrals over certain n-point microstructure functions.13,14 Of course, the full distribution function and hence the standard deviation are dependent on the size of the observation window relative to the particles. We begin by calculating the full distribution function of

The quantitative characterization of the microstructure of heterogeneous media, such as composite materials, colloidal dispersions, porous media, and cracked solids, is crucial in determining the macroscopic physical properties of such materials.1–5 One of the most important morphological descriptors is the volume fractions of the phases. While the volume fraction is macroscopically constant, it fluctuates on a local level. These fluctuations have been shown to be relevant to a number of problems, including scattering by heterogeneous media,6 transport through composites and porous media,7 the study of noise and granularity of photographic images,8–10 the properties of organic coatings,11 and the fracture of composite materials.12 To quantitatively characterize these fluctuations, Lu and Torquato13 defined the local volume fraction j~x! at position x for arbitrary, statistically homogeneous two-phase random heterogeneous media in any spatial dimension. The local volume fraction j~x! is defined to be the volume fraction of one of the phases, say phase 2, contained in some generally finite-sized ‘‘observation window’’ with position x. As illustrated in Fig. 1, the concentration j of phase 2 within a given observation window is a random variable ranging between 0 and 1, although the macroscopic volume fraction of phase 2 is constant, say f 2 . For random media, the standard deviation13 and higher moments14 of j have been characterized.15 In related work, the full cumulative distribution function of j for various continuum ~off lattice! models of random media was investigated.14 However, the disorder in these models prevents one from obtaining the full distribution function analytically; one must resort to numerical techniques. The local volume fraction is also well-defined for periodic heterogeneous media. At first glance, one might think that fluctuations in local volume fraction should not even exist for periodic media. However, while the positions of the particles are deterministically known, the observation window can still be placed randomly in the system so that j is a

FIG. 1. Four observation windows in a system of periodic rectangles. The lower left window achieves the maximum possible local volume fraction of phase 2, while the upper left window achieves the minimum. Notice that for all windows, the local volume fraction of phase 2 is simply the product of the local volume fractions of the constituent one-dimensional systems.

a!

Electronic mail: [email protected] Electronic mail: [email protected]

b!

0021-9606/99/110(6)/3215/5/$15.00

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© 1999 American Institute of Physics

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J. Chem. Phys., Vol. 110, No. 6, 8 February 1999

J. Quintanilla and S. Torquato

FIG. 2. The standard deviation of j for one-dimensional systems of rods. The thin lines give s j for periodic rods and are determined by Eq. ~6!. The thick lines give s j for fully penetrable rods and are given in Ref. 13. We see that the fluctuations in j are suppressed for periodic media relative to random media.

j for the one-dimensional system of periodic equal-sized rods along a line. We then calculate the standard deviation of j and compare with fully penetrable rods and totally impenetrable rods at the same macroscopic volume fraction f 2 . Consider an observation window of length L in a periodic system of rods of length f 2 whose centers are separated by a unit length. For convenience, we define l by L5n1l,

~1!

0
where n is an integer. Then the local volume fraction of phase 2 in the observation window has cumulative distribution function F(x)[Pr( j
H

0, x,m F ~ x ! 5 d 12L ~ x2m ! , 1, m
m
D512 d 22L ~ M 2m !

~2!

~3!

to be the probability that the local volume fraction is exactly M . The values of these four parameters are given by D5 u f 2 2l u ,

m5 @ n f 2 1max$ l2 f 1 ,0% # /L,

~4!

M 5 @~ n11 ! f 2 1min$ l2 f 2 ,0% # /L. Given the cumulative distribution function, we now can calculate the moments of j. The first moment of j is

^ j & 5 d m1DM 1

E

M

m

as expected. The variance of j has the form

s 2j 5 ^ j 2 & 2 ^ j & 2 5 d m 2 1DM 2 1

5

S

2Lx dx5 f 2 ,

~5!

E

M

m

2Lx 2 dx2 f 22

D

a2 a 2 , 2 b2b 2 L 3

~6!

b5max$ l, f 2 %

~7!

where a5min$ l, f 2 % , if f 2 <12l, while a5min$ 12l,12 f 2 % ,

In this expression, d is the probability that the local volume fraction is exactly m, and m and M are certain local volume fractions. We also define

d 5 u f 1 2l u ,

FIG. 3. The standard deviation of j for two-dimensional systems of squares. The thin lines give s j for periodic squares and are determined by Eq. ~14!. The thick lines give s j for fully penetrable squares and are given in Ref. 14. Again, the fluctuations in j are suppressed for periodic media.

b5max$ 12l,12 f 2 %

~8!

if f 2 >12l. In Fig. 2, we compare the standard deviation of j for periodic rods and fully penetrable rods13 at various volume fractions. Not surprisingly, the repetitive nature of periodic rods greatly suppresses the fluctuations in j compared to fully penetrable rods. Also, for periodic rods, s j vanishes when L is an integral multiple of the distance between the rod centers; this is intuitively obvious from geometrical considerations. Finally, we observe that the graph of s j is monotonically decreasing for L,1; this is expected since the window size is smaller than the distance between adjacent rod centers. We now consider a system of periodic rectangles in the plane, and a rectangular observation window aligned with the particles. We assume that the centers of the particles are spaced with unit length in both the horizontal and vertical directions; this assumption may be relaxed by appropriately scaling the following results. To study the distribution of j, we observe that the portion of the window covered by the particles is precisely the Cartesian product of the portions covered in each dimension; see Fig. 1. By taking certain integrals of the joint distribution function of these two inde-

J. Chem. Phys., Vol. 110, No. 6, 8 February 1999

J. Quintanilla and S. Torquato

pendent one-dimensional processes, we can obtain the cumulation distribution function of j and hence its moments. For the constituent one-dimensional processes, the parameters d i , D i , m i , and M i (i51,2) are defined. Without

0,

¦

F ~ x ! 5 d 2 12L 2

S D S S D S D S D

D

x 2 d 1L 2 2 d 2L 1 1 ~ x2m 1 m 2 ! 1 24L 1 L 2 , m 1m 2 m1 m2

S

5

m 1 m 2
D

1 a M1 1 2m 2 14L 1 L 2 x ln 12aL 2 ~ d 1 22L 1 m 1 ! 2 , M1 m1 m1 M 1

12D 1 D 2 14L 1 L 2 x ln

S

M 1 m 2
D

M 1M 2 2D 1 L 2 2D 2 L 1 2 ~ M 1 M 2 2x ! 1 14L 1 L 2 , a M1 M2

m 1 M 2
For square particles and square observation windows,

^ j & 5 f 2 5 ~ d m1L @ M 2 2m 2 # 1DM ! 2 and so the four parameters are determined by Also, the variance of j is given by

s 2j 5

x,m

2

d 14L x ln~ x/m ! 14 ~ x2m 2 !~ d L/m2L 2 ! , 2

2

m 2
12D 2 14L 2 x ln~ M 2 /x ! 1,

and L.

~14!

In Fig. 3, we plot the standard deviation of j for square observation windows for periodic squares and fully penetrable squares.14 We again observe the same qualitative behavior as in Fig. 2, although the suppression of the fluctuations is not quite as large in one dimension.

2

24 ~ M 2 2x !~ DL/M 1L 2 ! ,

~13!

f 1/2 2

~ 3 d m 2 12L @ M 3 2m 3 # 13DM 2 ! 2 9

2 ~ d m1L @ M 2 2m 2 # 1DM ! 4 . 0,

~9!

x>M 1M 2.

As expected, there are positive probabilities d 1 d 2 , d 1 D 2 , D 1 d 2 , and D 1 D 2 that the local volume fraction of phase 2 is exactly equal to m 1 m 2 , m 1 M 2 , M 1 m 2 , and M 1 M 2 , respectively. Under the special case of square observation windows in a system of periodic squares, the one-dimensional processes are mathematically equivalent, and hence the subscripts can be dropped. The distribution function thus reduces to

F~ x !5

loss of generality, we assume that M 1 m 2
x,m1m2

d1d214L1L2x ln

1,

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mM
x>M 2.

~10!

We now study the moments of j for these twodimensional systems. For rectangular particles and windows, the first moment of j is, as expected,

^ j & 5 ~ d 1 m 1 1L 1 @ M 21 2m 21 # 1D 1 M 1 ! 3 ~ d 2 m 2 1L 2 @ M 22 2m 22 # 1D 2 M 2 ! .

~11!

This is not surprising since j is the product of two independent random variables representing the fraction of coverage for the height and width of the window. We also find that the second moment of j is given by 1 9

^ j 2 & 5 ~ 3 d 1 m 21 12L 1 @ M 31 2m 31 # 13D 1 M 21 ! 3 ~ 3 d 2 m 22 12L 2 @ M 32 2m 32 # 13D 2 M 22 ! .

~12!

FIG. 4. The standard deviation of j for three-dimensional systems of cubes. The thin lines give s j for periodic cubes, while the thick lines give s j for fully penetrable cubes as given in Ref. 14. Again, the fluctuations in j are suppressed for periodic media.

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J. Quintanilla and S. Torquato

Finally, we calculate the full cumulative distribution function of j for a cubical window of side length L which is aligned with a system of periodic cubes. Again, we assume that the distance between adjacent cube centers has unit length. Under these assumptions, j is the product of

0,

x,m3

d 31

¦

S D

1

2L ~ x2m 3 ! 6L ~ x2m 2 M ! 2 2 2 3 d 26 d mL14m L 1 ! ~ ~ 2 d m22 d 2 m22Lm 2 16 d Lm 2 24L 2 m 3 2 d 2 M ! m2 m2M

S D S D S DS D S DS D

x M3 x mM 2 14L x ln 2 ln 14L 3 x ln 3 ln , m M x m x 12D 3 2

~15!

m M
2

S D

M3 2L ~ M 3 2x ! 4L 2 x ~ 3D12LM ! 2 2 2 ln 3D 16DLM 14L M 1 ! ~ M2 M x

14L 3 x ln2

S D

M3 , x

mM 2
x>M 3.

Using this distribution function, we can calculate the moments of j. We find that

^ j & 5 f 2 5 ~ d m1L @ M 2 2m 2 # 1DM ! 3

~16!

and

^ j 2& 5

m 3
4L 2 x ~ 3D12LM ! 4L 2 x ~ 3 d 22Lm ! x mM 2 ln 2 ln 1 M m M m x 3

1,

S D

x x 2L~x2m3! 4L2x~3d22Lm! ln 3 14L 3 x ln2 3 , ~3d 226dmL14m2L2!1 2 m m m m

d 3 13 d 2 D1

F ~ x !5

three independent random variables, each determined by the same d, D, m and M , which in turn are determined by f 1/3 2 and L. After integrating the joint probability density function, we find that

~ 3 d m 2 12LM 3 22Lm 3 13DM 2 ! 3 . 27

~17!

Again, these results are expected since j is the product of three independent random variables representing onedimensional coverage. In Fig. 4, we plot the standard deviation of j for cubical observation windows for periodic cubes and fully penetrable cubes.14 We again observe the suppression of fluctuations in j for periodic media relative to random media, although not as suppressed as in lower dimensions. In conclusion, we have analytically computed the full cumulative distribution function of the local volume fraction for periodic rods, rectangles, and cubes in a matrix. Using this information, we have computed the standard deviation of the local volume fraction. Not surprisingly, we have observed that the fluctuations are suppressed in periodic media relative to random media but, generally, are not zero. However, the magnitude of the suppression significantly decreases in higher dimensions. We also observe that the peri-

odic nature of these systems is evident in the graphs of s j . When the window size is an exact multiple of the particle size, the coarseness is exactly zero. The local maxima of s j are obtained at approximately the half-values. Hence, the length scale of the particles is observed in the behavior of the local volume fraction, independent of the volume fraction. Finally, we note that our analytical results will be of great value in a new application area. Specifically, in a recent study,16 it has been shown that local volume fraction fluctuations are crucial to determining coarse-grained macroscopic properties associated with finite-sized windows in heterogeneous media. The present analytical results for periodic heterogeneous media will allow exact evaluation of the coarsegrained macroscopic properties, in contrast to random heterogeneous media, where only numerical results are available.

S. Prager, Physica 29, 129 ~1963!. M. Beran, Statistical Continuum Theories ~Wiley, New York, 1968!. 3 G. W. Milton, Phys. Rev. Lett. 46, 542 ~1981!. 4 G. W. Milton, Commun. Math. Phys. 111, 281 ~1987!. 5 S. Torquato, Appl. Mech. Rev. 44, 37 ~1991!. 6 P. Debye, H. R. Anderson, and H. Brumberger, J. Appl. Phys. 28, 679 ~1957!. 7 S. Torquato and F. Lado, J. Chem. Phys. 94, 4453 ~1991!. 8 E. O’Neill, Introduction to Statistical Optics ~Addison–Wesley, Reading, MA, 1963!. 9 B. E. Bayer, J. Opt. Soc. Am. 54, 1485 ~1964!. 1 2

J. Chem. Phys., Vol. 110, No. 6, 8 February 1999 B. Lu and S. Torquato, J. Opt. Soc. Am. A 7, 717 ~1990!. R. S. Fishman, D. A. Kurtze, and G. P. Bierwagen, J. Appl. Phys. 72, 3116 ~1992!. 12 J. Botsis and C. Beldica, Int. J. Fract. 69, 27 ~1995!. 13 B. Lu and S. Torquato, J. Chem. Phys. 93, 3452 ~1990!. 14 J. Quintanilla and S. Torquato, J. Chem. Phys. 106, 2741 ~1997!. 10 11

J. Quintanilla and S. Torquato 15

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In Refs. 13 and 14, the local volume fraction of phase 1 was studied. However, we note that the moments of the local volume fraction of phase 1 can be simply obtained from the moments of j. For example, ^ (1 2 j ) 2 & 2 ^ 12 j & 2 5 ^ j 2 & 2 ^ j & 2 , and so the variances of the local volume fractions of phases 1 and 2 are the same. 16 S. Torquato, I. C. Kim, and D. Cule ~unpublished!.