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Transport properties of random arrays of freely overlapping cylinders with various orientation distributions

Manolis M. Tomadakis and Stratis V. Sotirchos Department of Chemical Engineering, University of Rochester, Rochester, New York 14627-1066

(Received 24 July 1992; accepted 22 September 1992)

We present dimensionless effective transport coefficients (formation factors) for arrays of cylinders of various orientation distributions, namely, cylinders randomly positioned and oriented with their axes parallel to a line, parallel to a plane, or in the three-dimensional space with no preferred orientation. Both cases of conducting cylinders dispersed in a nonconducting matrix and nonconducting cylinders dispersed in a conducting matrix are considered. The transport coefficients are computed by means of a random walk simulation scheme. A comprehensive survey of past studies on transport properties of random arrays of cylinders is also presented, and our results are compared with the predictions of various analytical approximations or bounds and with experimental data of the literature.

I. INTRODUCTION

Arrays of cylinders of various orientation distributions dispersed randomly in a continous matrix can serve as a physical model for various two-phase materials: fiber- reinforced composites, where both phases are solid, porous media with their fluid-saturated pores resembling random assemblages of cylindrical capillaries, and fibrous structures with a gas or liquid matrix phase are types of materials that fall in the above category. Solid-solid composite dispersions are encountered in the study of transport and other properties of fiber-reinforced ceramics,‘-4 conductive polymer matrix composites,5 fiber-reinforced plastics,6-8 metal matrix composites,’ and metal-loaded dielectrics. lo Beds of fibers surrounded by a fluid appear in studies of immobilized liquid fibrous membranes,“-‘3 fiber-glass or silica fibrous thermal insulation,1617 physiological systems,‘8’19 fibrous filters for aerosol filtration,*’ pulp suspensions,21 bulk solutions or colloidal dispersions in equilib- rium with fibrous matrices,22 nonwoven fiber mats in coalescers,23 and fibrous preforms undergoing densification.2b28 Finally, structures of randomly overlapping cylindrical capillaries are used in modeling transport and re- action and structure evolution in various processes, such as the gasification of char29-30 and graphite,31 char combus- tion,32 matrix acidization of oil wells,33 removal of pollut- ants from coal utilization streams,3637 densification of porous media,38 and in general models of fluid-solid reactions.39A

Because of their diverse applications, the transport behavior of ordered and random dispersions of various geometries has long attracted the interest of researchers working in a broad part of the scientific spectrum. Max- we11,45 Lord Rayleigh, and Einstein4’ made some of the first valuable contributions to the field. A large number of studies has followed their work, including some interesting reviews of the research progress in the area.48-52 Several of these studies have pointed out the existence of a mathematical analogy among physically different effective transport properties of multiphase media, such as bulk diffusivity, thermal and electrical conductivity, dielectric constant, and magnetic permeability,48*50-57 which renders results

obtained for any of these transport properties in reduced form, with respect to the tranport property of one of the conducting phases, directly applicable to the others. Some researchers have also discussed the conceptual relationship of the above transport properties with effective opti- ca154758-62 and elastic48,51957 coefficients.

A relatively large amount of theorectical and experimental work has been presented in the literature on the transport properties of arrays of cylinders. In most studies, the problem has been investigated in the context of a certain trasport property, but because of the aforementioned analogy among different transport processes, there is usually considerable overlapping among different independent studies. For instance, theoretical studies on the transport behavior of arrays of cylinders have been presented by investigators working independently in the areas of thermal conductivity,63P64 dielectric constant,65a7 effective diffusiv- it y, 68&9 and electrical conductivity.70 The majority of experimental investigations of the transport behavior of random arrays of cylinders available in the literature deals with structures where transport coefficients in the matrix and cylinder phases differ by less than an order of magnitude. Only a few experimental measurements are reported for the effective transport coefficients in random arrays of nonconducting or minimally conducting cylinders dispersed in a conducting matrix.“P7**72

The outcome of analytical studies on transport in multiphase media is usually of approximate nature, most often applying only to particle densities exhibiting insignificant overlapping. This drawback of most modeling studies was pointed out by Milton et a1.,13 who also stressed the need for an exact treatment of the problem of transport in arrays of overlapping cylinders. Many theoretical studies have been devoted mainly to the derivation of variational bounds for the transport properties. Variational principles were formulated and applied by Hashin and Shtrikman53 for the derivation of bounds for the effective magnetic permeability of macroscopically homogeneous and isotropic multiphase materials of arbitrary geometric characteristics. Hashin and Shtrikman’s pioneering work was followed by other researchers, who derived improved bounds for the effective transport coefficient. Such bounds refer to either

616 J. Chem. Phys. 98 (I), 1 January 1993 0021-9606/93/010616-11$06.00 @ 1993 American Institute of Physics

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multiphase media of a certain geometry74 or broader classes of composite materials,54P57 in which case they in- volve some properly defined but not readily calculated mi- crogeometry parameters.75’76

A thorough numerical investigation of the effective transport coefficient of random arrays of overlapping cylinders is presented in this study. Cylinders are dispersed randomly in a continous matrix with their axes parallel to a line or a plane, or oriented randomly in three dimensions ( l-, 2-, and 3-directional random arrays, respectively). The conductivity of either phase of the medium (i.e., the matrix or the cylinders) is considered negligible in comparison to that of the other phase. A discrete random walk procedure is employed to simulate the movements of the carriers of the transport property in the conducting phase, and the mean square displacement equations47’77-79 are used to compute the dimensionless effective transport coefficients (formation factors). Simulation results are compared with the predictions of several theoretical and experimental studies.

II. STRUCTURAL CHARACTERISTICS AND COMPUTATION OF THE EFFECTIVE TRANSPORT COEFFICIENTS

Among the various mechanisms available in the literature for drawing random chords in cubes or other convex bodies*‘-a problem equivalent to that of generating a random array of cylinders in the cubic unit cell used in our computer simulations-the p-randomness mechanism, in the terminology of Coleman,80 is the only one producing structures with spatially invariant characteristics, i.e., structures identical, in a statistical sense, to those obtained by distributing infinitely long cylinders randomly in space according to some orientation distribution. p-random structures are constructed by sampling points randomly from a uniform distribution on a fixed plane and assigning them random directions from a probability measure that is reflection invariant with respect to the surface of the convex body and bounded so that only points and directions defining a secant of the body are considered. Arrays of randomly overlapping cylinders result by treating these se- cants as axes of cylinders of radius Y. Figure 1 shows a cross section of the unit cell of such an array of unidirectional cylinders with matrix volume fraction equal to 0.45, and Fig. 2 a section of the unit cell of an array of 3-d cylinders with phase volume fractions equal to 0.5. A detailed description of the ,u-randomness construction procedure applied here and in our previous studies81982 is presented by Tomadakis.83

The matrix volume fraction, E, phase interface area per unit volume, S, and mean intercept length of the matrix and cylinder phases, zm and &, respectively, of random arrays of cylinders are related to the cylinder size and density of cylinders by the following equations”

e=l-+=exp(-rP); Sr=--2elne, (la,b)

a 4E -2 A=-=-. ;I, 44 -24 r Sr lne’

-=-- r Sr elne’ -- (2a,b)

M. M. Tomadakis and S. V. Sotirchos: Transport properties of overlapping cylinders 617

FIG. 1. Cross section of the unit cell of an array of unidirectional cylinders. ~=0.45.

I is the length of cylinder axes per unit volume of the medium (density of cylinders) and d, the volume fraction of the cylinders.

A Monte Carlo simulation procedure utilizing the mean square displacement, ( p), of a large number of random walkers introduced randomly in the unit cell is used to obtain the effective transport coefficients. The method is based on the integration of the equation describing Fick’s second law of diffusion, or its mathematical analogs for other transport processes, and application of the solution to get the expression for (c2). The resulting equations are of the form47977-79

FIG. 2. Section of the unit cell of an array of 3-directional cylinders. e=o.5.

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618

cP> c$ Pe’z ; Pej’y 2 _ .=.-_- (30)

where r is the travel (transport) time. (p) is used to obtain the orientationally averaged effective transport coefficients, pe, while the mean square component of the displacement in the j direction, (#), gives the effective coefficients associated with transport in that direction. Notice that pe is expressed in units of length2/time, and therefore, it stands for molecular diffusivity regarding mass transport, thermal diffusivity when heat transfer is considered, and so on. Formation factors, F and Fj, are obtained from the following equations:

F =P/P,; Fj=p/pej 9 (4a,b)

where p stands for the transport coefficient in the conductive phase, i.e., p=pm or p=pa accordingly. It should be noted that the subscript of direction, j, is dropped from Pej and <j during the presentation of our results because the directions of transport are clearly specified in the text.

Trajectory computations begin with sampling random points in the unit cubic cell repeatedly, until one that lies in the conducting phase is located and becomes the starting position for the trajectory of a random walker. The direction of the first step of the walker is determined through random direction cosines, and the position of the first po- tential collision with another walker is found by sampling the length of a continuum free path, /2, from the exponential distribution84~85~77

f(A) =; e-q (5)

In the context of molecular diffusion, this path distribution is strictly valid for carriers moving with constant velocity. However, a more accurate expression based on the Max- wellian distribution of velocities practically coincides with the exponential law.84 Experimental verifications of Eq. (5) are reported by Loeb” for the free paths of silver atoms diffusing in air or nitrogen and electrons moving through a gas in a beam. The continuum mean free path, 1, must fulfill the condition I& 2, with C? being the mean intercept length of the conductive phase. This condition guarantees that a sufficient number of carriers of the transport property is present in the medium so that their collisions with each other largely outnumber their collisions with the nonconducting walls. Different values of the parameter X/C? were tested, and those of the order of 0.02 or less were found to give results that were practically inde- - - pendent of the magnitude of this ratio. The value it/d =0.02 was thus chosen for the computations performed in this study. We elected to work with the largest possible - - value of A/d in order to keep the computational time as low as possible.

As long as the random walker completes the sampled steps without interference with the nonconducting phase or a boundary of the cell, a sequence of collisions with other walkers is generated by repetitive application of the above procedure. When a surface element of the nonconducting phase is found to lie in the walker’s way, its distance from

M. M. Tomadakis and S. V. Sotirchos: Transport properties of overlapping cylinders

the starting point of the current path is computed, and the next walker-wall collision is thus located. New direction cosines are then assigned to the random walker, to account for its diffuse reflection86’87 from the wall. Random walkers reaching the boundaries of the unit cell are reintroduced randomly in the conducting phase and let to continue their sojourn retaining their direction of travel. Specular reflection boundary conditions were also tested and seen to be suitable for simulations in a conductive matrix, but inap- propriate for computations in beds of conducting cylinders dispersed in a nonconducting matrix. A detailed discussion of the procedure for trajectory computations is given by Tomadakis,83 where other numerical techniques for computation of effective transport coefficients, based on dy- namic molecular simulation algorithmsg7 or on random walks employing the first passage time (FPT) probability distribution,88-g2 are also discussed.

In the absence of the nonconducting phase, the above described random walk simulation scheme yields p=4IV, with Y being the mean speed of the carriers, and this value of p is the one used in Eq. (4) to compute formation factors. Different values for the transport coefficient of the conducting phase (p) result when a nonexponential distribution of free paths is used in the random walk scheme or when the persistence of velocities77~s4~87~g3 during carrier- carrier collisions-i.e., the moderate tendency of succeed- ing paths to favor the direction of preceding ones-is taken into account. In a previous study,82 we investigated these effects on the computed effective transport coefficients in the context of bulk diffusion in random fiber structures. In order to study the effects of the form of the path distribution, computations were carried out for constant free path between molecule collisions. On the other hand, the depen- dence of the results on the persistence of velocities was examined by using biased random walks, in which the mi- nor angle formed by the vectors that represent the velocities of the carrier before and after collision (carrier- carrier) is arbitrarily bounded from above. In both cases, it was found that while both the self-diffusion coefficients in the gas phase and in the porous medium (p andp,) were different from those obtained for a no-memory random walk with exponential path distribution, their relative values [i.e., the tortuosity factor (for diffusion) or formation factor (in general)] remained invariant.

111. RESULTS AND DISCUSSION

A. Conductive matrix

1. Square array

We first computed the formation factor corresponding to transport perpendicularly to nonconducting, nonoverlapping, unidirectional cylinders ordered in a square array in order to validate our numerical code by comparing our results with those given by an analytical formula derived by Perrins et aLg4 Perrins et al. extended a method devised by Lord Rayleigh to obtain the values of the effective transport coefficient of both square and hexagonal arrays of cylinders for a broad range of values of the relative conductivity of the two phases. Their analytical result for

J. Chem. Phys., Vol. 98, No. 1, 1 January 1993



TABLE I. Formation factor of a square array of noxiconducting cylinders

E Simulations Perrins et al.

0.99 1.05 1.02 0.90 1.24 1.22 0.80 1.49 1.50 0.70 1.91 1.86 0.60 2.35 2.35 0.50 3.02 3.08 0.30 7.22 7.36

the formation factor regarding transport perpendicularly to a square array of nonconducting nonoverlapping cylinders is of the form

(6)

where C$ is the cylinder volume fraction, a=O.305 83, b= 1.402 96, and c=O.O13 36. The predictions of Eq. (6) are compared in Table I with our simulation results for various values of 4, and very good agreement is seen to exist between the two sets of values.

Equation (6) has been verified by Sangani and Yaog5 by means of a numerical method based on the many- particle interactions, Durand and Ungarg6 on the basis of the boundary integral element method, Kim and Tor- quato” through an extension of the first passage time probability-based simulation procedure, and Grove” through finite elements analysis. Perrins et aLg4 compared the predictions of Eq. (6) with their experimental results and those of other investigators, and very good agreement was noticed. A similar observation was also made by Du- rand and Ungar,g6 using experimental results from the literature.

The problem of transport in square arrays of cylinders has been considered by several other researchers, in addition to Perrins et al. Behrens” applied Born’s method of long waves to obtain analytical expressions for the thermal conductivities of various classes of composite materials, including the square array of cylinders. His result for nonconducting cylinders, F& ’ = 1 - 2f$/( 1 + f$), practically coincides with that of Perrins et al. for 4 ~0.5. Hasselman and Johnson” modeled the problem of a thermal barrier resistance at the matrix-inclusion interface, utilizing the Rayleigh and Maxwell theories. For the case of nonconducting cylinders oriented perpendicularly to the heat flow, their prediction coincides with that of Behrens. Shoutens and Roigg developed a model for the transverse electrical conductivity of a square array of nonconducting nonoverlapping fibers, but, because of various assumptions made in their analysis, their results do not agree with those of Per- rins et al. Finally, Milton et aL73 extended the work of Perrins et aLg4 to account for cylinder overlapping in square arrays, and obtained the corresponding effective transport coefficients for various finite values of the relative phase conductivity.

a 3-D Cylinders ., o : Our Results

b : Tsoi and Strieder c : Klemens d : Davies; Van Beek

I-. PC--+ 0

0

& 1

FIG. 3. Variation of the formation factor with the phase volume fraction, in the isotropic 3-directional random array of nonconducting cylinders.

2. Random arrays

Our computer simulation results for the formation factor of random dispersions of nonconducting, freely overlapping cylinders of all directionalities are presented in Figs. 3-7. Each data point on these figures represents a different realization of the two-phase medium with regard to cylinder positions, radii, and orientations. The relatively insignificant scattering of the data for each configuration of transport indicates that the formation factor exhibits a sys- tematic dependency on the phase volume fraction only, the effect of other realization differences being negligible. At the limit of E-+ 1, where the presence of the nonconducting cylinders becomes unimportant, the dimensionless effective transport coefficient approaches unity in all cases. The matrix volume fraction value below which the medium is im-

J - PC+0 Transport Perpendicularly to

2-D Cylinders ., o : Our Results

b : Moth-am and Taylor c : Tsai and Strieder d : Klemens

- Experimental Dota: o : Botemon et 01.

l2 0 : Penman

i d.2 ’ d.4 ’ d.6 ’ di-

MATRIX VOLUME FRACTION, E

FIG. 4. Variation of the formation factor with the phase volume fraction, perpendicularly to the plane defined by the directions of the cylinders in random arrays of bidirectional nonconducting cylinders.


620 M. M. Tomadakis and S. V. Sotirchos: Transport properties of overlapping cylinders

- PC-0 Transport Parallel to

2-D Cylinders L l , a : Our Results

b : Klemens c : Moth-am and Taylor d : Davies

g “\,C

lo= ‘\

2

‘.

e

1- ‘, II I ’ I r I * I ’ 0 0.2 0.4 0.6

MATRIX VOLUME FRACT,:; E

FIG. 5. Variation of the formation factor with the phase volume fraction, parallel to the plane defined by the directions of the cylinders in random arrays of bidirectional nonconducting cylinders.

permeable to transport, namely, the percolation threshold, ep, is seen to vary with directionality, in agreement with previous observations.44’81 ~~~

The results of other investigations on the problem of transport in random dispersions of nonconducting cylinders are presented in Figs. 3-6 and Table II, and will be discussed in the following sections.

a. Comparison with bounds. Tsai and Strieder74 applied variational principles to obtain both upper and lower bounds on the effective thermal conductivity of 3-d and 2-d arrays of randomly overlapping cylinders, for all possible values of the phase conductivities. For nonconducting cylinders, their lower conductivity bounds vanish and their upper bounds in conductivity become lower bounds in tor-

L

- PC-,0 Transport Perpendicularly to

*t,d 1-D Cylinders

l , a : Our Results b : Klemens c : Davies d : Mottram and Taylot e : Milton’s bound

f : Nielsen g : Peterson and Hermans

Impenetrable Cylin - h : Milton’s bound

q : Kim ond Torquato 1: X : Abukay et al.

I 1 t 8 I 2 I r 0

MA %X VO?“ME ;;ACT,;( 8

FIG. 6. Variation of the formation factor with the phase volume fraction, perpendicularly to the cylinders, in random arrays of unidirectional nonconducting cylinders

PC-, 0 Arrays of Cylinders

L .

fi5

100:

b cs

g

9

1OZ

5

17

Random: a : 3-D b : 2-D, Perpendicular c : 2-D, Parallel d : l-D, Perpendicular o : l-D, Parallel

----- - : Perpendicular

-I , a I t 1 4 , r

0 0.2 0.4 MATRIX VOLUME

FRACT,;: 0.6 cz

FIG. 7. Comparison of the formation factors of various arrays of nonconducting cylinders for all directionalities.

tuosity or formation factor. It is seen in Figs. 3 and 4 that Tsai and Strieder’s bounds are in agreement with our simulation results, and can further be used to determine the formation factor of dilute beds.

Milton54 obtained expressions for upper and lower bounds on the various transport properties of fiber- reinforced materials. In terms of the formation factor of a random array of nonconducting cylinders, Milton’s expressions are considerably simplified to give

1+ 2( l--E) -<F< 00.

&nl (7)

This bound is claimed to be valid for structures of any directionality provided that the material is statistically isotropic in the transverse plane. Parameter 5;, is a geometric characteristic of the matrix phase of the structure (the corresponding parameter for the cylinders is 5‘,= 1 - 6,) and must be calculated from the three-point correlation function.s4

Torquato and Beasley7’ expressed 5‘ in the form of a multidirectional integral involving lower-order n-point probability functions, and estimated its value for dispersions of freely overlapping unidirectional cylinders for various values of the phase volume fractions. Joslin and Ste1176 calculated independently the same values for <. They also examined the effect of the cylinder diameter polydispersity on c and found it to be insignificant.lW Smith and Tor- quato”’ obtained the values of [ for random arrays of unidirectional cylinders of varying penetrability, through Monte Carlo measurements of the n-point probability functions. Their results for the case of fully penetrable cylinders agree with those of the two previously mentioned investigations. Substituting these 5;, values into Es. (7) we obtained the lower bound that is plotted in Fig. 6. This bound is in very good agreement with our results, seen to provide the exact values of F for dilute dispersions of cylinders.




TABLE II. Inverse formation factor of random arrays of cylinders (~~-0).

l-d 1 1-d II 2-d 1 2-d II 3-d Source

3E- 1 2

( 1 1+g --I

”

2~-1+1.26(1+)*

2

E

3-26+0.535(1-&

2E- 1

E . . . . . . . . . Law of mixtures 3E- 1 . . .

2 5E- 1

4

. . . E

I-1nE . . . E

l-(2/3)lne Tsai and Strieder

. . . . . . . . . . . . Milton

. . . . . . . . . . . . Peterson and Hermans

. . . 2 2+e ~~ . . . 2

Mottram and Taylor

. . . . . . . . . . . . Nielsen

. . . 2E- 1 3E- 1 2

56-2 3

Davies; Van Beek( 3-d)

Smith and Torquato also obtained the values of f for structures of impenetrable (nonoverlapping) unidirectional cylinders, constructed through the nonequilibrium random sequential addition process. Their simulation results, for the range eaO.45, were substituted into Eq. (7) to compute Milton’s bound for such structures. This bound is also shown in Fig. 6, and it is seen to be in very good agreement with exact F values computed by Kim and Tor- quato through the app lication of Keller’s phase interchange theorem on their simulation results for infinitely conducting impenetrable l-d cylinders. Torquato and Lado lo2 obtained estimates of c by using the Kirkwood superposition approximation, but the resulting Milton’s bound diverged from the exact values of F at low matrix volume fractions. They also derived a low (particle) density expansion of 5; which was found to follow the same general trends as the simulation results of Sangani and Yaog5 and to provide a very good approximation to c for a very broad range of volume fractions.55”03 However, all studies gave practically identical results for dilute beds (l90.7).

b. Comparison with analytical approximations. Kle- mens63 derived an expression for the effective thermal conductivity of inhomogeneous media in terms of the Fourier components of the spatial variation of the conductivity, and applied it to composites consisting of spherical or cylindrical inclusions in a continous matrix. For insulating cylinders, Klemens postulated that his results would be valid for relatively dilute beds only, viz. e)O.5, because of the assumptions followed in his approach. Indeed, his results are seen (in Figs. 3-6) to be fairly accurate in the suggested volume fraction range, but they slightly violate the aforementioned variational bounds.

Davies67 followed an effective medium approach to de- rive approximate formulae for the dielectric constants of composite materials consisting of fiberlike inclusions em- bedded in a continous matrix. His results for the case of nonconducting fibers (Figs. 3-6) are seen to be accurate for very dilute beds only. It should be mentioned that Dav-

ies derived his expressions for geometries satisfying certain continuity conditions for the fiber and matrix phases, some of which are not obeyed by the structures examined in this study. An expression identical to that of Davies for 3-d fibers can be obtained from the work of Van Beek65 who derived an approximation for the dielectric constant of systems containing dispersed cylinders with random distribution of orientations using older approximations for dispersed ellipsoids. Ogston et aL6’ employed stochastic arguments to obtain an expression for the effective diffusivity through random arrays of fibers, in order to model transport of large molecules through solutions of chain polymers. The resulting approximation for zero size molecules was found to predict results much different from those found in our simulations.

Koch and Brady6* applied ensemble averaging techniques to the basic conservation equations in a fibrous medium to determine the asymptotic behavior of the bulk effective diffusivity in the limit of high porosities, both in the presence and absence of forced convection. Their for- mulas for the case of pure diffusional flow turn out to be-after correcting some errors in Tables I and II of their articleidentical to those of Davies.67 Mottram and Tay- lor-64 developed simple approximations for the effective thermal conductivity parallel to the fibers of a 2-d carbon cloth-reinforced composite and perpendicularly to the fibers of unidirectional or bidirectional composite materials. It is seen in Figs. 4 and 5, that Mottram and Taylor’s result for transport perpendicularly to 2-d fibers performs successfully in the most part of the phase volume fraction range. Their approximation for the formation factor parallel to the fiber layers in such structures is satisfactory for dilute beds only, which is also the case with their prediction for unidirectional fibers.

Nielsen7’ used previously developed equations from the theory of the elastic moduli of composite materials to calculate transport properties, such as electrical and thermal conductivities, of two-phase systems. His result for nonconducting unidirectional fibers is fairly accurate at



high matrix volume fractions even though it fails to follow the asymptotic behavior of the F vs E curve close to 1 (see Fig. 6). Peterson and Hermans carried out a detailed analytical treatment to the problem of the dielectric constant of suspensions of l-d cylinders perpendicular to the external field. Their approximation predicts the exact transport behavior of dilute beds, but it deviates signifi- cantly from the exact result and violates Milton’s bound even at relatively high values of the matrix volume fraction.

c. Comparison with experimental data. Bateman et al. l1 measured the bulk diffusivity of NO through various porous substrates and reported their results in terms of tortuosity factors. For flow perpendicularly to the fibers of a cellulosic filter paper substrate-a bidirectional random fibrous structure-they reported tortuosity factor 1.5 at 0.65 porosity, corresponding to a formation factor of 2.31 (F-~/E). Bateman et al. also reported a value of 1 for the tortuosity (or l/e for the formation factor) parallel to the uniform cylindrical pores of a track-etched polycarbonate substrate. This last result lends support to the validity of the experimental work of Bateman et al. since it is consistent with the law of mixtures for transport parallel to the interface of unidirectional structures. Penman’l measured the bulk tortuosity of many porous solids through steady state diffusion experiments. Among his results, a tortuosity value of 1.14 was reported for diffusion in steel wool of porosity e=O.93, corresponding to a value of 1.23 for the formation factor. Bateman et al’s and Penman’s results are shown in Fig. 4, and seen to be in very good agreement with the results of our study.

Abukay et aI.72 carried out reproducible measurements of the electrical resistivity of AI/B composites in a broad temperature range (78&O K). The fibers were well sep- arated from each other and considered totally nonconducting in comparison with the aluminum matrix, which was made using ribbons helically wound around the boron fibers. Abukay et al. claimed that because of this method of construction, strong electron scattering from the interfacial surfaces between the fibers and the surrounding matrix might exist, leading in turn to additional resistance to transport. Although it might be’ purely coincidental, the agreement of Abukay et al’s experimental result (shown in Fig. 6) with the results’ of our numerical simulations may be construed as an indication that the transport characteristics of arrays of freely overlapping cylinders may approximate successfully the corresponding characteristics of arrays of nonoverlapping cylinders with poor adhesion in the matrix-inclusion interface.

3. Effect of cylinder directionality on transport

All formation factor curves from Figs. 3 to 6 and Eq. (6) are summarized in Fig. 7 for the shake of comparison. Shown there are also simulation results for the formation factor parallel to the cylinders of random and square arrays of unidirectional, nonconducting cylinders. The results for the latter two cases are seen to agree exactly with the law of mixtures:

Pe=EPm+(l--E)Pc. (8) For nonconducting cylinders (pc=O), Eq. (8) becomes p,/p,=F= l/e. As we mentioned in the preceding section, Bateman et aL’si* experimental results for transport in unidirectional pores were also in agreement with the law of mixtures.

Figure 7 shows that unidirectional structures exhibit highly anisotropic behavior, as opposed to the almost isotropic behavior of the bidirectional structures. The latter was also deduced from thermal conductivity measurements of fiber-reinforced plastics by Harris et al. 6 Isotropic arrays of 3-directional cylinders are seen to exhibit smaller resistance to transport than bidirectional arrays for flow perpendicular to the plane defined by their cylinders. The highest resistance to transport is presented by unidirectional, random arrays in a direction perpendicular to the cylinders. In the same direction, square arrays of unidirectional cylinders behave, in the most part of the matrix volume range, more like 3-d random structures or 2-d structures parallel to the plane defined by the directions of the cylinders, rather than random unidirectional arrays. The most probable explanation for this behavior is the nonoverlapping placement of the fibers of a square array, which leads to a more open structure perpendicularly to the cylinders, in an average sense, than in the case of a random unidirectional array. Notice that the percolation threshold, ep, of a square array perpendicularly to the fibers is much lower than that of a random array (0.21 vs 0.33, respectively).

B. Conductive cylinders

The variation of the formation factor with the matrix volume fraction in arrays of conductive cylinders dispersed in nonconductive matrices is presented in Figs. 8-l 1 for all directionalities and configurations of transport. Realization effects at certain phase volume fractions are seen to be insignificant, as it was the case with nonconducting cylinders. The percolation thresholds of bidirectional and tridirectional arrays are now found to correspond to zero volume fraction of cylinders, as it is intuitively expected. This result was also shown by Burganos and Sotirchos42 in their study of Knudsen diffusion in randomly overlapping capillaries. The percolation threshold for transport perpendicularly to the conductive cylinders of a unidirectional array is located at the same matrix volume fraction as that for nonconductive cylinders, in accordance with previous observations.41

Most of the analytical expressions presented in the investigations discussed earlier in this work can also be trans- formed into simplified equations corresponding to arrays of conductive cylinders dispersed in a nonconductive matrix. The resulting expressions are presented in Table III and plotted in Figs. 8-l 1 for comparison with our exact results. Tsai and Strieder’s74 (Figs. 8 and 9) and Milton’s54 (Fig. 11) lower bounds are satisfied by our results, but they give satisfactory predictions only at the limits of the volume fraction range. The same observation also holds for a finite upper bound on the formation factor, F< 1 - 3/ln e, result-




3-D Cylinders . . a : Our Results

b : Tsoi and Strieder c : Van Beek

_ ------ : Davies

FIG. 8. Variation of the formation factor with the phase volume fraction, in the isotropic 3-directional random array of conductive cylinders dispersed in a nonconducting matrix.

ing from Tsai and Strieder’s expressions for 3-d conductive cylinders dispersed in a nonconducting matrix. Kle- mensys63 predictions are seen to be less successful1 than those for nonconducting cylinders, with the exception of transport parallel to the planes of cylinders in bidirectional structures. This configuration, along with that of transport in tridirectional arrays, is also where Davie&’ effective medium approach provides very successful1 predictions. Finally, Van Beek’8’ expression for tridirectional arrays is seen to predict the exact values of the formation factor for very dilute beds only.

_ Transport Perpendicularly to

L l , a : Our Results .

25 b E

$ 101

6 d.2 4.4 6.6 ’ $8 ’ MATRIX VOLUME FRACTION, E

FIG. 9. Variation of the formation factor with the phase volume fraction, perpendicularly to the plane defined by the directions of the cylinders in random arrays of bidirectional conductive cylinders dispersed in a nonconducting matrix.

Transport Parallel to

Z-D Cylinders L ., a : Our Results

.

8

100: b : Davies c : Klemens

b s

$

9

101

1: Pm- D

I 3 I * I b , a 0

MA%X VOoi4”ME ;iACl-,;( E 1

FIG. 10. Variation of the formation factor with the phase volume fraction, parallel to the plane defined by the directions of the cylinders in random arrays of bidirectional conductive cylinders dispersed in a nonconducting matrix.

C. Comparison of the transport behavior of all arrays

The simulation results from Figs. 3-11 for transport through random arrays of freely overlapping cylinders are brought together in Fig. 12 for comparison. Transport through arrays of conductive cylinders always presents more resistance than transport in the same direction through a conductive matrix of the same directionality for large fraction of conductive phase volume, a behavior that can be attributed to the concave character of the nonconductive walls in the former. Since the percolation thresholds of 2-d and 3-d structures of conductive cylinders are located at zero and are thus smaller than those of the corresponding structures of conductive matrix, the oppo-

1-D Cylinders P,- c Transport Perpendicularly

L to the Cylinders:

* 100 ., a : Our Results

8 b : Milton’s Bound

b c : Davies d : Klemens

P J id I

1 r I E , c , I , 1 0

MA;:,X VO?“ME i$ACTICt( E 1

FIG. 11. Variation of the formation factor with the phase volume fraction, perpendicularly to the cylinders, in random arrays of unidirectional conductive cylinders dispersed in a nonconducting matrix.



TABLE III. Inverse formation factor of random arrays of cylinders (p,,,-0).

l-d 1 1-d II 2-d II 2-d 1 3-d Source

3+- 1 . . . 54-l 34-l 44-l 2 4 2

Klemens 3

2E -’ ( 1 ‘+zc ... ‘.. . . . . . . Milton

. . . . . . . . . 42-l l+g+l+EInE

42-l *+~+1+(2/3)~ln~

Tsai and Strieder

. . .

24-l

. . . . . . . . . W/3 Van Beek

u+ (u2+9m’” u+(u2+~/3)‘” . . .

(.+2) 2+-1 (,2!q Davies

. . . . . . . . . . . . Law of mixtures

site situation prevails for low values of conductive phase volume fraction for these two structures. Of course, for transport parallel to unidirectional cylinders, the same ex- . . pression, I.e., the law of mixtures, relates the effective transport coefficient with the conductive phase volume fraction both for conductive and noconductive cylinders.

Another interesting observation is the highly anisotropic behavior of bidirectional arrays of conductive cylinders, which contrasts with the almost isotropic character of the corresponding arrays of nonconducting cylinders. Much higher anisotropy is exhibited by the unidirectional array of conductive cylinders. In either case of conductive and nonconductive cylinders, unidirectional cylinders present the highest resistance of all configurations of transport and cylinder directionality for transport perpendicularly to the cylinders, and the lowest for transport parallel to them.

KellerlW introduced the phase interchange theorem on

b : 2-D, Perpendicular c : l-D, Perpendicular d : l-D, Parallel e : 2-D, Parallel

FIG. 12. Comparison of the formation factors of various arrays of cylinders for all directionalities.

the effective conductivity (electrical, thermal, or other) of a composite medium

Pe(PI,P2)Pe(P21PI 1 =plP2 9 (9)

where 1 and 2 stand for the materials constituting the two- phase medium, and p,(p,,p,) for the effective transport property. Equation (9) states that the product of the effective transport coefficients of two geometrically identical two-phase media resulting from each other by interchang- ing the materials of their phases is simply equal to the product of the continuum transport coefficients in the two materials. By inspection of Eq. (9), it is readily concluded that Keller’s law can hold only for two-phase media in which only one of the two phases can exist in continuous (connected) form, such as a structure of unidirectional cylinders where the percolation thresholds of matrix and cylinders coincide. KellerlW presented a proof of his theorem for the case of a square array of cylinders only, but claimed that it would also apply to the average conductivity of a statistically homogeneous, isotropic (regarding transport perpendicularly to the cylinders) random disper- sion of cylinders of one. medium in another. Perrins et al. g4 gave a rigorous proof of Keller’s theorem for both square and triangular arrays of cylinders. Mendelson”’ presented a more general proof for any statistically homogeneous and isotropic (with respect to transport perpendicularly to the phase interface) two-phase system, for which the phase boundaries are infinite cylindrical surfaces with generators parallel to a certain axis.

For pI,p2#0 Keller’s law is written

pt (PbP2) =pg (p2,pd; Pe pc (PllP2) =; (P27PI).

(lOa,b)

It follows from Eq. (10) that the results we obtained for unidirectional structures of conductive or nonconductive cylin’ders can be used to determine the asymptotic behavior of the effective transport coefficient of the structures that result from phase interchange (nonconductive or conductive cylinders, respectively) for small perturbations of the



h I”” -I T-------l n-----Ar wlarly to 3-s I

k

0

a

10:

IrQnspori rerpencll‘

1-D Cylindc , l : F=p,/p,: p, - 0

E

3

& 2 1: I b I b I 8 I 1

0 0.2 “Oy”ME ;;A&: 1

MATRIX E

FIG. 13. Application of the phase interchange law on our results for unidirectional arrays of cylinders.

transport coefficient of the nonconductive phase from zero. For the case of conducting matrix (pe+O), Eq. (lOa) states that pm/p, (i.e., the formation factor) equals pJp, for the reverse structure as pm-O. In other words, while the presence of practically nonconducting cylinders in a conductive matrix reduces the overall conductivity by a factor of F, the presence of conductive cylinders in a practically nonconductive matrix increases the overall conductivity by the same factor. A similar result is obtained from Eq. (lob) for the case of conductive cylinders. The above results are plotted in Fig. 13, which also shows Milton’s bounds for the cases of conductive cylinders and conductive matrix. Milton’s bounds are consistent with Keller’s law, and as a result, they do not change when phase interchange is applied.

IV. SUMMARY AND CONCLUSIONS

The transport characteristics of dispersions of randomly overlapping cylinders of various orientation distributions were investigated by computing, through simulation, their dimensionless effective transport coefficients (formation factors). The two-phase structures consisted of cylinders dispersed randomly in a continous matrix with their axes parallel to a line or a plane or oriented randomly in the three dimensional space (l-, 2-, and 3-directional random arrays, respectively). Both extreme cases of nonconducting cylinders dispersed in a conductive matrix and conductive cylinders dispersed in a nonconducting matrix were considered. A step-by-step random walk procedure was employed to simulate the trajectories of the carriers of the transport property in the conducting phase, and the mean square displacement equations were used to compute the effective transport coefficients.

The formation factors were found to depend strongly on the directionality of the structures at low and interme- diate values of the conductive phase volume fraction, di-


verging to infinity as the percolation threshold of each dis- persion (i.e., the conductive phase volume fraction below which the structure becomes impermeable to transport) is approached. Arrays of 3-directional cylinders were seen to exhibit smaller resistance to transport than the corresponding bidirectional arrays against transport perpendicularly to their cylinder planes, the difference being much bigger for the case of conductive cylinders dispersed in a nonconductive matrix than it is for the other relative phase conductivity extreme. Transport in bidirectional arrays of nonconductive cylinders encountered slightly less resistance parallel to the planes of cylinders than perpendicularly to them, indicating that these structures exhibit a low degree of anisotropy. However, this was not the case with bidirectional arrays of conductive cylinders, where transport parallel to the planes of the cylinders was much faster.

Random unidirectional arrays of cylinders were seen to exhibit much higher resistance to transport perpendicularly to the cylinders than the corresponding 2- and 3-directional structures for both cases of conductive cylinders and conductive matrix, the differences being larger for conductive cylinders. Transport parallel to the cylinders in unidirectional structures was found to obey the well- known law of mixtures. The transport behavior of square arrays of nonconductive cylinders for transport perpendicularly to the cylinders resembled more that of tridirectional and bidirectional random arrays, rather than that of random unidirectional arrays. Comparison of the formation factors of arrays of cylinders of the same directionality but different relative phase conductivity showed that conductive cylinders presented higher resistance to transport than the corresponding structures of conductive matrix for high values of the fraction of conductive phase volume, while the opposite behavior prevailed at the other extreme. The only exception was observed for transport parallel to unidirectional cylinders, where the curvature of the walls is not a decisive factor, and the same expression, namely, the law of mixtures, predicts the transport behavior in both cases.

As expected, the computed values of formation factor satisfied lower and upper bounds reported in the literature for the studied structures, and most of these bounds were found to provide satisfactory estimates of the formation factor of dilute dispersions of cylinders (conductive or nonconductive) of all directionalities and dense arrays of conductive unidirectional cylinders. The computed formation factors for transport perpendicularly to nonconducting cylinders ordered in a square array were in execellent agreement with older analytical, numerical, and experimental estimates. Our simulation results for random arrays were found to be in very good agreement with some experimental data for transport in random arrays of nonconducting cylinders, reported in the literature. A few analytical approximations of the effective transport properties of the random structures that we concidered in this study performed satisfactorily throughout the phase volume fraction range, but all others provided acceptable results only at the limits of high or low matrix volume fraction.



ACKNOWLEDGMENTS

Acknowledgment is made to the Donors of the Petro- leum Research Fund administered by the American Chem- ical Society for support of this research. We also thank the Pittsburgh Supercomputing Center for providing super- computer time for the computations reported here.

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