PHYSICAL REVIEW B VOLUME 30, NUMBER 7

Space-filling constraint on transport in random aggregates

1 OCTOBER 1984

T. A. Witten and Yacov KantorCorporate Research Science Laboratories, Exxon Research and Engineering Company,

Route 22 East, Annandale, Ne~ Jersey 08801(Received ll May 1984)

Transport properties such as conductivity on scale-invariant structures such as random aggregates aregoverned by a spectral dimension exponent d. We investigate the behavior of d for branched structureswithout loops using geometric constraints on their current-carrying backbone. The requirement that thestructure fit into d-dimensional space imposes a new constraint on d. The distance b between branches ofthe backbone must be of the order of the span N measured along the backbone. Under mild scaling as-sumptions this distance 5 controls the conductance. We express d in terms of the fractal dimension D andthe scaling power 8 relating N to the geometric size L (iV- Ls): 8-D(2/d 1). T—his implies that d ( 3

for such branched structures with D ( 2.

Transport on tenuous random structures such as percolat-ing clusters or diffusion-limited aggregates appears to haveremarkably general scaling properties. Alexander and Or-bach' have argued that conductivity, diffusion, and scalarwaves on such structures may be described by a single newexponent called the fracton or spectral dimension d. Thisexponent describes the number of distinct sites $(r) visitedby a t-step random walk on the structure: S—t+' (for2 s

d(2). Remarkably, the spectral dimension appears tohave nearly the same value T for a wide class of structures,including percolation clusters at length scales smaller thanthe correlation length' and diffusion-limited aggregates. 4

Rammal and Toulouse and Leyvraz and Stanley' have ar-gued that this value may arise from some qualitative"homogeneous" characteristic of these structures.

In this Rapid Communication we consider transport onrandom branched structures, such as diffusion-limited aggre-gates or cluster aggregates. These aggregates are definedby irreversible growth processes, and have a negligiblenumber of loops at large scales, as illustrated in Fig. 1. Theabsence of loops puts constraints on the transport proper-ties, as we show below. Our argument requires certain scal-ing properties of the sort normally assumed for these scale-invariant structures. ' Given these properties, we mayrelate the spectral dimension to purely geometrical scalingproperties. Further, the spectral dimension and thefractal dimension' D relating mass M to geometricsize L (M —L ) are constrained by an inequalityD ~ il(2ld —1). A branched structure which violated thisinequality would be expected to overfill space of any finitedimension. To'our knowledge, such a relation betweentransport and the filling of space has not previously beenrealized. In view of our inequality d must be smaller than

at least when D(2. Since D (2 for both diffusion-limited aggregates and cluster aggregates in sufficientlysmall dimensions "' the arguments leading to d = T ap-

parently do not apply to these.Other authors' ' have identified relations between struc-

tural and transport properties for branched structures. Onesuch property is the point-to-point conductance. This prop-erty. is sufficient to determine the spectral dimension in abranched structure with a finite order of ramification. ' Butin the random structures of interest here the order of rami-fication is not known.

In a uniform medium the point-to-point conductancedetermines the conductivity of the medium and thence thespectral dimension. But for more general structures, thepoint-to-point conductance does not suffice to describetransport in other geometries. In a Cayley tree, for exam-ple, the point-to-point conductance is inversely proportionalto the number of bonds N in the path connecting them.

FIG. 1. Diffusion-limited aggregate (top) and cluster aggregate(Ref. 7) (bottom) in two dimensions. The absence of loops on allbut smallest scales is evident.

30 1984 The American Physical Society

4094 T. A. ~TTEN AND YACOV KANTOR 30

But the conductance between a point and a surroundingsphere is independent of the size of the sphere. It is thislatter conductance which is related to diffusion and to thespectral dimension. This point-to-sphere conductance is thequantity considered here. Our space filling constraint im-plies that the point-to-point and point-to-sphere conduc-tances should scale the same for physical structures,

'

in con-trast to the Cayley tree. Thus, our result justifies the widelymade assumption that various conductances may be treatedas equivalent in practice.

To describe transport on a branched structure, we choosea point on it at random, and draw a large spherical boundaryof radius L around it. The entire structure is assumed to bemuch larger than the bounding sphere. Then we may inves-tigate transport, e.g. , by measuring the conductance X(L)between the center and the sphere. The conductance' scaleswith L according to

Most of the original structure consists of dead ends, withno path to the boundary. Thus, in our conductance mea-surement, current flows only through a small subset of thestructure called the backbone. An example of such a back-bone in a diffusion-limited aggregate is depicted in Fig. 2.The backbone is itself a branched structure. But unlike theoriginal object, each new branch on the backbone means anew path to the boundary, each with its own branches andsubpaths. Unless the growth of the number of brancheswith increasing L is limited, the density of the backbone be-comes impossibly large. To avoid this, the distance betweenbranches must be large. We define N(L) as the averagechemical distance along the backbone from the center to theboundary. Equivalently, N(L) is the average distance alongthe full branched structure between two points at geometricseparation L. We suppose that N(L) scales as L'. Thispower 5 governs the point-to-point resistance in a branchedstructure. This exponent has been extensively investigatedfor percolation clusters: the "spreading dimension" of Ref.15 equals D/g, while the exponent v of Ref. 16 is the in-

verse of S.First we consider a regular nonrandom structure in which

all branches have the same length 6, and the length of anypath from the origin to the boundary is N Then any pathhas k=N/5 branches. This structure is a Cayley tree ofcoordination number three. The number of branches 8& at jsteps from the origin doubles at every step, so that 8&=2&.Thus, the mass M(N) of this tree 6$$, 8& grows ex-

ponentially with k. In order to fit into space of finitedimension, M must grow no faster than a power of N.Thus, k can grow no faster than lnN and 6 must be of orderN/lnN or larger.

This result may be extended to structures with branchesof arbitrary lengths. For any given L we can define an aver-age length 5& of jth generation branches. This average isobtained by dividing the sum of lengths of all jth genera-tion. branches by 8J, thus, if some of the branches are ab-sent they contribute zero length to the average. The mass

. M of this structure is g» 8~5~, and the average distance Nto the boundary is X»hj. This N is a different averagethan the N(L) used above. For arbitrary structures as forCayley trees the 4&'s must increase with N to avoid overfil-ling space. To see this we denote the largest of the 5& by6 and find the minimum mass M for the same 6 andEvidently the smallest mass is attained when all thebranches have the length b, . (Increasing one 6& at the ex-pense of a later one decreases the mass without changing¹)Thus, the largest of the average lengths 6& in anybranched structure must be of order N/lnN or larger, '7 as ina Cayley tree. (Clearly the b,

&are also bounded above,

since each 5J must be smaller than N. )Given this space-filling limitation on the 1engths, we may

estimate the conductance in actual random structures. Tomake this estimate, we must assume that the structure issufficiently homogeneous. That is the initial AJ setc. must be of the same order of magnitude as the max-imum 6j. Then the space-filling limitation implies that allthese branches have the characteristic length 5 between or-der N/lnN and order N. We presume that the point-to-point average N(L) is of the same order as N(L). This is-true as long as the number of branches along different pathsto the boundary does not vary too much. Now for Lthe resistance of the backbone must increase at least as fastas this 6 but not faster than N, ' and since both thesequantities scale the same (up to logarithmic corrections) weconclude that g-N ' —L ~. Using Eq. (1), we may re-late the scaling of X to the D and d of the entire structure:

5 = D(2/d —1)

Thus, d may be determined by purely geometric measure-ments, viz. , measurement of N(L) and M(L). This rela-tion has been obtained for finitely ramified structures. ' 'The formula is valid for the infinite percolating cluster in

high dimensions, ' where D=4, 5=2, and d=7. We have

shown that the formula is valid generally for branchedstructures.

The mass of any connected path must grow at least as fastas the radius L, but not faster than L; thus 1 ~ 5 ~ D.This sets bounds on d.

FIG. 2. Backbone of the diffusion-limited aggregate depicted in

Fig. 1. Note that only a small number of original branches survivein the backbone.

1 ~ d ~ 2D/(D+ 1) (3)

Thus, a branched structure with D & 2 must have d ( T.Aharony and Stauffer20 have suggested that the upperbound in Eq. (3) is the actual expression for d wheneverD ~2. They used an heuristic argument which applied toany fractal structure with D & 2 (although several coun-terexamples exist), while our bound is restricted to onlybranched structures but is not restricted to D & 2. In viewof our Eq. (2) this would imply that 5=1, i.e., N- L,whenever D ~ 2; the branches would be qualitativelyequivalent to straight lines. For diffusion-limited aggregates

30 SPACE-FILLING CONSTRAINT ON TRANSPORT IN RANDOM. . . 4095

in two dimensions" D =—1.7, so that Eq. (3) impliesd ~ 1.26. For cluster aggregation in three dimensions(Meakin 1984), the limit is about the same. For two--dimensional cluster aggregation with D =—1.5, the limit isstronger: d ~ 1.2.

These bounds on d are consistent with the available simu-lation results, and these can be used to get informationabout the exponent 5. Meakin and Stanley4 give estimatesof d for diffusion-limited aggregates ranging from 1.1 to1.45 in two dimensions, and from 1.2 to 1.64 in three. Us-ing these, we infer for diffusion-limited aggregates that5 & 1.4 in two dimensions and 5 & 1.67 in three.

It would be valuable to have accurate measurements ofthe exponent 5 relating N to L. This would give precisepredictions of d and allow the Aharony-Stauffer argument tobe checked. It would also be helpful to examine the statis-tics of backbone branches and of paths to the boundary inorder to check our assumptions of homogeneity. Our argu-ments should be equally applicable to the elastic response 'of a network composed of branched structures, such as sili-ca gel.

We thank D. Pearson, P. Pincus, and I. Webman forhelpful comments on the manuscript.

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16S. Havlin and R. Nossal, J. Phys. A 17, L427 (1984).Thus, e.g. , for a square lattice of spacing "a" in two dimensionshf(L) ~ rr(L/a)2~ mN~2;„~ mN2. But M(L) can be related tothe maximum length 5; M(L) ~ X&" c2JE «2"6 with k= N//h.Combining, we find mN2~~2Nlh. Since 2 «N, this means6 ~ N In2/In(mN).

~80ur conclusion about the resistance would break down if 5& in-creased without bound as j increased. Thus, in a structure whereall jth-order branches have length 2J, resistance would increaseonly logarithmically with N.

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