We study the properties of 2D fibre clusters and networks formed by deposition processes. We first examine the growth and scaling properties of single clusters. We then consider a network of such clusters, whose spatial distribution obeys some effective pair distribution function. In particular, we derive an expression for the two-point density autocorrelation function of the network, which includes the internal structure of a cluster and the effective cluster-cluster pair distribution function. This formula can be applied to obtain information about nontrivial correlations in fibre networks.Peer reviewe

Ala-Nissilä, Tapio

Alava, M. J.

Provatas, N.

arXiv

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This is an electronic reprint of the original article.This reprint may differ from the original in pagination and typographic detail.Author(s): Provatas, N. & Ala-Nissilä, Tapio & Alava, M. J.Title: Growth and Structure of Random Fibre Clusters and ClusterNetworksYear: 1995Version: Final published versionPlease cite the original version:Provatas, N. & Ala-Nissilä, Tapio & Alava, M. J. 1995. Growth and Structure of RandomFibre Clusters and Cluster Networks. Physical Review Letters. Volume 75, Issue 19. P.3556-3559. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.75.3556.Rights: © 1995 American Physical Society (APS). http://www.aps.org/All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, andduplication or sale of all or part of any of the repository collections is not permitted, except that material maybe duplicated by you for your research use or educational purposes in electronic or print form. You mustobtain permission for any other use. Electronic or print copies may not be offered, whether for sale orotherwise to anyone who is not an authorised user.Powered by TCPDF (www.tcpdf.org)VOLUME 75, NUMBER 19 PH YS ICAL REVIEW LETTERS 6 NovEMBER 1995Growth and Structure of Random Fibre Clusters and Cluster NetworksN. Provatas, ' T. Ala-Nissila, ' and M. J. Alava' University of Helsinki, Research Institute for Theoretical Physics, P.O. Box 9, FIN 00014-Helsinki, Finland2Laboratory of Physics, Helsinki University of Technology, Otakaari 1M, FIN 0215-0 Espoo, FinlandBrown University, Department of Physics, Box 1843, Providence, Rhode Island 02912and Tampere University of Technology, Department of Physics, P 0 Bo. x .692, FIN 3310-1 Tampere, Finland(Received 29 June 1995)We study the properties of 2D fibre clusters and networks formed by deposition processes. We firstexamine the growth and scaling properties of single clusters. We then consider a network of suchclusters, whose spatial distribution obeys some effective pair distribution function. In particular, wederive an expression for the two-point density autocorrelation function of the network, which includesthe internal structure of a cluster and the effective cluster-cluster pair distribution function. This formulacan be applied to obtain information about nontrivial correlations in fibre networks.PACS numbers: 81.15.Lm, 61.43.Hv, 81.35.+k, 82.70.KjThere are many phenomena in nature that can beviewed as deposition problems, brought about throughvarious transport mechanisms that bring particles to asurface. They include a multitude of processes such asdeposition of colloidal, polymer, and fibre particles [1—7]. Many deposition phenomena involve particles whosesize is large compared to their mutual interaction range,and so the main deposition mechanism is due to particleexclusion. Among the most studied in this class is therandom sequential adsorption model [1,2]. There particlesare deposited on a surface and either stick or are rejectedaccording to certain exclusion rules, with a maximumcoverage (the "jamming limit" ) less than unity. This isin contrast to multilayer growth [1,6,7].Processes of particle deposition may often begin fromcolloidal suspensions, solid particles suspended in a fluid.For some such systems, the interparticle repulsion isstrong enough to prevent multilayer growth [3]. However,the existence of dispersion forces can cause the particlesto Aocculate, or aggregate, and to precipitate out ofthe suspension [7,8]. For larger particles or clusters ofparticles, gravity often induces sedimentation out of thesuspension [9].A particularly interesting class of deposition-relatedproblems that has received little attention is the deposi-tion of fibres, fibre clusters, and the formation of fibrenetworks. Perhaps the most practical application of fibredeposition is that of paper making. During its formationpaper undergoes several stages, beginning as a colloidalsuspension of fibres and ending up as a deposition of fi-bres on a surface, when the quid is drained out. Therehave been many attempts to model the structure of fibrenetworks [4,5, 10—12]. Much of this work has focusedon the calculation of power spectra of ideal random fibrenetworks and their subsequent comparison with mass dis-tribution data obtained from paper-making experiments.In some sedimentation problems, such as in the makingof laboratory paper sheets, the fibres can fiocculatewhile still in solution [13]. This may occur for a400300—200-100-0—-100--200—-300—400 . I . I . I . i-400 -300 -200 -100 0X100 200 300 400FIG. 1. A fibre cluster made of 2000 fibres of dimensionA X ~ = 50 X 1. The cluster is constructed by depositingfibres randomly, keeping only those that form one connectedfibre cluster, beginning from an initial fibre.variety of reasons, ranging from mechanical fibre-fibreinteractions to hydrodynamic forces in the suspension [9].In such circumstances the resulting network may consistof clusters of fibres. Similar clusters are also expectedto occur in other deposition problems. This is one ofthe main motivations for our work where we present adetailed study of the statistical properties of depositionprocesses involving individual fibres and clusters.We start with a simple model of deposition of fibres oflength A and width cu on a 2D plane. Deposition beginsfrom an initial seed fibre. At each deposition event only afibre that overlaps at least one fibre already in the clusteris kept. This growth rule is motivated by the adhesivesticking of fibres. The process forms a connected clusterof N fibres that is statistically spherically symmetric.Thus it is possible to define an average maximum radiusR(N) Figure 1 shows . one configuration of a computer3556 0031-9007/95/75(19)/'3556(4) $06.00 1995 The American Physical SocietyVOLUME 75, NUMBER 19 PHYS ICAL REVIEW LETTERS 6 NavEMBER 1995pw(r) = mw(r) (2)o mdiv r drwhere m~(r) represents the mass per unit area at a radialdistance r from the center of the cluster. For large N,the normalized average mass probability density P~(r) —=(p~(r)) can be written as1yR N f(r/R(N)),Pw(r) = (3)generated N cluster on a lattice, where N = 2000. Wefind that for N )& 1 the radius satisfiesR(N) = BN~, (1)where the exponent P = 1/3, over the range of N ex-amined and for all fibre geometries simulated, while theconstant 8 depends only on fibre dimensions. Figure 2shows a plot of R(N) for two different types of fibre ge-ometries; A X cu = 20 X 1 and A X co = 50 X 1 show-ing the corresponding plots on a log-log scale. We notethat Eq. (1) also gives the number of fibres in a clusterdirectly from the radius.The exponent 1/3 in Eq. (1) also arises from the fol-lowing simple argument. Assume we have a clusterof average radius R(N) composed of N fibres of lin-ear dimension a = QAcu. Next if we deposit an addi-tional AN fibres onto the cluster its radius grows by ARand area by AA. The change in the cluster area AA =zazkN(2~Ra)/(~R2) = 2vrR/t. R, which in the contin-uum limit gives Eq. (1), with B = (3/27r)t/3a. The fac-tor 2~Ra/~R comes from the "active" zone at the edgeof the cluster, which is the only region where added fibresincrease the area.Another measure to characterize the N cluster is theradial mass probability per unit area. This quantity isiven bwhere y = fti uf(u) du and where the average impliesan ensemble average over all N cluster configurations.Figure 3 shows a plot of P~(r) for five different valuesof N while the scaling function f(r) is shown in the inset.The scaling function f is universal for the values of A andco studied here. The mass probability distributions havea peaklike component at the origin, continuing almostlinearly for the greater part of their width.At the edge of the cluster, there is a fast decaying regionthat also scales as shown in Fig. 3. This is associated withthe rough edge of the cluster, which we can examine in thecontext of kinetic roughening of growing interfaces [14].Defining fiuctuations of the radius as W(N) = ([R(N)—R(N)] )'/, where R(N) is the radius for one cluster, wefind it to grow as W —N, where x is consistent with1/3 as expected since W/R = const. The roughness ofthe cluster is expected to alter the percolation threshold ofa random fibre-cluster network from that of a distributionof solid disks [15].We now consider a disordered fibre network con-structed by deposition of N clusters. This model is com-pletely general and the only assumption is that there existsan effective spatial pair distribution function between anytwo clusters. One plausible application of such a networkis the description of sedimentation of fibre suspensionscontaining floes of fibres, as discussed above.Starting from a distribution of n,, fibre clusters, labeledby an index i, and where the ith cluster contains N(i)fibres, the mass distribution of the resulting networkis M(x) = g;='~ m~t;)(x —x;), where m~1;1 is the massdensity and x; the coordinate of the center of the ithcluster. The two-point density autocorrelation function of6.0- 0.007-5.5-0.006-0.005-~ 0.004-z4.5— 0.0034.0-0.0020.0013.5 I I I I I I I3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5ln(N)0.0-400 -300 -200 -100 0 100 200 300 400rFIG. 2. A plot of R(N) on a log-log scale. In the bottomcurve A X ~ = 20 X 1, while for the top curve A X ~ =50 X 1. The solid lines are drawn for comparison and havea slope of I/3.FIG. 3. The radial mass probability per unit area, for N =1500, 2000, 2500, 3000, and 4000 fibres, with P X w =50 X 1. The inset shows the scaling function f(x) vs x =r/R3557VOLUME 75, NUMBER 19 PH YS ICAL REVIEW LETTERS 6 NovEMaER 1995this network is given by [10](lM(k) l') (4)A )'where A is the area over which the 2D network isdeposited, F is the inverse Fourier transform, and theaverage is taken over all network configurations. TheFourier transform of M(x) is given byne nen=l m=1where I~(„) is the Fourier transform of miv(„)(x) evaluatedover the area of the N(n) cluster. The coordinate in Iv(„)is defined relative to the center of the nth cluster. Takingthe ensemble average of Eq. (5) we make the assumptionthat the description of individual clusters is independentof the positions of their centers. Thus( 2vrik (x„—x„,) p p' y ( 2vrik (x„—x,„))~ N(n) x &(m) ~X (Iiv(„)Iiv(m)) .Next, making the assumption that the internal structureof any given cluster is independent of the others, we write(Iiv(„)I~( )) = (Iiv(„))(I~( )), noting that (Iiv(„)) impliesan ensemble average over all miv(„)(x). This average canbe split into two parts: over all configurations where thenth cluster contains N(n) fibres and over the number N(n)itself. Breaking up the average in this way we definemiv(r) =—(miv(„)(x)) to be the average mass density of anN —= (N(n)) cluster. Using Eq. (3) and noting that thetotal average mass in an N cluster is Nwy, where ~~ isdefined as the mass per fibre, we can show thatmiv(r) = R(N)f(r/R(N)) . (7)myB3Thus we have (Iiv(„))(I~( )) = lJiv(lkl)l, where Ji(i(lkl)is the Fourier transform of miv(r). To evaluate theexponential term on the right-hand side of Eq. (6) weconsider two cases: n = m and n 4 m. The first givesunity, while the second depends on the type of effectivecluster-cluster interaction between clusters n and m.Defining Ax„—= x„—x, , we note that the ensembleaverage over all cluster-network realizations is equivalentto integrating e ' '" over all Ax„, weighted bya cluster-cluster pair probability density g(Ax„, ). Wealso assume, by isotropy, that g(Ax„) is independentof the indices n and m, and radially symmetric [16].The average of the exponential on the right-hand side ofEq. (6) can thus be written as2vri k hx„„,)—-,„—f& e ' '""g(lxl)dx, n 4 rn1, n=m.Combining the equations above we finally obtain(lM(k)l ) = n, lJ (k)l4Giv(r) = R (N)B II (k)kJo(2vrkr) dk2+0where II(k) (a dimensionless quantity) is given byII(k) = f (z)zJo(2~kR(N)z) dzII (k)g(k)kJo(27rkr) dk, (10)and where 10 is the Bessel function of order zero. Inobtaining Eq. (10) we normalized G(r) by dividing by thesquare of g wI [10,17] and noted that both II (k) and g (k)are radially symmetric.Equation (10) is our main result for the fibre network.The first component in it describes the density-densitycorrelations of a uniformly random distribution [g(r) =0] of fibre clusters. For this case it is easy to show thatEq. (10) becomes Giv(r) = R(N)I'(r/R(N)) where I'(r)is Gaussian for small r and rapidly goes to zero as r» 1.This dependence on R(N) in Eq. (10), allows us to extractthe typical fibre-cluster size directly from the correlationfunction.The second term in Eq. (10) arises due to an effectivepair distribution function acting between clusters. Thisfunction can lead to nontrivial long-range correlationsin the fibre network. Such correlations could arisefrom hydrodynamics in the sedimentation of particlesfrom suspensions [9]. We have examined a modelpair distribution given by g(k) ~ k x. Figure 4 showsthe second term of Giv(r) for g = 1.1, 1.3, and 1.6and R(N) = 3, where distance is measured in unitsof B For r ). R(N) [where the first term of Giv(r)becomes negligible], we find Giv(r) —r, where thecorrelation exponent n ( 1 decreases continuously as yis increased from 0 to 2. Since the function 11(k) = 0for k ) 1/R(N), we expect any effective pair distributionsatisfying g(k) —k x for small k to give essentially thesame power law correlations at large r.In conclusion, we have presented a model for thestatistical properties of disordered 2D fibre networks+ ' ' IJ (k)I'g(k), (9)where g(k) is the Fourier transform of g(lxl). Equa-tion (9) comprises two terms. The first represents thespectral density of n,, individual fibre clusters. Substitut-ing this term into Eq. (4) gives the two-point density cor-relation function of n,, individual N clusters. The secondterm arises due to effective cluster-cluster interactions.The number of clusters n, can be expressed in terms ofa parameter g, defined as the number of fibres per unitarea [17]. In particular, we have n, = rtA/N Writ.ingN in terms of the average cluster radius and substitutingEq. (9) into Eq. (4), the following Giv(r) is finally ob-tained:3558VOLUME 75, NUMBER 19 PHYSICAL REVIEW LETTERS 6 NOVEMBER 1995~ ~ ~~~~~~~~ g-10--3I0ln(r)FIG. 4. The correlation function of Eq. (10) with g(k) =~, where y = 1.1, 1.3, 1.6 from bottom to top. In the larger limit power law behavior r, with o. = 0.92, 0.71, 0.4,emerges.formed by deposition processes. The radius of fibreclusters scales with the number of fibres. Similarly,the density profiles of a cluster obey scaling laws, witha universal scaling function for the cases studied here.Starting from the scaling form, we have derived thetwo-point density-density correlation function for a fibrenetwork due to deposition of clusters, whose spatialdistribution is given. In the case of uniform, randomdeposition of fibre clusters correlations exist over therange of the mean cluster size. However, an effectivecluster-cluster interaction can give rise to long-rangepower law correlations. In this way the final structore ofdeposited fiber networks may give us information aboutthe effective interactions between fibers in a suspension.We expect that our Gg(r) can be compared withexperiments on sedimentation of fiberlike particles insuspensions. A prime example of such a process is theformation of laboratory paper sheets, whose final massdensity can be measured by the beta-radiogram technique[11]. Our calculations will be compared with long-range correlations observed in such radiogram data in anupcoming publication.This work has been supported in part by the Academyof Finland through the MATRA program.[1] J.W. Evans, Rev. Mod. Phys. 65, 1281 (1993).[2] V. Privman, in Annual Reviews in Computational Physics,edited by D. Stauffer (World Scientific, Singapore, 1995),Vol. 3.[3] G. Y. Onoda and E.G. Liniger, Phys. Rev. A 33, 715(1986); A. Schmit, R. Varoqui, S. Uniyal, J.L. Brash, andC. Pusiner, J. Colloid Interface Sci. 92, 25 (1983);J. Federand I. Giaever, J. Colloid Interface Sci. 7S, 144 (1980).[4] M. Deng and C. T. J. Dodson, Paper: An EngineeredStochastic Structure (Tappi Press, Atlanta, GA, 1994).[5] K. J. Niskanen and M. J. Alava, Phys. Rev. Lett. 73, 3475(1994).[6] P. Nielaba and V. Privman, Phys. Rev. E 51, 2022 (1995).[7] N. Ryde, H. Kihira, and E. Matijevic, J. Colloid InterfaceSci. 151, 421 (1992).[8] C. A. Murray and D. G. Grier, Am. Sci. S3, 238 (1995).[9] S. Schwarzer, cond-mat@babbage. sissa. it, no. 9503091.[10] L. Haugland, B. Norman, and D. Wahren, Svensk Papper-stidning no. 10, 362 (1974).[11] B. Norman and D. Wahren, Svensk Papperstidning arg.75, 807 (1972).[12] O. J. Kallmes and G. Bernier, Tappi J. 47, 694 (1964).[13] R. J. Kerekes and C. J. Schell, J. Pulp Paper Sci. 18, J32(1992).[14] A. -L. Barabasi and H. E. Stanley, Fractal Concepts inSurface Growth (Cambridge University Press, Cambridge,1995).[15] B. Lorenz, I. Orgzal, and H. -O. Heuer, J. Phys. A 26, 711(1993).[16] This assumption is made here only for simplicity, andcould be relieved in situations where the cluster-clusterinteractions are asymmetric.[17] This quantity is proportional to mass per unit area, knownas the basis weight in paper physics literature, see Ref. [4].3559

Growth and Structure of Random Fibre Clusters and Cluster Networks

Aaltodoc

This is an electronic reprint of the original article.
This reprint may differ from the original in pagination and typographic detail.
Author(s): Provatas, N. & Ala-Nissilä, Tapio & Alava, M. J.
Title: Growth and Structure of Random Fibre Clusters and Cluster
Networks
Year: 1995
Version: Final published version
Please cite the original version:
Provatas, N. & Ala-Nissilä, Tapio & Alava, M. J. 1995. Growth and Structure of Random
Fibre Clusters and Cluster Networks. Physical Review Letters. Volume 75, Issue 19. P.
3556-3559. ISSN 0031-9007 (printed). DOI: 10.1103/physrevlett.75.3556.
Rights: © 1995 American Physical Society (APS). http://www.aps.org/
All material supplied via Aaltodoc is protected by copyright and other intellectual property rights, and
duplication or sale of all or part of any of the repository collections is not permitted, except that material may
be duplicated by you for your research use or educational purposes in electronic or print form. You must
obtain permission for any other use. Electronic or print copies may not be offered, whether for sale or
otherwise to anyone who is not an authorised user.
Powered by TCPDF (www.tcpdf.org)
VOLUME 75, NUMBER 19 PH YS ICAL REVIEW LETTERS 6 NovEMBER 1995
Growth and Structure of Random Fibre Clusters and Cluster Networks
N. Provatas, ' T. Ala-Nissila, ' and M. J. Alava
' University of Helsinki, Research Institute for Theoretical Physics, P.O. Box 9, FIN 00014-Helsinki, Finland
2Laboratory of Physics, Helsinki University of Technology, Otakaari 1M, FIN 0215-0 Espoo, Finland
Brown University, Department of Physics, Box 1843, Providence, Rhode Island 02912
and Tampere University of Technology, Department of Physics, P 0 Bo. x .692, FIN 3310-1 Tampere, Finland
(Received 29 June 1995)
We study the properties of 2D fibre clusters and networks formed by deposition processes. We first
examine the growth and scaling properties of single clusters. We then consider a network of such
clusters, whose spatial distribution obeys some effective pair distribution function. In particular, we
derive an expression for the two-point density autocorrelation function of the network, which includes
the internal structure of a cluster and the effective cluster-cluster pair distribution function. This formula
can be applied to obtain information about nontrivial correlations in fibre networks.
PACS numbers: 81.15.Lm, 61.43.Hv, 81.35.+k, 82.70.Kj
There are many phenomena in nature that can be
viewed as deposition problems, brought about through
various transport mechanisms that bring particles to a
surface. They include a multitude of processes such as
deposition of colloidal, polymer, and fibre particles [1—
7]. Many deposition phenomena involve particles whose
size is large compared to their mutual interaction range,
and so the main deposition mechanism is due to particle
exclusion. Among the most studied in this class is the
random sequential adsorption model [1,2]. There particles
are deposited on a surface and either stick or are rejected
according to certain exclusion rules, with a maximum
coverage (the "jamming limit" ) less than unity. This is
in contrast to multilayer growth [1,6,7].
Processes of particle deposition may often begin from
colloidal suspensions, solid particles suspended in a fluid.
For some such systems, the interparticle repulsion is
strong enough to prevent multilayer growth [3]. However,
the existence of dispersion forces can cause the particles
to Aocculate, or aggregate, and to precipitate out of
the suspension [7,8]. For larger particles or clusters of
particles, gravity often induces sedimentation out of the
suspension [9].
A particularly interesting class of deposition-related
problems that has received little attention is the deposi-
tion of fibres, fibre clusters, and the formation of fibre
networks. Perhaps the most practical application of fibre
deposition is that of paper making. During its formation
paper undergoes several stages, beginning as a colloidal
suspension of fibres and ending up as a deposition of fi-
bres on a surface, when the quid is drained out. There
have been many attempts to model the structure of fibre
networks [4,5, 10—12]. Much of this work has focused
on the calculation of power spectra of ideal random fibre
networks and their subsequent comparison with mass dis-
tribution data obtained from paper-making experiments.
In some sedimentation problems, such as in the making
of laboratory paper sheets, the fibres can fiocculate
while still in solution [13]. This may occur for a
400
300—
200-
100-
0—
-100-
-200—
-300—
400 . I . I . I . i
-400 -300 -200 -100 0
X
100 200 300 400
FIG. 1. A fibre cluster made of 2000 fibres of dimension
A X ~ = 50 X 1. The cluster is constructed by depositing
fibres randomly, keeping only those that form one connected
fibre cluster, beginning from an initial fibre.
variety of reasons, ranging from mechanical fibre-fibre
interactions to hydrodynamic forces in the suspension [9].
In such circumstances the resulting network may consist
of clusters of fibres. Similar clusters are also expected
to occur in other deposition problems. This is one of
the main motivations for our work where we present a
detailed study of the statistical properties of deposition
processes involving individual fibres and clusters.
We start with a simple model of deposition of fibres of
length A and width cu on a 2D plane. Deposition begins
from an initial seed fibre. At each deposition event only a
fibre that overlaps at least one fibre already in the cluster
is kept. This growth rule is motivated by the adhesive
sticking of fibres. The process forms a connected cluster
of N fibres that is statistically spherically symmetric.
Thus it is possible to define an average maximum radius
R(N) Figure 1 shows . one configuration of a computer
3556 0031-9007/95/75(19)/'3556(4) $06.00 1995 The American Physical Society
VOLUME 75, NUMBER 19 PHYS ICAL REVIEW LETTERS 6 NavEMBER 1995
pw(r) = mw(r) (2)
o mdiv r dr
where m~(r) represents the mass per unit area at a radial
distance r from the center of the cluster. For large N,
the normalized average mass probability density P~(r) —=
(p~(r)) can be written as
1
yR N f(r/R(N)),Pw(r) = (3)
generated N cluster on a lattice, where N = 2000. We
find that for N )& 1 the radius satisfies
R(N) = BN~, (1)
where the exponent P = 1/3, over the range of N ex-
amined and for all fibre geometries simulated, while the
constant 8 depends only on fibre dimensions. Figure 2
shows a plot of R(N) for two different types of fibre ge-
ometries; A X cu = 20 X 1 and A X co = 50 X 1 show-
ing the corresponding plots on a log-log scale. We note
that Eq. (1) also gives the number of fibres in a cluster
directly from the radius.
The exponent 1/3 in Eq. (1) also arises from the fol-
lowing simple argument. Assume we have a cluster
of average radius R(N) composed of N fibres of lin-
ear dimension a = QAcu. Next if we deposit an addi-
tional AN fibres onto the cluster its radius grows by AR
and area by AA. The change in the cluster area AA =
zazkN(2~Ra)/(~R2) = 2vrR/t. R, which in the contin-
uum limit gives Eq. (1), with B = (3/27r)t/3a. The fac-
tor 2~Ra/~R comes from the "active" zone at the edge
of the cluster, which is the only region where added fibres
increase the area.
Another measure to characterize the N cluster is the
radial mass probability per unit area. This quantity is
iven b
where y = fti uf(u) du and where the average implies
an ensemble average over all N cluster configurations.
Figure 3 shows a plot of P~(r) for five different values
of N while the scaling function f(r) is shown in the inset.
The scaling function f is universal for the values of A and
co studied here. The mass probability distributions have
a peaklike component at the origin, continuing almost
linearly for the greater part of their width.
At the edge of the cluster, there is a fast decaying region
that also scales as shown in Fig. 3. This is associated with
the rough edge of the cluster, which we can examine in the
context of kinetic roughening of growing interfaces [14].
Defining fiuctuations of the radius as W(N) = ([R(N)—
R(N)] )'/, where R(N) is the radius for one cluster, we
find it to grow as W —N, where x is consistent with
1/3 as expected since W/R = const. The roughness of
the cluster is expected to alter the percolation threshold of
a random fibre-cluster network from that of a distribution
of solid disks [15].
We now consider a disordered fibre network con-
structed by deposition of N clusters. This model is com-
pletely general and the only assumption is that there exists
an effective spatial pair distribution function between any
two clusters. One plausible application of such a network
is the description of sedimentation of fibre suspensions
containing floes of fibres, as discussed above.
Starting from a distribution of n,, fibre clusters, labeled
by an index i, and where the ith cluster contains N(i)
fibres, the mass distribution of the resulting network
is M(x) = g;='~ m~t;)(x —x;), where m~1;1 is the mass
density and x; the coordinate of the center of the ith
cluster. The two-point density autocorrelation function of
6.0- 0.007-
5.5-
0.006-
0.005-
~ 0.004-z
4.5— 0.003
4.0-
0.002
0.001
3.5 I I I I I I I
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
ln(N)
0.0
-400 -300 -200 -100 0 100 200 300 400
r
FIG. 2. A plot of R(N) on a log-log scale. In the bottom
curve A X ~ = 20 X 1, while for the top curve A X ~ =
50 X 1. The solid lines are drawn for comparison and have
a slope of I/3.
FIG. 3. The radial mass probability per unit area, for N =
1500, 2000, 2500, 3000, and 4000 fibres, with P X w =
50 X 1. The inset shows the scaling function f(x) vs x =
r/R
3557
VOLUME 75, NUMBER 19 PH YS ICAL REVIEW LETTERS 6 NovEMaER 1995
this network is given by [10]
(lM(k) l') (4)
A )'
where A is the area over which the 2D network is
deposited, F is the inverse Fourier transform, and the
average is taken over all network configurations. The
Fourier transform of M(x) is given by
ne ne
n=l m=1
where I~(„) is the Fourier transform of miv(„)(x) evaluated
over the area of the N(n) cluster. The coordinate in Iv(„)
is defined relative to the center of the nth cluster. Taking
the ensemble average of Eq. (5) we make the assumption
that the description of individual clusters is independent
of the positions of their centers. Thus
( 2vrik (x„—x„,) p p' y ( 2vrik (x„—x,„))~ N(n) x &(m) ~
X (Iiv(„)Iiv(m)) .
Next, making the assumption that the internal structure
of any given cluster is independent of the others, we write
(Iiv(„)I~( )) = (Iiv(„))(I~( )), noting that (Iiv(„)) implies
an ensemble average over all miv(„)(x). This average can
be split into two parts: over all configurations where the
nth cluster contains N(n) fibres and over the number N(n)
itself. Breaking up the average in this way we define
miv(r) =—(miv(„)(x)) to be the average mass density of an
N —= (N(n)) cluster. Using Eq. (3) and noting that the
total average mass in an N cluster is Nwy, where ~~ is
defined as the mass per fibre, we can show that
miv(r) = R(N)f(r/R(N)) . (7)
myB3
Thus we have (Iiv(„))(I~( )) = lJiv(lkl)l, where Ji(i(lkl)
is the Fourier transform of miv(r). To evaluate the
exponential term on the right-hand side of Eq. (6) we
consider two cases: n = m and n 4 m. The first gives
unity, while the second depends on the type of effective
cluster-cluster interaction between clusters n and m.
Defining Ax„—= x„—x, , we note that the ensemble
average over all cluster-network realizations is equivalent
to integrating e ' '" over all Ax„, weighted by
a cluster-cluster pair probability density g(Ax„, ). We
also assume, by isotropy, that g(Ax„) is independent
of the indices n and m, and radially symmetric [16].
The average of the exponential on the right-hand side of
Eq. (6) can thus be written as
2vri k hx„„,)—-
,
„—f& e ' '""g(lxl)dx, n 4 rn
1, n=m.
Combining the equations above we finally obtain
(lM(k)l ) = n, lJ (k)l
4
Giv(r) = R (N)B II (k)kJo(2vrkr) dk
2
+
0
where II(k) (a dimensionless quantity) is given by
II(k) = f (z)zJo(2~kR(N)z) dz
II (k)g(k)kJo(27rkr) dk, (10)
and where 10 is the Bessel function of order zero. In
obtaining Eq. (10) we normalized G(r) by dividing by the
square of g wI [10,17] and noted that both II (k) and g (k)
are radially symmetric.
Equation (10) is our main result for the fibre network.
The first component in it describes the density-density
correlations of a uniformly random distribution [g(r) =
0] of fibre clusters. For this case it is easy to show that
Eq. (10) becomes Giv(r) = R(N)I'(r/R(N)) where I'(r)
is Gaussian for small r and rapidly goes to zero as r» 1.
This dependence on R(N) in Eq. (10), allows us to extract
the typical fibre-cluster size directly from the correlation
function.
The second term in Eq. (10) arises due to an effective
pair distribution function acting between clusters. This
function can lead to nontrivial long-range correlations
in the fibre network. Such correlations could arise
from hydrodynamics in the sedimentation of particles
from suspensions [9]. We have examined a model
pair distribution given by g(k) ~ k x. Figure 4 shows
the second term of Giv(r) for g = 1.1, 1.3, and 1.6
and R(N) = 3, where distance is measured in units
of B For r ). R(N) [where the first term of Giv(r)
becomes negligible], we find Giv(r) —r, where the
correlation exponent n ( 1 decreases continuously as y
is increased from 0 to 2. Since the function 11(k) = 0
for k ) 1/R(N), we expect any effective pair distribution
satisfying g(k) —k x for small k to give essentially the
same power law correlations at large r.
In conclusion, we have presented a model for the
statistical properties of disordered 2D fibre networks
+ ' ' IJ (k)I'g(k), (9)
where g(k) is the Fourier transform of g(lxl). Equa-
tion (9) comprises two terms. The first represents the
spectral density of n,, individual fibre clusters. Substitut-
ing this term into Eq. (4) gives the two-point density cor-
relation function of n,, individual N clusters. The second
term arises due to effective cluster-cluster interactions.
The number of clusters n, can be expressed in terms of
a parameter g, defined as the number of fibres per unit
area [17]. In particular, we have n, = rtA/N Writ.ing
N in terms of the average cluster radius and substituting
Eq. (9) into Eq. (4), the following Giv(r) is finally ob-
tained:
3558
VOLUME 75, NUMBER 19 PHYSICAL REVIEW LETTERS 6 NOVEMBER 1995
~ ~ ~
~
~
~
~
~
~
~ g
-10-
-3
I
0
ln(r)
FIG. 4. The correlation function of Eq. (10) with g(k) =
~, where y = 1.1, 1.3, 1.6 from bottom to top. In the large
r limit power law behavior r, with o. = 0.92, 0.71, 0.4,
emerges.
formed by deposition processes. The radius of fibre
clusters scales with the number of fibres. Similarly,
the density profiles of a cluster obey scaling laws, with
a universal scaling function for the cases studied here.
Starting from the scaling form, we have derived the
two-point density-density correlation function for a fibre
network due to deposition of clusters, whose spatial
distribution is given. In the case of uniform, random
deposition of fibre clusters correlations exist over the
range of the mean cluster size. However, an effective
cluster-cluster interaction can give rise to long-range
power law correlations. In this way the final structore of
deposited fiber networks may give us information about
the effective interactions between fibers in a suspension.
We expect that our Gg(r) can be compared with
experiments on sedimentation of fiberlike particles in
suspensions. A prime example of such a process is the
formation of laboratory paper sheets, whose final mass
density can be measured by the beta-radiogram technique
[11]. Our calculations will be compared with long-
range correlations observed in such radiogram data in an
upcoming publication.
This work has been supported in part by the Academy
of Finland through the MATRA program.
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3559


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