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 pa

GrowthandStructureofRandomFibreClustersandClusterNetworks

UniversityofHelsinki,ResearchInstituteforTheoreticalPhysics,

P.O.Box9,FIN–00014UniversityofHelsinki,Finland

2

LaboratoryofPhysics,HelsinkiUniversityofTechnology,Otakaari1M,FIN–02150Espoo,Finland

3

BrownUniversity,DepartmentofPhysics,Box1843,Providence,R.I.02912,U.S.A.,andTampereUniversityof

Technology,DepartmentofPhysics,P.O.Box692,FIN–33101Tampere,Finland

(September3,1995)

1

N.Provatas1,T.Ala–Nissila1,3,andM.J.Alava2

arXiv:cond-mat/9511043v2 20 Nov 1995

Westudythepropertiesof2D breclustersandnetworksformedbydepositionprocesses.We rstexaminethegrowthandscalingpropertiesofsingleclusters.Wethenconsideranetworkofsuchclusters,whosespatialdistributionobeyssomee ectivepairdistributionfunction.Inparticular,wederiveanexpressionforthetwo–pointdensityautocorrelationfunctionofthenetwork,whichincludestheinternalstructureofaclusterandthee ectivecluster–clusterpairdistributionfunction.Thisformulacanbeappliedtoobtaininformationaboutnontrivialcorrelationsin brenetworks.81.15.Lm,81.35.+k,61.43.Hv,82.70.Kj

Therearemanyphenomenainnaturethatcanbeviewedasdepositionproblems,broughtaboutthroughvarioustransportmechanismsthatbringparticlestoasurface.Theyincludeamultitudeofprocessessuchasdepositionofcolloidal,polymerand breparticlesManydepositionphenomenainvolveparticleswhosesizeislargecomparedtothetheirmutualinteractionrange,andsothemaindepositionmechanismisduetoparti-cleexclusion.AmongthemoststudiedinthisclassistheRandomSequentialAdsorptionmodel[1,2].Thereparticlesaredepositedonasurfaceandeitherstickorarerejectedaccordingtocertainexclusionrules,withamaximumcoverage(the“jamminglimit”)lessthanunity.Thisisincontrasttomultilayergrowth[1,6,7].

Processesofparticledepositionmayoftenbeginfromcolloidalsuspensions;solidparticlessuspendedina uid.Forsomesuchsystems,theinterparticlerepulsionisstrongenoughtopreventmultilayergrowth[3].How-ever,theexistenceofdispersionforcescancausethepar-ticlesto occulate,oraggregate,andtoprecipitateoutofthesuspension[7,8].Forlargerparticlesorclustersofparticles,gravityofteninducessedimentationoutofthesuspensionAparticularlyinterestingclassofdeposition–relatedproblemswhichhasreceivedlittleattentionisthedepo-sitionof bres, breclusters,andtheformationof brenetworks.Perhapsthemostpracticalapplicationof bredepositionisthatofpaper–making.Duringitsformationpaperundergoesseveralstages,beginningasacolloidalsuspensionof bresandendingupasadepositionof -bresonasurface,whenthe uidisdrainedout.Therehavebeenmanyattemptstomodelthestructureof -1

brenetworksMuchofthisworkhasfocusedonthecalculationofpowerspectraofidealrandom brenetworksandtheirsubsequentcomparisonwithmassdis-tributiondataobtainedfrompaper–makingexperiments.Insomesedimentationproblems,suchasinthemak-ingoflaboratorypapersheets,the brescan occulatewhilestillinsolution[13].Thismayoccurforavarietyofreasons,rangingfrommechanical bre– breinteractionstohydrodynamicforcesinthesuspension[9].Insuchcir-cumstancestheresultingnetworkmayconsistofclustersof bres.Similarclustersarealsoexpectedtooccurinotherdepositionproblems.Thisisoneofthemainmoti-vationsforourworkwherewepresentadetailedstudyofthestatisticalpropertiesofdepositionprocessesinvolvingindividual bresandclusters.

Westartwithasimplemodelofdepositionof bresoflengthλandwidthωona2Dplane.Depositionbeginsfromaninitialseed bre.Ateachdepositioneventonlya brethatoverlapsatleastone brealreadyinthecluster,iskept.Thisgrowthruleismotivatedbytheadhesivestickingof bres.TheprocessformsaconnectedclusterofN breswhichisstatisticallysphericallysymmetric.Thusitispossibletode neanaveragemaximumradiusR(N).Fig.1showsonecon gurationofacomputergeneratedN–clusteronalattice,whereN=2000.We ndthatforN>>1theradiussatis es

R(N)=BNβ,

(1)

wheretheexponentβ≈1/3,overtherangeofNex-aminedandforall bregeometriessimulated,whiletheconstantBdependsonlyon bredimensions.Fig.showsaplotofR(N)fortwodi erenttypesof brege-ometries;λ×ω=20×1andλ×ω=50×1showingthecorrespondingplotsonalog–logscale.WenotethatEq.alsogivesthenumberof bresinaclusterdirectlyfromtheradius.

Theexponent1/3inEq.(1)alsoarisesfromthefollowingsimpleargument.1Assumewehaveaclus-terofaverageradius√R(N)composedofN bresoflin-eardimensiona=

ThisresulthasbeenalsoindependentlyderivedbyJ. Astr¨om(J. Astr¨om,ProGraduavhandling, AboAkademi(1989)(unpublished).

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