shortopeneuler

PHYSICALREVIEWE92,043023(2015)DirectrelationsbetweenmorphologyandtransportinBooleanmodelsChristianScholz,1,2FrankWirner,1MichaelA.
Klatt,3DanielHirneise,1GerdE.
Schr¨oder-Turk,4,3KlausMecke,3andClemensBechinger1,512.
PhysikalischesInstitut,Universit¨atStuttgart,Pfaffenwaldring57,70569Stuttgart,Germany2Institutf¨urMultiskalensimulation,N¨agelsbachstrae49b,Friedrich-AlexanderUniversit¨atErlangen-N¨urnberg,91052Erlangen,Germany3Institutf¨urTheoretischePhysik,Friedrich-AlexanderUniversit¨atErlangen-N¨urnberg,Staudtstrae7B,91058Erlangen,Germany4MurdochUniversity,SchoolofEngineering&IT,Maths&Stats,90SouthStr.
,MurdochWA6150,Australia5Max-Planck-Institutf¨urIntelligenteSysteme,Heisenbergstrae3,70569Stuttgart,Germany(Received22July2015;published30October2015)Westudytherelationofpermeabilityandmorphologyforporousstructurescomposedofrandomlyplacedoverlappingcircularorellipticalgrains,so-calledBooleanmodels.
MicrouidicexperimentsandlatticeBoltzmannsimulationsallowustoevaluateapower-lawrelationbetweentheEulercharacteristicoftheconductingphaseanditspermeability.
Moreover,thisrelationissofaronlydirectlyapplicabletostructurescomposedofoverlappinggrainswherethegraindensityisknownapriori.
WedevelopageneralizationtoarbitrarystructuresmodeledbyBooleanmodelsandcharacterizedbyMinkowskifunctionals.
Thisgeneralizationworkswellforthepermeabilityofthevoidphaseinsystemswithoverlappinggrains,butsystematicdeviationsarefoundifthegrainphaseistransportingtheuid.
Inthelattercaseouranalysisrevealsasignicantdependenceonthespatialdiscretizationoftheporousstructure,inparticulartheoccurrenceofsingleisolatedpixels.
TolinktheresultstopercolationtheoryweperformedMonteCarlosimulationsoftheEulercharacteristicoftheopencluster,whichrevealsdifferentregimesofapplicabilityforourpermeability-morphologyrelationsclosetoandfarawayfromthepercolationthreshold.
DOI:10.
1103/PhysRevE.
92.
043023PACSnumber(s):47.
56.
+r,61.
43.
Gt,47.
61.
kI.
INTRODUCTIONTheowofliquidsthroughporousmediaisofconsiderableimportanceinmanyscienticareas,suchasgroundwaterpollution,secondaryoilrecovery,orbloodperfusioninsidethehumanbody[1,2].
Althoughtheliteratureonporousmediahasbeengrowingrapidlyoverthelastdecades,itisstillnotfullyunderstoodhowtransportpropertiesofliquidsthroughporousmaterialscanberelatedtothemicrostructureevenforsingle-phaseow.
Analyticalresultsexistforregularstructures[3],andrigorousboundshavebeenproposedforrandommedia[4].
Yet,itisstillanopenquestion,inparticularformanyrandomsystems,whichstructuralpropertiesdeterminethepermeability,i.
e.
,theabilityofamaterialtoconductuidow.
Thepermeabilitykofaporousmedium,whichisperhapsthemostfundamentalowproperty,relatestheowrateQandtheappliedpressureP,accordingtoDarcy'slawQ=kAηP,(1)whereAisthecross-sectionalareaofthematerialandηtheviscosityoftheuid.
InEq.
(1)oneassumesalinearrelationbetweenowandpressure.
ThisisstrictlyspeakingonlyvalidforlowReynoldsnumberow(Re0.
999).
Adilutesuspensionofcolloidalparticlesofdiameterσ=3μmisinjectedintothismicrouidicdeviceastracerparticles.
FormacroscopicexperimentsthepermeabilityaccordingtoEq.
(1)istypicallydeterminedbyapplyingaxedowrateandthenmeasuringthepressuredropacrosstheporousstructure.
However,forsoft-lithographicchannelstopreventfeedbackandleakage,onlysmallowratesmustbeapplied,whichrequireshigh-sensitivitymotorizedsyringes.
Additionally,anaccuratedeterminationofthepressuredropbetweeninletandoutletisrequired,whichisdifculttoachievewithmacroscopicpressuretransducers.
Forthisreasonweuseparticletrackingvelocimetry[29–31]todeterminetheaveragevelocityoftheinjectedcolloidaltracerparticlesasafunctionoftheappliedpressurePineachchannel.
Here,incontrasttothexedow-ratemethod,theappliedhydrostaticpressurecanbetunedaccuratelybyvaryingthewaterlevelinthetworeservoirs.
Theouterchannelsactasreferencestocalibratetherelationshipbetweenparticlevelocityanduidvelocity.
Asshowninpreviousexperiments[32–34],theaverageparticlespeedwithinthinrectangularchannelstypicallydeviatessignicantlyfromtheaverageuidvelocity,dependingontheparticlediameter,theheightofthechannelandtheparticles'gravitationalheight[35,36].
However,themeanuidvelocityvandmeanparticle043023-2DIRECTRELATIONSBETWEENMORPHOLOGYAND.
.
.
PHYSICALREVIEWE92,043023(2015)(a)AresLQRef1QStructQRef2Δh=ΔPρgRef1StructRef2(b)ΔPR0RinRporRoutR0(c)050100150200250300050100150200ΔP[Pa]u[μm/s]Ref1Ref2StructFIG.
1.
(Coloronline)Summaryofexperimentalmethod:(a)Amicrouidicsamplewithaporousstructureinthecenterofthemiddlechanneliscreated.
Adilutesuspensionofcolloidalparticlesisinjectedintothedeviceandanexternalhydrostaticpressureisappliedwiththehelpoftworeservoirs.
Theowratethroughthechannelismeasuredfromthevelocityofthecolloidaltracers.
Unpatternedreferencechannelsforcalibrationareaddedparalleltothestructuredchannel.
Toaccountforpossibledeviationsinthechannelheighttworeferencechannelsareusedtocrosschecktheresults.
(b)EquivalentcircuitdiagramofhydraulicresistancesRhyd=PQ=ηLkA.
(c)Meanparticlevelocityumeasuredinthetworeferenceandthemiddlechannelcontainingtheporousstructureasafunctionoftheappliedpressure.
Thedashedlinesaretstothedatapoints.
Theaverageparticlevelocityuislowerinthestructuredchannelduetothelowerpermeabilityoftheporouspart.
Theuidvelocityinthestructurechannelisnoticeablyslowerthaninthereferencechannels.
Similaruidvelocitiesarefoundinthereferencechannels,however,differencesarecausedbydifferentlength(L≈500μm)andheight(h≈0.
5μm)ofthereferencechannelsandmustbetakenintoaccount.
Theerrorbarsaresmallerthanthesymbolsizes.
velocityuinachannelareproportional[37],sothatv=cdu.
(5)Theuidowprobleminthereferencechannels,whichareassumedtobeinniteparallelplates,canbesolvedanalyticallyforagivenpressuredropP.
Thepermeabilityk0ofthereferencechannelsisequaltoh2/12[38].
Therefore,thecalibrationfactorcdcanbeobtainedbycalculatingthetheoreticalvalueforvwithEq.
(1)andmeasuringu.
Sincek0∝h2,thecalibrationisverysensitivetotheheightofthestructure(anditsspatialvariations)[38].
Wedoublecheckthecalibrationforeachsamplewithtworeferencechannelstoaccountfordifferencesbetweenindividualsamples,whichcanoccurduringthelithographicprocess.
Figure1(c)showsanexemplaryplotofuvs.
Pforthethreechannelsofonesample.
Asillustrated,theuidvelocityinthechannelcontainingtheporousstructureisconsiderablyslower,whileinthereferencechannelsitissimilar[smalldifferencesarecausedbythedifferentlength(L≈500μm)andslightlydifferentheightofthechannels(h≈0.
5μm),whichresultfromtheproductionprocess].
ByconsideringthemiddlechannelasaseriesofhydraulicresistancesRhyd=PQ,asdepictedintheequivalentcircuitdiagraminFig.
1(b),wecandeterminethepermeabilityofthecentralstructuredirectlyfromtheowrate.
Ifallowratesareknown,therelativepermeability(normalizedtothek0ofthereferencechannel)ofastructureisgivenbykstructk0=LstructLrefQrefQstruct(Lin+Lout),(6)whereLrefisthetotallengthofthereferencechannelandLin/outarethelengthoftheinletandoutletofthestructurechannel[seeFig.
1(a)].
FromEq.
(5)wehaveQref/struct=cd*uref/struct*A,whichallowsustodirectlydeterminethepermeabilityfromaparallelmeasurementofuinthereferenceandstructurechannels.
Itisimportanttonote,thatcrossovereffectsbetweenthedifferentsectionsofthemiddlechannelareneglectedhere,i.
e.
,theindividualsegmentsareassumedtobewellconnected.
IV.
BOOLEANMODELSBooleanmodelsarewellestablishedmodelsforporousmaterialsfromstochasticgeometry[39–41].
There,porousstructuresarecomposedofoverlappinggrainswithrandompositionandorientation(i.
e.
,pointsinaplanearechosenrandomlyinaPoissonpointprocess).
Ateachpointagrainisplacedandinthecaseofanisotropicgrainsorientationsarealsochosenrandomlyfromauniformdistribution.
Inthisarticle,weconsidermodelsofrandomlyoverlappingmonodispersecircles(ROMCs)andrandomlyoverlappingmonodisperseellipses(ROMEs)withisotropicrandomorien-tation.
Wealsosimulatesystemsofoverlappingmonodisperserectangles(ROMRs)withrandomorientation,whichallowsustominimizediscretizationerrors,becausesuchgrainscanbe,incontrasttospheres,directlyrepresentedbypolygonsofarbitraryprecision.
Eachstructureisparametrizedbythetype,aspectratioandnumberNornumberdensityρ=N/L2ofgrains,whereListhelinearsystemsize.
ExamplesofthesemodelsareshowninFig.
2forgrainpercolation[Figs.
2(a)–2(c)]andvoidpercolation[Figs.
2(d)–2(e)].
Thewhitephasecorrespondstotheconductingphase.
Exchangingthetwophasesresultsintotallydifferentporespacemorphologies.
Fortheexperimentalandnumericaldeterminationofconductivityandpermeability,wecreateverealizationsofROMCandROMEstructuresonaquadratictwo-dimensional(2D)latticewithlinearsizeL=4000inpairsofequalopenporosityφo,i.
e.
,thevolumefractionofonlythesample-spanningpartoftheconductingphase.
Thecircleshavearadiusofr=34inunitsoflatticesitesandtheellipseshavealongandshortsemiaxisofa=96andb=12.
Inthemicrouidicsamplesthisequalsr30μmanda84μm.
Themorphologicalpropertiesoftheresultingstructuresand043023-3CHRISTIANSCHOLZetal.
PHYSICALREVIEWE92,043023(2015)abcdefFIG.
2.
Booleanmodelsofoverlappinggrainsfor(a)circles,(b)ellipses,and(c)rectangles.
AsetofNpointsisselectedrandomlyinaplane(totalareaL2)(Poissonprocess).
Ateachpointagrainwithrandomorientationisplaced.
EachstructureischaracterizedbyitspointdensityN/L2.
Thewhitephasecorrespondstotheconductingphase.
(a)–(c)arestructures,whichweclassifyasvoidpercolation,whereasthestructures(d)–(f)exhibitgrainpercolation.
theirpercolatingphase(markedwithanindexoforopen)aresummarizedinTableI.
V.
MINKOWSKIFUNCTIONALSMinkowskifunctionalsaremorphologicalmeasures,cor-respondingtovolumeandsurfaceintegralsofgeometricsets,whichareparticularlyusefulforcharacterizingrandomstructures[39,42–47].
IntwodimensionstheMinkowskifunctionalsofacompactsetAaregivenbyW0(A)=Ad2r,(7)W1(A)=12Adr,(8)W2(A)=12A1Rdr,(9)whereRistheradiusofcurvature.
Fromthisdenition,theMFinthecontinuumcanbeidentiedwithareaV,perimeterP,andEulercharacteristicχofaset:V=W0,P=2W1,andχ=W2/π,sothatthevaluesforaunitdiskareWi=π.
Ona2DlatticethenormalizationofWiischosendifferently,sothatthevaluesforaunitpixelareWi=1,i.
e.
,V=W0,P=4W1,andχ=W2.
AschematicillustrationoftheMFisgiveninFig.
3.
TheEulercharacteristicisatopologicalconstant,whichintwodimensionsisequivalenttothenumberdifferencebetweenconnectedcomponentsandholesinaset.
Thisquantityisparticularlyusefulforthecharacterizationofpercolatingstructures,becauseformanyrandomsets,χbecomeszeroclosetothepercolationthreshold,i.
e.
,thenumberofconnectedcomponentsofbothphasesareapproximatelythesame[42].
ForBooleanmodelsinthecontinuumtheMFsofindividualgrains(localMFswi)andthemeanMFsofrealizationsofthemodel(globalMFsWi)withmeandensityρ=N/L2arerelatedby(a)(b)1423512(c)FIG.
3.
SchematicillustrationoftheMinkowskifunctionals:(a)TheareaVoftheconductingphaseisshowninwhite,(b)theperimeterPcorrespondstothelengthoftheblackboundary,and(c)theEulercharacteristicχisthenumberdifferenceoftheconnectedcomponentsofeachphase,whichinthecaseshownwouldbe25=3.
TheopenEulercharacteristicχo(similartoopenporosityφoandopenperimeterSo)doesnotcountanyinclusions,i.
e.
,χo=15=4,astheinclusioninclusterNo.
5wouldbeneglected.
W0(ρ)/L2=1eρw0,(10)W1(ρ)/L2=ρw1eρw0,(11)W2(ρ)/L2=ρw2(2w1)24ρeρw0.
(12)However,porousstructures,inparticularwhenobtainedfromexperimentaldata,areoftenrepresentedasdiscretizedbinarydatasetsonalattice.
ForsuchdataitisconvenienttodenetheMFsinadiscretesystem.
ForBooleanmodelsonalatticewitheight-pointconnectivity(horizontal,vertical,anddiagonalneighbors)therelationsareW0(ρ)/L2=1eρw0,(13)W1(ρ)/L2=eρw0(1eρw1),(14)W2(ρ)/L2=eρw0(1+2eρw1eρ(2w1+w2)).
(15)Inbothcasestheseequationsareinvertible.
SuchaninversionhasbeenusedtodetermineBooleanmodelswithgraincompositionswithmatchingwitoreconstructnaturalporousmedia,suchasFontainebleausandstone[5].
NumericallytheMinkowskifunctionalscanalsobecalcu-latedforthepercolating(open)phase.
Fromthisweobtainφo,So,andχo.
VI.
KATZ-THOMPSONMODELIntheliteratureitiscontroversiallydiscussedwhetherpermeabilityandconductivityhavedifferentorequalscalingexponents[2].
IntheKatz-Thompsonmodelequalscalingexponentsareassumed,whichgivesarelationshipbetweenconductivityandpermeabilitybasedonargumentsfrompercolationtheory.
Thisequalityisrelevantinourcase,sincemanyanalyticalresultsontheporescaleareonlyobtainedfortheconductivity,butnotthepermeabilityofporousmedia.
ThelengthscalethatdeterminesthepermeabilityintheKatz-Thompsonmodelisidentiedasthecriticalporediameter[seeFig.
4(b)].
Duetothequasi-2Dgeometryofour043023-4DIRECTRELATIONSBETWEENMORPHOLOGYAND.
.
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PHYSICALREVIEWE92,043023(2015)TABLEI.
Tableofpropertiesoftheanalyzedstructures,correspondingtheBooleanmodelsofrandomlyoverlappingcirclesandellipses:NumberofgrainsN,porosityφ,openporosityφo,totalperimeterofthevectorizedimageS,perimeteroftheopenclusterS0,perimeterofthepixelizedimageassumingeight-pointconnectivitySd8,Eulercharacteristicofthevectorizedimageχ,Eulercharacteristicofpercolationclusterχo,Eulercharacteristicassumingeight-pointconnectivityχd8,criticalporediameterintwodimensionsDc,criticalporediameterinthreedimensionslc,effectivenumberofgrainscalculatedfrominvertingMinkowskifunctionalsofthevectorizedimageNeffandthepixelizedimageNeffd8,reconstructednumberofgrainsNrec8frominversionoftheMinkowskifunctionalsofthegrainphase(equalsNeffd8forS1–S10),permeabilityksim/clc2fromLBsimulations,conductivityσ/σ0fromFEMsimulations.
StructureNφφoSSoSd8χχoχd8DclcNeffNeffd8Nrec8kexp/cl2cksim/cl2cσ/σ0S145920.
3650.
298367820281380444740167220(d4)1952.
4212.
4214593547254720.
01580.
02730.
0118S239680.
4180.
40136563733751644436181395(d4)5311.
50683999469746970.
0470.
04140.
0764S327040.
5510.
549326905321646401440442635(d4)41621.
98482553289428940.
1370.
15750.
2107S416320.
7010.
700249486248098313041628724(d4)60751.
30681496166816680.
3850.
38070.
4405S57540.
8500.
850137404137123180900485520(d4)474100.
01886887407400.
6410.
66530.
7031S620640.
6510.
26650841816900961468266445(d4)7045.
9575.
9572014247424740.
006960.
01120.
00851S721760.
6390.
40052254327161663136375180(d4)7906.
2456.
2452151265926590.
020.
01740.
0135S818400.
6840.
549474276332151577379463146(d4)5006.
2756.
2751714210821080.
03850.
03550.
0392S913870.
7510.
70039195933678747998454275(d4)8641.
25481283151115110.
1180.
11930.
1501S107710.
8540.
850247904241577313061250352(d4)23167.
42987128058050.
35980.
33330.
3787S1150170.
6680.
2783687881421024514336883786605.
855.
852547283354570.
0160.
0190.
01228S1248350.
6580.
40036700921180145017068057465310.
7282579288050750.
02280.
0210.
02602S1352730.
6820.
5503565482745524381167747917456.
056.
052496276152890.
0400.
0340.
03498S1461740.
7420.
70034691631743642886111221179109535.
9382600283368880.
2100.
1320.
16322S1587260.
8510.
85027808427683035058218341842181447.
97826852829102840.
3040.
4250.
47783S1624050.
3880.
27054424137089166550114061114136313.
738133821581624810.
011060.
0170.
0227S1726020.
4170.
40056951954162169796917931808175212.
918135861598826910.
043660.
0490.
0636S1838820.
5510.
54964493864173780040338653881381918.
298138111587341540.
09640.
1310.
1724S1958220.
7010.
70064772864772881565363386348629021.
548133101481686980.
1860.
2970.
3779S2090950.
8500.
85050437150437265258186558657861727.
6681194112756187080.
74160.
5360.
6388043023-5CHRISTIANSCHOLZetal.
PHYSICALREVIEWE92,043023(2015)102101100102101100lcσ/σ0k/clc2ROMCvoidROMEvoidROMCgrainROMEgrainKatz-ThompsonFIG.
4.
(Coloronline)Cross-propertyrelationbetweenperme-abilitykdeterminedbyLBsimulationsandconductivityσobtainedfromFEMsimulations.
Theconductivityisnormalizedbythebulkconductivityσ0.
Thepermeabilityisnormalizedbythesquareofthecriticalporediameterlcandaconstantc=1/12,determinedfromthelimitlc→handφ→1,wherethesystemequalstheowproblembetweeninnitelylargeparallelplateswithspacinglc.
Inset:Determinationofthecriticalporediameterlcfromparallelsurfaces.
samplesthiscriticaldiameterisgivenbylc=min(Dc,h),(16)whereDcistheactual2Dcriticaldiameter.
Ifthisdiameterisgreaterthantheheighthofthestructure,hconnestheowandbecomestherelevantlengthscale.
ForourstructureswedeterminedlcdirectlyfromtheimagesofthestructurescomputingtheEuclideandistancetransform(EDT).
ForeachpointintheconductingphaseofthesampletheEDTassignsthedistancetotheclosestpointonthesurface.
Fromthislccanbeeasilyidentied[48,49](seeinsetofFig.
4).
Theconstantc=1/12ischosentotthedilutegrainlimit,wherecl2c=h2/12.
InFig.
4therelativepermeabilityisshownindepen-denceoftheconductivity,bothdeterminednumericallyfromlattice-Boltzmann(LB)andnite-element(FEM)simulations,respectively.
ThepredictionoftheKatz-Thompsonmodelisdepictedasadashedline.
Inparticularforlargerφ,thedataagreesverywellwiththeKatz-Thompsonmodel,withonlyaslightdeviationofthepermeabilitytowardslowervaluesthanpredicted.
Additionallyevenforlowpermeabilitiesthepredictionsdeviatebylessthanafactoroftwoforallbutonestructure.
Forφ→1,thisagreementisnotsurprising,sincetheheightofthestructureismuchsmallerthanthedistancebetweentheobstacles,whichleadstoanequivalenceoftheowandtheconductanceproblemsincetheweightofthedifferentpathwaysisthesameforowandconductance.
Astheowproleislocallyequivalenttoowconnedbetweeninnitelylargeparallelplatesthesystemcanbethoughtofasanetworkofhydraulicconductorswithconductivityproportionaltoh3.
Inbothcases,ahomogeneouscurrentorowisonlydisturbedbyisolatedobstacles,whichonlylocallyinuencesthe(hydraulic)conductanceoftheporousstructure,withoutchangingthehydraulicradius,whichremainsontheorderoftheheighth.
However,thisisnotthecaseclosetothepercolationthresholdφc.
KatzandThompsonarguethatσandkfollowsimilaruniversalpowerlawsclosetothecriticalporositywithanaccuratechoiceofthecriticalporediameter.
Accordingto00.
20.
40.
60.
8100.
20.
40.
60.
81φc≈0.
32φc≈0.
66ROMCROME(a)φk/clc200.
20.
40.
60.
8100.
20.
40.
60.
81φc≈0.
34φc≈0.
68ROMEROMC(b)φk/clc200.
51102101100(c)(φφc)/(1φc)k/clc200.
51102101100(d)(φφc)/(1φc)k/clc20.
20.
40.
60.
8102101100(e)φok/clc20.
20.
40.
60.
8102101100(f)φok/clc2102101100102101100(g)αv=1.
27(1χo)/Nk/clc2102101100102101100(h)(h)αv=1.
27αg=2.
05(1χo)/Nk/clc2FIG.
5.
(Coloronline)Experimentally(closedsymbols)andnumerically(opensymbols)determinedpermeabilitykofvoid(top)andgrainpercolation(bottom)vs.
differentmorphologicalproperties:(a),(b)porosityφ,asexpectedk/cl2cvanishesaroundφcandgoesto1forφ→1;(c),(d)rescaledporosity,datapointscollapsewithsomedeviationsduetonite-sizeeffects;(e),(f)openporosity,forvoidpercolationROMCstructureshavehigherpermeabilities,forgrainpercolationROMEstructureshavehigherpermeabilityatequalφo;and(g),(h)Eulercharacteristic:datacollapsesontoasinglecurveforvoidpercolation,butforgrainpercolationdeviationsarefound.
Thedashedlinesin(g)and(h)aretstoEq.
(17)withonefreeparameterα.
Errorbarsareonlyshowniflargerthansymbolsize.
043023-6DIRECTRELATIONSBETWEENMORPHOLOGYAND.
.
.
PHYSICALREVIEWE92,043023(2015)FIG.
6.
(Coloronline)(a),(b)Currentdensitymagnitudefromnite-elementsimulationsoftheLaplaceequation(i.
e.
,conductivity)normalizedtothetotalmaximumcurrent.
(c),(d)Fluidvelocitymagnitudefromlattice-Boltzmannsimulations.
Forellipticalgrainstheoverlapleadstopronouncedstagnantparts,whilecirculargrainsformmorecompactobstacles,asillustratedin(e),(f)respectively.
Withinsuchstagnantpartsastrongdecreaseofboththecurrentandtheowvelocityisobserved.
Howeverthedecreaseofjappearstobefaster,asobservedinRef.
[50].
someauthorsthisisonlytruefortwodimensions[12–14].
Otherwise,nonuniversalpower-lawexponentshavetobeconsidered.
ToobtainfurtherinsightintothisproblemweshowthecurrentdensityandtheowvelocitymagnitudeforrepresentativeROMCandROMEstructuresinFig.
6.
Eventhoughtheeldssharesomemorphologicalfeatures,suchastheprincipalowpaths,thedecayofthecurrentmagnitudeintodeadendsappearstobefasterthanthatoftheuidvelocity.
Thisfeatureisalsoobservedwhenconsideringthedistribu-tionofcurrentsorowvelocityrespectively.
AsshowninFig.
7thecurrentdistributiondecaysfasterforbothstructures.
However,thedistributionsappeartobequalitativelysimilar,ROMCROME102101100107106105104103102101100v/vmax|j/jmaxp(v)|p(j)FIG.
7.
(Coloronline)Comparisonofvelocitymagnitudedistri-butionp(v)(solid)andcurrentmagnitudedistributionp(j)(dashed)inROMC(black)andROME(red)structuresshowninFig.
6.
Bothdistributions,i.
e.
,p(v)andp(j),followsimilartrends,however,thevelocitydecayswithasignicantlyfasteramplitude.
whichcouldexplainthesurprisinglyaccuratepredictionoftheKatz-Thompsonconjecture.
VII.
TRANSPORTANDMORPHOLOGYTransportinBooleanmodelsiseitherdescribedbyper-colationtheoryorbyeffectivemediumapproximations[22].
Inpercolationtheoryoneassumesthatclosetoφctransportpropertiesaredescribedbypowerlawsandfarawayfromφceffectivemediumtheoriesareapplied.
Inbothcasesthedataagreesqualitativelywiththisassumption.
InFig.
5boththeexperimentally(closedsymbols)andnumericallyobtainedvalues(opensymbols)forthepermeabilitieskareplottedversusseveralquantitiesforvoidandgrainpercolation.
AsshowninFig.
5(a)andFig.
5(b)thepermeabilitiesvanishclosetoφcandapproachthevalueofanunpatternedchannelforφ→1.
Duetothenitesizeofthesamplesthemeasuredpermeabilitiesscattermoreandmoreasthepercolationthresholdisapproached.
Duetothenitesystemsizesomestructureshaveporositiesbelowφcandarestillconductive(technicallyforthesemodelsφcisonlywelldenedforinnitesystems).
Thisbecomesparticularlyclearwhenkisplottedvs.
therescaledporosity[seeFigs.
5(c),5(d)],asmotivatedbyArchie'slaw.
Themeasuredpermeabilitiescollapseontoasinglecurvewithintheexperimentalaccuracy.
However,inthecaseofnegativerescaledporosities,Eq.
(4)obviouslycannotbeapplied.
Independenceoftheopenporosityforvoidpercolation,ROMCstructureshaveahigherpermeabilitythanROMEstructuresforequalφo[seeFig.
5(e)].
Inthecaseofgrainpercolationthissituationisreversed[seeFig.
5(f)].
Thisfactcanbeexplainedfromthemorphologyofthevelocityeldsasshownbelow.
Comparedtocirclestheellipsesformmoreelongatedinterconnectedobstacleswithasignicantamountofstagnantpartsbetweengrains.
Thisreducesthepermeabilitysignif-icantly.
Inthecaseofgrainpercolationtheellipsesform043023-7CHRISTIANSCHOLZetal.
PHYSICALREVIEWE92,043023(2015)moredirectpathwaysfortheow,whichexplainsthehigherpermeabilitycomparedtoROMCstructures.
Theexperimentalandnumericalresultssupportthatk=clc21χoNα(17)forthevoidpercolationmodels[7]whereαisafreeparameter.
Equation(17)canbejustiedfromthevelocitymagnitudedistributionsinsidetheporousstructuresshowninFig.
6.
WhencomparingROMCandROMEstructuresitbecomesclearthatcirculargrainsformmorecompactobstaclesatequalφocomparedtoellipticalgrains[compareFigs.
6(e)andFig.
6(f)],becausetheprobabilitytooverlapandformmoretortuouspathwaysaswellasdeadendswherenoowoccursislargerforelongatedellipsesatequalgraindensities.
Thisfactiscapturedbythefactor(1χo)/N,whichcanbeinterpretedasthenumberdensityofobstaclesformedbyjoinedgrains.
AlldatapointsforvoidpercolationarewelldescribedbyattoEq.
(17)withαv=1.
27[seeFig.
5(g)],whichisclosetothecriticalexponentμ=1.
3intwodimensionsforArchie'slaw.
OneparticularshortcomingofEq.
(17)istheexplicitdependenceonthegrainnumberN,whichmightbeunknownorill-denedformanyporousmaterials,wheretheformationprocessisunknown.
Also,inthecaseofgrainpercolationtheuseofNinthedenominatorofEq.
(17)giveseventhewronglimitk=0forN→∞.
ThereforewereplaceNbyaneffectivegrainnumberN,whichisderivedfromthemorphologicalcorrelationsoftheMinkowskifunctionalsofBooleanmodels[Eqs.
(10)–(12)].
InthecontinuumoneobtainsN=P24πAφχφ(18)andfrom(13)–(15)onthe2DlatticeN=L2lnχL2φ1S4L2φ21S4L2φ2+21S4L2φ1.
(19)InthecaseofvoidpercolationNis,asexpected,typicallyclosetotheactualvalueofN(seeTableI).
InthiscaseNcaneitherbecalculatedfromvectorizedimages(thebinaryimagesofthestructuresarevectorizedusingamarchingsquaresalgorithm)usingEq.
(18)ordirectlyfromtherasterimagesusingEq.
(19).
AsshowninTableI,similarvaluesareobtained,withslightlybetteragreementofNandNforthelatticeequation.
ForgrainpercolationtheroleofbothphasesisinvertedandNdoesobviouslynotcorrespondtoN,butinsteadisusedtodeneaneffectivegrainnumber.
AsshowninFig.
5(h)thepredictionofthepermeability,i.
e.
,theblackdashedline,whichwegotforvoidpercolation,isquiteaccurateforROMEstructuresathighporositiesbutsignicantlyoverestimateskforROMCstructures.
Thereasonforthisdeviationmightbetheoccurrenceofmanyverysmallisolatedobstacles(seeFig.
8)withasizeofafewpixels,whichsignicantlyinuenceNwithoutstronglyinuencingk.
SuchisolatedobstaclesarefoundmorefrequentlyforROMEthanforROMCstructures.
LookingagainatFig.
5(h),thedata012345100101102103log10(X)(px2)p(X)(a)012345100101102103log10(X)(px2)p(X)(b)FIG.
8.
(Coloronline)Distributionforobstaclesizesp(X)for(a)voidpercolationand(b)grainpercolation.
OpenbarscorrespondtoROMEandlledbarscorrespondtoROMCstructures.
Thehistogramsforgrainpercolationshowalargenumberofverysmallobstacles.
pointsfollowasimilartrendasinFig.
5(g).
However,thescatteringissignicantlystronger.
AtofEq.
(17)tothemeasureddata,whichisshownasagreendashedline,yieldsanexponentαg=2.
05.
Thequantitativedeviationfromαvcouldindicatethatthemotivation,whichgaverisetoEq.
(17),isobviouslynotdirectlyapplicableinthecaseofNforarbitrarystructures.
Thefactor(1χo)/Nwasinterpretedasthenumberdensityofobstacles,whichareformedbyjoinedindividualgrains.
Thisinterpretationisnotthatstraightforwardforgrainpercolation,sincetheobstaclesareformedbywhatisleftafterremovinganumberNofcircularorellipticalareas.
However,eveninthiscasetheequationdoesnotfailqualitatively,sothatweexpectareasonablepredictionofkforanystructurescomposedofoverlappinggrains.
VIII.
LOWGRAINDENSITYThesuccessoftherelation[Eq.
(17)]betweenpermeabilityandEulercharacteristicofBooleanmodelsraisesthequestionwhethercertainresultscanbeobtainedanalyticallyoratleastsemiempirically.
Intheregimeoflowgraindensityanalyticalresultsareavailablefortheconductivityσ[51].
Inthisregime,i.
e.
,forφ→1theEulercharacteristicoftheconductingphasebecomesχo→χasthegraindensityissolowthatindividualgrainsdonotoverlap.
ConsequentlywecanexpandEq.
(10)–(12)andobtainχN=11+4aE1b2a224π2ab(1φ)+O(φ2),(20)whereaandbarethelongandshortsemiaxisoftheellipseandEistheellipticalintegralofthesecondkind.
Hereweapproximatethecircumferenceoftheellipse4aE(1b2a2)≈π2(a2+b2)andobtainχN≈1(a+b)22ab(1φ).
(21)Thisresultisindeedequivalenttotheexactresultforσofaconductivesheetwithasmallnumberofcircularobstacles(i.
e.
,thedilutelimit)intwodimensions[51]andthusσσ0≈χN.
(22)Forlargeobstacledensitieshowever,noanalyticalresultsareavailable.
Intheregionclosetoφc,whereBooleanmodels043023-8DIRECTRELATIONSBETWEENMORPHOLOGYAND.
.
.
PHYSICALREVIEWE92,043023(2015)fallintotheuniversalityclassof2Dlatticepercolation,itisassumedthattheconductivityisdescribedbypowerlaws,aspreviouslystated[22].
However,whetherthesameistrueforχoisnotobvious.
Inthefollowingwepresentanumericalanalysisofthedependenceofχoclosetoφc,intheintermediaterangeandfarawayfromφcforBooleanmodelsandlatticepercolation,independenceofgrainshapeandsystemsize,toevaluatethepossibleuniversalbehaviorofχoandlinktheresultstopercolationtheory.
IX.
CRITICALBEHAVIORClosetoφctransportpropertiescanchangedramatically,duetothefractalbehaviorofthepercolatingcluster.
Resultscanthendependdramaticallyonthesystemsize.
Inpercolationtheoryauniversalcriticalexponentisassumedfortheconductivity.
BecauseoftherelationbetweenφcandχoinEq.
(17),thequestionariseswhetherχoalsoshowsacriticalbehaviorclosetoφc.
Toanalyzethedependenceofχoonthesystemsizeandφclosetoφcwegeneratetwodifferenttypesofstructuresthatminimizediscretizationerrors:sitepercolationonalatticeandROMRstructuresinthecontinuum.
Thepercolationprobabilityofsitepercolationsystemsclosetoφcisdescribedbyauniversalexponentβ=5/36,whichisknownanalyticallyintwodimensionsfromconformaleldtheory(weusethestandardnotationforcriticalexponentsascommonintheliterature,e.
g.
,Ref.
[26]).
Booleanmodelsofrandomlyoverlappinggrainsfallintothesameuniversalityclass,sothecriticalexponentsareequal.
However,thepercolationthresholdisnonuniversalanddependsonthedetailsofthegrains.
InthecaseoftheEulercharacteristictheproblemislesswellunderstood.
AlthoughformanyrandomeldstheEulercharacteristiccanbedeterminedanalytically,thisisnotthecasefortheEulercharacteristicoftheopenphase,i.
e.
,thepercolatingcluster.
Forfractals,thescalingofarea,perimeterandEulercharacteristicbehavedifferently.
Whiletheareascaleswithonecriticalexponent,suchasinthecaseofthepercolationprobability,theperimeterhasoneadditional,andtheEulercharacteristichastwoadditionalscalingexponents,whicharenotindependent[52,53].
Theamplitudesofthescalingrelationsareingeneralnotknownanalytically.
Therefore,weanalyzethescalingbehavioroftheEulercharacteristicnumerically.
First,wecalculate(1χo)/NforROMRstructuresindependenceoftherescaledporosity.
Calculationsareper-formedforsystemsizesL=10a,20a,50a,100awhereaisthelengthofthelongsideoftherectangles.
Thesimulationsarerepeatedfordifferentaspectratios1:1,1:2,1:4and1:10.
Anensembleof2500samplesforthesmallestandsixsamplesforthelargestsystemsizewassimulated,which,exceptforthelargestsystemsize,resultsinanegligibleerrorofthemeanforalldatapoints.
AsshownintheresultingcurvesinFigs.
9(a)–9(d),weobserveforallsystemstwodistinctregimeswithsignicantlydifferentφdependence.
First,acriticalregimeclosetoφcandsecond,aneffectivemediumregimewherethedatapointscollapsefordifferentaspectratios,butasignicantlydifferentslopeofthecurvecomparedtothecriticalregimeisobserved.
Noqualitativechangeinthisbehaviorisobservedfordifferentsystemsizes.
However,duetotheratherlimitedsystemsizeoftheROMRstructures,itisnotobviouswhetherthesetworegimespersistforinnitesystemsize.
Therefore,wealsosimulatesitepercolationona2Dsquarelattice,forwhichalinearsystemsizeofL=214latticesitescanbeachievedwithanensembleof26realizationsandupto216forthesmallestsystemsizeL=25.
Here,Nmustbereplacedby(1p)L2,which,pbeingtheprobabilitythat102101100102101100eectivecritical(a)(φφc)/(1φc)1χoNL=10a1:1L=10a1:2L=10a1:4L=10a1:10102101100102101100(b)(φφc)/(1φc)1χoNL=20a1:1L=20a1:2L=20a1:4L=20a1:10102101100102101100(c)(φφc)/(1φc)1χoNL=50a1:1L=50a1:2L=50a1:4L=50a1:10102101100102101100(d)(φφc)/(1φc)1χoNL=100a1:1L=100a1:2L=100a1:4L=100a1:10103102101100102101100(ppc)5/36(e)(ppc)/(1pc)1χo(1p)·L2L=25L=26L=27L=28L=29L=210L=211L=212L=213L=214101102103104101.
8101.
7101.
6101.
5(f)L1χo(1pc)·L2(pc,L)SitePercolationL5/48101102103101.
8101.
6101.
4101.
2101100.
8100.
6(g)L1χoN(φc,L)1:11:21:41:10L5/48FIG.
9.
(Coloronline)BehavioroftheEulercharacteristicofthepercolatingclusterχofor(a)–(d)voidpercolationofdifferentROMRsystems(aspectratios1:1,1:2,1:4,1:10)asafunctionoftheporosityφand(e)sitepercolationona2Dlatticefunctionoftheoccupationprobabilityp.
FortheROMRstructuressystemsizesofL=10a,20a,50a,and100ahavebeensimulated.
ForsitepercolationL=22214.
(f)Finite-sizescalinganalysisofχoforsitepercolationatthepercolationthresholdfordifferencelinearsystemsizeL.
(g)Finite-sizescalingofχoindependenceofthelinearsystemsizeLforROMRstructures.
043023-9CHRISTIANSCHOLZetal.
PHYSICALREVIEWE92,043023(2015)asiteisconducting,correspondstothevolumeoccupiedbyobstacles.
Inthissystem,asshowninFig.
9(e),wendexactlythesamebehaviorasforROMRstructures,forallsystemsizes.
Thisfurthersupportsourassumptionthattheoccurrenceoftwoscalingregimespersistsforothergrainshapesandlargersystemsizes.
However,adirectdeterminationofthescalingexponentfromnite-sizesystemsimulationsisnotfeasible[26].
Instead,anestablishedmethodtoextractcriticalexponentsnumericallyisusedfromnite-sizescaling[25,26].
Here,weonlygiveabriefdescriptionofthemethod:IfwecombineEq.
(4)andEq.
(17),weget(1χo)/N∝(φφc)β,whereβ=μ/α.
Becausethecorrelationlengthξdivergesatφcaccordingtoapowerlaw,wecanusethatξ∝(φφc)νtoobtain(1χo)/N∝ξβ/ν.
Sinceξisinniteatφ=φc,Lbecomestheconninglengthscaleofthesystemandweobtain(1χo)/N∝Lβ/ν.
Thisassumptionisnowtestednumerically.
Here,ourresultsmightdifferforsitepercolation[Fig.
9(e)]andROMRstructures[Fig.
9(f)].
Forsitepercolationweclearlyobserveapowerlawwithanexponentofβ/ν=5/48inperfectagreementwithpercolationtheory.
Thissuggeststhatβ=5/36indeeddescribesthescalingoftheopenEulercharacteristic.
However,forROMRstructurestheslopeofthecurveissignicantlydifferentforsmallLandonlyatlargeLbecomesconsistentwithascalingofLβ/ν(seedashedline).
ApparentlytheROMRsystemismoresensitivetonite-sizeeffects.
However,largerLarecomputationallytooexpensive,sothatL<256forsquaregrainsandworseforlargeraspectratios,sincetheamountofgrainsataxedφdivergeswiththeaspectratio.
Nevertheless,ournumericalresultssupportthefollowinginterpretation:WhenweconnectthescalingoftheopenEulercharacteristictotheexperimentalobservationthecriticalex-ponentseemstobeirrelevantwithrespecttoourexperimentalsystemsizes.
InthecriticalregimefromEq.
(4)percolationtheorywouldyieldk∝1χoNμ/β,(23)whichweobviouslydonotobserveintheexperiment.
InsteadourinterpretationisthatthepowerlawfromEq.
(17)isrelatedtotheeffectiveregime.
SinceinthisregimetheopenEulercharacteristiccollapses,wearguethatthesameshouldbetrueforconductivityandpermeability,howeverwithasignicantlydifferentexponent(comparedtoβ)withavaluecloseto1.
Amoredetailedanalysisofthisproblem,however,requireseithernewsimulationsorexperimentalstudiesofconductivityorpermeabilityofstructureswithmuchlargersystemsize.
Consequently,asignicantimprovementofthecomputationaland/orexperimentaleffortisrequired.
Sofar,only20structurescouldbemeasuredandsimulated.
Forahighernumberofsamplesareliableautomationoftheexperimentwouldberequired.
Forlargestructuresclosetoφctheresolutionoftheexperimentmustbeimprovedbyatleastoneorderofmagnitude.
Thesameistrueforthecompu-tationaltimerequiredtosolvetheconductivityorpermeabilityproblem.
X.
SUMMARYWehaveanalyzedthepermeabilityandconductivityofporousmicromodelscomposedofrandomlyoverlappinggrains(Booleanmodels).
Wehaveanalyzedvoidandgrainpercolationforoverlappingcirclesandellipses(i.
e.
,structureswherethevoidisconductiveandstructureswherethegrainsareconductive).
InallcasestherelationbetweenpermeabilityandconductivityiswellpredictedbytheKatz-Thompsonmodel.
InthecaseofvoidpercolationthepermeabilitycanbededucedfromtheEulercharacteristicofthepercolatingclusternormalizedtothetotalnumberofgrains,whichrequiredaprioriknowledgeofthegraindensity.
ForgrainpercolationasimilarapproachisstudiedbasedonthedenitionofaneffectivegrainnumberN,whichiscalculatedfromtheglobalMinkowskifunctionalsofthestructures.
ThisapproachworksqualitativelyforROMEstructures,butoverestimateskforROMCstructures,duetothesensitivityofNontheoccurrenceofisolatedpixels,whichotherwisedonotstronglyaffectk.
Forvoidpercolationinthelowgrain-densitylimititcanbeanalyticallyshownthattheformationfactorisgivenbytheEulercharacteristic.
ThecriticalbehavioroftheEulercharacteristicofthepercolationclusterχoforφ→φcisanalyzednumericallytolinkourresultstopercolationtheory.
Forthe2Dsquarelattice,wendthatχoscaleswiththecriticalexponentβonlyveryclosetoφc.
Furtherawayfromφc,aneffectiveregimeisfoundforbothsquarelatticeandBooleanmodelswherethevaluesofχooverlapfordifferentsystems,i.
e.
,differentgrainshapesandsystemsizes,justifyingtheapplicabilityofourmodeltomanydifferenttypesofstructures.
Aremainingquestionistheapplicabilitytofullythree-dimensional(3D)porousmedia.
Inprinciple,3Dmodelsareaccessibleexperimentally,e.
g.
,via3Dprinting,andhavealsobeenstudiednumerically[54].
AlsoMFsarewellunderstoodinthe3Dcase,e.
g.
,theEulercharacteristicalsovanishesclosetoφc.
Therefore,itisreasonabletoassumethatEq.
(17)couldholdinthreedimensions,eventhoughthereisnointuitiveinterpretationsimilartothe2Dcase.
Itmustbeexpectedhowever,thatmeasurementsandsimulationsaresignicantlymorechallengingandcomputationallymoreexpensive.
ACKNOWLEDGMENTSWethankJanG¨otzforthesupportonLBsimulationsforinvertedBooleanstructures.
WealsoacknowledgefundingbytheGermanScienceFoundation(DFG)throughGrantsNo.
ME1361/12andNo.
SCHR-1148/3.
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