Morris Dep artment of Physics University of or onto 60 St Ge or ge Str et or onto Ontario M5S 1A7 Canada CS 4570Qj pattern formation CS 6220Mk atigue brittleness fracture and crac ks Abstract Columnar join ts are threedimensional fracture net orks t ID: 68953
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EurophysicsLettersPREPRINTOrderanddisorderincolumnarjointsLucasGoehringandStephenW.MorrisDepartmentofPhysics,UniversityofToronto,60St.GeorgeStreet,Toronto,Ontario,M5S1A7,CanadaPACS.45.70.Qj{patternformation.PACS.62.20.Mk{Fatigue,brittleness,fracture,andcracks.Abstract.{Columnarjointsarethree-dimensionalfracturenetworksthatformincoolingbasaltandseveralothermedia.Thenetworkorganizesitselfintoordered,mostlyhexagonalcolumns.Thesamepatterncanbeobservedonasmallerscaleindesiccatingstarch.Weshowhowsurfaceboundaryconditionsinthedesiccationofstarcha ecttheformationofcolumnarjoints.Underconstantdryingpowerconditions,we ndapowerlawdependenceofcolumnarcross-sectionalareawithdepth,whileunderconstantdryingrateconditionsthiscoarseningiseventuallyhalted.Discontinuoustransitionsinpatternscalecanbeobservedunderconstantexternalconditions,whichmaypromptareinterpretationofsimilartransitionsfoundinbasalt.Starchpatternsarestatisticallysimilartothosefoundinbasalt,suggestingthatmaturecolumnarjointingpatternscontaininherentresidualdisorder,butarestatisticallyscaleinvariant.Columnarjointing,inwhichshrinkagefracturesarrangethemselvestoleavebehindapatternofmainlyhexagonalpillars,hasfascinatedscientistsandnaturalistsforcenturies[1{3].TheGiant'sCausewayinNorthernIrelandandtheDevil'sPostpileinCaliforniaarefamousexamples.Columnarjointsoccurinlava\rows,sandstone,mud,coal,glass,starch,andice[3{11],whilea2Dvariantmaycauseice/sand-wedgepolygonsinpermafrostonEarthandMars[5,12].Columnarjointsareformedthroughtheself-organizationofshrinkagecracksinathree-dimensionalmaterial,aseitherheatormoistureisremovedfromonesurface[6,13,14].Brittlefractureisinitiatedbythestressbuildupatthissurface[14],andtheresultingcracksprogressivelyextend,trackingashrinkagefrontasitmovesintothebulk[15,16].Atanygiventime,activefracturesarecon nedtoanearlyplanarlayerbetweentheremainingcompliantmaterial,andthefullyfracturedmaterial[15,16].Theevolutionofthisquasi-2Dpolygonalpatternthroughtimeisrecordedinthedepthdependenceoftheresultingprismaticcolumns.Often,ingeophysicalexamples,thispatternonlybecomesapparentwhentheinterioroftheformationisexposedbyerosionandweathering.Asaresult,previousgeophysicalstudiesofthefracturepatternhavebeenlimitedtoexposedplanarsurfaces.Columnarjointingisasurprisinglygeneralphenomenon,wellknowninigneousrocks(bothterrestrial[13]andlunar[17]),butalsoseeninsedimentary[5,10],andmetamorphicrocks[5],aswellasinman-madeandbiologicalmaterials[4{9].ExamplesinbasaltandcornstarchareshowninFig.1.Jointscanrangeinsizefrommicronsizeddi usivelycooledcolumnsinvitri ed,impure,ice[11]tometersizedevaporativelycooledcolumnsinbasalt[14].c\rEDPSciences 2EUROPHYSICSLETTERSFig.1{Examplesofcolumnarjointing.(a)ThecolonnadeoftheDevil'sPostpile,CA,USA.(b)ExposedsurfaceoftheDevil'sPostpile,showingthequasihexagonalfracturepattern,asitoccursinbasalt.(c)MicroCTx-raytomographyimageofcornstarchcolonnadewith36m3voxelresolution,and2.5cm/side.(d)Cross-sectionofatomogramatadepthof18mm.(e)Atypicalcornstarchcolonnade(showninverted)studiedintheseexperiments,grownwithaconstantevaporationrate.Otherpolygonalfracturepatternscon nedtoaplanemayevolveandorderbysimilarprocesses,ifamechanismforthelateralmotionofcracksexists.Forexample,fracturedfreeze-thawpolygons,withmobileedges,areobservedinArcticandAntarcticpermafrost[18,19].SimilarfeatureshavebeenimagedonMars,suggestingthepresenceofMartianpermafrost[19,20].Recentphase eldmodelsofcolumnarjointinghavebeenconstructedtoapplyequallytoallthesecases[12].Thisgeneralitysuggeststhatadetaileddescriptionofcolumnarjointing,basedoncontrolled,repeatableexperiments,couldhaveapplicabilitytoanumberof elds.Todate,duetotheconstraintsongeophysicalobservation,thereexistsnoquantitativedescriptionofhowcolumnarjointsorder.Intheabsenceofsuchexperimentalor elddata,anumberoftheorieshavebeenproposed[12,13,21{23].However,thereisnoconsensusonFig.2{Experimentalmethods.Anautomatedscalewasusedtofeedbackcontroltherateofevapo-ration.Afterdrying,thefracturepatternismeasuredusingacombinationofMicroCTx-raytomog-raphyanddestructivesampling. LucasGoehringandStephenW.Morris:Orderanddisorderincolumnarjoints3exactlyhoworwhythesejointsorder.Norisitunderstoodhowsurfaceboundaryconditionsandmaterialpropertiescontributetothecolumnarstructureandscale.Inthispaper,wepresentthe rstever3Ddatadescribingcolumnarjointing,speci callydescribingtheorderingandcoarseningprocessesobservednearthefreesurface.OurexperimentaltechniquesarebasedonthoseofMuller[6{8],whoindependentlyre-discoveredthejointinginstarchespreviouslynotedbyHuxley[3]andbyFrench[4].ThesetechniqueshavealsorecentlybeenadoptedbyToramaruandMatsumoto[9],whoinvestigatedtherelationshipbetweendesiccationrateandthestarchpatternata xeddepth.250Wheatlampswereusedtodry1:1slurriesof100%purecornstarchandwater.Tracesofbleachwereusedtosterilizetheexperiment.Westudiedsamples1-100mmdeep,driedincircular\rat-bottomeddishes;evaporationrateswerebetween10-40mg=hcm2,andsamplestypicallydriedbetween1and28days.Watercontentwasmeasuredbyweighingthesamplesonceperminuteonanautomatedscale,asshowninFig.2.Desiccationratesweresetbyregulatingtheheatingandventilationappliedtothetopsurfaceofthestarch.Inonecaseoverheadlampsata xedstarch-lampdistancesuppliedacontinuous,constantdryingpower.Thedryingpowercouldbechangedbetweenexperimentsbyvaryingthelamp-starchdistance.Inoursecondsetupweusedthemeasuredsampleweight,m(t),todriveafeedbackloopwhichcontrolledthedutycycleoftheheatlampsandasmallfan.Weusedthisfeedbacktokeeptheevaporationrate,dm=dt,constant.Weshallrefertothesemethodsasconstantpowerandconstantrateconditions,respectively.ThislevelofcontrolhasnotbeenavailableinthepreviousexperimentsofMuller[6{8]orToramaruandMatsumoto[9].Asastarchslurrydries, rstgenerationcracksappear,penetratingthroughtheentiresampledepthandbreakingthesampleintolarge,disconnectedpolygons.Later,much nersecondarycracksinitiateatthetopsurface,andpropagateintothesample.Thesecondarycrackfrontleavesbehindcolumnarjointswithineachlargepolygon.Thistwo-stepfractureensuresthatcontainersizeandshapedonota ectthecolumnarjointpattern.WehaveusedMicroCTx-raytomography,showninFig.1c,d,toproducefully3Dvi-sualizationsofthepatternofcolumnarjointingindesiccatingstarchunderseveraldryingconditions.Wehavealsorecordedthispatternincross-sectionbymeasuringthecounterpartsleftonthedryingcontainersafterthestarchisremoved,andbysawingsamplesopeninordertodestructivelyobservethepatternatdi erentdepths.Theviolentnatureofthisdestructivesamplingconstrainsitsusetosamples�1cmheight,limitingdatacollectioninsomecases.Fig.3ashowstheevolutionoftheaveragecolumnarcross-sectionalareaasafunctionofthedepth.Underconstantpowerconditions,theevaporationratedropssteadily,asitbecomesmorediculttodrivewaterfromtheinteriorofthesample.Therateofchangeofevaporationsuggeststhatwatertransportispartiallydrivenbyawickingprocess,asisseeninthedryingofothersuspensions[16],andnot,asMuller[6]assumed,entirelybydi usion.Infact,bothdi usiveheattransportandconvectioninthecracksareactiveinbasalt[24].Inourstarch,watertransportisbywickingthroughthebulkofthematerial,ratherthanviathecracks.Theseconditionsresultinaslowlydeceleratingdryingfront,andapower-lawcoarseningofthecolumnararea.Thislimitedpowerlawbreaksdowninseveralsuggestiveways.Thefracturescaleislimitedbya nesurfacecrackpattern,whichiswell tbyaddingasmallconstanttothepowerlaw.Thissuggeststhatthematurepatternscaleisindependentofthesurfacefracturescale.Inconstantrateexperiments,afteraninitialtransient,thedryingfrontmovesatconstantspeed,andthecoarseningofthepatternisalmostentirelyhalted.Thetransientcoarseningisindistinguishablefromthatobservedusingconstantexternaldryingconditions.Indeep(�5cm)samples,suddentransitionsinscalewereobservedtooccur(seeFig.3b).Thesetransitionsaresharp,quitereproducible,andseparatecolonnadesofverydi erent 4EUROPHYSICSLETTERSFig.3{Thecoarseningbehaviourofcornstarchwasstudiedinseveraldryingsituations.(a)Gener-ally,forsampleswithconstantpower(solidsymbols),acrudepowerlawcoarseningofcross-sectionalareawithdepthwasseen,withexponentsof1.6to2.2(thelineshownhasaslopeof1.6).Asex-pected[13],fasterdryingratesproducesmallercolumns.Thecolumnsshownwithinvertedtrianglesweregrownwithaninitialdryingrateof10mg=hcm2,squaresrepresentexperimentswithaninitialdryingrateof32mg=hcm2.Forsamplesdriedwithconstantrateconditions(opensymbols),thecoarseningishalted.Incasesofdeepersamples(squares),asuddentransitioninscalewasobserved,withoutanaccompanyingdiscontinuityinthedryingrate.(b)Thistransition(betweendashedlines)isasharpdiscontinuityinscale,andisassociatedwithincreasedpatterndisorder.(c)showstherelativedisorder,asmeasuredbythestandarddeviationoftheareadistribution(A),dividedbythemeanoftheareadistribution(A).Fig.4{Patterncoarseningprocesses.(a)sketchestheessentialfeaturesofcoarseningthroughfracturetermination,themotilityofthecrackedges,andcolumncreationfromexistingvertices.Theuppersurfaceiscoveredwitha nenetworkoffractures,whichwehaveomittedforsimplicity.Thegreyregiondemonstrateswhereaburstofcrackterminationcausesasuddenchangeinpatternscale.(b1-9)showthepatternfeaturesinrealspace,andareselectedfromatomogramvolume- llingimage.Eachpanelshowsa4.5x3.5mm2cross-sectionfromapproximatelythecentreofthevolume,atdepthsincreasingby360mperpanel,beginningatadepthof10.0mm.Theblackcirclesarebubbles,andthefaintstripesaretomographicartifacts.Notehowmobilethecolumnedgescanbe.Columnmergeroccursbetweentwopairsofcolumnsnearthecentreofpanel1,andbetweenpanels3and4,inthelowerrightcorner.Eitherasinglecrack,oravertexcanceasetopropagate,andthemergedcolumnrapidlyreadjustsitsshapeandcross-sectionalareatomatchitssurroundings.Betweenpanels3and4,anewcolumnappearsfromavertexnearthecentreoftheimage. LucasGoehringandStephenW.Morris:Orderanddisorderincolumnarjoints5scales.Withinthetransitionregion,thefracturepatternshowsincreaseddisorder,asshowninFig.3c.Theinspectionoftomogramsshowsthatmergersoftwoorthreecolumns,throughtheterminationofacommonfractureorjunction,aretheonlyeventsthatcanleadtopatterncoarsening.Fig.4showsseveralmechanismsofcolumnevolution.Newcolumnswereocca-sionallycreatedatexistingvertices,butnocolumnswereseentovanishbyconstrictingintoavertex.Theseobservationsimplythatthetransitionregionsarezonesofgreatlyenhancedmergerrate.Thisnaturallyincreasesthevarianceoftheaveragecolumnareaduringthetransition.Patterndisorderisincreasedduringasuddenscaletransition,butsomeorderingprocessactsecientlytoreturnthevalueoftherelativevariationinareatoitspre-transitionvalueof0.35withinonedatapoint(2mm)oftheendofsuchatransition.Similarorderingoccursnearthedryingsurface.Weinvestigatedthisorderingbehaviourina28mmdeep,continuallycoarsening,starchcolonnade,tostudyhowadisorderedsurfacecracknetworkmatured.TheplotsinFig.5showthedepthdependenceoffourstatisticsweusedtoquantifydisorder.Weobservetheevolutionofaninitiallydisorderedsurfacefracturepatternintoawell-orderedcolonnade.Toquantifythisorderingwehavestudiedthedistributionsofareas,jointangles,andthenumberofneighboursasafunctionofdepth,measuredfromthedryingsurfaceaftercompletedesiccation,incross-sectionsofthetomogram.Notethatthenumberofneighbours,unlikethenumberofsides,whichistraditionallyreported,mustaverage6forapolygonaltilingoftheplanethatavoidsX-junctions[25].Allfourstarchstatisticsreachastatisticallysteadystateafter1cmofevolution,main-taining xed,butlargevaluesthereafter.Thisindicatesthatconsiderabledisorderremainsinthepattern.Furthermore,inallfourcases,theplateauvaluesofthestarchstatisticsmatchthevaluesfortheGiant'sCauseway[2],amature,well-orderedbasaltpattern[13].Inthisexperiment,itcouldbearguedthattheresidualdisorderisdictatedbythecontinuedcoarsen-ing.However,weobservestatisticallysimilardescriptionsofthelimitingpatterninboththeconstantrateexperiments,forwhichcoarseningise ectivelyhalted,andtheexposedfeaturesofthenon-coarseningbasaltoftheGiant'sCauseway.Thesimilarityinstatisticsfromtwoverydi erentsystems,desiccatedstarchandcooledbasalt,combinedwithevidenceofastrongorderingprocessesawayfromthecommonlimitingpattern,suggeststhatresidualdisorderisintrinsictothequasihexagonalpatternofcolumnarjointing.Thisiscontrarytothefrequentlyencounteredassumptionthatcolumnarjointingtendstowardsaperfecthexagonallatticeincross-section[13,21].However,suchanassumptionisbasedontheargumentthathexagonalfracturemaximizestheelasticenergyrelease[21].Instronglynon-equilibriumsituations,suchasthisone,energyargumentsarenotnecessarilyvalid.Ourresultscanbeusedtoexplainapuzzlingfeatureofbasalticcolumnarjoints.Asinglebasalt\rowcancontainseveraladjacentcolonnades,withdi erentscales,separatedbydisorderedzones(entablature)oftenlessthan1mwide[26,27].Somecolonnade-colonnadeboundariesoccurwhencoolingfrontstravellingfromthetopandbottomofa\rowcollide.However,othertransitionsremainunexplained,exceptthroughproposedcatastrophiceventssuchasintermittent\rooding[27,28].Wesuggestthatentablatureandscalechangesinbasaltcouldoccurevenwithoutsuddenchangesintheexternalconditions.Instarch,weobservedtransitionsinscaleusingconstantpowerexternalconditions,inwhichthedryingfrontisslowly,continuously,deceleratingwithdepth.Starchcolumnsareapproximately100timessmallerindiameterthanbasalticcolumns.Directlyscalingthe1cmwidescaletransitionsinstarch,we nda1mscalethatcorrespondsreasonablywellwiththewidthoftheentablaturesobservedinbasalt.Suchtransitionscouldbetheresultofdynamicalinstabilitiesofthefracturepattern,whichoccurwhenastablerangeofpatternscalesisexceeded.Ananalogousinstabilityoccursindirectionallydriedthin lms,inwhichaperiodicarrayofcrackscan 6EUROPHYSICSLETTERS0.30.40.5sA/mA0.60.811.2sN0510152025depth (mm)506070% 105o-135o510152025depth (mm)101520sq(a)(b)(c)(d)Fig.5{Statisticalcomparisonofatomographystudyofa28mmdeepcornstarchsampledriedwithconstantpower(datapoints,correspondingwithdatashowninuprighttrianglesinFig.3a),andO'Reilly's1879surveyoftheGiant'sCauseway[2](dashedlines).(a)Therelativevariationincolumncross-sectionalarea.(b)Standarddeviationofthedistributionofnumberofneighbours(N).(c)%ofY-joints,herede nedasthosewithin15ofa120joint.(d)Standarddeviationinthedistributionofjointangles(),indegrees.continuetopropagateunderarangeofdryingconditions[29].Outsidethisrange,thearraybecomesunstable,andmakesatransitiontoanewspacing[22,29].Ourdynamicalinstability-drivenentablaturehypothesisistestablebyobservingthestriaewidthsinbasalticcolonnades.Striaeare'chisel'marksonthesidesofcolumns,whichrecordindividualfractureadvancesasthecoolingfrontadvances[5].Byobservingthevariationofthestriaewidthacrossacolonnade-colonnadescaletransitionitshouldbepossibletodeterminehowthecoolingconditionschanged.Acontinuousevolutionacrosssuchaboundary,ratherthanamatchingdiscontinuityinstriaescale,wouldcon rmtheabsenceofcatastrophicchangesintheexternalconditions.Ourexperimentshavefocussedonexploringthe3Dstructureofcolumnarjointingincornstarch.Wehavedirectlyobservedtheoperationofastrongorderingprocessthatresultsintheeventualsaturationofthepatternstatistics.Thismaturestateofthepatterncontainsconsid-erabledisorder,whichwesuggestisduetothestronglynonequilibrium,complexdynamicsofthejoints.Analternateexplanationofresidualdisorderisthatthesystem'sgeometryevolvestowardequilibriumbutgetsstuckinalocalminimumofthefreeenergy[23].Thisscenarioisdiculttocompletelyruleout,butseemsunlikelygiventheobservedpersistentmobilityofthejoints,whichdonot\ructuateaboutequilibriumpositions.SeveralnewdynamicalmodelsofcolumnarjointinghaverecentlybeenproposedbyJagla[12,22],andJaglaandRojo[23],someofwhichseemtocapture,qualitatively,thedisordereddynamicswe nd.Wehaveobservedcoarseningofthestarchcolonnadeasitpenetratesoursamples.Thiscoarseningproceedsthroughtheterminationofcracktipsalongcolumnsidesorjoints.Thecoarseningcanbehaltedinconstantrateexperiments,revealinganimportantrelationshipbetweenfractureadvancerateandpatternscale.Theobservationofdiscontinuoustransitionsinscale,however,implyitisnotaone-to-onerelationship.Rather,asoccurswithsimilar2Dpatterns[29],theexactpatternscalewilldependonthesystem'sdynamicalhistory,aswell LucasGoehringandStephenW.Morris:Orderanddisorderincolumnarjoints7asonthecurrentexternallyimposedconditions.Thisalonemaysigni cantlymodify eldinterpretationsofcolumnarjointing,aswehaveoutlinedabove.However,theappreciationofquasihexagonalfractureasanon-equilibriumdynamicalsystemcangomuchfurther,andaquantitativeunderstandingofthispatterncouldprovidevolcanology,cryophysics,planetaryphysics,andpatternphysicswithnoveldiagnostictools.Theseconsiderationsmotivateacontinuinginterestinthisbeautifulphenomenon.WethankMarkHenkelmanandtheMouseImagingCentreforsupplyingaccesstomicro-tomographyequipment,andZhenquanLin,A.MarkJellinek,Pierre-YvesRobin,E.AlbertoJagla,andR.PaulYoungforhelpfuldiscussions.REFERENCESSirR.B.S.R.S.,Phil.Trans.R.Soc.London,17(1693)708.[2]O'Reilly,J.P.,Trans.R.IrishAcad.,26(1879)641.[3]Huxley,T.H.,Physiography:AnIntroductiontotheStudyofNature,(MacMillanandCo.,London)1881,p.204.[4]French,J.W.,Trans.Geol.Soc.Glasgow.,17(1922)50.[5]DeGraff,J.M.,andAydin,Geol.Soc.Am.,99(1987)605.[6]Muller,G.,J.Geophys.Res.,103(1998)15239.[7]Muller,G.,J.Volcanol.Geotherm.Res.,86(1998)93.[8]Muller,G.,J.Struct.Geol.,23(2001)45.[9]Toramaru,A.,andMatsumoto,T.,J.Geophys.Res.,109(2004)B02205.[10]Seshadri,K.V.,Jour.Geol.Soc.India,49(1997)452.[11]Menger,F.M,Zhang,H.,Caran,K.L.,Seredyuk,V.A.,andApkarian,R.P.,J.Am.Chem.Soc.,124(2002)1140.[12]Jagla,E.A.,Phys.Rev.E,69(2004)056212.[13]Budkewitsch,P.andRobin,P.-Y.,J.Volcanol.Geotherm.Res.,59(1993)219.[14]Peck,D.L.andMinakami,T.,Geol.Soc.Am.Bull.,79(1968)1151.[15]Aydin,A.andDeGraff,J.M.,Science,239(1988)471.[16]Dufresne,E.R.,Corwin,E.I.,Greenblatt,N.A.,Ashmore,J.,Wang,D.Y.,Dins-more,A.D.,Cheng,J.X.,Xie,X.S.,Hutchinson,J.W.andWeitz,D.A.,Phys.Rev.Lett.,91(2003)224501.[17]Jones,E.M.,(Apollo15LunarSurfaceJournal;HadleyRille.http://www.hq.nasa.gov/oce/pao/History/alsj/a15/a15.rille.html(1996)).[18]Lachenbruch,A.H.,Geol.Soc.Am.Spec.Paper.,70(1962)69p.[19]Sletten,R.S.,Hallet,B.,andFletcher,R.C.,J.Geophys.Res.,108(2003)8044.[20]Mellon,M.T.,J.Geophys.Res.,102(1997)25617.[21]Mallet,R.,Philos.Mag.,50(1875)122.[22]Jagla,E.A.,Phys.Rev.E.,65(2002)046147.[23]Jagla,E.A.,andRojo,A.G.,Phys.Rev.E.,65(2002)026203.[24]Hardee,H.C.,J.Volcanol.Geotherm.Res.,7(1980)211.[25]Gray,N.H.,Anderson,J.B.,Devine,J.D.,andKwasnik,J.M.,Math.Geol.,8(1976)617.[26]Grossenbacher,K.A.andMcDuffie,S.M.,J.Volcanol.Geotherm.Res.,69(1995)95.[27]Long,P.E.andWood,B.J.,Geol.Soc.Am.Bull.,97(1986)1144.[28]DeGraff,J.M.,Long,P.E.,andAydin,A.,J.Volcanol.Geotherm.Res.,38(1989)309.[29]Shorlin,K.A.,deBruyn,J.R.,Graham,M.andMorris,S.W.,Phys.Rev.E,61(2000)6950.