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RheologicaActamanuscriptNo. (willbeinsertedbytheeditor) Fractures in Complex Fluids: the Case of Transient Networks ChristianLIGOURE SergeMORA · Received:date/Accepted:date 3 1 0 Abstract We present a comprehensive review of the current state of fracture phe- 2 nomenaintransientnetworks,awideclassofviscoelasticfluids.Wewillfirstdefine n what is a fracture in a complex fluid, and recall the main structural and rheological a J propertiesoftransientnetworks.Secondly,wereviewexperimentalreportsonfrac- 5 tures of transient networks in several configurations: shear-induced fractures, frac- 1 turesinHele-Shawcellsandfractureinextensionalgeometries(filamentstretching rheometryandpendantdropexperiments),includingfracturepropagation.Thetenta- ] tiveextensionoftheconceptsofbrittlenessandductilitytothefracturemechanisms t f in transient networks is also discussed. Finally, the different and apparently contra- o s dictorytheoreticalapproachesdevelopedtointerpretfracturenucleationwillbead- t. dressedandconfrontedtoexperimentalresults.Rationalizedcriteriatodiscriminate a therelevanceofthesedifferentmodelswillbeproposed. m - Keywords Fracture transientnetworks gels Non-linearviscoelasticity complex d fluids brittleness d·uctility fracturenuc·leatio·n fracturepropagation; · n · · · · o PACS 83.80.Kn PACS83.85.Cg 83.60.Df 62.20.Mk 62.20.Fe c · · · · [ 1 v 0 1 4 3 . 1 C.Ligoure 0 LaboratoireCharlesCoulomb-UMR5221 3 Universite´Montpellier2andCNRS 1 PlaceE.Bataillon.F-34095MontpellierCedex,FranceE-mail:[email protected] : v S.Mora i LaboratoireCharlesCoulomb-UMR5221 X Universite´Montpellier2andCNRS r PlaceE.Bataillon.F-34095MontpellierCedex,FranceE-mail:[email protected] a 2 ChristianLIGOURE,SergeMORA Listofsymbols G shearmodulus 0 τ relaxationtime η viscosity γ shearstrain γ˙ shearrate ε˙ elongationalstrainrate σ stress σ shearstress xy N orσ Firstnormalstressesdifference 1 N W Griffithenergycost L cracksize L Griffithlength c F interfacialcohesiveenergyperunitarea s t delay(waiting)fracturetime b t averagedelay(waiting)fracturetimepredictedbytheThermallyactivatedcracknucleationmodel 1 (cid:104) (cid:105) t averagedelay(waiting)fracturetimepredictedbytheActivatedbondrupturemodel 2 (cid:104) (cid:105) t averagedelay(waiting)fracturetimepredictedbytheSelfhealingandactivatedbondrupturenucleationmodel 3 (cid:104) (cid:105) σ failurestress c σ characteristicstressinvolvedintheThermallyactivatedcracknucleationmodel 1 σ characteristicstressinvolvedintheActivatedbondrupturemodel 2 σ characteristicstressinvolvedintheSelfhealingandactivatedbondrupturenucleationmodel 3 G strainenergyrelease V tipvelocity k Boltzmannconstant B T temperature ξ typicaldistancebetweenjunctionsinanetwork ρ density Q injectionrate δ lengthofasolvophobicend De Deborahnumber Γ Interfacialorsurfacetension 1 Introduction Theabilityofviscoelasticfluidstofracturehasbeenrecognizedinthesixties[Hut- ton(1963)]butremainsmuchlessdocumentedthatthebreakdownofsolidmaterials. The starting point for discussing fracture in viscoelastic fluids is coming up with a rigorousdefinitionoffracture.Thelayman’sdefinitionoffailureproposedforsolid materialsbyBuehler[Buehler(2010)]-failureoccurswhentheloadbearingcapac- ityofthematerialunderconsiderationissignificantlyreducedorcompletelylostdue toasuddendevelopment-isnotapplicableforfluids,whichaccommodatearbitrar- ilylargedeformationsafterafinitetime.AtamorefundamentallevelBuehlerargues thatfractureofamaterialduetomechanicaldeformationcanbeunderstoodascon- FracturesinComplexFluids:theCaseofTransientNetworks 3 versionofelasticenergyintobreakingofchemicalbondsorheat.Thisdefinitionis moreappropriateforfluidsbutdoesnotallowtodiscriminateclearlythefragmenta- tionofliquidsduetosomehydrodynamicinstability[EggersandVillermaux(2008)] andthefractureoffluidsreminiscentofthefractureofsolids.Forinstance,liquidjets serveasaparadigmforhydrodynamicinstabilityleadingtodropbreakupthroughthe Rayleigh-Plateaucapillaryinstability:inthiscase,intermolecular(elastic)energyis convertedatthesurfacevicinityalone.Atentativedefinitionforthefractureoffluid couldbethefollowing:”thefractureofafluidduetomechanicaldeformation,includ- ing flow, can be understood as a dissipation of bulk elastic energy into breaking of physicalorchemicalbonds.”Thisdefinitionexcludesthehydrodynamicinstabilities observedalsoininviscidorNewtonianfluidsofthefieldoffluid’sfracturebecause elasticenergythatcomeinplayisofinterfacialnaturesolely. Complex fluids denotes a (too) huge class of condensed-phase materials that posses mechanical properties intermediate between ordinary liquids and ordinary solids[Larson(1999)].Manyofthemare”solids”atshorttimeand”liquid”atlong time,hencetheyareviscoelastic;thecharacteristictimerequiredofthemtochange from”solid”to”liquid”variesfromfractionsofasecondtohours.Numerousexten- sivestudiesontheruptureofentangledpolymermeltsinextensionhavebeencarried outinthepastfortyyears.Thisallowstodistinguishtwozonesofeitherviscoelas- ticruptureorelasticfractureintheMalkin-Petriemastercurveintendedtoillustrate thedifferentextensionalresponseswithincreasingstrainratesofentangledpolymer melts[MalkinandPetrie(1997)].Thesetwotypesofrupturehavebeenrecentlyre- considered by Wang [Wang and Wang (2010)]: a yield-to-rupture failure transition is observed: the yield failure (called also ductile failure by other authors [Ide and White(1976,1977,1978)]shouldbeduetotheyieldingofentanglements,whereas at higherextensional strainrates, thespecimen breaksup in apurely elasticregime (withoutanyflow). Amongvisoelasticfluids,self-assembledtransientnetworks,thatconsistsofre- versibly cross-linked polymers in solutions constitute model systems for the physi- cist, with well defined structural properties and simple linear rheological behavior. The aim of this review is to embrace the current aspects of the phenomenology of transientnetwork’sfracturesinacompressivemanner.Wefirstpresentwhataretran- sient networks: systems and rheology. Second, we report on fracture’s experiments and computer simulations of transient networks in several configurations: shear ge- ometry,Hele-Shawcellsandextensionalgeometry.Thisalsoincludes,fractureprop- agationandtheextensionoftheconceptsofbrittlenessandductilityforviscoelastic fluids. Then, we describe the several theoretical approaches developed to describe cracknucleationintransientnetworks,andhowtheycomparetoexperiments,before toconclude. 2 Transientnetworks:systemsandrheology Self-assembledtransient(oftencalledphysical)networksareaclassofcomplexma- terials forming spontaneously 3D networks at thermodynamical equilibrium, that can transiently transmit elastic stresses over macroscopic distances. Transient self- 4 ChristianLIGOURE,SergeMORA assemblednetworksarecommoninbothnaturalandsyntheticmaterials.Theycon- sistofreversiblycross-linkedpolymersinwhichweakinteractionssuchashydrogen bonds, hydrophobic interactions, van der Waals forces, or electrostatic interactions are responsible for cross-links formation. Because of the transient character of the junctions, and so of the thermodynamics equilibrium state of these systems at rest, theydon’texhibitanyagingnoryieldstresscontrarytootherclassesofsoftout-of- thermodynamicequilibriumviscoelasticmaterialslikedenseparticulatesuspensions (colloidalglasses),orthermoreversiblegels[Larson(1999)].Oneofthemajorissue oftransientpolymernetworksistoconveyusefulrheologicalpropertiestosolutions, such as increased viscosity, gelation, shear-thinning or shear-thickening. They can be used as controlled drug delivery systems [Sutter et al (2007)], rheological reg- ulators in polymer blends [Kim et al (2004)], coatings, food, and cosmetics, or as matrix materials for tissue engineering [Kim and Mooney (1998)]. Self-assembled transient networks consist mostly of binary solutions of associative polymers, or ternary solutions of associative polymers and self-assembled surfactant aggregates even if supramolecular reversible networks formed by mixtures of small molecules associatingbydirectionalinteractionshasbeenalsoreported[Cordieretal(2008)]. Associating polymers are macromolecules with a part that is soluble in a selective solvent (often water), the so-called backbone or spacer to which two or more moi- etiesthatdonotdissolveinthissolvent,thestickers,areattached.Thestickersmay berandomlydistributedalongthebackboneormaybegroupedinblocks.theassocia- tionofsuchpolymersinsolutionhasbeenstudiedextensively,andmanyreviewscan befoundintheliterature,seeforexample[Larson(1999);WinnikandYekta(1997); Berretetal(1997);MengandRussel(2006);Chassenieuxetal(2011)].Notethatthe restrictive definition of transient networks that we propose, excludes simple entan- glednormalorlivingpolymersolutionsandmelts,whereentanglementsofpolymer chainsplaytheroleoftransientcross-linksandexhibitelasticyielding[Malkinand Petrie (1997); Boukany et al (2009)]. Telechelic polymers are often used as model linkers because they are architecturally simple: they consist of a long solvophilic mid-block with each end terminated by a solvophobic short chain (a sticker) [Se- menov et al (1995)]. The stickers incorporate into the solvophobic domains of the aggregates and can bridge them via their solvent-soluble mid-block resulting in an attractive interaction between the aggregates. . The nature and the morphologies of theaggregatesformingthenetwork’sjunctionsareversatile:(i)telechelicpolymers in binary solution that self-assemble spontaneously into non-interacting flower-like micelles at low concentration and form three dimensional networks above a perco- lationconcentration[Annableetal(1993);Sereroetal(2000);Wertenetal(2009); Seitzetal(2006)],(ii)surfactantvesicles[Leeetal(2005)],(iii)lyotropiclamellar phases [Warriner et al (1997)] (iv) wormlike micelles [Ramos and Ligoure (2007); Lodgeetal(2007);Nakaya-Yaegashietal(2008);Tixieretal(2010)],(v)spherical micelles[Appelletal(1998);Tixieretal(2010)],(vi)oil-in-water[BaggerJorgensen etal(1997);Filalietal(1999)]or(vii)water-in-oilmicroemulsiondroplets[Oden- waldetal(1995)],(viii)photocrosslinkablenano-emulsions[Helgesonetal(2012)]. Therheologyoftransientnetworksisdeterminedbytheamountandthelifetime of the bridges. In the simplest case when the bridging chains are flexible and un- tangled, each bridging chain contributes about 1 k T per unit volume to the elas- B FracturesinComplexFluids:theCaseofTransientNetworks 5 tic modulus G according to the theory of rubber elasticity [Green and Tobolsky 0 (1946); Tanaka and Edwards (1992e); Yamamoto (1956)]. The typical value of the shear modulus for these networks ranges between few Pa to several ten thousands Pa.Theelasticresponserelaxeswhenthesolvophobicblocksescapefromthecore. Oftentheescapeischaracterizedbyasinglerelaxationtimesothattheterminalre- laxation of the viscoelastic properties is characterized by a single Maxwell process wellseparatedfromthefasterinternalmodesthatcharacterizetheconformationalre- laxationofthechains[Annableetal(1993);Sereroetal(2000);Micheletal(2000); Filali et al (2001); Tabuteau et al (2009a); Hough and Ou-Yang (2006)]. Far above thepercolationconcentration,theviscoelasticrelaxationtimeτ isrelatedtotheav- erage lifetime of a connection [Green and Tobolsky (1946)] which in turn depends on the breakage probability of a cross-link. The average lifetime of bridge will de- pend on its chemical nature, the external conditions and the physical state of the cross-links. It can vary through few ms [Tixier et al (2010)], few seconds [Michel et al (2000); Filali et al (2001)], up to hours [Serero et al (2000); Skrzeszewska et al (2010); Seitz et al (2006)]. The simplicity of the linear viscoelastic behavior ofmostofself-assembledtransientnetworksisincontrastwiththeirhighlycomplex non-linearresponsethatcanvarywidely[Pellensetal(2004b)].Amongallsystems, the experimental model transient network consisting of oil-in-water microemulsion droplets reversibly linked by telechelic polymers [Filali et al (2001); Michel et al (2000)] is perhaps the unique one which exhibits the steady shear flow curve of a pureMaxwellfluidforboththeshearstressandthefirstnormalstressdifferenceun- tilitbreaks[Tabuteauetal(2009a)].Inmostothersystems,thesteadyshearviscosity exhibitsthreeflowregimes.AboveaNewtonianplateau(linearregime),theviscosity increasesconsiderably(shearthickening)priortotheonsetofshearthinningathigher shearrates[Annableetal(1993);Xuetal(1996);Otsubo(1999);BerretandSerero (2001);Pellensetal(2004a);Tripathietal(2006)].However,theshearthickeningre- gionisnotalwayspresent[Micheletal(2000);Tixieretal(2010);Tirtaatmajdaand Jenkins(1997);Mewisetal(2001);Skrzeszewskaetal(2010)].Alargenumberof constitutivetheoreticalmodelsarebasedonthetemporary-networkkineticmodelfor telechelic polymers network theory [Tanaka and Edwards (1992e)] by applying the original ideas formulated by Green and Tobolsky [Green and Tobolsky (1946)] and Yamamoto [Yamamoto (1956)] and have been developed to tentatively capture the mainfeaturesofnon-linearrheologicalpropertiesofthesenetworks[Marruccietal (1993);AhnandOsaki(1995);vandenBruleandHoogerbrugge(2000);Vaccaroand Marrucci(2000);Tripathietal(2006);Hernandez-Cifreetal(2003)]. 3 Fractureexperiments Several geometries have been considered to explore fracture mechanisms in transientnetworks:elongationalflowsusingeitherextensionalrheometer[Tripathietal (2006)], pendant drop experiments [Tabuteau et al (2009a, 2011)], shear flows in rheometriccells[BerretandSerero(2001);Tabuteauetal(2009a);Tixieretal(2010); Skrzeszewska et al (2010)] or flows in Hele-Shaw cells [Zhao and Maher (1993); Igne´s-Mulloletal(1995);Vladetal(1999);MoraandManna(2010)]. 6 ChristianLIGOURE,SergeMORA 1200 1 mm 0 ms 900 ) a P ( 600 crack s 22 ms s e r z t s 300 y σ 47 ms yz σ N 0 0 0.3 0.6 0.9 1.2 shear rate γ˙ (s−1) 310 ms Fig. 1 From [Tabuteau et al (2009a)]: (Left) shear stress (σyz) and first normal stresses difference (σN=σzz σyy)versusshearrateforabridgedmicroemulsionwithashearmodulusG0=1210Paand − arelaxationτ=0.8s.Thecontinuouslinesarefits(Maxwellmodel).(Right)Developmentofafracture, occurringatthesurfaceofthesample,forashearrateequalto0.9s−1,correspondingtoacriticalfirst normalstressesdifferenceσN=1190Pa. 3.1 Shear-inducedfractures 3.1.1 Stationaryshearrate Whensubmittedtoaconstantshearrateγ˙,themeasuredshearstressσ of”brittle” xy transientnetworksfirstincreasessmoothlywiththeshearrate[Molinoetal(2000); BerretandSerero(2001);Tabuteauetal(2009a);Skrzeszewskaetal(2010);Tixier etal(2010)].However,aboveacriticalshearrateγ˙ τ 1,whereτ istherelaxation − ∼ time of the network, the flow curve exhibits a sharp discontinuity (Figure 1) and theviscositydecreasesabruptly.Belowthisvalue,thebranchoftheflowcurvecan be Newtonian and shear thickening [Berret and Serero (2001)], or only Newtonian [Molino et al (2000); Tabuteau et al (2009a); Skrzeszewska et al (2010)], or New- tonian and shear thinning [Tixier et al (2010)] depending on the experimental sys- tem under consideration. Note that for ”brittle ”transient networks, the first normal stresses difference N =σ σ drops suddenly for the same critical shear stress 1 xx yy − [Tabuteau et al (2009a)]. Authors of [Molino et al (2000)] were the first to suggest thatthesuddendropofthestressintheflowcurveofatransientnetworkisthesign ofafracturepropagation. However,thefirstunambiguousdemonstrationofshear-inducedfracturesintran- sientnetworkswasdonein[BerretandSerero(2001)],byusingaflowvisualization technique in a plate-plate transparent shearing cell. At low shear rate, the velocity FracturesinComplexFluids:theCaseofTransientNetworks 7 profileishomogeneous:thevelocitydecreaseslinearlyfromtherotatingwalltothe stationary one. Above the critical shear rate, the stationary velocity field within the gap of the cell exhibits a discontinuity, that defines a zone of fracture as the part of the fluid submitted to a high shear rate ( 10 times the applied rate). This has ∼ been confirmed by Skrzeszewska et al [Skrzeszewska et al (2010)] by Particle Im- age Velocimetry: the fracture zone has an irregular shape and is rather wide on the order of a few hundred micrometers. With increasing overall shear rate, the width of the fracture zone increases. The fracture zone can happen everywhere in the gap andisdifferentforeveryexperiment.Fornon-adhesivegels,thefracturezoneoccurs generally at one of the wall [Berret and Serero (2001)], and so appears as sliding, thatisobservedinotherclassesofcomplexfluidslikepastesorconcentratedemul- sions[Meekeretal(2004)],whereasforadhesivegels[Skrzeszewskaetal(2010)], orroughwallsurfaces[Tixieretal(2010)],thefractureoccursinthebulk.Adirect opticalobservationoftheshearfractures[Tabuteauetal(2009a)]showsthatabove thecriticalstress,cracksopenupallaroundthesampleandgrowrapidly.Itisworth notingthatthefracturesaretilted45 fromtheshearplane,perpendiculartothedi- ◦ rection of the maximum extension (Figure 1). Interestingly, for two very different systems[Tabuteauetal(2009a);Skrzeszewskaetal(2010)]thathaveperfectlyNew- tonianflowcurve,beforefractureoccurs,thefractureshearstressinthesteadystate flowcurvescalesasσ G ,whereG istheshearmodulusofthenetwork.Incon- xy 0 0 ∼ trasttosolids,herethefractureshealoverrapidlyaftertheshearrateisswitchedoff, andanewexperimentcanbeperformedafterafewminuteswithquantitativelythe samebehavior[Tabuteauetal(2009a);Skrzeszewskaetal(2010)]. Abovethecriticalfractureshearrate,strongfluctuationsoftheshearstresshave beenobserved[Sprakeletal(2009a);Tixieretal(2010);Ramosetal(2011)],thatcan leadtoanapparentshearplateau[Sprakeletal(2009a)]reminiscentofshear-banded flows observed in a wide variety of soft materials such as solutions of entangled wormlikemicelles,colloidalsuspensionsandentangledpolymersolutions[Fielding (2007);Olmsted(2008)].Suchstrongfluctuationsoftheshearstresshavealsobeen observed in solutions of entangled wormlike micelles at high shear rate and have beenassociatedalsowitharupture-likebehaviorasevidencedbyParticleTracking Velocimetry [Boukany and Wang (2007)]. By using a novel class of transient net- worksmadeofsurfactantmicellesoftunablemorphology[Tixieretal(2010)](from spheres to rods to flexible worms) linked by telechelic polymers, and coupling rhe- ologyandtime-resolvedstructuralmeasurements,RamosandLigoure[Ramosetal (2011)] clearly show that true shear-banding is not associated with strong fluctua- tionsoftheshearstress.Indeed,fluctuationsoftheshearstressareentirelycorrelated to the fluctuations of the degree of alignment of the micelles that can probe a frac- tureprocess.Sprakeletal[Sprakeletal(2009a)]arguethattheintermittentbehavior observed in the stress response is due to repeated microfracture-healing events in thematerial.Thecumulativedistributionofthetotalstressdrops∆σ duringafrac- turedisplaysacharacteristicpower-lawbehavior,P(>∆σ)∝δσ 0.85,withP the − stressprobabilitydistribution.Theexponentisclosetothevalueof0.8reportedfor true stick-slip motion [Feder and Feder (1991)]. Attempts to explain such scaling behaviorinvolvetheconceptofself-organizedcriticality. 8 ChristianLIGOURE,SergeMORA Fig.2 From[Sprakeletal(2009b)]:Transientshearstressresponseinashearstart-uprun.Bottompanel showsthecorrespondingevolutionofthevelocityprofile,inwhicheachlinerepresentsthelocalfluid velocity(vx(y,t))atagivenpositioninthegradientdirection.Snapshotsofthesimulationbox(flowfrom left-rightwithavelocitygradientfromtop-bottom)illustratethehomogeneousinitialconfiguration(left) andthefinalfracturedstate(right). Responsiveparticledynamicssimulations[Sprakeletal(2009b)]havealsoshown shear-induced fractures (Figure 2) in transient polymer networks. Moreover these simulations have revealed a transition from shear banding to fracture upon increas- ingtheoverallpolymerconcentration.andemphasizethedifficultytodefineanun- ambiguous criterion to discriminate between banding and fracture. In the literature, bandingisusuallydefinedassituationswherethevelocityprofileinthegradientdi- rectioniscontinuousbutkinked,whereasitisdiscontinuousattheplaneoffracture. Itisworthmentioningthanthetheissueofvelocitychangewithinthegapisinfact more complicated [Manneville (2008)] in addition to ”classic banding” coexistence of yielded and unyielded regions with finite and zero rates/velocities have been observed in systems like colloidal glasses [Besseling et al (2010)]. Unfortunately the spatialaccuracyistypicallylimitedtothemicrometerrangeandcannotallowtodis- tinguishbetweennarrowhighshearbandsandfractureplanes.Howevertheauthors of[Sprakeletal(2009b)]showthatadiscontinuityinthevariationofthefirstnormal stressesdifferenceN withtheshear-rateisanunambiguoussignatureofafracture, 1 since normal forces should be largely reduced as all connections between the two shearbandsacrossafractureplanearebroken.ThediscontinuityofN atthethresh- 1 old has been observed experimentally for fracturing transient networks [Tabuteau etal(2009a);Tixieretal(2010)],incontrastwithshear-bandingregimes[Tixieretal (2010)],whereN increaseswithγ˙asexpectedtheoretically[Spenleyetal(1993)]. 1 FracturesinComplexFluids:theCaseofTransientNetworks 9 3.1.2 Fractureatconstantappliedstress Anotherapproachtostudythefailureofgelsistoapplyaconstantstressandtofollow theresultingshearrateasafunctionoftime[Skrzeszewskaetal(2010)].Abovethe criticalfracturestressσ G ,fractureoccursimmediatelyasrevealedbyadramatic 0 ∼ increaseofthemeasuredsharerate.Belowthecriticalfractureshearstress,adelayed timeisobservedbeforefracturing,aphenomenonwelldocumentedinsolidmaterials [Zhurkov (1965); Santucci et al (2007)] and observed also in colloidal suspensions [Sprakeletal(2011);Divouxetal(2010)],wormlikemicelles[Olssonetal(2010)] andsoftelastomers[Bonnetal(1998)].Thedelayfracturetimet variessignificantly b fromoneexperimenttoanotheroneindicatingthestochasticnatureofthefracture mechanism.Withdecreasingstress,theaveragedelaytimeincreases<t >rapidly. b Severaltheoreticalinterpretationshavebeenproposed,thatwillbereviewedinSec- tion5).Theypredictdifferentdelayfracturetimes.However,theintrinsicstochastic natureofthefracturemechanismimpliesaverylargeuncertaintyof<t >thatdoes b notallowtoconcludedefinitivelyabouttherelevanttheory. Fig.3 From[Skrzeszewskaetal(2010)]:Delayedfractureofatransientpolymernetworkformedby telechelicpolypeptides,fordifferentappliedstresses.Afteracertainwaitingtimewhichvariesfromone experimenttoanother,thegelbreaks,leadingtoaveryrapidincreaseoftheoverallstrain. 3.2 FracturesinHele-Shawcells Fracture-like flow instabilities that arise when a fluid is injected into a Hele-Shaw cellfilledwithanaqueoussolutionoftelechelicpolymershasbeenreportedbysev- eral authors [Zhao and Maher (1993); Igne´s-Mullol et al (1995); Vlad et al (1999); 10 ChristianLIGOURE,SergeMORA Fig.4 From[ZhaoandMaher(1993)]:Fourtypicalpatternsofa2.5%solutionofhexadecylend-capped polymersofmolecularweight50700atinjectionrates(a)Q=1.0mL/min,(b)Q=1.0mL/min,(c)Q=5.0 mL/min,and(d)Q=20mL/min.Thecrossoverfromtheviscous-fingeringpatterntothefracturingpattern isseenin(a)and(b). MoraandManna(2010)].ZhaoandMaher[ZhaoandMaher(1993)]showedusing aradialHele-Shawcell,thatfortransientnetworksofassociativepolymers,thereex- istsathresholdinjectionrate,belowwhichthepatterninstabilityistypicallyviscous fingering (Saffman-Taylor instability) [Bensimon et al (1986)], and beyond which, thepatternsresemblefracturepatternsobservedinbrittlematerials(Figure4).Note that analogous fracture patterns have been also observed in other classes of com- plexfluidslikeclaysuspensions[Lemaireetal(1991)]orlyotropiclamellarphases [Greffieretal(1998)].Thedifferencebetweenafingeringpatternandafracturepat- tern is drastic and can be quantified by calculating the mass fractal dimension of a givenpattern:itdropsdrasticallyfrom 1.70to 1.0nearthetransitionthreshold. ∼ ∼ Interestingly, the fingering/fracture transition is not observed for the corresponding entangledhomopolymerssolutions,indicatingthatthefingering-fracturingtransition isadirectmanifestationofanassociating-networkeffect.Acharacteristic,Deborah number D =τ /τ (τQ)/(b2L ) can be defined, where τ =1/γ˙ is the e thin f cell thin thin ∼ inverseoftheshearthinningrateofthetransientnetworkandnotitsrelaxationtime τ, Q is the injection rate, b the thickness of the cell and L , some characteristic cell lengthscaleoftheradialcell.Thetransitionfromfingeringtofracture-likebehavior