Fractures in complex fluids: the case of transient networks

2.9 MB·English

by  Christian Ligoure

RheologicaActamanuscriptNo.
(willbeinsertedbytheeditor)
Fractures in Complex Fluids: the Case of Transient
Networks
ChristianLIGOURE SergeMORA
·
Received:date/Accepted:date
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Abstract We present a comprehensive review of the current state of fracture phe-
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nomenaintransientnetworks,awideclassofviscoelasticfluids.Wewillfirstdefine
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what is a fracture in a complex fluid, and recall the main structural and rheological
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J propertiesoftransientnetworks.Secondly,wereviewexperimentalreportsonfrac-
5 tures of transient networks in several configurations: shear-induced fractures, frac-
1 turesinHele-Shawcellsandfractureinextensionalgeometries(filamentstretching
rheometryandpendantdropexperiments),includingfracturepropagation.Thetenta-

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tiveextensionoftheconceptsofbrittlenessandductilitytothefracturemechanisms
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in transient networks is also discussed. Finally, the different and apparently contra-
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s dictorytheoreticalapproachesdevelopedtointerpretfracturenucleationwillbead-
t. dressedandconfrontedtoexperimentalresults.Rationalizedcriteriatodiscriminate
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therelevanceofthesedifferentmodelswillbeproposed.
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- 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 · · · ·
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C.Ligoure
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LaboratoireCharlesCoulomb-UMR5221
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Universite´Montpellier2andCNRS
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PlaceE.Bataillon.F-34095MontpellierCedex,FranceE-mail:[e-mail]
:
v S.Mora
i LaboratoireCharlesCoulomb-UMR5221
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Universite´Montpellier2andCNRS
r PlaceE.Bataillon.F-34095MontpellierCedex,FranceE-mail:[e-mail]
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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 tran-
sientnetworks: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 ob-
served 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