Foammicrorheology: fr om honeycombsto random foams

Andrew M. KraynikEngineeringSciencesCenter MS 0834

SandiaNationalLaboratoriesAlbuquerque,New Mexico 87185–0834 USA

DouglasA. ReineltDepartmentof Mathematics

SouthernMethodistUniversityDallas,Texas 75275–0156 USA

April 9, 1999

We dedicate this paper to Henry M. Princen for his seminal work in foam rheology.Henry was born in Eindhoven; his Dutch name is Thijs.

Both of his parents were born in ’s Hertogenbosch.

Abstract

Foamincludesabroadrangeof materialsfrom shaving creamto theflexible polyurethanethatcushionsour seats.This overview coverstwo decadesof researchon therheologyof liquid foamfrom a micromechanicalpoint of view. Thesehighly structured,multiphasefluids exhibit richrheologicalresponsethatcanberelatedto geometryandmechanicsat thecell level. Propertiesofinterestincludetheshearmodulus,yield stress,andnon-Newtonianviscosity. Theoriesbasedonthesimpleliquid honeycombin 2D illustratecell-level mechanismsthatarealsoimportantin 3D.Theseincludeenergy storagein expandingsurfacesthatcauseselasticity, irreversibletopologicaltransitionswithin thefoamstructurethatproduceyield phenomena,andtheinterplaybetweencelldistortionandfilm-level viscousflow that is responsiblefor viscoelasticity. Static3D structuresrangingin complexity from the Kelvin cell to the elegant Weaire-Phelanstructureto randompolydispersefoamsarecalculatedwith the SurfaceEvolver, a computerprogramdevelopedbyK.A. Brakke. Excellentagreementwith experimentaldataonfoamstructureandshearmodulusisdemonstrated.Simulationsinvolving largequasistaticdeformationsof Kelvin andWeaire-Phelanfoamsin simple shearingflow are comparedwith 2D results. The geometryand rheologicalconsequencesof Plateaubordersin wet foamsare described.The fluid mechanicsof bubblesgrowing in a viscousfluid revealsthe evolution of foam structurethat controlsbehavior in thesolidstate.Simulatedsoapfroth structureis usedasatemplateto developfinite elementmodelsofcellularsolids.Themicromechanicalapproachthatis describedhasestablishedafirm theoreticalfoundationfor developingstructure-property-processingrelationshipsfor foamedpolymers.

1

1 Intr oduction

The flexible polyurethanethat cushionsour seatsand the extrudedthermoplasticsthat insulateourhomesarefamiliar examplesof foamedpolymersthat drive large andprofitablebusinesses.Theseandmany otherfoams,which maybe liquid or solid, mundaneor esoteric,supporta broadrangeofapplicationsthat motivate the developmentof foam scienceandfoam technology(Hilyard & Cun-ningham,1994;Prud’homme& Khan, 1996;Gibson& Ashby, 1997;Sadoc& Rivier, 1999). Theinherentmultidisciplinarycharacterof this field resonateswith thethemeandphilosophyof PPS-15.

Foamsareprotypicalengineeredmaterials.Their cell-level structureandconstituentsdeterminemacroscopicproperties.While many commercialproductsarecellularsolids,theirstructuretypicallyevolves in the fluid statewheregasbubblesgrow in viscousliquids. Theseheterogeneoussystemsundergo variousdegreesof expansionandshearduring processing.Understandingthe rheologyofthesecomplex fluidsin thesecomplex flowsis necessaryto designprocessingequipmentandestablishproceduresto controlthemicrostructureandperformanceof thefinal solidfoam.Wewill focusonthedevelopmentof microrheologicalmodelsthatprobetheseconnectionsandprovide a firm theoreticalfoundationfor understandingstructure-property-processing relationships.We would like to know,for example,what foamedmaterialandcell structureareoptimal for a givenapplicationandhow tocontrol theevolution of thatstructureduringprocessing.Theschematicin Fig. 1 illustratestheflowof a gas-chargedmelt throughadie duringthermoplasticfoamextrusion(Kraynik, 1981)

In broadoutline, theoreticaldevelopmentsin this field progressfrom the simpleststructurestomorecomplicatedones,as indicatedby the subtitleof this article “...from honeycombsto randomfoams.” Like mostcomplex materials,thestructureof foamsinvolvesmultiple lengthscales;andthefunctionof foamsofteninvolvesmany physicalphenomena.Foamscienceandfoamtechnologydrawchemists,physicists,mathematicians,andengineersfrom many disciplines. Foammicromechanicsspanstraditionalfluid mechanicsandtraditionalsolid mechanics(Kraynik, 1988;Weaire& Fortes,1994;Kraynik et al., 1999).Foamscontinueto presentsignificantscientificandtechnicalchallenge;andthischallengegoeshand-in-handwith substantialopportunity.

Figure1: Schematicof thermoplasticfoamextrusion.

2 Structure of dry soapfr oth

From a geometricpoint of view, one can argue that a dry soapfroth hasthe ‘simplest’ cell-levelstructureof any 3D foam. The liquid volumefraction

�is zeroin thedry limit. Liquid drainageis

absentandgasdiffusionbetweencellscanbeneglectedunderstaticconditions.In ahypotheticaldryfoam, the thin liquid films degenerateto mathematicalsurfacesthatbecomethe facesof polyhedralcellsandthePlateauborderscollapseto form cell edges.Theaveragecell volume � determinesthe

2

Figure2: Kelvin cell, Weaire-Phelanfoam,andrandommonodispersefoamwith 125cells.

only characteristiclengthscale������� . Thefoamstructurecanbemodeledasa continuousnetwork ofsurfaces,bothsidesof which have uniform surfacetension� . Theequilibriumconditionsknown asPlateau’s laws(Plateau,1873;Taylor, 1976)requireeachfaceto haveconstantmeancurvature.Threefacesmeetat equaldihedralanglesof � ��� alongcell edges;andfour edgesjoin at equaltetrahedralanglesof ������� ����� �� ����������! #"%$ � at eachcell vertex.

The SurfaceEvolver, which hasbeendeveloped,maintained,and freely distributed by Brakke(1992), hasbecomethe standardcomputerprogramfor calculatingthe minimal surfacesin a 3Dfoam; andit is capableof solving many otherproblems.The Evolver is availableover the internetfrom http://www.susqu.edu/facstaff/b/brakke/evolver/.

Unlessotherwisestated,all of thestructuresin this articlearespatiallyperiodic.This meansthatspacecanbefilled with a representative volumeof foam(unit cell) thatcontains& cells.Thespecialcase,& equalsone,is reservedfor theKelvin cell (Kelvin, 1887)shown in Fig. 2. TheKelvin cell istheonly structureknown thatsatisfiesPlateau’s laws andformsaperfectlyorderedfoamin whichallcellshave identicalshapeandorientation.

Kelvin believedthathisminimal tetrakaidecahedron(14-hedron)dividedspaceinto equal-volumecellswith theleastpossiblesurfacearea.Weaire& Phelan(1994)usedtheSurfaceEvolverto computetheelegantcounterexamplethatis alsoshown in Fig. 2. TheWeaire-Phelan(WP) foamhas0.3380%lesssurfaceareathantheKelvin foam.It containseightdifferentmonodispersecells: two pentagonaldodecahedraandsix 14-hedrathathave twelve pentagonalfacesandtwo hexagonalfaces.TheWPfoam belongsto a classof two dozenor so structuresknown as tetrahedrallyclose-packed (TCP)to crystallographersandFrank-Kasperto metallurgistsandmaterialscientists(Rivier, 1994). All ofthe TCP foamscontaintwo or morepolyhedraof four types. Eachof theseunique ' -hedra( '���!(��")(��*!(��+ ) containstwelve pentagonsand ' � � hexagonsthatdo not occupy adjacentfaceson aparticularpolyhedron.

The Kelvin cell andTCP foamsareusefulmodelsof 3D structurebut they do not possessthetopologicaldisorderof randomsoapfroths. Matzke (1946)usedaneye dropperto carefullypreparefoamsthathebelievedto bemonodisperseandclassifiedall of thedifferentpolyhedrathatheobservedundera microscope.Dataon six hundredcellsfrom thefoaminterior aresummarizedin Fig. 3. TheKelvincell doesnothavethemostcommonface:apentagon;andTCPfoamsdonothavequadrilateralfaces,whicharealsovery common.

3

Figure3: Distributionsof polyhedrawith ' facesandpolygonswith , sides.

We have performedcomputersimulationsto calculatethe microstructureof random,monodis-persefoams. Softwaredevelopedby JohnM. Sullivan (University of Illinois) wasusedto calculateVoronoi polyhedra. The initial Voronoi seedsweregeneratedby two methods:randomsequentialadsorption(RSA) andrandomclosepackingof hardspheres(RCP).In RSA, a randomlygeneratedseedis acceptedin theunit cell only if thedistanceto existing seedsis greaterthanthespherediam-eter, which is chosento be aslarge aspossibleto pack & spheresin a cubewith spatiallyperiodicconstraints.Relatively loosepackingsof monodispersesphereswith volumefractionsup to

�.- �/�0 1�2*wereproduced.Moleculardynamicssimulations,usingsoftwarewrittenby FrankB. VanSwol (San-dia NationalLabs),producedmuchdenserconfigurationswith

�.- �/�0 1+3" . ThestandarddeviationsonVoronoipolyhedravolumeswereabout4653�7�0 8�� for RSAand 4953�/�0 :� " for RCP. TheSurfaceEvolverwasusedto producerandommonodispersefoamsby relaxingtheVoronoi structuresunderthecon-straintof equalcell volumes(seeFig. 2). This processinvolvesmany topologicaltransitionsbeforestructuresthat satisfyPlateau’s laws areachieved. Typical distributions of polyhedrawith ' facesandpolygonswith , sides,arealsoshown in Figs.3. Matzke foundaveragesof 13.70faces/celland5.124edges/face.Tensimulationsbasedon RCPwith &;����2�2� produced13.86faces/celland5.134edges/face.Five simulationsbasedon RSA with &;�<*0� gave 13.95faces/celland5.140edges/face.The computeddistributions andaveragesfor randommonodispersefoamsare in substantialagree-mentwith theexperimentaldata. Notice that triangularfacesarecommonamongtheVoronoi cellsbut very rarein therelaxed foams.Matzke did not find any triangularfaces;nor did hefind a singleKelvin cell, althoughothershave. Wefounda few Kelvin cellsamongour randomstructuresbut theywerevery rare,which is not surprisingsincepentagonalfacesaresocommon.

We usethe term polydisperseto describefoamsthat containcells of differentsize,i.e., volumein 3D andareain 2D. Randompolydispersefoamshave alsobeensimulated.Thesefoamscontaina broaderdistribution of ' -facedpolyhedrathan monodispersefoams: smallercells tend to havefewer facesandvice versa. But surprisingly, the distribution of , -sidedfacesis very similar to themonodispersecaseeven whenthe individual cell volumesvary by over a factorof ten. This wouldnotoccurif thepolydispersefoamcontainedmany extremelysmallcellsbecausetetrahedrawith fourtriangularfaceswouldbecommon.

4

Figure4: WetKelvin foamwith� �/�0 :�) , PlateauBordersegments:

� �/�0 :�) (=�0 :�2�) .3 Structure of wet foam

Even thoughthefilms andPlateaubordersin real foamshave finite thickness,we will considersit-uationswherethefilm thicknessis zero. This is a reasonableassumptionsincethe thicknesssetbycolloidal forcesin thin liquid films is often muchsmallerthanthebubblesizeandthe radiusof thePlateauborders,which scalesas

� ����> � ����� . TheSurfaceEvolver wasusedto calculatethegeometryof thewet Kelvin foamwith

� �/�0 :�) shown in Fig. 4. ThePlateauborderinterfaceshave tension�andthefilms have tension ?� from two interfaces.ThebubbleandthePlateauborderhave volume(1-� �@� and

� � , respectively. Thefigurealsocontainsan isolatedsegmentof Plateauborder, whichclearlyindicatesthattheborderis relatively thick when

� �/�0 :�) , andthatthevolumeof liquid is sig-nificant in the ‘node’ that forms at the junction of Plateauborders.When

�BA , theborderscanbeconsideredlong andslender, andthevolumeof thenodecanbe neglected;however,

� �/�0 :�) is notsmallenoughto giveaccurateresults.Thesecommentsonfoamstructurearealsorelevantto thestrutsin a solid foamwith opencells. Becauseof thePlateauborders,accuratecalculationsof geometryoffoamswith severalcellsarecomputationallyintensive andhave notbeenpursued.

In sharpcontrastwith thedry limit, a perfectlyorderedwet foamcanhave morethanonestablestructuresincePlateau’s lawsarenotvalid when

�is finite. A secondstructurecorrespondsto bubbles

compressedon a face-centered-cubic(FCC) lattice, which relatesto closestpacked spheres.Thepolyhedronassociatedwith FCCpackingis therhombicdodecahedron.Someof thenodes(Plateauborder junctions)of the wet rhombic dodecahedroninvolve eight Plateaubordersinsteadof four.More detailson thestructureof wet foamscanbefoundin thesectionon their rheology.

4 Micr orheologyof 2D foam

In 2D, Plateau’s lawsrequirepolygonalcellswhoseedgesarecirculararcsthatmeetatequaldihedralanglesof � � � . The cell edgescorrespondto liquid films with zerothickness.A perfectlyordered,monodispersefoamcontainsidenticalregularhexagonsanda polydispersehexagonalfoamcontainsirregular hexagonsbut it alsohasperfecttopologicalorder. Understaticequilibriumconditions,allof theedgesarestraightandall of theinternalcell pressuresareequalin hexagonalfoams.Becauseall verticesare threefoldin a dry foam, Euler’s law requiresthat the averagenumberof sidespercell is exactly six. A randomfoamcontainsseveraldifferenttypesof polygonsandis topologicallydisordered;thecell edgesarecurvedandthepressuresaredifferent.

Thenonlinear elasticresponseof apolydisperse hexagonalfoamis isotropic (Khan& Armstrong,1986;Kraynik et al., 1991).In simpleshear, for example,theshearstress4 is givenby

4C�DFE

G �H E > �?" (D � �JIKL

��;IM �N�0 1�2�0 �

�OIM (1)

5

Figure5: Simpleshearingflow of aperfect2D foam.Theshearstressandfirst normalstressdifferencearescaledby �P�3Q whereQ is theinitial edgelength.

whereD

is theshearmodulus,E

is shearstrain,and � is theaveragecell ‘volume.’ In general,theshearmodulus,stress,andenergy densityall scaleas �R�2�SIT where U is thedimensionand �VIT isthecharacteristiclengthscale.Equation(1) indicatesthatasoapfroth with smallercellsis stiffer; butthe cell-sizedistribution is inconsequentialaslong asthe cells arehexagonal. Weaireet al. (1986)computedtheshearmodulusof randomfoamsandalwaysfoundsmallervaluesfor

D, whichledthem

to conjecturethat(1) is anupperboundon theshearmodulusof adry 2D foam.Princen(1983)investigatedsimpleshearingflow of a perfect2D foam. The orientationthat he

chose(seeFig. 5) givesperiodicresponsewith thesmallestpossiblestrainperiodE3W �X�� L � . The

rheologicalbehavior exhibitsseveraluniversalfeaturesof dry foamsunderquasistaticflow, regardlessof whetherthey are2D or 3D, orderedor random.Thestress-straincurvesarepiecewisecontinuous,which correspondsto elastic-plasticbehavior. Eachbranchof thecurve representslarge-deformationelasticresponseof afoamwith fixedtopology, i.e., thebehavior is reversibleandcell neighborsdonotchange.EachbranchterminateswhenthefoamstructureviolatesPlateau’s laws. Stability is restoredby a cascadeof local topologychangescalledT1sthatresultin astablefoamstructurewith differentcell neighbors.Thejumpsin stressandstructurearenot reversible.

Equation(1) is valid for all polydispersehexagonalfoamsup to a critical strainwherethelengthof someedgegoesto zeroandproducesa fourfold vertex. This definesan elasticlimit andcreatesthe unstablesituationthat provokesT1s. In Princen’s case,thenew structurelooks exactly like theoriginal structureafter every T1. The perfectstructureandparticularorientationthat he analyzedareresponsiblefor othermicrorheologicalartifactsthatareevidentin Fig. 5 but not representative ofrandomfoams.Theseincludelargestressfluctuationsandstrain-periodicbehavior. It is evencommonto getnegative shearstressfor otherorientationsof aperfectfoam(Kraynik & Hansen,1986).

For comparison,Fig. 6 containsthestress-straincurve for a randompolydispersefoamwith 256cells(Herdtle,1991).Thefluctuationsaresubstantiallysmallerbut havenotdisappeared.Thisoccursbecausea T1 cascadecaninvolve many cells. A singleedgelengthgoingto zerowith straintriggersa local T1 thatonly involvesfour cells. Thesubsequentstructuremaynot relaxto a stablefoam;this

6

Figure6: Shearstressfor a randompolydispersefoamwith 256cells (Herdtle,1991). Thestressisscaledby theenergy densityof aperfectfoam.

triggersanotherT1 andsoonuntil stability is eventuallyrestored.Onecananticipate‘smooth’curvesthatasymptoticallyapproacha plateauwhen & is very large;but largehasyet to bequantified.Thisplateaudeterminesthe dynamicyield stressor flow stressof the foam. The stress-straincurve canexhibit anovershootwhentheinitial structureis verydifferentfrom thestationarystate.

Princen(1983)alsostudiedsimpleshearingflow of awetfoamwith finite liquid fraction�. When�

is sufficiently large,thestressandstructureevolvesmoothlywith strainbut theaveragestressis zero(Reinelt& Kraynik, 1990).In this incongruoussituation,thefoamflowsandhasashearmodulusbutno dynamicyield stress.It seemsunlikely thatthisbehavior couldoccurin a randomwet foam.

Kraynik & Hansen(1987)developedan ad hoc model for viscouseffectswhen�

is small andtopologicaltransitionsarefast.Theexcesstensionin afilm Y� is givenby

Y�[Z\� �^] Q`_a � a > ( ] Qb�NcJ� IM _E ��� (2)

where] Q is thecapillarynumber, c is theviscosityof thecontinuousliquid phase,a is thedimen-

sionlessfilm length, _a is the film stretchrate,and _E is the foam shearrate. The excesstensionispositive whena film is stretchingandvice versa. In contrastwith thequasistaticcase,differentfilmtensionsproduceunequalvertex anglesthat vary with time. This simplemodelproducesrelaxationphenomenathatcanbefound in moresophisticatedtreatments.Whenthecapillarynumberis large,theresponsenever becomestime periodicbut cell distortioncontinuesto grow andfilms continuetothin. Thehighly distortedmicrostructureis especiallyevidentin thefirst normalstressdifference& � .Thebehavior for large

] Q providesa plausiblemechanismfor shear-inducedfilm ruptureandfoambreakage.Distortionanddamageof themicrostructurecouldbehighly undesirablein afoamprocess.

Whenaperfectfoamis sowet thattopologicaltransitionsaresmooth,theslow film-level viscousflow canbe modeledwith the asymptotictheoryof Mysels et al. (1959) that describesa thin filmbeingpulledfrom or pushedinto aPlateauborder. Theexcesstensionis givenby

Y�[Zd� Y] QMe

(3)

wherethelocal capillarynumber Y] Q is basedon therelative film speed.Theresultingfoamviscositycgf is givenby

cgfhZNc ] Q � Ie (4)

which indicatesshear-thinningbehavior (Reinelt& Kraynik, 1990).

7

(a) (b)

(c) (d)

0

0.5

1

1.5

σ xy

a

b

c

d

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.50

0.5

1

1.5

γ

N1

Figure7: Simpleshearingflow of awet foamwith� �/�0 :��* and

] Q)�� (Santos,1998).

SantosandReinelt(Santos,1998)usedafinite-differencemethodthatinvolvesoverlappinggridsto solve the Stokesequationsfor a wet foam with

�Vi �0 8 . The foam is stabilizedby a disjoiningpressurej thatpreventsthelocal film thicknessk from goingto zero;

jV� �`l k � � (5)

where l is theHamaker constant.Their approachis ableto resolve thetime evolution of thePlateaubordershapeandthe film-level geometryfor fast topologicaltransitions(seeFig. 7). Calculationssuchastheseprovide importantinformationonthedevelopmentof foamgeometryduringprocessing.This includesanisotropy in cell shapeaswell asthedistributionof materialthroughoutthecontinuousphase,whicheventuallysolidifiesto form acellularsolid.

Martinez& Kraynik (1992)usedaboundaryintegral methodto studytheeffectof viscousforceson foam expansionfrom dilute gasbubblesto highly crowded cells. The initial structureconsistsof circular bubbleson a hexagonallattice suspendedin a viscousfluid. Startingfrom a typical gasfraction,

�9m �� � � �7�0 1 , thefoamundergoesuniformexpansionatconstantvolumetricexpansionraten. The only parametersin this low Reynoldsnumberflow are

�9manda capillary numberdefined

as] Qo�CcJ� IM n ��� where � is the instantaneousbubblevolume. When

] Q A , surfacetensioneffects dominateand the bubblesexpandas circles until they nearly touch at

�6mqp �0 1� ��+2� . Theresultsfor

] Q0�� arevery different,asshown in Fig. 8. When� m �7�0 1+ , the bubblesexhibit slight

deviationsfrom circularshapeanddevelop ‘flat spots’betweenneighbors.By�9m �/�0 1r , honeycomb

8

Figure8: Foamexpansionat] Q)�� : �9m �/�0 1*!(=�0 1+!(=�0 s$%(=�0 1r!(=�) 1�!(t�0 1��* .

Figure9: Final foamstructureat�9m �/�0 1�2* : ] Q0�7�0 :�) (=�0 8��0(� :�2� .

microstructurehasemerged; mostof the liquid is found in thick films andthe Plateaubordersarehighly curved. The hexagonalmicrostructurecontinuesto evolve as

�9mvu . The cells canneverbecircularwhen

�9mxw �0 1� ��+2� , but they canbehighly distortedfor smaller�9m

whenviscousforces,asmeasuredby

] Q , are large. Figure9 shows the effect of] Q on liquid distribution betweenthe

films andPlateauborderswhen�9m �/�0 1�2* . Thefilm thicknessincreaseswith

] Q andasymptoticallyapproachesa maximumvaluethatcorrespondsto regularhexagonalbubbles.Santos(1998)hasusedthefinite-differencemethodto studyfoamexpansionandfoundthesamebehavior.

A simpleshell model for foam expansionis basedon a circular bubbleexpandingin a ring ofviscousfluid. Theeffective foampressure,Bf is givenby

� ,.fh� �h� ,.y � �z �{H|c ��9m

�9m n(6)

where,}y is theinternalgaspressureandz

is thebubbleradius.Thecoefficient ofn

is theexpansionviscosityof the foam ~9f . Even thoughthe shell modelclearly cannotcapturethe evolving bubbleshapewhen

] Q and�9m

arelarge,Eq.6 doesprovide anexcellentestimateof theexpansionviscositywhen

] Q is � � �� .It would bevery difficult to form anopen-cellfoamif theprocessremainedin thelargecapillary

numberregimeup until thefoamsolidified. Simulationsthatcombinesimultaneousfoamexpansionandshearwould shedlight on processingflows suchas thermoplasticfoam extrusion(seeFig. 1).

9

Commercialscaleproductionof flexible slabstockpolyurethanefoam involvesexpansionof a thicklayer of well-mixed prepolymersthat have beendepositedon a moving belt. Simultaneousfoamexpansionandextensioncauseanisotropiccellsthatareorientedalongthe‘rise’ direction.

Pozrikides& coworkers (Li et al., 1996)andLoewenberg et al. (1999)aredevelopingbound-ary integral capabilitiesthatenablesimulationsinvolving hundredsof bubblesor dropsin 2D andafew dozenin 3D. Researchon adaptive meshrefinementalgorithmsalsoshows greatpromise.Thismethodologyis necessaryto dealwith highly curvedinterfacial regionsandthin film formation,bothof whichoccurin highly expandedfoamflows.

5 Micr orheologyof 3D foam

TheSurfaceEvolver hasbeenusedto applyhomogeneousdeformationsto spatiallyperiodicfoams,computethe minimal structures,and thenevaluatethe completestresstensor. The shearmodulus,large-deformationelasticbehavior, andquasistaticresponsein simpleshearingflow havebeenstudiedfor wetanddry foams(Reinelt& Kraynik, 1993,1996,1999;Kraynik & Reinelt,1996a-1996d).

5.1 Dry foam

5.1.1 Shearmodulus

Theshearmodulusof a dry foamscalesas �R�2��Ie . TheKelvin foamandvariousmonodisperseTCPfoams(Weaire-Phelan,Friauf-Laves,andBergman)havecubicsymmetry, whichmeansthattherearetwo independentshearmoduli givenby

D� � ��� �@� ��� ��> �@� �(

D> � ���@� (7)

wherethe ���:� areelasticconstantsdefinedby Love (1994)andNye (1985). An effective isotropicshearmodulus

Dcanbe obtainedby averaging

Dover all foam orientations.The averagecanbe

calculatedin many ways,e.g., at constantstrain(Voigt) or at constantstress(Reuss).TheVoigt andReussaveragesgive relatively looseupperand lower boundsfor

D. The shearmoduli of various

monodispersefoamsaregiven in Table1 whereD

is reportedas the meanof the bounds,and therange. The Kelvin foam is significantlymoreanisotropicthanthe TCP foams. The T foam, whichhas81 cells andmany facesof differentorientation,is essentiallyisotropic. Basedon

D, the TCP

foamsaresignificantlystiffer thantheKelvin foam.Wehave alsocalculatedtheshearmoduli of sev-eral randomfoamswith 64 cells; theseincludedmonodisperseandpolydispersefoams.Theseshearmoduli exhibit modestanisotropy, presumablybecauseof systemsize,but all valuesfall in a narrowrange: �0 s$ r<�\�0 :��r . Perhapsevenmoreimportant,thereis no significantdifferencebetweenresultsfor the monodisperseandpolydispersefoams. Recallingthat � is simply the averagecell volume,this preliminaryfinding suggeststhat theshearmodulusof a randomdry foammaybeinsensitive tocell-sizedistribution. This would bea very simpleresult. It is alsointerestingthat theaverageshearmodulusof aKelvin foamprovidesanexcellentestimateof theshearmodulusof therandomfoams.

Princen& Kiss (1986)measuredtheshearmodulusof concentratedoil-in-wateremulsionswithpolydispersedrop-sizedistributions. They usedtheir datato develop an empirical correlationthatincludesthedependenceon liquid volumefraction (seeFig. 15). Their correlationextrapolatesto ashearmodulusof 0.82in thedry limit, whichagreeswell with oursimulations.

Theoreticalevidencesupportingthepossibilitythatpolydispersityhaslittle or no influenceontheshearmodulusof a dry foamcanbe found in 2D and3D. Cell-sizedistribution hasno influenceontheelasticbehavior of polydispersehexagonalfoams(Kraynik et al., 1991).Theshearmodulusof a

10

D�

D>

DKelvin 0.5706 0.9646 0.7814� 0.0256Weaire-Phelan(A15) 0.8902 0.8538 0.8682� 0.0002Friauf-Laves(C15) 0.8448 0.8860 0.8693� 0.0002Bergman(T) 0.858 0.856 0.857Random 0.78 � 0.08

Table1: Shearmoduli of dry foams

bidisperseWeare-Phelanfoamdependsvery little on the relative cell sizeandis neithera minimumnoramaximumfor themonodispersecase(Kraynik & Reinelt,1996c).

Figure10: Uniaxial extensionof a Kelvin foam.

5.1.2 Lar ge-deformation extension

The mostsymmetricdistortionof a Kelvin foam involves uniaxial extension;Fig. 10 includestheevolution of cell geometryand tensilestresswith Hencky strain � . The areaof the ‘square’ facethat is being pulled decreaseswith increasingstrain but remainsfinite in the limit � u �@� � �/�0 12*3" .Thereis no stablesolutionthatmaintainscontactbetweentheoriginal cell neighborsbeyond �@� � , theelasticlimit. The stress-straincurve exhibits a maximumwell beforethe turning point at �@� � . Theunstablesolutionson the curve below the turning point have the sametopologybut highersurfaceareathantheir stablecounterparts.Theendpointat ��� ��

� �2����/�0 12�0 , wherethetensilestressis zero,correspondsto a rhombicdodecahedronon anFCClattice,which is unstablein thedry limit.

5.1.3 Simpleshearingflow

In contrastwith linear elasticbehavior, microrheologicalresponsein simpleshearingflow dependsstrongly on the complexity of the foam structure. The stressand structureof a Kelvin foam arepiecewisecontinuousfunctionsof shearstrain

E, asshown in Fig. 11. This foamorientationproduces

strain-periodicbehavior with thesmalleststrainperiodE W � � ��� ��t����> . Eachdiscontinuityin thestress-

straincurve correspondsto topologicaltransitionsthat areprecededby shrinking faces. As in 2D(seeFig. 6), eachT1 cascadereducessurfaceenergy, resultsin cell-neighborswitching,andprovidesa cell-level mechanismfor irreversibleyield behavior during foamflow. Therearetwo T1 cascades

11

Figure11: Simpleshearingflow of aKelvin foam;cell shapesforE

= 0, 0.30,0.60(beforeT1), 0.62,0.80,0.98(beforeT1), 1.0,

E W � G ��� .

Figure12: Simpleshearingflow of aWeaire-Phelanfoam,E W �� .

Figure13: Simpleshearingflow of a randommonodispersefoamwith 72cells.

12

per cycle; eachis triggeredby oppositeedgesof shrinkingquadrilateralfacesgoing to zerolength,which leadsto unstableedgeconnectivity. Thesestandard transitionsare very different from thepoint transitionin extension,whereall edgeson a faceshrink together. Symmetryimposesstrongrestrictionsonthetypeandoutcomeof topologicaltransitionswhenthefoamhasperfectorder;KelvincellsbegetKelvin cells.

The Weaire-Phelan(WP) foam haslesssymmetryandexhibits morediversetopologicaltransi-tions thantheKelvin foam in simpleshearingflow. Sincethereareeight differentpolyhedrain theunit cell, the Weaire-Phelanfoam is far lessconstrained.Individual polyhedrachangetype asT1cascadesproducefoamsthatcontainamuchgreatervarietyof polyhedraandfacesthanexistedorig-inally. This occursbecauseindividual faces,which begin aspentagonsor hexagons,cangainor loseoneor two edges.Thefirst T1 cascadealoneinvolvespolygonsrangingfrom trianglesto octagons,whichassembleto form many differentpolyhedra.

Figure12 indicatesthatstress-strainfluctuationsarealsolargefor theWeaire-Phelanfoam,whichis still a relatively smallsystem.ThemostelementarylocalT1 involvesfivecellsandaparticularcellcanbealteredseveral timesduringa T1 cascade(Schwarz,1964). Consequently, thedisturbancetothefoamstructureandthecorrespondingjumpsin stressandenergy arelarge.

For someorientations,Weaire-Phelanfoamsreturnto their original topology. In othercasestheybecomeKelvin cellsbut subsequentT1 cascadesproducetopologicaldisorder. A T1 cascadebeginswith acell edgegoingto zerolengthasstrainincreases.Theprocessby whichedgesvanishin Kelvinfoamsandin TCPfoamscanbesmoothandcontinuous,or abruptanddiscontinuous.Whenmultiplecellsareinvolved,abruptonsetis oftenconnectedwith symmetry-breakingbifurcationsthatprovideamechanismfor Kelvin cellsto disorder. Thebifurcationscanalsocausestrainlocalizationin whichaT1 cascaderesultsin layers,two Kelvin cellsthick, slidingpastoneanother.

Figure13 containsthestress-straincurve for a randommonodispersefoamwith 72 cells. Someof thepolyhedrain the initial structurehave shortedgesthatgo to zerolengthat small strains.Thisreducesthe elasticlimit and leadsto strain-hardeningelastic-plasticbehavior. The foam structureappearsto besufficiently complex to reducestress-strainfluctuationsandpreventnegativeshearstressat largestrains.Thesepromisingtrendsalsooccurin 2D (compareFigs.6 and13).

Gopal& Durian(1995)haveusedamultiple-lightscatteringtechniquecalleddiffusing-wavespec-troscopy (DWS) to studynonlinearbubbledynamicsduringfoamflow. They observe localizedstick-slip likerearrangementof bubblesthatundoubtedlyrefersto T1 cascades.Simulationof largerandomfoamswill eventuallyprovide connectionswith theDWSexperiments.

5.2 Wet foam

5.2.1 Shearmodulus

Figure14containswetKelvin cellsandwet rhombicdodecahedra(RD) with thesameliquid content:� �/�0 :� " . Thereis an examplewhereeachstructurehasisotropicstress.The shearmoduli of bothfoamsaregraphedin Fig. 15 andcomparedwith theempiricalcorrelationof Princen& Kiss (1986).Both foamsarestableover someoverlappingrangeof

�. The smallershearmodulusof the Kelvin

foamD� decreasesrapidly as

�approaches0.11. This is an indication that the smaller(original

quadrilateral)films areshrinkingto zeroareaprior to beingconsumedby surroundingPlateauborders.When thesebordersconverge to form eight-way junctions, the bubbleslose contactwith all nextnearestneighborson theBCC latticeandthewetKelvin cell becomesunstable.

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Figure14: A wetKelvin cell onaBCClatticeis beingstretchedto form awet rhombicdodecahedronon anFCClattice,

� �/�0 :� " .

Figure15: Shearmoduli of wetKelvin foamsandwet rhombicdodecahedra(RD).

Figure16: Tensilestressandenergy for a wet Kelvin cell beingstretchedto form a wet rhombicdodecahedron,

� �/�0 :��+ . Samedeformationasin Fig. 14.

5.2.2 Lar ge-deformation extension

As discussedin Section5.1.2above, a BCC latticecanbestretchedto form anFCClattice. Simarly,a wet Kelvin cell can be stretchedto form a wet rhombic dodecahedron.Figure 14 depictsthisprocess. The small film on ‘top’ of the Kelvin cell shrinksand eventually vanishesas the foamstretches.This provokesa topologicaltransitionthat resultsin a wet RD that hastwelve films andsomefour-way andeight-way Plateauborderjunctions(nodes). Furtherstretchingtakes the foam

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to FCC structureandisotropicstress.Figure16 containsrepresentative graphsof tensilestressandenergy densityfor

� �/�0 :��+ . TheshearmodulusD� of eachstructurecanbeevaluatedfrom theslopeof

thestress-straincurve neartheendpoints.Theenergy maximumthatoccurswhenthestresschangessigndeterminestheenergy barriersbetweentheundeformedstructures.Thesebarriersareequalwhenbothundeformedstructureshave thesameenergy, whichoccurswhen

�Bp �0 :��+3" . ThewetKelvin cellhaslowerenergy thanthewetRD when

�vi �0 :��+3" andvice versa.Theprocessjust describedappearsto becompletelyreversible;a wet RD canbecompressedto

a wet Kelvin cell. Bubblesthatwereseparatedby aneight-way junctionbecomeneighborswhenthefoamis compressed.This reversetopologicaltransitionis initiatedwheninterfaceson oppositesidesof the eight-way nodecomeinto contact.The film that forms uponfirst contactgrows in areawithfurthercompression.

Figure17: Simpleshearingflow of awetKelvin foamwith� �/�0 :��+ . Thebottomrow givesadifferent

view of a thin film on a Kelvin cell vanishingandinterfacesseparatingto form aneight-way nodeonawetRD.

5.2.3 Simpleshearingflow

Simpleshearingflow of wet Kelvin foamsinvolvesbothof thetopologicaltransitionsjust described,but they are not reversible. Figure 17 containsrepresentative structuresand a stress-straincurve.Insteadof two T1 cascadespercycle, like thedry case,therearefour distincttopologychangesin thewet case;thefoamalternatesbetweenwet Kelvin andwet RD. Thefirst andthird transitionsinvolvethe‘same’shrinkingfilm asthedry foam. Thesecondandfourth transitionsareprovokedby contactbetweenoppositeinterfacesof aneight-wayjunction.Differentfrom thereversibleuniaxialextension,the energy dropsat eachtransitionin simpleshear. The magnitudeof thesejumpsis muchsmallerfor wet foamsthandry foams.Thesameis truefor theaverageshearstress,whichcorrespondsto the(viscometric)yield stressof thefoam.Consistentwith measurementsof yield stressby Princen(1985)theaveragestressdecreasesvery rapidlywith

�.

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When�

is verysmall,weanticipatethatthewetRD structureswill beunstableintermediatesthatleadto wet Kelvin structures.Therewill be two Kelvin branchesin thestress-straincycle, just likethedry case.

5.3 Foamexpansion

Theinterplaybetweenviscousforcesandsurfacetensionandits influenceon theevolving foammi-crostructureduringfoamexpansionis fundamentalto thescienceandtechnologyof foamprocessing.Thenaturalstartingpoint for a 3D analysisinvolvesuniform expansionof a viscousfluid containingsphericalbubbleson a BCC lattice. ChoosinganFCClatticewould of coursepermit tighterpackingof sphericalbubblesbut it would leadto instabilitiesinvolving eight-way nodeswhen

�6mapproaches

unity. Thiswouldcomplicateattemptsto studyotherphysicalphenomenathatcouldcontrolthecom-mercially importantlow-densityregime. Theseincludescalartransportof heatandmass,chemicalreaction,andrheologicalcomplexity of thebulk fluid andtheinterface.

Questionsof stability aside,sphericalbubbleson a BCC lattice make first contactwith eightnearestneighborsandform theprecursorsof hexagonalfaceswhen

�9m �/� L ��� r��/�0 1+2r � . Understaticconditions,which fall in thedomainof theSurfaceEvolver, six next nearestneighborsmake contactwhen

�9m?p �0 1r2� ; thiscompletesthefourteenfacesof theKelvin cell. Thestableequilibriumstructuresbeyondthispoint have beencalculated.

Figure 18: Uniform expansionof bubbleson a BCC lattice evolving into Kelvin cells;] Q0�/�0 1* ,�9m ���0 1r �0(=�0 1� �0(=�0 1�2*! :�) 1��r (Loewenberg et al., 1999).

Neglecting fluid inertia, the importantparametersin the viscousfree-surfaceflow problemin-clude: the volumefraction

�9m, � the ratio of gasviscosity to liquid viscosity, andthe capillary

] Qdefinedas

] Qb�dcJ� Iey n ��� (8)

where �9y is thebubblevolumeandn

is thevolumetricexpansionrateof thefoam.Many qualitative featuresof theflow canbeanticipatedfrom the2D analysis.A shellmodelmay

provideagoodestimateof thefoamexpansionviscositywhen] Q is large.Hydrodynamicinteractions

betweenneighborswill causebubbledistortionwhen�9m

is justbelow 0.680andlarge] Q will promote

theformationof thick liquid films.

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Loewenberg et al. (1999)have madesubstantialprogresson this problemby usingthebounday-integral methodto solve the Stokes equations.Thus far, they have only consideredthe case �.��becausethenumericalimplementationis simpler. Resultsof their simulationsfor

] Q0�/�0 1* arecon-tainedin Fig. 18,which illustratestheevolution of bubbleshapefrom spheresto Kelvin cells.

Figure19: Bubbleshapefor�9m �/�0 1� � : whenexpansionat

] Q)�/�0 1* terminatesandafter20 timeunits.Thebottomrow containstwo views of theequilibriumstructure.

Oncethefoamexpansionhasstopped,thebubbleshapewill continueto relaxtowardequilibrium.Thisprocessis illustratedin Fig.19. Thefoamexpandsuntil

�9m �/�0 1� andthenthebubbleshaperelaxesfor twentytimeunits.Thebottomrow containstwo viewsof theequilibriumstructurecalculatedwiththeSurfaceEvolver; theshapeof a representative liquid region is alsoportrayed.

6 Micr omechanicsof solid foamswith opencells

Thisarticlehasfocusedonthemicrorheologyof liquid foams.Weconcludewith anillustrativeexam-ple from theworld of cellularsolids. In natureandindustry, liquid foamsundergo phasechangestoproducesolid foamssothecell-level structureof theformerheavily influencesthelatter. This justifiesandmotivatesusingthe geometryof a soapfroth andrelatedmaterialsastemplatesfor developingmicromechanicsmodelsof cellularsolids.

Theprimarymicrostructuralfeaturefoundin low-densityopen-cellfoams,suchasflexible poly-urethane,is thenetwork of slenderstrutsthatmeetmostoftenat four-way joints. We aredevelopingfinite elementmodelsby usingbeamelementsto discretizethemicrostructure.Theundeformedstrutsareassumedto haveuniformcrosssectionandastraightaxis,but theseapproximationscanberelaxed.Thelocationof joints andtheconnectivity of strutsarebasedon foamgeometriescalculatedwith theSurfaceEvolver. Figure20containsarandom,monodispersesoapfroth andthecorrespondingmodelof anopen-cellfoam.Thegeneralpurposefinite elementprogramABAQUSis usedto performlarge-deformationsolidmechanicscalculationof thefoamstructureandstrut-level forces.This information

17

is thenusedto evaluatethemacroscopicstressof the foam. Thegraphin Fig. 20 containsa typicalstress-straincurve for an unconfinedfoam subjectedto uniaxial compressive stress.The strutsareshapedlikePlateaubordersandarecomposedof linearelasticmaterialwith Youngsmodulus� . Thenonlinearresponseis causedby large-deformationgeometriceffects.

Figure20: Soapfroth andcorrespondingopen-cellfoam. Undeformedanddeformedfinite elementmesh.Stress-straincurve for unconfineduniaxialcompression.

Magneticresonanceimagingandmicro-X-RayCT arebeingusedto analyzethecell-level struc-tureof polyurethanefoam(Pangrleet al., 1998).Thesedatacanbecomparedwith statisticsobtainedfrom simulationsof foam structure. Beyond SurfaceEvolver resultsfor minimal soapfroths, fluidmechanicscalculationsof foamstructureevolution in process-relatedflowspromisemuch-neededin-formation. Combineall of this with finite elementmodelsfor themechanicsof cellularsolidsandacomprehensive packagebeginsto emerge.Themicromechanicalapproachhasestablishedafirm the-oreticalfoundationfor developingstructure-property-processing relationshipsfor foamedpolymers.

ACKNOWLEDGEMENTS

We thankKenBrakke for developingandmaintainingtheSurfaceEvolver. AMK alsothanksMitziBower for valuableassistancewith the graphics. Sandiais a multiprogramlaboratoryoperatedbySandiaCorporation,a LockheedMartin Company, for thetheU.S.Departmentof Energy undercon-tract#DE-AC04-94AL85000.Thiswork wasalsosupportedby theDow ChemicalCompany underaCooperative ResearchandDevelopmentAgreement(CRADA).

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