a mathematical model for an expanding foam l. w
TRANSCRIPT
A MATHEMA TICAL MODEL FOR AN EXPANDING FOAM
L. W. Schwartz & R. V. Roy
Departmentof MechanicalEngineeringUniversityof Delaware
Newark,Delaware,USA
June2002
A theoretical and numerical model is presentedfor the shape evolution of the thin liquid filmsseparating thegasbubblesin a foam. Themotion is dueto capillary action, surfacetension gradients,and the overall expansion of the foam. The expansion is the result of the increasein gascontentwith time. Process modeling is accomplished via the solution of threecoupled partial differentialequations. Two time scales are included in the model: a processtime anda drying or curing time.It is demonstrated that the amountof surfactant is the dominant control mechanism for the final filmthickness. If sufficient surfactant is present,the films will be shown to dilate uniformly in space. Anumberof known featuresof expanding foamsarereproducedby themodel.
1. Intr oduction
This paperpresentsa theoreticalandnumericalmodelfor predictingtheshapeevolutionof theliquid region in anexpandinggas/liquid foam. Resultsaredirectly relevant to rapidly-expandingfoamsthatultimatelysolidify, suchasa polyurethane(PU) foamandotherplasticfoams. Certainfeaturesof the presentmodelingeffort arealsoapplicableto aqueousfoamsandemulsions,aswell asto flocculationof dropletsandbubbles.
Foamsform an importantintermediateclassof matter, wherebyliquids andgasescanbeblendedtogetherto form a low-density substancethatcanpersistfor long times. Foamsfindapplicationin many industries including foodstuffs, personalcareproducts,petrochemicalsanddefense.Sometimesthegenerationandpersistanceof foamsis undesirable;animportantexampleof this is thefoamingof lubricating oils.
A two-dimensional geometryanda high degreeof symmetry have beenassumedin thepresentmodel.Becauseof this,thephysics is considerablysimplifiedandonly asmallportionof the flow, termeda “unit cell,” needsto be analyzed.While an actualfoam is, of course,three-dimensional, most of the liquid is to be found in the region where threethin liquidfilms, or lamellae,meetat mutualanglesof 120 degrees. Theseare the so-calledPlateauborders. ThePlateaubordersform thecross-sectionsof thestrutswheremuchof the liquidaccumulates.For industrial foams,the liquid ultimately solidifies so that the strutsbecomestructuralelements.Thustheproblemof predictingthepartitionof liquid betweenthefilmsandthe Plateaubordersis primarily a two-dimensional one. Becausethe cells areassumedto be small, gravity doesnot have a significant influenceof the flow betweenthe thin films
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andtheborders.Thusgravity is omittedfrom thepresentstudy. Clearly gravity doescausesignificantdrainage,alongthe interconnectedsystemof struts,in a non-solidifying aqueousfoamor soapfroth [1].
The main goal of the presentstudy is to presentthe form of a minimal model for thiscomplicatedprocess.By demonstratingthecomputability of themodelto bepresentedhere,we will have established thebasicframework thatwill allow a quantitative description andabetterunderstandingof a numberof industrially importantprocesses.Sucha modelcanbeinterrogatedto predicttheoverall effectsof particularchangesin oneor morephysicalinputvariables. The modeldevelopedherecanbe criticized for its crudenessin certainaspects;somesuggestions for improvementwill begivenin theconcludingsection.
The flow is driven primarily by a given rateof gasgenerationor injection; this requiresthe liquid films betweenPlateaubordersto stretch.At eachinstantof time theprofile shapechangesin responseto the newly-imposeddeformationand the fact that the time-evolvingsystemis notatmechanicalequilibrium. Thestretchingstopsafteraperiodof time,calledtheprocesstimeby us,corresponding,for example,to thefoammixturefilling theavailablespacein acontainer. However, thefilm profilecontinuesto evolve,undertheactionof hydrodynamicforces,until theliquid hassolidified.
Physicalquantitiesin themodelincludeviscosity, surfacetension, andvariousparametersassociatedwith thepresenceof surfactant.In generaltheeffectof surfacetension(capillarity)is to draw liquid backinto the Plateaubordersbecausethesebordersarehighly-curved andthusthe (capillary)pressureis lower therethanin the thin films. Sincethe liquid volumeisconserved,this backflow thinsthefilms thatseparatethegasbubbles.In industrial parlance,thethin films areoftenreferredto aswindows. Thenumericalsolution to themodelproblem,as formulatedhere,will demonstratethat, if litt le or no surfactantis present,windows willbecomevery thin andwill break.This breakingmayoccureitherduringtheexpansion or thefully-expandedstagesof themotion.Bothsituations canbeobservedin realfoams.
With increasingamountsof surfactant,the windows will becomethicker. We will showthatthisisdue,fundamentally, toasurface-tension-gradientstress(sometimescalledaMarangonistress)thatstiffensor hardensthe liquid-gasinterface. This stresschangesthenatureof thebackflow and diminishesit very substantially. Thereis a critical amountof surfactantforwhich thebackflow will bemaximallyretarded.If thisamountis present,theinterfacewill bealmostcompletelyhardened.Undertheseconditions, during theexpansionphase,the inter-facewill beshown to dilateuniformly. Thatis, eachelementof theinterface,or thethin film,will lengthenby thesamefractionalamountduringagiventimeinterval. With thiscritical sur-factantlevel, anassumedsimpleNewtonianliquid thatis curinguniformly in spacewill havethethickestpossiblewindows,for agiven setof processconditions. Surfacehardeningin soapfilms is discussedby Myselsetal [2]. Themodelresultsto begivenin thispaperaresimilar totheexperimental observationsthatthey report.In prior work anassumedcompletely-hardenedsurfacefor a foam wasusedin analyticalcalculationsthat led to predictionsof foam exten-sionalviscosity, i. e. theviscosityof a foamduringanimposedpurely-extensionalmotion [3].Webelieve thatthefirst numericaldemonstration of surfacehardening,dueto surfactant,was
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doneby usseveralyearsagofor a thin liquid coatingonasolidsubstrate[4].
Themathematical modelthatwe develop hereconsistsof threecouplednonlinearpartialdifferentialequationsaswell asappropriateboundaryconditions. The threeunknown func-tions are the film profile, the surfactantdistribution, and the interfacial speeddistribution.Eachof thesefunctionsvary in spaceandtime. The modelis an adaptationandsignificantextensionof a modeldevelopedby usto explain thebehavior of drainingsoapfilms [5]. Fora soapfilm on a verticalwire framethatis drainingundertheinfluenceof gravity, it hasbeenobservedexperimentally thatrapidthinning occursat low surfactantlevelswhile highsurfac-tantconcentrationsresultin thick films thatdrainquiteslowly. In addition, thesetwo regimeshave characteristicallydifferentprofile shapes.All thesefeatures,asobservedby Mysels[2],weresuccessfullyreproducedandexplainedby theearliermodel. In addition, thatmodelre-producedtheoccurrenceof very thin, socalledblack, films thathavebeenobservedto appearduring late stagesof draining. This validationof the basicmodel, in a relatedproblem,isimportantsinceit is unlikely that accuratetime-dependentin-situ profile measurementscanbemadewithin a three-dimensionalsoap-waterfoam.It is evenlesslikely thatsuchmeasure-mentscouldbemadewithin expandingindustrial foams.
As will be shown below, thereareother featuresthat canbe includedin the theoreticalandnumericalmodelthat will influencethe final shapeandthicknessof the windows in anexpandedfoam. Theexpansionhistory is an input functionthat includesthespecificationofboththeaveragegrowth speedaswell astheform of thespeedversustime relationship.Thefinal volume fraction,essentiallyequivalentto thefinal densityof a dry industrial foamprod-uct,canalsobevaried.A diffusion constantis alsospecifiedto modeltherateof spontaneousspreadingof thesurfactantalongtheinterface.Diffusionis adissipativeprocessandrelativelylargevaluesof thediffusion constantcanbeshown to relaxsurfacehardeningandultimatelyleadto thinnerfilms. Therateof drying/curing is anotherinput variable.This variabledeter-minestheelapsedtimethatis requiredfor thefoammatrixmaterialto solidify. Themolecularrepulsionthat leadsto stable“black” film thatis commonly observedin soapfilms is alsoin-cludedin themodel.Thiseffect,usuallytermeddisjoiningpressure, is importantfor films thatbecomeexceedinglythin. It is ourbelief thattheprimaryroleplayedby disjoiningpressureisthecontrolof window breakagewhich it tendsto prevent. Thusit is potentiallyimportantinmodelingtheproductionof flexible foamproductswheremostof thewindowswill ultimatelybreak. The effect would be muchlessimportant for rigid foam productsbecausethesearecharacterizedby relatively thick unbrokenwindows.
Thispaperalsoincludessomeresultsobtainedwith thenumericalalgorithmthathasbeendevelopedbasedonthepresenttheory. Certainknown featuresof PUfoamPlateauborderandwindow shapeshave beenreproduced.Theseinclude,for example,theobservedoccurrenceof localizedthin regionsor dimplesnearborder-film junctures[6].
Thenext sectiondemonstratesthereductionin thenumberof controlling parametersthatmaybeaccomplishedusingdimensionalanalysis.Sections3 through5 specifytheformsofinput functionsthat ultimately determinethe shapeof the final dry film. Theseincludetheinitial conditions, the function that controlsthe rateof stretchingof the liquid films, anda
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viscosity-versus-timelaw. Thesystemof coupledpartialdifferentialequationsandtheasso-ciatedboundaryconditionsaregiven in section6. The argumentdemonstratingthat strongsurfactanteffect will result in spatiallyuniform stretchingof the thin films is given in sub-section6.2. Somesampleresults,usingthenumericalmodel,aregiven in sections7 and8.Section8 alsopresentsamodelfor generalizedNewtonian rheologyof theliquid.
2. Model parameters
Within thepresentmodel,a numberof parametersinfluencethefinal shapeof the liquidregion. Five parametersare includedin the disjoining pressuremodel. For thick films forwhichwindow breakingis notanissue,theseparameterswill have little or noeffect. We willconsidertheremainingparametersfirst anddiscussthedisjoiningeffectafterward.
Let usassumethatwewishto solvefor aparticularvalueof thefinal film thickness,suchastheminimum thickness, for example.Alternatively, we couldconsidersomesuitably definedaveragewindow thickness. Let one of thesebe denotedas
��. The modelassumesthat
��dependsonanumberof inputparametersaccordingto���������� ���������������������������! #"��$�&%&�(')��*+��,-���.0/21 �435176�/Here
�denotessomefunctionaldependence.Determinationof theactualform of thisfunction
is a principal objective of the presentwork. It will needto be found numericallythroughsolutionof themodelequations. Theparameterson theright of equation(2.1)are:
(i)��
is the(essentiallyarbitrary)initial gasareafraction.Themaximumpossiblevalueof�8
is�:9<;
0.907for anassumedinitially circularhexagonalpatternof bubbles.This is theplanarclose-packingvalue,i. e. thevalueat which circularbubbleswould touchoneanother. Thus��
mustbesmallerthan� 9
. Providedthebubblesarenot toocloseto eachother, theirmutualinfluencewill besmallandthecircular initial conditioncanbe justified. They will continueto grow asapproximatecirclesuntil they getcloseto eachotherandviscousandothereffectsbecomeimportant. However, sincethe theorywe usepostulatesthat the liquid areais longandthin,
�=mustnotbetoosmall.Goodvaluesfor
�<arein theapproximaterange(0.5,0.85)
Within this range,theactualvalueof��
canbeshown to haveonly a smalleffect on thefinalprofile shape.In the early stagesof growth, while the separationbetweenbubblesremainsrelatively large,thebubblesgrow without interactionwith oneanother.
(ii)��
and�&%!�#'
:��
is theviscosity of theliquid at thestartof thecalculation.A “book” valuefor PU liquids is 100poiseor greater[7].
�$%&�('is a characteristictime for curingor drying. In
thesimulation theviscosity�
is takento increaseexponentially in time. Thetimeconstantforthis processis
�$%!�#'. For timesmuchlarger than
�$%!�#', theviscosity will beeffectively infinite
andthefinal profileshapewill havebeenachieved.
(iii)�>
and*
:��
is the initial, assumeduniform, value of surfacetension. The surface
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tensionis reducedlocally by thepresenceof surfactant.For simplicity we assumethatall ofthesurfactantmoleculesremainon theliquid-air interfaceandthattheirnumberis conserved.Wealsoassumea linearrelationof theform�?�@��<AB*C�4D?ABD0E/ �4351F3G/where
Dis thelocalconcentrationof surfactantandis acalculatedfunctionof timeandspace.*
andD0
are constants.Sincethe actual functional relation between�
andD
is smooth,sucha point-slope linear relation is valid provided
�doesnot deviate too much from
��.
Without loss of generality, the initial uniform concentrationD<
can be taken equalto one.Thenthe surfactant“strength,” that is the reductionin surfacetensionper unit of surfactantconcentration,is givenby thevalueof
*.*
is measuredin thesameunitsassurfacetension.
(iv)��
is theinitial lengthof thecomputationalcell. Thecomputationalcell is aportionof theunit cell. The latter is thebasicbuilding block of thetwo-dimensionalexpandingfoam. Thelength
�H, andtheothercomputational andunit-cell dimensions,aredefinedin Figs.1 and2.��
is, in fact,proportional to thestartingbubblesize,asshown in Fig. 2.
(v)��
and.
:��
is the averagevalueof drawing speed.The left endof the computationalcell is consideredto befixedin space.As thefoamexpands,theright endof thecell, whichis a symmetry line (seeFig. 1), movesto the right at speed
�8Iwhoseaveragevalueduring
theexpansion phaseis��
. As thecell expands,thegasfraction�
increasesbecausethetotalareaof the unit cell increaseswhile the liquid arearemainsconstant.
.is an exponentthat
determinestheshapeof thespeedversustime function.
(vi)�����! #"
is theprocesstime, thetimerequiredfrom thestartof themotionfor expandingfoamto completely fill the availablespacein a closedcontainer, for example. In general,
�!���! ("is
smalleror muchsmallerthan��%&�('
.
(vii),�
is the initial valueof thediffusion constantfor surfactant.Thediffusive mechanismis assumedto besimilar to Brownianmotion. For this reason,thevalueof this constantwilldecreaseasthebulk viscosity
�increases.Surfactantmoleculesarein contactwith theunder-
lying bulk liquid andan increasein bulk liquid viscosityshouldimpedesurfactantmobilityaswell. The inversedependenceof diffusivity on viscosity is given by Einstein’s Law forBrownianprocesses[8].
Dimensionalanalysisallows thenumberof parametersto bereducedby three.In dimen-sionlessform, thelaw (2.1) is���� �@�KJML ������� � *�� ������ �����! ("N 9 � �&%!�#'�����! #" � ,� N 9��O ��.QP �4351SRT/
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where�TJ
now denotessomeotherfunction.Thecharacteristictime in (2.3) isN 9 � RG������� 1 �4351VUK/Thecombination W:XZY �[ �=�\)�� is acapillarynumberbasedonaveragedrawing speed
�]. In
theprogramthespeed�=
is notuseddirectly. Insteadit is replacedby themaximum windowlength
��^�_a`accordingto ��8� �]^=_&`bAB�������! (" �(351ScT/d1
Thenthecapillarynumberis W:X � N 9R)�����! #"fehg�]^�_a`fAi6�j �4351SkT/where g�]^�_a`l���]^=_&`)\G�� . g�]^=_&` will berelatedto thefinal gasfraction.
Thedimensionlessparameter*m\)�Q
controlsthestrengthof thesurfactanteffect. It hasastronginfluenceon the final window thicknessandshape.The presentassumption, that thesurfactantis insoluble in thebulk liquid, makesthesurfactantveryeffective in controllingthestretchingbehavior of the liquid-gasinterface. It will beseenthat relatively small valuesof*m\)��
will havealargeeffectonthedynamicsof thefilm. Thiseffectivenesswouldbereducedwere surfactantsolubility includedin the model. The two time ratios in (2.3) can alsobeexpectedto havesomeeffect on thefinal shapeof thesolidifiedfilm. Finally,
,n N 9�\G� O Y g,is a dimensionlessmeasureof diffusionof surfactanton theinterface.Relatively largevaluesof g, will decreasetheeffectof thesurfactantandareexpectedto leadto thinnerfinal films.
Similar to thesituation for soapfilms [5], additionaleffectsareexpectedherewhenpor-tions of the films or windows becomevery thin. Theseeffectsare incorporatedin a contri-bution to the total pressurein the liquid. This contribution is calleddisjoining pressure andrepresentsa combination of Van der Waal’s andothermolecularforcesandwould alsobeinfluencedby thepresenceof surfactants[9]. Thedisjoiningpressurefunctionwasintroducedin the1930’sby thechemicalphysicistsFrumkinandDeryaguinin orderto betterunderstandtheorigin of thestaticcontactanglethatappearswherea sessileliquid dropletmeetsa solidsubstrate[10]. Notethat,becauseof thewaydisjoiningpressureis defined,apositivedisjoin-ing pressureis anegative contribution to thepressureappliedto theliquid. Thecharacteristicthicknessscalefor disjoiningpressureo is
� 9 andweusethefive-parametermodel
o �(��pE� 9 /H�@qrJrstGL � 9� PvuHw J ACL � 9� P8x:w J4yz L � 9� A W P{1 �4351F|G/The constantshave the following ranges:
q-Jm}�~��n� ��� ��6T��� 9 �{~���~�� W�� 6.
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A repulsive pressure( o ��~) is developedfor
� � � 9 . A representative valuefor� 9 is 10
nanometers. This repulsivepressurewill tendto preventwindow breakage.Themagnitudeofthedisjoining effect is controlledby theprefactor
q-Jwhich hasthedimensions of pressure.
Settingthis parameterequalto zerowill remove the disjoining effect entirely. In that case,correspondingto little or nosurfactanteffect,windowswill breakin thecomputersimulation.Thetwo-termversionof thedisjoiningmodel(i. e. W = 0) hasbeenusedin a varietyof thin-film modelingstudies[11, 12, 13, 5]. The disjoining pressureis attractive, or destabilizing,for
� 9 � � � � 9 \ W andbecomesrepulsive (stabilizing)for����� 9 \ W .
� ; � 9 is a stableenergy minimum for this function,corresponding,for soapfilms, to theso-calledblack filmthickness.The parameterW is introducedherefor the first time. The two cases,W = 0 andW �
0, correspondto thetwo basictypesof o functionsshown schematicallyin Teletzke etal [14]. Inclusionof W addsfurthergeneralityto themodel.
3. Initial Shape,ReferenceParabola and Expanding Foam“Density”
Themodelassumesa two-dimensionalgeometry, correspondingto aplanesectionpassedthrougha three-dimensionalfoam including thin films andPlateauborders. With the sym-metry shown in Fig. 1, only a unit cell problemneedsto be solved. For circular bubblesin hexagonalpacking,the maximum areafraction correspondingto bubblesjust touchingis�r98;�~�1V�T~K|
. Weassumeaninitial areafraction�= � �:9 . Thecorrespondingradii are � 9 and� . Theinitial unit cell is shown in Fig. 2.
While we usea circular bubble to specify the startinggasfraction�H
, the calculationemploysthesmall-slopeapproximationandactuallyusesaparabolapassingthroughtthepoint�4�������E/
thatmeetsthe30
symmetryline (at � = 0) in arightangle.Thedistance��
is changedsomewhatby this replacement.Specifically, weusetheparabola
g�n� g�]� 63�� R � g� A�6�/ O �R�176�/ratherthanthecirculararc g����3�� g����� UlA�� g� A�6�/ O �R�1F3G/where � � �E������d/<����l� g� � g�0� g��/M1Theinitial thicknessusedby thealgorithm
��is relatedto theinputvolumeor areafraction
��
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by ��M� � L�� � 9�� A�6 P �R�1SRT/Thereis a small differencemadeby using the parabolicapproximation ratherthan the
circular arc. The maximumgasfraction for circles that just touch is��9�; ~�1V�T~K|
. Thisnumberis reducedby aboutoneper cent to 44/49
;0.898when the parabolicarc is used
in the definition. This differenceis much lessthanthe differencebetweentwo- and three-dimensionalgeometrieswherethemaximum-packingvolumefraction
;�~�1F|Gc.
Thecalculationextendstheliquid domainto theright at a prescribedratewhile theliquidarea,composedof the calculationdomain
~�� � ��� �¡�$/and the smaller(shaded)30-60-
90 trianglein Fig. 2, is held constant.The gasfraction in the larger triangleincreasesas�
increases.Thereis alsoa well-definedparabolicarc that meetsthe � -axis andenclosesthesameliquid area.Theequationof this referencecurve is
g�h�#I#¢f� 63�� R g� e g� A g�]�!I#¢�j O �R�1VUK/where
g�]�!I#¢b� � 68� 3G£c � R g�]� 6�3c¤g� O 1 �R�1FcG/For an expansionthathappensslowly, without surfactanteffect, andundercertainothercir-cumstances,the referencearc will closelyapproximatethe final interfaceshape.Sincethe“film” thicknessis identicallyzero,thereferencearcwouldrepresentawindow thatwill break.
Oneof thecalculatedquantitiesthatis monitoredasthesimulation proceedsis theratio ¥ ¢definedas ¥ ¢b�@¦b§©¨«ª(¬E¨«%d\�¦f"�I(§§ �R�1SkT/where ¦b"�I(§§�� � R3 L � ���$/[� �h®���$/� R P O �R�1F|G/is the areaof theunit cell and
�>®�¡�$/is the thicknessat � �¯~
asshown in Fig. 2.¦:§©¨«ª(¬E¨«%
isa constant,independentof time, andis the areaunderthe initial curve (3.2) plus the areaofthesmallerinitial 30-60-90triangle. ¥ ¢ maybereferredto simply asthe foamdensity. Thisinterpretationwouldgive thecorrectnumericalvalueof thedensityif theliquid hasaspecificgravity of oneandthe gasdensityis consideredto be negligible. If this werethe case,thedimensionlessvalue ¥ ¢ wouldbethefoamdensityin tonnes/m° .
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4. Drawing Speed
The speedat which the right endof the computational window, asshown in Fig. 1, ismoving to theright is an input function. This is therateof stretchingof theliquid portionoftheunit cell. Weusethefunction� �¡�$/<�@�H L 6v��± ������! #" PH² �~-���v�i�����! #"&/ ��Uh176�/where
�����! #"is theprocesstime, thetime takenfor thefoamto fill theavailablespace;i. e. for�M�³�����! #"
theright endis stationary. Since�����<^=_&`
at�]�������! #"
, theconstantis
±��µ´ �]^�_a`���¶¸·¹ Ai6r1 ��Uh1F3G/Therateof expansion
�=I ���$/is theactualvariableusedastheright-endboundaryconditionfor
thesurfacespeed�¸� � ���$/ . It is
�=I����$/�� ¥ �¥ � ��.0± �������! #" L 6v��± ������! #" PH²®w J 1 ��Uh1SRT/It is assumedthatexperimentsareperformedandthe rateof volumetric expansion¥»º \ ¥ � ina laboratorycolumn,for example, is recorded.If the rateof expansion wereobserved to beconstant,for example,andthenumberof cellsdoesnot change,then,since º¯¼ � ° , theex-ponent
.would be 1/3. Perhapsmoreconsistentwith two-dimensionalmodeling,the areal
expansionratecouldbetakento beconstant,in which case.+��6�\T3
. In eitherof thesecases,the speedwould decreasewith time. In a completelypredictive model, the function
�8I��¡�$/would bea calculatedratherthanan input function. It dependson therateof gasgenerationdueto reactionand,to a lesserextent,on thefoamtemperaturehistoryusingthegaslaw.
5. Viscosification
Aqueousfoamswill continueto evolve in time indefinitely, albeitataslow rate.Industrialfoams,on the otherhand,aremadeto solidify ratherquickly. This solidification is accom-plished,within the model,by a large increasein viscosity. Thus the portion of the modeldiscussedin thissectionis only requiredfor foamsthatsolidify.
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At early timesthe liquid viscosity is relatively small. Throughoneor severalprocesses,the matrix materialbecomesa solid. Informationon the chemistryof this set-upcould beincludedin a morecomprehensive processmodel. It could involve polymerization, lossofsolvent(“drying”), thermalcuringor othereffects. Thecommonfeatureis that theviscositywill increasein time,ultimatelyby many ordersof magnitude.A solidcanbethoughtof asaliquid of infinite viscosity.
Thesimplestmodelis onein which theviscosity�
is a functiononly of time. We choosetheone-parametermodel �H�¡�$/<����>½d¾5¿ L ��&%&�(' P �4c5176�/where
�&%!�#'is calleda“drying time” andis aninputparameter. Weprescribetheratio
��%!�#'�\����E�# #"astheinputquantity. With theassumedexponential dependence,when
�is a significantmul-
tiple of�a%!�#'
, theviscositywill behighenoughthattheliquid will ceaseto moveandthefinalfilm shapeis givenby thesimulation. Weexpectthat
�E%!�#'will beseveraltimeslargerthan
�4�E�# #".
To theextendthatthis is true,oncetheextensionhasstopped,theeffectof viscosificationcanbeabsorbedby changingthetimescale.Thismeansthattheprofilefor aliquid of constantvis-cosityat any timecouldbethefinal profile for anappropriatelyselectedvalueof
��%!�#'. During
thepulling phase,on theotherhand,theprofile shapeis influencedbothby viscousdrainingandthe rateof pulling; thusthereis a nonlinear interactionbetweenthe two effects that isdeterminedby thenumericalsimulation. In generalwe expectthat rapiddrying will leadtothicker films becausecapillary-drivenbackflow into thePlateaubordersis inhibited.
With themodelin (5.1)theliquid is still Newtonian.Zhang,Davis & Macosko [6] suggestthat PU materialis shear-thinning becauseof ureaprecipitation. Shearthinning is perhapsnot theright word sinceit presupposesa shear-dominatedflow. Extensional thinning is moreappropriatefor mobile interfaces(smallsurfactanteffect) sincetheflow is largelyextensionalfor thatcase.In eitherevent,theviscosity is a decreasingfunctionof strainrateandwill thenalsovary in spaceaswell astime. This effect, usuallyreferred to asgeneralizedNewtonian[ seeBird et al [15]], canbeconsideredwithin the lubrication model. This generalizationisconsideredin section8 whereaso-calledEllis modelis addedto thenumericalmodel[16,17].
Another alternative could be called a “paint drying model.” In this case,the viscosityincreasesasa particularcomponentof the mixture, i. e. a solvent, is lost from the liquid.Sincethelossis throughtheinterface,thinnerportionscanbeexpectedto dry faster. Thustheviscosity in the thinnestportionof thefilm will increasemostrapidly. This will alsoleadtothickerfilms, webelieve. Sincetheviscosity is a functionof thebulk concentrationof agivenelementof liquid (i. e. how dry it is) andsincetheseelementsmove, it wouldbenecessarytoincludethebulk concentrationasanadditionalunknown in themodel.Thishasbeenmodeledfor thesimpler problemof paintdrying [18].
For a Newtonianor generalizedNewtonianliquid, the viscosity dependsat moston thestateof drynessor cureandtheinstantaneouslocal rateof deformation.It is possiblethatthe
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viscosity couldalsodependon theprevioushistory of themotion. Polymerizationcouldaddelasticityto the matrix material. A one-dimensional visco-elasticmodelof Maxwell-type issuitableto approximatea flow thatis primarily extensional. It will requirethatthelocal stateof strainbe monitoredalongthe film asthe calculationproceeds.This informationcanbeextractedfrom thenumericalalgorithm.
6. Mathematical Model; Differ ential equationsand boundary conditions
Threecouplednonlinearpartial differential equationsdeterminethe temporaland spa-tial variationof theprofile
�[� � ���$/ , thesurfacespeed�¸� � ���$/ andthesurfactantconcentrationD<� � ���$/ . Theseequationsarederived, startingfrom the Stokesequationsfor slow motion of
viscousfluids, andthe pressureandkinematic free-surfaceconditions, by Schwartz & Roy[5]. Themajorsimplificationemployedthereis theuseof thesmall-slopelubricationapprox-imation. Mathematicalderivationsof thebasicapproximation aregivenby Benney ([19]) andAtherton& Homsy([20]). Theboundaryconditionsusedhere,at thePlateauborderandthemoving end,are,however, quite different from the earlierwork. It may alsobe notedthatinertial, i. e. finite-Reynoldsnumber, effectsarequitenegligible for thepresentproblem.
6.1Theh-evolutionequation
Theevolutionequationfor thefilm shapeis��À��ÁAfÂÃ`Ä�ÁAZ�!�Ä�>/&`fA �RG� Å � ° ´ ��`�`�`8� o `�³¶[Æ ` 1 �k�176�/wheresubscripts signify partial differentiation. The first equality is the conditionof massconservationfor theassumedconstant-density liquid.
Âis thearealflux, definedasÂ�� � ���$/H��Ç�È�É `�Ê ÀÌË Í � � ��ÎÏ���$/ ¥ Î
wherethecoordinateÎ
is measuredfrom thefilm centerlineand Í is thevelocity componentin the � direction.Thefirst termontheright of (6.1)containstheconvectivecontribution
��[`andthedomain-stretchingeffect
�Q�=`. Thesecondtermis thepressuregradientcontribution
toÂ
andconsistsof thecapillarycomponent,usingthesurfacecurvaturegradient��`�`�`
, andthedisjoining componentusingthefunction o givenin equation(2.7).
Equation(6.1) canbe madedimensionlessusing�Ð
asthe characteristiclengthfor both�and � and
N 9 ��4RG�����E/�\)��asthecharacteristic time from (2.4). Thedimensionlessvari-
ablesareformed,of course,by dividing theoriginaldimensionalquantitiesby theappropriate
11
referencequantity. Severalphysicalparametersin theoriginal systemareseento beremovedby this transformation.Usingthenew dimensionlessvariables,(6.1)be��À[�ÒA¸�#�l�>/!`bA�Ó� ° �(��`�`�`v� o `®/&ÔÌ`Z1 �k�1F3G/
Thetwo left-end(Plateauborder)boundaryconditionsare:(i) Theslopeis givenfor all timeat � ��~ ,�h`��4~����$/��ÒAÖÕ�×)Ø[��Ù�\)kK/81 �k�1SRT/(ii) Thesmaller30-60-90trianglearea
¦ÄÚin Fig. 2 is relatedto theflux at � �¯~
which isdenotedby
Â��4~K/. Thus Â��4~T/<�ÒA ¥ ¦fÚ¥ � p�¦bÚM���$/H� � O �~����$/3 � R 1 �k�1VUK/
Themoving-endboundaryconditionsarethesymmetrycondition��`»��f�¡�$/d���$/H��~ �k�1FcG/andtheflux condition Â��4� ���$/E���$/<���[�4� ���$/E���$/v��I��¡�$/ �k�1SkT/where
��Iis given by (4.3) for
� � �4���! #" and�=I
= 0 for�l}Û�¡�E�# #"
. The initial shape�[� � ��~K/ is
givenby equation(3.1).
6.2Theequationfor thesurfacespeed
Considerfirst the simplestcase,a viscousliquid sheetbeing drawn out by a constanttensionperunit width
N. If nootherforcesarepresent,thenormalstressÜ J!J , onasheetwhose
thicknessis almostuniform,satisfiesthestress-strainraterelationÜ J!J<� N \T�n��UT����`where
Nis the (half) film tension[15]. The quantity
UT�is sometimescalled the Trouton
viscosity for this extensional process.�
is the (half) film thickness.The stretchingvelocityis zeroat thePlateauborderandis a specifiedfunctionof time
��I��¡�$/at theline of symmetry.
12
Thus,for forceequilibriumN �
constant,and�
satisfies�(�Q��`®/&`Ä�@~ �k�1F|G/andtheboundaryconditions�¸�4~5���$/H��~�� �¸����$�$/<�Ý�=I ���$/M1 �k�1V£ X ���d/Since
�Q��`is constantalongthesheetor film, it followsimmediatelythattherateof elongation
variesinverselywith thethickness�. Thusthin regionsof thefilm will thin mostrapidly and
it will notbepossibleto extendthefilm, or expanda foam,to any appreciabledegreebecausefilm thicknesseswill becomecatastrophicallysmall. In fact, it is the presenceof surfactant,andtheresultingsurfacetensiongradientforces,thatmakesfoamexpansionpossible.
Following Schwartz& Roy [5], we useinsteada morecompletemodelthat includessur-facetractiondueto nonuniform surfactant,finite capillarypressure,anddisjoining pressure.Thesurfacetractionis Ü ��Þ �Þ �within the small-slopeapproximation [8]. Using (2.2), the tractionis Ü �ßAb* D<`
andweperformaforcebalanceonanelementof film. Insteadof (6.7),thestretchspeednow satisfiesUT���(�Q��`�/&`8�������[��`�`�`8� o `�/=AB*àD0`b��~-1 �k�1S�T/In dimensionlessvariables,using
�Masthereferencelengthfor � and
�and
N 9 ���4RG�0��E/$\)��asthecharacteristictime,asbefore,(6.9)becomes�(�Q��`®/&`Ä�ÛA RU �(�n��`�`�`8��� o `�/0���4*m\®��E/$D�`:1 �4k�1Ì6�~K/We have madethe approximation
�Á;á�Ï. This is equivalent to statingthat the fractional
changein surfacetension, dueto the presenceof surfactant,is small. For systemsof smallphysical dimensions, suchas thoseconsideredhere, it is possible to have large valuesofÞ �[\ Þ � eventhoughthechangein
�is small. Notethat,since(6.10)is anelliptic ratherthan
a parabolic differentialequation,no initial conditioncan,or should,bespecifiedfor�¸� � ���$/ .
This followsfrom theconsistentneglectof inertialeffectsin thepresenttheory.
The dynamicalbehavior of the stretchingliquid films becomessimple in an importantlimit ing case.The physical reasonfor this simplification is clear. Assumea uniform initialdistribution of surfactant.If thesurfactanteffect is strong,thestretchedsurfacemustalsohaveanalmostuniform distribution, becauseany nonuniformity will give rise to a strongtractiontendingto restoreuniformity. This implies that, within the small-slopeapproximation,the
13
rateof dilation ¥ �M\ ¥T� is constantor�¸� � �$�$/l��± � , where
±is someconstant.This uniform
dilation, with afixedpointat � ��~ , is referred to mathematically asanaffinetransformation.Resultsof a separatecalculation,usingtheaffine stretchingassumption, will begivenbelowandcomparedwith resultsof the morecompletetheory. This simpleargumentneglectstheweakeningeffect of surfacediffusionof surfactant.It will bedemonstratedusingtheresultsof thecomputersimulation,shown below in section7, thatthis is thecorrectlimit ing behaviorwhenthesurfactanteffect is strong.
6.3Thesurfactant concentration convection-diffusionequation
As in Schwartz& Roy [5], thesurfactantconcentrationsatisfiesD�À0�ÁAZ�!�bD�/&`8��,+���$/]D�`�`r1 �4k�1Ì6T6�/Becausethisequationis linearin
D, theunitsof
Dareirrelevantsincetheboundaryconditions
arealsohomogeneous. TheseboundaryconditionsareD�`»�~����$/���D�`»��f�¡�$/d���$/H��~â1 �4k�1Ì6�3T/The first of thesefollows from the observation that the left end of the free surfacelies onan inclined symmetryaxis, as in Fig. 1. On that inclined axis, ¥ D�\ ¥»ã ��~
where ã is arclengthalongthe interface. Sincethe slopethereis finite,
D�`Ö�ä~thereaswell. The right-
endboundaryconditionis alsoD=`+�å~
, irrespective of whetherthe boundaryis moving orstationary, sincethis boundaryis alsoa line of symmetry. The initial condition is that thesurfactantis uniformly distributed, i. e.
D<� � ��~K/v�ÝD���Ýædç®. ã � . Without lossof generality,D=
canbetakenequalto one.
It maybeverified,usingthedifferentialequationandboundaryconditions, that the totalquantityof surfactantis conserved.Consider¥¥ � Çâè É ÀÌË D ¥K� � Ç�è D�À ¥K� ��D<��v/ ¥ �¥ � �4k�1Ì6�RK/usingLiebnitz’ rule for differentiatingunderthe integral sign. Substituting the differentialequationin theintegralon theright andnotingthat ¥ �<\ ¥ �]�Ò�¸�4�Ð���$/H�Ò�8I ���$/ , wefind¥¥ � Ç�è É ÀÌË D ¥T� �ÁAÃ�¸��v/HD<�4�</0�i�¸�~K/�Dv�~K/0��,ÒÓ©D�`»�4�</]A�D�`»�~K/&Ô»�³D<�4�</v�-��</Ð1 �4k�1Ì6�U�/But
�¸�4~T/=0 from (6.8) andthefirst andlast termson the right of (6.14)areseento cancel.
14
Thus,with theboundarycondtions(6.12),wehaveÇ è É ÀÌË D ¥T� ��D08��M�@ædç®. ã �81 �4k�1Ì6�cT/Thevalueof theconstantis onein dimensionlessvariables.
7. Somesampleresults
Figures3 through6 show the film profile�, the surfacespeed
�andthe surfactantdis-
tributionD
at a particulartime. Thesurfactantstrengthis varied,yielding a family of curvesfor eachof thesequantities. We arbitrarily selectthe time instantwhen the profile lengthhasdoubledduring the stretchingphaseof motion. In the finite-differencecalculations, thepoint spacing� is held constantasthe liquid film is stretched.Thusadditionalpointsareaddedcontinuously as the simulation proceeds.The surfactantlevel is given, in effect, byspecifying
*m\)�>which will besymbolized by ê in thefigures.Apart from this quantity, the
non-dimensional inputparametersarelistedin thefollowing table:
Table I: Parameter valuesusedto createsimulation results
0.85�
Initial gasfraction3.0
��^�_a`G\G��Maximum cell length
100.����! ("a\ N 9
Dimensionlessprocesstime0.01
qrJ�\h���d\G��E/Disjoiningprefactor
0.0001� 9 \G�� Disjoining thickness
0.1, N 9 \G� O
Surfactantdiffusion constant4.
�Stabledisjoining exponent
3.�
Unstabledisjoiningexponent0.5 W Disjoiningstabilizationconstant1.
.Exponentin stretchinglaw
�<\G�8Ð�Á�a6v� W �$/ ²10.
�a%!�#'�\®�����! #"“Drying” timeratio
Thesystemof threecoupledpartialdifferentialequationsin section6 is solved numericallyby themethodof finite differences.Thecomputercodeis written in theFortranlanguage.Thecomputation is startedusing100equally-spacedpointsto representthesurfaceordinates,thesurfactantdistribution andthesurfacespeed.Pointsareadded,asthestretchingproceeds,sothat300pointsareultimatelyusedfor eachvariable.Time marchingis doneimplicitly in thesensethata systemof algebraicequationsis solved for the incrementalchangesat eachtimestep.A completeevolutionhistory, for agivensetof inputparameters,canbeobtainedin less
15
thanoneminuteusinga 1.2 GHz desktopcomputerandthe Linux public-domainoperatingsystem.
It is instructive to suggestsomerepresentative physical values,similar to thosegiven in[7]. Assume
�H= 0.01cm,
�>= 30 dynes/cmand
�[= 100poise= 100gm/(cmsec).Then
thecharacteristictime isN 9
= 0.1secusingequation(2.4). Thetabular valuefor theprocesstimeis then10secwhichseemsto beareasonablevalue.Thecharacteristictimefor dryingorsolidification is then100sec.Thedisjoining thickness
� 9 is takento be10w»ë cm, equivalentto a 20 nm total thicknessfor the“two-sided”film. This is a satisfactoryvalue,at leastin anorder-of-magnitudesense.This valueis largeby a factorof betweentwo andfour for blacksoapfilms, however [21]. For the initial gasfraction andmaximumcell lengthgiven in thetable,thefinal gasfraction is 0.978;thusthefoam“density” is 0.022.Thelargegasfractionmeansthatthis is a “high-quality” foam.
Figure3 is asnapshotof thefilm profile,�Ï\)�v
versus� \G�H , atatimeduringtheexpansionwhenthe film lengthhasdoubled,i. e. when
�=3)�M
. Note that the vertical scaleis exag-geratedby abouta factorof three.Fourseparatecurvesareshown, correspondingto differentlevels of the surfactantstrengthparameterê ��*m\®�Ï
that appearsin equation(6.10). ê =0 is thecaseof zerosurfactantstrength;consequently, in this case,thereis no surfaceshearstressresultingfrom spatialnonuniformity in thesurfactantdistribution
Dv� � ���$/ . An additionalcurve,labelled“affine” is alsoshown. It is calculatedbasedontheassumption thatthesurfacestretchesuniformly, asdiscussedin section6.2.Thisassumption holdswhenthesurfactantef-fect is relatively strong.In thiscasethemotionmaybedecomposedinto a linearcombinationof affine stretchinganda motion thatsatisfiesa no-slipcondition on the liquid-gasinterface.Thefilm stretchingcanthenbecalculatedusingonly thesingleevolutionequation��À0�ÛA¸�#�l�>/!`bA �R)� e � ° ��`�`�` j ` �4|5176�/where
�is givenby �-� � ���$/H���^=_&`5���$/ �� ���$/ 1 �4|51F3G/
Therecognitionthatflow with asurfactant-ladeninterfacemustsatisfyano-slipcondition onthat interface,andis thereforeanalogousto film flow on a solid substrate,may be found inMyselset al. [2].
The informationin Fig. 3 is replottedin Fig. 4 usinga logarithmic scaleto moreclearlyillustratethe importanteffect of the surfactant. The film thicknessesfor no surfactant( ê =0) andthe caseof ê = 0.001canbe seento differ from oneanotherby at leasta factorof100. In fact, for ê =0, the computation would have failed, with the thickness
�going to
zeroat somepoint, hasthe relatively weakdisjoining pressureterm not beenpresentin thethicknessevolution equation(6.1). Note alsothe closeagreementbetweenthe resultof theaffinecalculationandtheprofile for ê = 0.001.It is perhapssurprisingthatthisrathermodest
16
surfactanteffect is sufficient to effectively immobilize theinterface.
Figure5 shows the interfacial speedvariation�¸� � \G�vd/ at the same“snapshot” time for
whichthe�
profilesareshown. Notethatthecalculatedspeedvariation for thelargestê valueis virtually astraightline, confirmingthevalidity of theaffinestretchinghypothesis.Notethat�
canbecomenegative, implying backflow into thePlateauborder, when ê is small. Over acertainportionof thefilm length,therefore,liquid is beingremovedin boththe forwardandbackwarddirections. This is theorigin of thethicknessminimum or “dimple” thatcanoftenpersistuntil the film hasdried. Dimples in the final dry profiles,for small ê valuescanbeseenin Fig. 7.
Figure6 shows the distribution of surfactantD<� � \G�Md/ at the samesnapshottime for the
four differentvaluesof ê . Thesedistributionsarecalculatedusingthe convection-diffusionequation(6.11). In thenumericalmodelthediffusion constant
,is reducedastheviscosity
increaseswith time, consistentwith Einstein’s law for diffusion [8]. This reductiondoesnotappearto be an important effect, however. Sincesurfactantis assumedto be conserved, theareaundereachcurve in Fig. 6 shouldbe the same. Someerror in the total quantity ofsurfactantis apparentin thefigure. This error is about2 percentandarisesbecausethefourseparaterunswereterminatedmanuallyandthe stoppingtime wasnot exactly the sameineachcase.Notethat,for thelargest ê value,thesurfactantis distributedalmostuniformly, asexpected.
Thefinal shapes,afterthefilm hasbeenpulledto threetimesitsoriginallengthandallowedto dry, are shown in Fig. 7. The threepreviously-usedvaluesof surfactantstrengthê alldisplaya thicknessminimum. Resultsfor two larger ê valuesarealsoshown, demonstratingthatthe“dimple” canbeavoidedif desired.Moreover, it is clearthat increasingê from 0.01to 0.1 hasalmosta negligible effect on the final film profile. Thusit follows that thereis amaximumwindow thicknessthatcanbeachievedusingsurfactant.Oncethis level is reached,thereis nousefulpurposeservedby furtheraddition of surfactant.
The foam expansion processis illustratedmoredramaticallyin Fig. 8. Here the initialshapesof four circularbubbles,at a gasfractionof 0.85,arecomparedwith thefinal pattern.Thevalueof ê is 0.0001.Thebubbleshapeswererecoveredfrom theunit-cell resultsby useof theappropriatereflections,translations,andrepetitions. Thefully-expandedbubbleprofileis takenfrom theoneshown in Fig. 7 for thesamevalueof ê .
8. Incorporation of certain Non-Newtonianflow effects
Equations(6.1)and(6.9)arebasedonanassumedNewtonianrheologyfor theliquid. Theviscosity changesonly globally with time to simulatesolidificationor drying. However, itis expectedthat in a PU foam,for example,the liquid phaseactuallyexhibits morecomplexrheologicalbehavior. Suchbehavior mayaffectthecell openingcharacteristicsof thefoamviaincreasedfilm thinning andconsequentwindow breakage.Duringtheexpansionof aPUfoam,
17
polymericureais known to precipitateto form a separatephasewithin thepolymernetwork.Thereis evidencethat,at somepoint during the reaction,the resultingfoammatrix consistsof asuspensionof ureaparticleswith strongshearthinning andextensionalthinningbehavior.Thelocationsof stressconcentrationwhichoccurat thedimplesof thefilm duringexpansionareaccompaniedby a local lowering of the viscosity, andhenceby a further weakeningofthefilm. Therefore,onemayconcludethatureaprecipationplaysa contributory role in cellopening.SeeZhang,Davis & Macosko [6] for furtherdiscussionandexperimental evidence.
In orderto incorporatethis effect into the model,we choosethe generalizedNewtonianEllis modeldiscussedby Bird et al. [15]. The modelcanbe expressedusingthe followingrelationshipbetweenthe(shear)viscosity
�andtheshear-stressÜ :�H� � �$�$/<� 3)��J6v�Òì Ü \ Ü Tì í w J �£�176�/
where Ü , and î areconstants. The vertical barssignify absolute value.�<J
is spatiallyconstantat any giventime andis, in fact,givenby theprevious solidificationlaw (5.1). NotethattheNewtonianmodelis recoveredby setting î �ï6
. Shear-thinningbehavior is obtainedwhen î �ß6
. Unlike the previous simplermodelgiven above, the viscosity now variesinspaceaswell astime.
It canbeshown thatevolutionequation(6.1) for thefilm shapebecomes��À[�ÒA¸�#�l�>/!`ÄA ¥ ÂÃð¥T� �KÞ ÂÃðÞ � � � °RG��JÄñ � � /0���[��`�`�`8� o `�/ �£�1F3G/wheretheprefactor ñ � � / is given by
3 ñ � � /H�Û6v� Rî ��3 L �Ü ì �[��`�`�`v� o `hì P í wJ �£�1SRT/
Againequation(6.1) is recoveredby settingî �Á6 .In orderto extendequation(6.9) to thegeneralizedNewtoniancase,we postulatethatthe
extensional viscosity is givenbyUT�
, theso-called“Troutonviscosity” for aNewtonianliquid.Equation(6.9)governingthesurfacespeednow becomesU>���Z�Q��`�/!`Ï�m�[��[��`�`�`Ï� o `�/»A�*àD�`Ã�@~5� 3)��J� �Á6�� Rî ��3 L �Ü ì �[��`�`�`8� o `�ì P í w
J �£�1VUK/wherenow theextensionalviscosity is alsoa functionof bothspaceandtime. Note that theNewtoniancasecan alsobe obtainedfrom the Ellis model in the limit Ü óò ô
. Then aNewtonianrheologyis approachedwith viscosityequalto
3)�vJ.
18
Figure8 shows theeffect of increasingtheparameterî , andhencemakingthe liquid in-creasinglynon-Newtonian,while keepingall otherparametersconstant.Parametervaluesaresomewhat different from thoseusedfor the Newtoniancasediscussedabove. Thesediffer-encesproduceonly minor quantitative changes,whencomparedwith moredramaticeffectsdueto localizedviscosity reductions.In particular, thesimulationof Fig. 8used
� ^=_&`G\G��M��U,�����! #"&\ N 9
= 800,�&%&�('�\®�����! ("
= 2.5and ê = 0.0017.Thefigureshowsfinal dry film profiles.As îis increased,theflat partof thefilm becomesmarkedlythinnerandthedimplesbecomedeeper.Thuswhenthis non-Newtonianeffect is considered,windows will have a greatertendency tobreak. It maybe thatan open-cellor partially open-cellfoamwill resultwhena closedcellfoammight otherwisehavebeenanticipated.
9. Concluding remarks
Wehavedevelopeda theoreticalandnumericalmodelfor thegrowth, shapeevolution andultimatesolidification of a foam. Although the modelembodiesmany simplifying assump-tions, we believe that it is currently capableof makingquantitative predictionsconcerningthe influenceof changesin gasinjection rates,surfactantstrength,fluid rheology, andotherprocessvariables.At a minimum, thecurrentmodelshouldserve asanappropriatestimulusfor furtherwork.
It is necessaryto pointoutsomelimitationsof thepresentmodel.It dealsprincipallywiththedescriptionof fluid dynamicbehaviorasdrivenby a known rateof gasproduction. In amorecompletelypredictive modelof anexothermallyexpandingfoam,simplifiedmodelsofthechemicalandthermalprocesses,thatcanbeusedto predictthegasproductionrate,couldbe incorporated.Thepresentassumption that the liquid volumeremainsconstantcould thenalsoberemovedandreplacedby acalculatedmasstransferhistory.
Thepresently-usedassumption, thatthesurfactantis insoluble in thebulk liquid, is felt tobeanoversimplificationthatis likely to bemoreappropriatefor soapsurfactants(fatty acids)that it is for PU foam constituents. A more completemodel could incorporatetransferofsurfactantbetweenthe interfaceandthebulk liquid. A simpleway of capturinga portionofthe effect of transferof surfactantbetweenthe interfaceandthe bulk liquid film is to useasimple “reservoir” model.A singletermcanbeaddedto equation(6.11)sothatit becomesD�À��ÁAZ�!�bD�/&`M��,õD�`�`fAB,-öE�D?ABD0ö&/ ���176�/where
,-öis a typeof “bulk diffusionconstant”and
D]öis a referenceor reservoir value.If the
interfacialconcentrationD
becomeslessthanD�ö
, thebulk suppliessurfactantto the interfaceat a rateproportionalto
,âö; converselysurfactantis lost to thebulk whenever
Di�ÁD]ö. This
changeis straightforwardto implement.This morecompletemodelhasbeenusedpreviously[22]. It maybecriticized,however, becausefor very thin films thereservoir supplyof surfac-
19
tant may be insufficient to maintaina constantvalueofD]ö
. A morecompletemodelwouldconsiderthe bulk concentration
D�öasan unknown, to be found throughthe useof another
convection-diffusionequation[23].
The significantgeometricsimplification incorporatedin the presentmodel needsto beappreciated.We have replaceda complex three-dimensional matrix structurewith a two-dimensional idealizationandhave alsopostulated a high degreeof symmetry. It is possibleto extendthe modelto incorporatecertainthree-dimensional features.Gravity drainage,forexample,wouldneedto beincluded,alonganetwork of struts,for anaqueousor slow-dryingfoam. Thenthesmall-scaleproblemtreatedherecouldstill beappropriate,providedthat theliquid massconservation conditionsare suitably modified; the total liquid areain the two-dimensional problemwould no longerbe constant.It is alsopossible in principle to replacethepresentNewtonianor generalizedNewtonian liquid with a moregeneralmodelthatwillallow theviscosity to vary in spacein responseto changesin boththerateof deformationandthe local deformationhistory. This latterdependenceis usuallyreferred to asviscoelasticity.Viscositycanalsobemadeto vary in spaceasafunctionof theevolving mixturecomposition.Notethattheinitial sizeof anexpandingbubbleis assumedto beaknown quantity. Wearenotcurrentlyattemptingto modeltheinitial stagesof nucleationandgrowth of bubbles,includingany possiblegaseousexchangemechanismsuchasOstwaldripening, thatcanoccurthen.
Onegoal is the identificationof unrealisticassumptionsin thepresentmodelin ordertosuggestthe mostimportantmodelimprovements.Model predictionscanbe comparedwithmicrographsof actualexpandedfoams.Variousdiagnosticexperiments thatcanbedesignedandperformedto validateandimprove the presentmodel. We believe that the model is, orwill shortly become,capableof yielding improved understandingthat canleadto improvedprocessingandnew products.Benefitsmayincludethepossibility of very low densityfoams,aswell aspotentialcostsavingsby careful controlof expensive mixtureconstituentssuchassurfactants.
Acknowledgment
This work wassupportedby ICI andHuntsman-ICI Polyurethanes.Thesupport,andper-mission to publishtheseresults,is gratefullyacknowledged.
References
[1] D. Weaire,S.Hutzler, G. Verbist,andE. Peters.A review of foamdrainage.Adv. Chem.Phys., 102:315–373,1997.
20
[2] K. Mysels,K. Shinoda,andS. Frankel. SoapFilms, Studiesof Their Thinning. Perga-mon,New York, 1959.
[3] L. W. SchwartzandH. M. Princen.A theoryof extensional viscosityfor flowing foam.J. Coll. InterfaceSci., 118:201–211,1987.
[4] L. W. Schwartz, R. A. Cairncross,and D. E. Weidner. Anomalousbehavior duringleveling of thin coatinglayerswith surfactant.Phy. of Fluids, 8(7):1693–1695, 1996.
[5] L. W. Schwartz andR. V. Roy. Modeling drainingflow in mobile andimmobile soapfilms. J. Coll. InterfaceSci., 218:309–323,1999.
[6] X. D. Zhang,H. T. Davis,andC.W. Macosko. A new cell openingmechanismin flexiblepolyurethanefoams.J. Cellular Plastics, 35:458–476,1999.
[7] G. Woods.TheICI PolyurethanesBook. JohnWiley, Chichester, 1990.
[8] L. D. LandauandE. M. Lifshitz. Fluid Mechanics. PergamonPress,Oxford,1959.
[9] J.N. Israelachvili.IntermolecularandSurfaceForces. AcademicPress,London,1992.
[10] N. V. Churaev and V. D. Sobolev. Predictionof contactangleson the basisof theFrumkin-Derjaguinapproach.Adv. in Coll. andInterfaceSci., 61:1–16,1995.
[11] V. S.Mitl in andN. V. Petviashvili.Nonlineardynamicsof dewetting: Kinetically stablestructures.Phy. Let.A, 192(5-6):323–326,1994.
[12] L. W. SchwartzandR. R. Eley. Simulationof dropletmotionon low-energy andhetero-geneoussurfaces.J. Coll. InterfaceSci., 202:173–188, 1998.
[13] A. Sharma.Equilibrium anddynamicsof evaporating or condensingthin fluid domains:thin film stability andheterogeneousnucleation.Langmuir, 14:4915–4928,1998.
[14] G. F. Teletzke, H. T. Davis, and L. E. Scriven. Wetting hydrodynamics. Revue dePhysiqueAppliquee, 23(6):989–1007,1988.
[15] R. B. Bird, R. C. Armstrong,andO. Hassager. Dynamicsof PolymericLiquids. Wiley,New York, 1977.
[16] D. E. Weidner. Numerical simulation of coating flows. PhD thesis,University ofDelaware,1993.
[17] D. E. WeidnerandL. W. Schwartz. Contact-linemotionof shear-thinningliquids. Phy.of Fluids, 6:3535–3538,1994.
[18] M. H. Eres,D. E. Weidner, andL. W. Schwartz. Three-dimensionaldirect numericalsimulation of surface-tension-gradienteffectson the leveling of an evaporatingmulti-componentfluid. Langmuir, 15(5):1859–1871,1999.
21
[19] D. J.Benney. Longwaveson liquid films. J. of Math.andPhy., 45:150–155,1966.
[20] R.W. AthertonandG.M. Homsy. Onthederivation of evolutionequationsfor interfacialwaves.Chem.Eng. Comm., 2:57–77, 1976.
[21] I. B. Ivanov. Thin liquid films; fundamentals and applications. Marcel-Dekker, NewYork, 1988.
[22] P. L. Evans,L. W. Schwartz, andR. V. Roy. A mathematicalmodel for craterdefectformationin adryingpaintlayer. J. Coll. InterfaceSci., 227(5):191–205,2000.
[23] V. G. Levich. PhysiochemicalHydrodynamics. Prentice-Hall,Inc., Englewood Clif fs,1962.
22
Gas
ComputationalCell
x
L(t)
Figure 1: Definition sketch of stretchedfilms and Plateaubordersat time t. Initially thecomputational cell was smallerand the initial length was
� �4~K/à� ��. At the initial time
the gasbubbleshapeswere circular arcs. The six-fold symmetryis maintained during thestretching.
23
ho
L o
b o
gas
liquid
R*
R
30o
o
Figure 2: Initial (�
= 0) configurationof the unit cell. � is the startingradiusof a two-dimensionalbubble.
�v� � d\T3 is usedasthereferencelengthin dimensionlesscalculations.� 9 is themaximum circular radiusof bubblesthat just touch. Thestand-off distanceis�T�� 9HA � .
24
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.5 1 1.5 2
h/L
x/L
0.00010.001
0.00001G=0
0
0
affine
Figure3: Dimensionlessfilm thicknessversuslengthfrom border. This is a snap-shotat thetime instantwhen
�<\G�vb�ï3. Four differentsurfactantlevelsarecomparedcorrespondingto
differentvaluesof ê . Theprofile for thehighestsurfactantlevel, ê �õ~�1V~T~�6, is closeto the
one labelled“affine.” The affine profile is calculatedusingthe assumption that the surfacespeedis givenby Í \ Í ^�_a`l�C� � \G�</ where Í ^=_&` is a constantand
���� ���$/. Accordingto the
presentmodel,the affine caseis a “hardened”or no-slip interfaceandproducesthe thickestpossiblefilm. Reducingthestrengthof thesurfactanteffect resultsin thinnerfilms.
25
0.0001
0.001
0.01
0.1
1
0 0.5 1 1.5 2
h/L
0
G=00.000010.00010.001affine
x/L0Figure4: Film thickness
�Ï\)�vversusposition � \G�v asin thepreviousfigure.Herea log scale
is usedto moreclearlyillustratethethicknessvariationof thefilm andhow it dependson thesurfactantlevel ê . The thinnestfilm would eventually breakwere it not for the disjoiningpressurethatis includedin themodel.
26
-0.01
-0.005
0
0.005
0.01
0.015
0.02
0 0.5 1 1.5 2
x/L0
0.00010.001
G=00.00001
U
Figure5: Dimensionlesssurfacespeed�
versus� \)�8 for severalsurfactantlevels ê .�v\)�HÐ�3
atdimensionlesstime��;�~�1Sc
correspondingto the�
profilesshown in Figs.3 and4. For thethehighestsurfactantlevel shown, theprofile is virtually linear, justifying theso-calledaffineassumption. For littl e or no surfactant,thereis backflow towardsthe Plateauborder. Thisbackflow canevenoccurduringtheexpansion phase,asdemonstratedhere,if thesurfactanteffect is quiteweak.Whenbackflow occursduringtheexpansionphase,theflow mustchangedirectionat a certainposition. This mayleadultimatelyto formationof a thicknessminimumor “dimple” in thefinal profile.
27
0.45
0.46
0.47
0.48
0.49
0.5
0.51
0.52
0.53
0 0.5 1 1.5 2
x/L0
Γ
0.00001G=0
0.00010.001
Figure 6: SurfactantdistributionD
versus � \G�v for different levels of surfactanteffect;�<\G��m��3. The total quantityof surfactantis normalizedto one. Slight differencesin the
averagevalueshown herearebecausethe runswerestoppedat slightly differenttimes. Forthestrongestsurfactanteffect shown, thesurfactantdistributioncanbeseento bealmostuni-form.
28
0.0001
0.001
0.01
0.1
0 0.5 1 1.5 2 2.5 3
h/L
x/L
0
0
G=0.0001
G=0.001
G=0.1 G=0.01
G=0.00001
Figure7: Final (dry) profile shapesfor the parametervaluesgiven in TableI. Note that thefilm thicknessscaleis logarithmic. The surfactantstrengthparameterê �å*m\)��
is variedfrom 10w»÷ to
6�~ w J . Increasingthe surfactantstrengthincreasesthe final window thickness.The“dimple” is removedfor ê valuesof 0.01or greater.
29
-10
-5
0
5
10
-15 -10 -5 0 5 10 15
-10
-5
0
5
10
-15 -10 -5 0 5 10 15
Figure8: Four adjacentbubblesconstructedfrom reflectionsandrepetitionsof the unit-cellshapefunction. Parametervaluesaregivenin TableI; ê = 0.0001.Top: Initial statewith gasfraction
�= 0.85. Bottom: Final stateof expandedfoam. Two Plateaubordersareseenat
the interstitial cornersof thealmost-hexagonalbubbles.Thegasfraction is now 0.978. Thelengthunit in bothpicturesis
�M.
30
1
10
0 50 100 150 200 250 300 350 400
h [m
icro
n]
x [micron]
= 100 [poise] = 10 [dynes/cm^2]
= 1.0 = 1.2 = 1.4 = 1.8
α
µτ0
0
Figure9: Final(dry) film profilesusingshearandextensional thinningrheology. Themodifiedmathematicalmodel is given in section8. For î = 1, the flow is Newtonian. Increasingîreducestheviscosityin thosepartsof thefilm wherethestressesarelarge.Thinner“windows”anddeeperdimplesaretheresultof this viscosityreduction.Additional parametervaluesaregivenin thetext. Notethatboth � and
�aremeasuredin micronshere.
31