Studying foam dynamics in levitated, dry and wet foams using diffusing wave spectroscopy

Studying foam dynamics in levitated, dry and wet foams using diffusing wave spectroscopy

Nathan Isert

^a

, Georg Maret

^a,∗

, Christof M. Aegerter

^b

aFachbereichPhysik,UniversitätKonstanz,Universitätsstrasse10,78457Konstanz,Germany

bPhysik-Institut,UniversitätZürich,Winterthurerstrasse190,8057Zürich,Switzerland

h i g h l i g h t s

•Coarseningdynamicsoffoamsstud- iedbymulti-specklediffusingwave spectroscopy.

•Magnetic levitation allows a large rangeofliquidfractionstobestudied forhours.

•CrossoverfromNeumanntoOstwald regimeatliquidfractionaround30%.

•Indry foamsintermittentburstsof activitysignalcollectivebubblerear- rangements.

•Inwetfoamsbubblesmoveballisti- callywithrandomcollisions.

g r a p h i c a l a b s t r a c t

a r t i c l e i n f o

Keywords:

Foamdynamics Levitation

Diffusingwavespectroscopy

a b s t r a c t

Weusediffusingwavespectroscopytostudythemicroscopicdynamicsoffoams.Thesefoamsarelev- itateddiamagnetically,suchthatveryhighliquidfractionscanbeachieved.Wefindthatatlowliquid fractionthedynamicsisdominatedbylocalrearrangements,whereasathighliquidfractionthemove- mentofbubblesisballisticandlargescalerearrangementsareabsent.Thischangeinthemicroscopic dynamicscoincideswithachangeinthescalingofcoarseningonincreasingtheliquidfractionthatwe havefoundearlier.

1. Introduction

Thedynamicsoffoamshasbeenstudiedintenselyduetothe factthattheyareinterestingmodelsystemsfornon-equilibrium dynamicsaswellasimportantinseveralindustrialapplications [1].Howeverduetothedrainageoffluidinresponsetogravity[2], thesestudieshavebeenmostlylimitedtofoamscontaininglittle liquid,i.e.atliquidfractionsbelow20%[3–5,7,6,8,9].Recently,we

∗Correspondingauthor.Tel.:+497531884151;fax:+497531883090.

E-mailaddress:Georg.Maret@uni-konstanz.de(G.Maret).

havestudiedfoamsatveryhighliquidfraction(upto50%)using diamagneticlevitationtocircumventtheproblemofdrainage.In thisway,thelifetimeofwetfoamswasstronglyincreased[10].

Alternatively,experimentsareplannedtobecarriedout onthe internationalspacestation,whichwouldalsoallowthestudyof verywetfoams[11].

Thestudyofwetfoamsisofparticularinterestbecauseatliquid fractionsaround30%atransitioninthefoamdynamicsispredicted duetothefactthatattheseliquidfractionsthegasbubblesstart tolosetheircontacts[11–13].Thisnotonlychangesthedynamics ofgasexchangebetweenbubbles,whichisofcrucialimportance forthecoarseningbehaviour,butalsothemovementofbubblesis

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qualitativelychangedleadingtoaliquidlikebehaviourofthefoam.

Here,wewilladdressthequestionofthelocalbubbledynamicsin diamagneticallylevitateddryandwetfoamsusingdiffusingwave spectroscopy (DWS)[14,15,8].In particular usingmulti-speckle DWS[18]basedonafastcamera,wearestudyingthetimedepend- enceoflocalmovementofthescatterersontimescalesofseveral seconds[17].Wefindlargelocalrearrangementsindryfoams[8]

contrastingwithamorecontinuousdynamicsinverywetfoams.

Before discussing the results, we will discuss the basis of diamagnetic levitationand the potentiallandscape for residual accelerationsinoursetup[19,20],aswellasthefundamentalsof DWS withanemphasis onmulti-speckleDWS[14,15,17]. Then wewillpresentresultsontheaverageddynamics,whichcorre- spondswellwithourearlierresultsonthetransitionincoarsening dynamics[10].Finally,wewilldiscussthespatio-temporallylocal dynamicsforbothwetanddryfoams.

2. Diamagneticlevitation

Duetothediamagnetismofwateritispossibletostablylevi- tateitinastrongenoughmagneticfieldgradient.Asdescribedin [21,22],twoconditionshavetobefulfilledinordertoachievea stablelevitation.Thesecorrespondtoapotentialminimuminboth thevertical(z)andtheradial(r)directions.Thispotentialenergy densitylandscapecanbeobtainedforagivensolenoidtobe:E_pot(r, z)=₀H²(r,z)+gz.Here,isthemagneticsusceptibilityand isthedensityofthematerialstudied.Inordertohaveaminimum inthispotentialenergy,themagneticsusceptibilityhastobeneg- ative,suchthatonlydiamagneticsubstancescanbelevitated.In addition,levitationisachievedatacertainheight(determinedby theratioofto),wheretheradialgradientofthemagneticfield ofthesolenoidhastobepositive.Inourexperiments,weusea superconductingsolenoidcapableofreachingamaximumfieldof 18Tandaverticalroomtemperatureborewithadiameterof4cm [19].Thefielddistributionofthesolenoidleadstopotentialenergy landscapesforlevitationshowninFig.1[19,20].Theresidualaccel- erationsinthecentralareacanbeeasilyobtainedfromtheslope ofthisplotandarelessthan0.001ginalldirectionsintheinner- mostvolumeof1cm³,i.e.betterthanthejitterinparabolicflight experiments.Thereforeweuseasample-cellofaheightof1.2cm andadiameterof1.7cmtoensureahomogeneouslevitationofthe entiresample.

Thesamplesweuseconsistofamixtureofwaterandsodium dodecylsulfate(SDS),containing8%ofSDSbyweightthatisfoamed withnitrogengasusingtwoconnectedsyringesconnectedbyathin tube[23,19].Thefoamsthuscreatedhaveanaverageinitialbubble sizeof100␮mandalargepolydispersityofmorethan50%,ascan beseeminFig.1ofRef.[10].Theliquidfractioniscontrolledby theinitialratioofgastoliquidinthetwosyringes,whichcanbe controlledtoanaccuracyof1–2%.Onfoamingthesample,thegasis compressed,leadingtoasystematicunderestimationofthefoams liquidfractionfromthecontrolledinitialliquidfraction.Theliquid fractionsquotedinthefollowingalwaysrefertotheinitialliquid fraction.Fromanestimationofthepressureexertedonthegasafter foamingtheunderestimationoftheliquidfractionisaround5%.

3. Diffusingwavespectroscopy

Diffusingwavespectroscopyisaversionofdynamiclightscat- teringthatcanbeappliedtomultiplyscatteringsamplesandgives informationaboutmovementsonthenm-scaleinsuchsamples [14,15].Forthispurpose,weilluminatethesamplewithaCoher- entVerdi CW laserat a wavelengthof 532nmand a powerof 100mW.Thetransmittedlight iscollectedwithanopticalfibre andaphotomultiplierandacorrelator,whichisaveragedovera

-40 -20 0 20 40 60 80 100 120 140

0.0 0.5 1.0

U(m2/s2)

z (cm)

-10 0 10

-0.0010 -0.0005 0.0000 0.0005 0.0010 0.0015 0.0020

U(m2/s2)

r(mm)

Fig.1. Potentialenergylandscapeinthevertical(z)andradial(r)directions.Ascan beseen,uptodistancesofafewmminbothrandz,theresidualaccelerationsfrom thesepotentialsarelessthan0.001g.

fewminutesforsinglespecklemeasurementsandwithaLaVision highspeedstar5(HSS5)cameraataframerateof1000fpscaptur- ingthedynamicsofseveralspecklesforatimeframeofupto10s.

SinceDWSisamultiplelightscatteringtechnique,theinformation obtainedonthescattererdisplacements,whilebeingonverysmall spatialscales,isaveragedoverthewholesample.Insinglespeckle measurements,itisalsoaveragedoverlongtimesinordertohave reliablestatisticsfordeterminingthecorrelationfunction g₂(t)=Aexp(−2k²ır(t)²L²/l^∗2)+1, (1) wherek=2/isthewavenumberoftheincidentlight,l^*isthe transportmeanfreepathofthelightandListhesamplethick- ness[15].ır(t)²isthemeansquaredisplacementofthescatterers, whichisgivenbyır(t)²=6DtincaseofBrownianmotionwithdif- fusionconstantDandbyır(t)²=(l^∗k)²/30·t² forshearflow withshearrate[16].

In ordertoaccessfastrearrangementsin a foam,wherethe dynamicsofthesampleonlytakesplaceinshortbursts,thetempo- ralresolutionofasinglespeckleDWSexperimentisnotsufficient.

Forthispurpose,weusemulti-speckleDWS,wherewefollowmany specklesusingafastcamera,asshowninFig.2[17].Averagingthe correlationfunctionsofmanyspecklesspatially,eachindividual pixelistreatedasasinglecorrelatorforwhichtemporalaveraging canbereducedtojust onesecond,whilespatialaveragingpro- videsthestatistics.Tostudyfastdynamicssuchastheyareinduced bybubblerearrangementeventstherecordedframeswerepost- processedusingthismultispecklestatisticswithashiftingstarting timeandaconstanttimeaverage.Thechangeindynamicscanbe attributedtotheshift,whichinthiscasewas50frames,i.e.50ms.

Fig.2.Specklepatternofafoamwith20%liquidfractionatanageof100minviewedwithahighspeedcamera.Thedifferentimagesshowsubsequentframesspacedby 10msofthesamespecklepattern.Thiscancoverthedynamicsofthechangesinthespecklepatternasevidencedbythecircledspecklethatappearsanddisappearsinthe coveredtime-frame.

SDS fo!IITI 34% liquid fraction

0.8

02

Pig. 3. Correlation function from multiple speckles obtained with a fast camera. The different curves correspoo'ld to different times in the foam on the scale of seconds showing large fluctuations In the temporal dynamics of the scarterers.

Therefore it becomes possible to identify large rearrangements of bubbles in dl)' foams. These variations can be seen in Fig. 3, where the correlation functions averaged over many speckles taken at dif- ferent but small averaging times reveal large fluctuations of the DWS correlation time

t'o.

Before we discuss these fluctuations of

t'o

we turn first to the ageing behaviour of

t'o

averaged over longer times

(but still short compared to the ageing time scale).

4. Results

As

discussed above, it is predicted that foams show a qualita- tive change in the coarsening dynamics with increasing the liquid fraction above roughly 30%. We have shown this by studying the diffuse transmission from levitated foams as a function of foam age and liquid fraction, where we have found a transition between two distinct coarsening regimes [

1 0].

The diffuse transmission, giving a measure of the mean free path (

(24] and hence the mean bub-

ble size r

[2S,26]

shows a change in age scaling exponent from

1/2

to 1/3, which needs several decades of dynamic range to be separated experimentally. Therefore at intermediate liquid frac- tions, only an intermediate exponent could be determined

[10].

When studying the local dynamics via DWS and the correspond-

ing decay time of the correlation function, the spread in scaling exponent is much bigger. As can be seen from Eq.

( 1 ), the charac-

teristic decay time •o depends on ( 2 and 1/(8il). In the dl)' case the dynamics is Brownian,

(8il(t)) oct,

probably due to tiny ran- dom motions originating from thermally fluctuating film surfaces and/or random relaxation events due to distributed stresses [1 0].

At higher liquid fractions (8il(t)) progressively crosses over to bal- listic motion

(8il(t)) oc

r 2. probably due to the progressive onset of vel)' slow convective motions which give rise to small relative displacements of individual bubbles [10]. Note that, whatever the shear rater. diffusive motion will always dominate (8r2(t)) at times t«r,R:D/(f

²r²) (16]. Therefore, as small (convective) shear may

appear more easily at higher liquid fractions, the temporal cross over of (8il(t))

<X t²

at large t to

(8il(t))

oct at small t moves with increasing liquid fraction through the time window of the DWS experiment set by (Sr

²(t)) R: (2_f(Lk)2.

We thus expect that r

₀

crosses overfrom r

₀ ocl"²/(k²L²D)(dl)' case) to r₀R: rc(wet case) and only

at higher shear rates when r, moves out of the small t side of the DWS time window we expect r

₀ oc 1/(krL).

In

order to arrive at the scaling of

t'o

with age, we use the fact that I' scales with bubble size r. Therefore, in the diffusive regime, r

₀

scales as r

l

and hence

t'D

oc age3/2 for dry foam.

For wet foams

and r

₀ R:

t'c we obtain

t'o

oc i'2/(Dk2L2) oc

r2

ocage2/3. Such a change in age-exponent from 3/2 to2/3 is indeed observed in Fig. 4,

where the

0,1

:§:

0,01

~0 1E-3

• 9%

• 15%

18%

• 22%

27%

• 30%

32%

• 38%

• 41'%

43

10 100 1000 10000

Foam age (min)

F'ig. 4. DWS correlation times obtained from integrating g2 for several foams as a function of foam age. At low liquid fraction, the correlation time scales with age³1² (full line), whereas at high liquid fraction it scales as age²1³ (dashed line). Around liquid fractions of 30~. close to the transition. two regimes corresponding to the limiting cases can be seen.

age dependence of the DWS correlation time is shown for several foams ranging in liquid fraction from 9% to 47%. At intermediate liquid fractions (between 27% and 32%), the two regimes of scaling can be seen with distinct exponents as

t'c

crosses the DWS time window.

The results of Fig. 4 come from single speckle DWS and thus average the dynamics over long time frames. As discussed above, when averaging over shorter times, large fluctuations can be seen, which we study in the following using multi-speckle DWS

[17].

Here, we obtain the decay time from determining the correlation

function from

several speckles and averaging those over a

time frame of SO ms. With this we obtain good enough statistics for a determination of g2 and the corresponding decay time on a SO ms time scale. Since we obtain these data from movies of up to a minute, we can study the time dependence of the decay time that in

Fig. 4 was averaged over completely. This is shown in Figs. S and 6

for two foams with liquid fractions of 19% and 34%, respectively.

In addition,

the time dependence of the decay time is shown for several foam ages. showing the effect of coarsening on the dynam- ics. For comparison, the figures also contain the average decay time, which shows the same dependence as that from single speckle DWS shown in

Fig. 4.

Comparing the dynamics for the two different liquid fractions, one observes that for relatively dry foams, the dynamics is dom-

inated by large, intermittent bursts

of

activity, in between which

the dynamics is very slow, corresponding to an almost stational)'

~ 0,1

..

0 Cll

E

>.

a) 0,01

(,) Cll

"0

1E-3

0 10 20 30 40 50 60

time (s)

F'ig. 5. Temporal dependence of the DWS decay time for a foam of 19% liquid frac- tion for several ages of the foam. In addition to an overall increase of the average decay time due to the coarsening of the foam, a marked change in the dynamics can be seen. At late age, the dynamics Is dominated by intermittent burst of activity cor- responding to large scale rearrangements of bubbles. which decrease in frequency as the foam coarsens.

0 20 40 60 1E-3

0,01 0,1

575min 357min 131min 30min

decaytmeτD(s)

time (s )

Fig.6. TemporaldependenceoftheDWSdecaytimeforafoamof34%liquidfraction forseveralagesofthefoam.IncontrasttotherelativelydryfoamofFig.5,the dynamicsisherenotdominatedbysingleevents,butratherbyrandomfluctuations.

Inaddition,thetimescaleofthedecaytimeismuchfaster,connectedtotheballistic motionofbubblesbetweencollisions.

1 10 100 1000

0,01 0,1 1 10

numberofbubblerearangements

foamage(min) SDSfoam19%liquidfraction

Fig.7.Therearrangementfrequencyinafoamof19%liquidfraction.Thefrequency scaleswiththefoamage^−1/2,asindicatedbythesolidline.Therearrangement frequencyisgivenins⁻¹

foam.Thefrequencyoftheseburstsdecreaseswithageofthefoam suchthatintheoldestfoaminFig.5,theburstsofactivityarewell separated.Thefrequencyofsuchlargerearrangementsinthefoam canbeobtainedfromthistimedependenceandisshowninFig.7 asafunctionofage.Thestraightlineindicatedinthefigurecor- respondstoascalingoftherearrangementrate∝age^−1/2,which issimilartowhatDurianetal.[27]foundinshavingcream.This wouldcorrespondtovonNeumanncoarseningdynamicswhenthe overalldynamicsprobedbyDWSissolelyduetolargescalerear- rangements,whichgiverisetoanoveralldiffusivedynamicswith ır²∝tasisindeedobserved[10].

Incontrast,thedynamicsofthedecaytimefor wetfoamsis markedlydifferentas therearenolargescalebursts ofactivity andtheoverallheterogeneitiesarerathersmooth.Also,theoverall increaseinthedecaytimeismuchslowerasgivenbytheslower coarseninglawvalidinthewetlimit.Theoveralldynamicshere israthergivenbytheballisticmotionofbubblesinbetweencolli- sionevents,whichtakeplaceattheshorttimescalespickedupby DWS[10].Thustherearenolargescalerearrangements,whichis clearlyreflectedinthetimedependenceofthedecaytimesandno rearrangementfrequencycanbedetermined.Again,thisdynam- icspointsstronglytoastateofthewetfoam,wherebubblesare separatedandperformballisticmotion.

5. Conclusions

Inconclusion, wehave shownthat thecoarseningdynamics offoamsofvariousliquidfractionscanbestudiedingreatdetail usingmulti-speckleDWS.Thefoamsarelevitatedallowingalarge rangeofliquidfractionstobestudied,inparticular,wecanstudy foamsofliquidfractionscorrespondingtoseparatedbubblesrather thansolid foams.Aswe havefoundpreviously[10],ata liquid fractionofabout30%,thecoarseningdynamicsshowsamarked changecorrespondingtotheseparationofbubbles.Herewestudy thistransitioninthelocaldynamics,wherewefindthatthedecay timeofg2 shows achangeinthedynamicsasfoundpreviously betweenavonNeumannlike[12]toanOstwald-likeregime[13].

Atintermediateliquidfractionhowever,thelargerspreadinthe correspondingscalingexponentsallowsforadeterminationofa cross-overindynamicsinasinglefoamwithage,astheincreasein bubblesizewithcoarseningwillleadtoaneventualcontactofthe bubblesandthereforeachangeinthecoarseningdynamics.

Inaddition,wehavestudiedthenon-localdynamicsatinterme- diatetimescalesusingmulti-speckleDWS[17],wherewefindthat atlowliquidfractions,thedynamicsisdominatedbyintermittent burstsofactivity,whichweassociatetolargescalerearrangements ofbubbles,whosefrequencydecreaseswithfoamage[27].Forwet foams,incontrast,nosuchrearrangementscanbeseenandthe overalldynamicscorrespondstothatofballisticallymovingslowly convectingbubbles.

Acknowledgments

ThisworkwasfundedbyDFGinthecontextoftheIRTG667on SoftMatterPhysicsofModelSystemsaswellastheLandesstiftung Baden-WürttembergandtheSwissNationalScienceFoundation.

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