Life and death of not so "bare" bubbles

4.4 MB·

by  Lorène Champougny

Life and death of not so “bare” bubbles
Lor`ene Champougny1, Matthieu Roch´e1, Wiebke Drenckhan1, and Emmanuelle. Rio1
1Universit´e Paris-Sud, Laboratoire de Physique des Solides, UMR8502, Orsay, F-91405
January 16, 2017
7
1
0 Abstract
2
Inthispaper,weinvestigatehowthedrainageandruptureofsurfactant-stabilisedbubblesfloatingatthesurface

n of a liquid pool depend on the concentration of surface-active molecules in water. Drainage measurements at the
a apex of bubbles indicate that the flow profile is increasingly plug-like as the surfactant concentration is decreased
J
from several times the critical micellar concentration (cmc) to just below the cmc. High-speed observations of

3 bubbleburstingrevealthatthepositionatwhichaholenucleatesinthebubblecapalsodependsonthesurfactant
1 concentration. Onaverage,theruptureisinitiatedclosetothebubblefootforlowconcentrations(¡cmc)whileits

locus moves towards the top of the bubble cap as the concentration increases above the cmc. In order to explain
]
t thistransition,weproposethatmarginalregenerationmayberesponsibleforbubbleruptureatlowconcentrations
f but that bursting at the apex for higher concentrations is driven by gravitational drainage.
o
s 1 Introduction
of the nucleation point influences the dispersion of these
.
t aerosols in the surrounding atmosphere.
a
m Numerous natural and industrial processes involve the Bubble drainage and rupture are a priori coupled
presence of bubbles which rise to a liquid surface, where since a thinner film is expected to be more prone to
-
d theydrainandfinallyrupture. Forexample,despitetheir bursting[11]. For surfactant-free films, for example, Vrij
n
small size compared to typical geophysical scales, bub- & Overbeek[12] showed that the typical time required
o
bles floating at the surface of the oceans can have sur- for a van der Waals driven thickness instability to grow
c
[ prisingly large effects at the scale of the entire planet by is proportional to the film thickness to the power of 5,

mediating mass transfert between the sea and the atmo- thusleadingtotheearlierruptureofthinnerfilms. Inthe
1
sphere. Thetinydropletsexpelleduponbubblebursting absenceofsurfactants,i.e. inthecaseof“bare”bubbles,
v
0 were observed to contribute significantly to the global theliquid/airinterfacesarestress-freeandthefilmthick-
1 productionofseasprayaerosols,whichinturnaffectsthe ness h at the apex is found to decay exponentially with
6 climateonEarth[1]. Conversely,bubbleburstingcanalso time[4,6,7]. Asexpectedfromintuition,viscoussilicone
3
inject surface material into the water column[2]. At a oil bubbles were observed to puncture spontanously at
0
smallerscale,theaerosolsproduceduponbubblerupture the apex, where the film is the thinnest[4].
.
1 at the surface of gasified beverages were shown to have The presence of surfactants, even in tiny amounts, al-
0
a strong impact on the drinker’s sensations since they tersthebubblethinningdynamicsinawaythatisnotyet
7
1 contain most of the flavors[3]. The drainage and subse- fully understood. On the one hand, interfacial stresses
: quent bursting of bubbles at the surface of a liquid pool due to surface tension gradients or surface viscosity[13]
v
havethereforeattractedsomeattentionintheliterature, can arise and slow down the drainage dynamics. On the
i
X mostly in the case of “bare” viscous bubbles[4, 5, 6, 7] other hand, additionnal thinning mechanisms, such as
r or in the presence of minute quantities of surface active marginalregeneration,canappearinthepresenceofsur-
a
agents[8, 9] mimicking seawater composition. factants. Marginal regeneration, as introduced by My-
The rupture scenario of bubbles floating at a liquid sels et al.[14] consists in the creation of thin “pinching”
surface can be summarized as follows. Due to gravity or zones in a film, where it connects to a meniscus. The
capillarysuction,theliquidflowsdowninthebubblecap, origin of these pinching regions remains unclear: they
leading to a thinner and thinner film. A hole eventually were shown to arise from capillary suction in a film with
nucleates in the film and propagates. The rim surround- a rigid boundary condition at the interfaces[15], but are
ing the hole can destabilise into small droplets, which not expected in films with stress-free interfaces[16]. It
are ejected, or even fold and produce daughter bubbles was also proposed that marginal regeneration may stem
by air entrapment[10]. The question of when and where fromasurfacetensiongradientbetweenthefilmandthe
abubbleburstsisrelevanttoaerosolgenerationsincethe meniscus[17, 18]. Whatever its source, marginal regen-
bubblelifetimesetstheaveragecapthicknessatbursting, eration contributes to the overall film thinning by the
and thus the aerosol size distribution, while the position rising of thinner patches generated in the vicinity of the
1
meniscus[14] and is thought to play an important role in
the drainage of bubbles made from very diluted surfac-
tant solutions[8, 9]. Additionally, the observation that
tap water bubbles preferentially puncture close to their
foot – i.e. where the thinner patches are produced – led
Lhuissier & Villermaux[8] to suggest that marginal re-
generationmaybeattheoriginofbubblebursting. Nev-
ertheless, it is still unclear how the location of bursting
dependsontheconcentrationofsurface-activeimpurities
in water.
In this article, we investigate the influence of sur-
factant concentration on the drainage mechanism in
surfactant-stabilised bubbles and its relation to film
white light source
bursting. Weperformexperimentsonbubblesmadefrom
TTAB (tetradecyl trimethylammonium bromide) solu-
tionsofvariousconcentrations,bothaboveandbelowthe optical
fibre
criticalmicellarconcentration(cmc). Thecorresponding
experimental setup is presented in Section 2. In Section spectrometer
3,wereportourmeasurementsonthedrainagedynamics
lens
at the apex and interpret them using an extrapolation
h
length b, which characterizes the flow profile resulting
air
from the stress balance at the liquid/air interfaces. Sec-
tion 4 contains our resultson the position ofthe rupture
nucleationpoint, whichweputintoperspectivewiththe plastic soapy solution
cylinder (a)
drainage measurements, before concluding in Section 5.
2 Experimental setup
2.1 Bubble generation
Figure1: (a)Schemeofthesetupusedtostudyairbub-
blesatthesurfaceofaliquidpool. Thebubbleofvolume
The surfactant used in this study is TTAB (tetradecyl
V is generated by injecting air just underneath the liq-
trimethylammonium bromide, purchased from Sigma-
uid surface with a controlled flow rate Q . Thickness
air
Aldrich), a cationic surfactant of critical micellar con-
measurements at the apex are performed with a white-
centration cmc = 3.6 mmol/L. Aqueous TTAB solu-
light interferometric technique, while the bubble is kept
tionsofconcentrationcrangingfrom0.01and10cmcare
immobile using a plastic cylinder. (b) Time evolution of
made with ultrapure water (resistivity > 18.2 MΩ·cm)
abubblestabilizedbyTTABatc=10cmc,showingthe
obtained from a Millipore Simplicity 185 device. The
color change due to bubble drainage. In this sequence,
samples are prepared in plastic containers in order to
the bubble is blown at a flow rate Q =4 mL/min and
air
avoid the electrostatically driven adsorption of TTAB
has a final volume V =0.8 mL.
onto glass surfaces. The solutions are sealed and used
within one week of preparation.
The solution is poured into a petri dish of 5 cm in
diameterand5−10mmindepthuptothebrim. Allpetri
dishes are rinsed with ethanol and then ultrapure water
before use. As sketched in Figure 1a, a bubble is blown
by injecting air just below the solution surface, through
a needle (Terumo) of internal diameter 1.2 mm. The air
flowrateQ andtotalinjectedvolumeV arecontrolled
air
using a syringe pump (KD Scientific). Once the target
volume V has been reached, air injection stops and the
needle is either left in place or delicately removed. The
bubble is then left to drain until it bursts.
2
2.2 Bubble observation and thickness
measurements c = 1 0 c m c 4 b u b b le 1
From generation to bursting, a side-view of the bubble 4 cc == 30 .c8 m c cm c (cid:11)(cid:3)3 bb uu bb bb llee 23
is recorded with a color camera (U-Eye), allowing bub- (cid:1)(cid:2)(cid:15)(cid:11)(cid:3) 3 h(cid:1)(cid:2)(cid:15)12 (cid:1)
ble lifetime measurement as well as observation of the (cid:13)(cid:1)h 0
motions of the colored fringes that result from the inter- (cid:6)(cid:13) 0 2 0 4 0 6 0 8 0
ference between the successive reflexions of white light (cid:5)(cid:9)(cid:12) 2 t (s ) (cid:1)
(cid:7)(cid:8)
onbothinterfacesofthebubblecap. Atypicalimagese- (cid:11)(cid:1)(cid:14)
quencecorrespondingtobubblegenerationanddrainage (cid:4)(cid:8)(cid:10)
1
is presented in Figure 1b for a 10 cmc TTAB solution.
The bubble bursting is recorded at frame rates ranging
from 10 000 to 25 000 frames per second using a high-
0
speedcamera(PhotronFastcamSA3),allowingtolocate 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0
the rupture nucleation point. T im e t (s )
Inordertoobtainamorequantitativeinsightintobub-
ble drainage, the film thickness at the apex is measured Figure 2: Time evolution of the film thickness h at the
as a function of time using a white light interferomet- bubble apex for various TTAB concentrations (colors)
ric technique, as sketched in Figure 1a. Polychromatic and a fixed air flowrate Qair = 4 mL/min. For each
light emitted by a halogen lamp is carried by an optical concentration, we superimpose the data corresponding
fiberplacedverticallyabovethebubbleandfocusedonto to three different bubbles (symbols) and compare the
the bubble top by a lens. The reflected light spectrum drainage curve to the prediction in the limit of rigid liq-
is collected by the same optical fiber and analysed by uid/air interfaces (Equation (1), solid lines). The inset
a spectrometer (USB 400 Ocean Optics), of bandwidth shows the reproducibility of the drainage curve for three
350−1000nm. Thefilmthicknessattheapexisobtained different bubbles generated in the same conditions from
by fitting the spectrum with help of the NanoCalc thin a 10 cmc TTAB solution.
film metrology software (Mikropack, Ocean Optics).
A good spectrum contrast is observed only if the bub-
tantconcentrationsbelow0.8cmccouldnotbeachieved
ble surface is exactly perpendicular to the optical fiber.
because the bubbles burst immediately upon contacting
Thicknessmeasurementsthusrequireaccuratecontrolof
the edge of the plastic cylinder holding them in place.
thebubbleposition. Thisisachievedbyblowingthebub-
For a fixed concentration c = 10 cmc, we also explore
ble in the area enclosed by a plastic cylinder (see Figure
the influence of the air flowrate Q at which the bub-
air
1a), ontheedgeofwhichthebubblemeniscusispinned.
ble is inflated. Figure 3 shows the drainage curves mea-
Finepositionadjustmentscanbemadethankstoax−y
suredattheapexof1mLbubbles,forQ rangingfrom
air
horizontaltranslationplateonwhichthepetridishisset.
2mL/min(correspondingtoagenerationtimeof30s)to
60 mL/min (corresponding to a generation time of 1 s).
WestressthatthetimerepresentedinFigure3isthereal
3 Bubble drainage
timeminusthebubblegenerationtime,sothatt=0cor-
respondstothebeginningofbubblefreedrainage,i.e. to
3.1 Experimental results and compari-
the end of bubble inflation.
son to existing models
Inordertodiscussthedrainagedynamicsobservedin
The film thickness h at the apex of V = 1 mL bubbles Figure2,letuscomputetheBondnumberBo≡ρgR2/γ,
generated at a flow rate Q = 4 mL/min is measured whichcomparesthehydrostaticpressurescaleρgR(with
air
as a function of time for different TTAB concentrations ρtheliquiddensityandg thegravitationalacceleration)
c=0.8,3and10cmc. Thethicknessmeasurementstarts to the capillary pressure scale γ/R (with γ the surface
at the beginning of the free drainage phase, once the tension) for a bubble of radius R. Introducing the capil-
(cid:112)
bubble has reached its final volume. larylength(cid:96) ≡ γ/ρg,theBondnumbercanbesimply
c
The insert in Figure 2 shows the drainage curve at rewritten as Bo = (R/(cid:96) )2. In our experiments, the ra-
c
the apex of three bubbles generated in the same condi- dius of 1 mL bubbles is R ≈ 9 mm, while the surface
tionsfroma10cmcTTABsolution. Theinitialthickness tension of TTAB solutions [19] ranges from 38.5 mN/m
and the drainage dynamics turn out to be reproducible. for c = 3 and 10 cmc to 41.8 mN/m for c = 0.8 cmc,
These data sets are represented again in Figure 2 along yielding Bond numbers of Bo = 21 and Bo = 19 respec-
with the drainage curves obtained for lower surfactant tively. Inallcases,wehaveBo(cid:29)1,meaningthatbubble
concentrations (c = 0.8 and 3 cmc). In each case, the drainage is mainly driven by gravity.
data corresponding to three different bubbles are super- Inthepresenceofsurfactants,itseemsnaturaltocom-
imposed(symbolsinFigure2). Measurementsforsurfac- pareexperimentaldatatoagravitationaldrainagemodel
3
y = r - R
6 2 m L /m in g b
3 m L /m in
(cid:11)(cid:3) 5 46 mm LL //mm iinn air h(θ,t) y = 0
(cid:13)(cid:13)(cid:1)h(cid:1)(cid:2)(cid:15) 4 113 050 mmm LLL ///mmm iiinnn air θ R
(cid:5)(cid:9)(cid:12)(cid:6) 3 46 00 mm LL //mm iinn (cid:1) liquid uθ(y,θ,t)
(cid:7)(cid:8)
(cid:11)(cid:1)(cid:14)
(cid:4)(cid:8)(cid:10) 2 Figure 4: Scheme of the bubble with spherical coordi-
1 natesusedinthemodel. Theenlargmentontherightin-
troducesthenotationsusedtodescribetheliquidflowin
0 thefilm. Inparticular,wedefinetheextrapolationlength
0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0
b ≥ 0 at which the parabolic velocity profile u (y,θ,t)
θ
T im e t (s )
becomes zero.
Figure 3: Time evolution of the film thickness h at
the bubble apex for a fixed TTAB concentration (c = for intermediate – i.e. partially rigid – boundary condi-
10 cmc) and various air flowrates Q plotted in log- tions at the bubble liquid/air interfaces.
air
linear scale. For eachflowrate, theinitial time t=0 cor- Note that, despite the presence of marginal regenera-
respondstotheendofbubbleinflation,whenthebubble tion features at all the surfactant concentrations tested,
has reached its final volume. the model proposed by Lhuissier & Villermaux[8] for
marginal regeneration driven drainage of bubbles fails
with large interfacial stresses. In this limit, the velocity to account for our experimental data, as shown in the
at the interfaces tends to zero, leading to a rigid bound- ESI (section 1).
ary condition at the interfaces. The film thickness h at
the apex then follows an algebraic decay with time t, of 3.2 Drainage model with partially rigid
the form [8, 13]
interfaces
h ηR
h(t)= (cid:112)1+0t/3T with Tr = ρgh2, (1) Weseektodescribethegravitationaldrainageofahemi-
r 0 spherical bubble of radius R greater than the capillary
where h0 is the initial film thickness and η the liquid length(cid:96)c,assketchedinFigure4. Computingtheactual
viscosity. This model is represented by the solid lines surface stress at the interfaces of a surfactant-stabilised
in Figure 2. The experimentally observed drainage dy- bubble is not an easy task, since it results from viscous
namics is however significantly faster than predicted by surface stresses related to interfacial velocity, and from
Equation (1) or by more general functions of the same surface tension gradients, which in turn depend on the
form, as developed in the ESI (section 2). Consequently, local surfactant concentration [20]. Instead, we propose
we now examine the opposite limit of stress-free inter- a more geometrical description of the flow profile in the
faces. bubblecap,basedontheintroductionofanextrapolation
In this limit and for the gravitational drainage of a length b, which is approximated as being independent of
bubble of radius R, Koˇc´arkov´a et al. predict an expo- time and position. As represented on the right panel in
nential decay [4, 6] and propose the expression [7] Figure 4, b is defined as the distance from the interface
where the tangential velocity field u – which has the
η θ
h(t)=h0 exp(−at/Tsf) with Tsf = 2Rρg. (2) classical parabolic shape – goes to zero.
The bubble radius R≈9 mm being much larger than
They showed both experimentally and theoretically that thetypicalfilmthickness,weusethelubricationapprox-
the thinning rate a in Equation (2) is a decreasing func- imation. The tangential velocity field u (y,θ,t) is then
θ
tion of the Bond number Bo and tends to a≈0.1 in the governed by Stokes’ equation [13]
limit of Bo (cid:29) 1 corresponding to our experimental con-
ditions. The characteristic time Tsf = η/2Rρg is of the η∂yyuθ =−ρgsinθ, (3)
order of 10 µs for our aqueous TTAB solutions, hence a
characteristic drainage time T /a of the order of 0.1 ms where the coordinate y = r − R is defined by trans-
sf
in the limit of large Bond numbers. We observe exper- lating the usual radial coordinate r so that the liquid
imentally that drainage typically proceeds over tens of film is located in the interval y ∈ [−h/2,h/2]. Integrat-
seconds, whichisclearlyincompatiblewiththehypothe- ing Stokes’ equation (3) with the symmetry condition
sisofstress-freeinterfaces. Inthefollowing,wethuspro- ∂ u = 0 and the definition of the extrapolation length
y θ
poseasimplemodelforgraviationaldrainageaccounting u (y = h/2 + b) = 0, we can compute the tangential
θ
4
velocity profile in the film
ρg (cid:34) (cid:18)h (cid:19)2(cid:35)
u =− sinθ y2− +b . (4)
θ 2η 2
The time evolution of the film thickness h is then given
by the mass conservation in the film, which reads in the
lubrication approximation [21]
1
∂ h+ ∂ (sinθhu¯ )=0, (5)
t Rsinθ θ θ
( a )
whereu¯θ isthecross-sectionallyaveragedvelocityatan- 1
gle θ
1 (cid:90) h/2 ρg (cid:34)h2 (cid:18)h (cid:19)2(cid:35)
u¯ (θ,t)≡ u (y,θ,t)dy =− sinθ − +b .
θ h θ 2η 12 2 0
−h/2 (6) hh / 0 ,1
Under the assumption that the extrapolation length b
does not depend on the angle θ, the mass conservation
(Equation (5)) finally becomes c = 1 0 c m c
c = 3 c m c
c = 0 .8 c m c
ρg (cid:40) (cid:34)h3 (cid:18)h (cid:19)2(cid:35) 0 ,0 1
∂ h− 2cosθ −h +b 0 ,0 0 ,5 1 ,0 1 ,5 2 ,0
t 2ηR 12 2
t /T

(cid:18)h2 (cid:19)(cid:27)
− sinθ∂ h +2hb+b2 =0. (7)
θ 2 ( b )
1
Out of the two terms between the braces, only the left
one remains when Equation (7) is evaluated at the bub-
ble apex (θ = 0). Consequently, the spatial derivative
disappears from Equation (7), leading to the ordinary
differential equation h / 00 ,1 23 mm LL //mm iinn
h 4 m L /m in
ρg (cid:18)h3 (cid:19) 6 m L /m in
∂ h+ +bh2+b2h =0. (8) 1 0 m L /m in
t ηR 6 1 5 m L /m in
3 0 m L /m in
4 0 m L /m in
Forb(cid:28)h/2,werecovertheequationgoverningthebub- 6 0 m L /m in
0 ,0 1
ble drainage under the assumption of rigid liquid/air in- 0 ,0 0 ,5 1 ,0 1 ,5 2 ,0 2 ,5
terfaces,whosesolutionisgivenbyEquation(1),andthe
t /T
flow in the film is a classical Poiseuille flow with zero ve-
locityattheinterfaces. Intheoppositelimitofb(cid:29)h/2, Figure 5: Rescaled drainage curves at the bubble apex
Equation (8) becomes linear – provided that b is con- for (a) various TTAB concentrations c and a fixed air
stant in time – and yields an exponential decay of the flowrate Q = 4 mL/min and (b) various air flowrates
air
film thickness Q and a fixed TTAB concentration c = 10 cmc. The
air
thicknessandtimearerespectivelynormalizedbytheini-
ηR
h(t)=h0exp(−t/T) with T ≡ ρgb2. (9) tial thickness h0 and the characteristic time T obtained
by fitting the data with Equation (9) (solid red line).
The flow in the film is then getting closer to a plug flow.
Finally, when b ∼ h/2, all terms must be kept in Equa-
tion (8), which has to be numerically integrated.
3.3 Comparison to experimental data
In the following, we make the ad-hoc assumption that
b(cid:29)h/2,meaningthattheflowprofileinthefilmiscloser
to a plug flow than to a Poiseuille flow. The experimen-
tal data presented in Figure 2 are fitted with Equation
5
(9), using the initial thickness h and the characteristic
0
time T as adjustable parameters. For each TTAB con-
centration, the fitted values of h and T are reported in
0
Figure 6 (a and c) and used in Figure 5a to rescale the
5 8 experimental data. The same is done for the experimen-
4 6 taldatafromFigure3, obtainedforvariousairflowrates
) 3 ) Q , and the rescaled data are presented in Figure 5b.
(µmh0 12 (a ) (µmh0 24 (b ) Tanhadierdresausltfuinngctviaolnusesofofthhe0aainrdflTowarraeteplQotatier.dIinnFboigtuhreFi6gb-
0 0 ures 5a and b, the prediction of Equation (9) (solid red
1 1 0 1 1 0 1 0 0
c (c m c ) Q (m L /m in ) line) is then in good agreement with experimental data,
air
5 0 5 0 except for the end of the drainage dynamics, where the
4 0 4 0 thicknessdecreasesfasterthanexpectedfromthemodel.
3 0 3 0 For a fixed surfactant concentration, the initial film
) )
(sT 2 0 (sT 2 0 thickness h0 is found to rise by a factor 6 when increas-
1 0 1 0 ing the flowrate from 2 to 60 mL/min (Figure 6b). This
(c ) (d )
0 0 variation of h0 can be understood qualitatively from the
1 1 0 1 1 0 1 0 0 factthatthebubblealreadydrainsduringitsgeneration.
c (c m c ) Q (m L /m in )
air Thus, thesmallertheflowrate, thelongerthegeneration
time and the thinner the film when the bubble reaches
Figure 6: Values of the initial thickness h and the char-
0
its final volume (t=0 in Figures 3 and 5b). By analogy
acteristic timescale T obtained by fitting the experimen-
withthegenerationofflatverticalfilms,whosethickness
tal data with Equation (9), as a function of the surfac-
is known to increase with the pulling velocity [14], we
tant concentration c (a and c, respectively) and the air
canalsoanticipatethatthethicknessattheapexofbub-
flowrate Q (b and d, respectively).
air
bles will increase with the air flow rate, which controls
the rate of production of fresh interface during bubble
generation. Both phenomena probably contribute to the
observed dependency of h with Q .
0 air
For a fixed air flowrate, Figure 6a shows that the ini-
tial film thickness h at the apex rises when the TTAB
0
concentration c is increased, going from h ≈1.7 µm for
0
c = 0.8 cmc up to h ≈ 4 µm for c = 10 cmc. This
0
variation is likely due to an increased tangential stress
8 8
at interfaces during bubble generation at higher TTAB
6 6 concentrations. For instance, it has been shown that the
) )
b (µm 4 b (µm 4 tehnitcrkanineesmsoefnfltastpveeerdt,icbaultfilmalssonowtitohnlythiencsruerafsaecsewvitishcothues
2 2
(a ) (b ) stress [22] and the surface tension gradient [23] at the
0 0
1 1 0 1 1 0 1 0 0 liquid/air interfaces, for a given pulling velocity.
c (c m c )
Q (m L /m in ) Figures6canddfinallyindicatethatthecharacteristic
air
1 0 1 0 timescaleT fordrainagedependsonlyweaklyonboththe
8 8 surfactant concentration c and the air flowrate Q , and
air
2b / h0 46 2b / h0 46 rceamnabiensdeodfutcheedofrrdoemr othfe30tims.eTschaeleexTtrgaipveonlaitnioFnigleunrgetsh6cb
2 2 and d through the relation
(c ) (d )
0 0
1 1 0 1 1 0 1 0 0 (cid:115)
c (c m c ) Q (m L /m in ) ηR
air b= , (10)
ρgT
Figure 7: Values of the extrapolation length b and ratio
2b/h deduced from the characteristic timescale T and wherethebubbleradiusisR≈9mm. Theresultingval-
0
initial thickness h (Figure 6), as a function of the sur- uesofbasafunctionoftheTTABconcentrationandthe
0
factant concentration c (a and c, respectively) and the air flowrate are respectively displayed in Figures 7a and
air flowrate Q (b and d, respectively). b. The initial flow profile in the film is characterised by
air
the ratio 2b/h , which we plot as a function of the sur-
0
factantconcentrationandflowrateQ inFigures7cand
air
d, respectively. The hypothesis of a dominant plug flow
(2b/h (cid:29) 1) is well justified for c = 0.8 cmc and small
0
6
Figure 9: Image sequences recorded with a high-speed
camera at 25 000 frames per second, showing typical
bursting events for TTAB concentrations of (a) c =
0.1 cmc and (b) c = 10 cmc. In both cases, the ini-
Figure 8: Definition of the rupture angle θ , which
rupt
tial time is defined as the time when the hole nucleation
characterizesthepositionoftherupturenucleationpoint,
is observed and the corresponding error is ±0.04 ms.
and the angle θ , corresponding to the limit between
min
the meniscus (in black) and the thin bubble cap.
a circle, allowing us to define θ as the angle between
rupt
theprojectionofthenucleationpointontothecircleand
flowrates,whileitseemstobejustvalidatedforthemost
concentrated solutions and flowrates Q (cid:62) 4 mL/min, thehorizontaldirection. Similarly, wealsointroducethe
air
angle θ (where “min” stands for “minimum”) corre-
where2b/h isoforder2−3. Thedecreaseof2b/h with min
0 0
sponding to the limit between the meniscus (in black)
increasing the TTAB concentration and flowrate corre-
and the thin film.
sponds to a decrease of the plug flow contribution to the
flowinthefilmandthustoanincreasedtangentialstress Because of the bubble symmetry with respect to the
at the liquid/air interfaces. verticalaxis,theruptureangleθrupt variesintheinterval
[θ ,π/2]. In the following, the position of the rupture
It can be noted from Figure 7 that the extrapolation min
nucleationpointwillthusbecharacterisedbythereduced
length, of the order of 6 µm, does not change much
rupture angle
with surfactant concentration and air flowrate, and that
θ −θ
the variations of 2b/h0 thus essentially stems from the Θ= rupt min, (11)
contribution of h0. A comparable value for the ex- π/2−θmin
trapolation length had been obtained in the work by
whichrangesfrom0, whentheholenucleatesatthefoot
Saulnier et al. on vertical C E -stabilized films studied
12 6 of the bubble, to 1, when the bursting is spontaneously
during their generation [24]. Thickness measurements
initiated at the apex.
of PEO/SDS-stabilized films were also interpreted by
Adelizzi & Troian [25] using extrapolation lengths of the
ThereducedruptureangleΘismeasuredsystematically
orderofafewmicronsandBerget al. [26]foundextrap-
as a function of TTAB concentration, ranging from 0.01
olationlengthsaboutoneorderofmagnitudesmallerfor
to10cmc. AsshowninFigure9,theangleΘisobserved
SDS-stabilized vertical films.
tospanthewholerangeofpossiblebehaviours,fromΘ=
0(Figure9a)toΘ=1(Figure9b). Variousexperimental
4 Bubble bursting conditions are tested: (i) leaving the needle in place or
(ii) removing it with care after bubble blowing and (iii)
We now turn to bubble bursting, which we characterise enclosing the whole setup in a sealed glove box.
bymeasuringthepositionoftherupturenucleationpoint The results obtained in these different situations are
andthebubblelifetime. Inthissection,westudybubbles displayed in Figure 10a, where each point is an average
of fixed volume V = 0.8 mL, corresponding to a radius over 4 to 20 bubbles and the error bar stands for the
R ≈ 8 mm, and the generation flow rate remains equal standarddeviation. ForTTABconcentrationsabovethe
to Qair =4 mL/min. Note that the plastic cylinder used cmc, the position of the rupture nucleation point turns
toholdthebubbleinplacefordrainagemeasurementsis out to be reproducible and is located at the bubble apex
removed. as displayed in the image sequence in Figure 9b. Below
the cmc, however, the dispersion of data points becomes
4.1 Position of the rupture nucleation quite large and the position of the rupture nucleation
pointisnolongerreproducible. Onaverage, thereduced
point
ruptureangleΘdecreaseswhencdecreases,showingthat
Since the bubble has a (hemi)spherical symmetry, we more and more rupture events are observed close to the
characterise the position of the rupture nucleation point bubble foot (Θ = 0) when the surfactant concentration
byitslatitudeθ ,asshowninFigure8. Usingtheim- diminishes. Finally, note that no significant difference
rupt
age analysis sofware ImageJ, the bubble cap is fitted by was found between the data sets corresponding to the
7
lifetime of concentrated bubbles is consistent with our
observationthatthetangentialstressincreaseswithsur-
1 .0
factantconcentration,henceasloweddowndrainage(see
Q Paragraph 3.3).
gle 0 .8 At low surfactant concentrations (c < cmc), the bub-
n
a 0 .6 ble lifetime is relatively scattered, in agreement with ob-
e
r servations reported in the literature on very dilute so-
tu
p lutions [8, 9]. Bubble lifetime reproducibility seems to
ru 0 .4 improve above the cmc. Again, we do not observe any
d
uce 0 .2 significantdiscrepencybetweenthedatasetscorrespond-
d ing to the experiments performed in different conditions
e ( a )
R (in a glove box, with the needle left in place or removed
0 .0
after generation).
0 .0 1 0 .1 1 1 0
T T A B c o n c e n tra tio n c (c m c ) 4.3 Discussion
1 2 0 The experimental results presented in Paragraph 4.1
w ith n e e d le have highlighted two distinct behaviours, depending on
1 0 0 w ith o u t n e e d le the surfactant concentration c:
) in g lo v e b o x
s
(
t 8 0 • for c > cmc, the rupture nucleation point is repro-
e
ducible and located at the bubble apex ;
tim
life 6 0 • for c < cmc, the bubble nucleation point is rather
le scattered, whatever the precautions taken, but is
b 4 0
b
u getting closer and closer to the bubble foot when
B
the concentration is decreased.
2 0
( b )
Theburstingofconcentratedbubblesattheapexisquite
0
intuitive,sincethefilmisexpectedtorupturewhereitis
0 .0 1 0 .1 1 1 0
thethinnest,whichisa priori atthebubbletopbecause
T T A B c o n c e n tra tio n c (c m c )
of liquid drainage towards the bubble foot. The tran-
sition towards the low concentration regime (c < cmc,
Figure 10: (a) The reduced rupture angle Θ, charac-
with Θ coming closer to 0) is in agreement with the re-
terising the position of the rupture nucleation point (see
sults of Lhuissier & Villermaux [8] and Modini et al. [9]
Equation(11))and(b)thelifetimeτ ofbubbles,counted
who respectively observed that tap water bubbles and
from the beginning of bubble generation, are plotted as
salty bubbles with traces of surfactant (<0.1 cmc) pref-
functions of TTAB concentration c. The different sym-
erentially burst close to their foot.
bols represent different experimental conditions.
However, Debr´egeaset al. [4]noticedthatviscoussili-
coneoilbubbles–i.e. intheabsenceofsurfactant–burst
atthetop. Thus,thereisadiscontinuitybetweenthebe-
differentexperimentalconditionsmentionedabove(with
haviourofbubblesmadeofpureliquid(siliconeoilforin-
needle, without needle, experiments in glove box).
stance) and bubbles containing minute quantities of sur-
face active agents (like tap water), even though the flow
4.2 Bubble lifetime profile is similar in both situations. We indeed showed
in Paragraph 3.3 that the flow profile in the bubble cap
The bubble lifetime τ is defined as the time lapse be- is getting closer and closer to a plug flow (which is the
tweenthebeginningofbubbleblowingandthebursting. limit expected with stress-free interface) when the sur-
In the following, we only take into account the bubbles factant concentration is decreased. This evidences that
that burst after the end of generation, corresponding to the position of the bursting point is no longer controlled
a minimal lifetime of 12 s for 0.8 mL bubbles blown at by gravity-driven drainage at low surfactant concentra-
Qair =4 mL/min. tions. Note that the high viscosity of silicone oil bubbles
Thebubblelifetimeτ isplottedasafunctionofTTAB mayhelpdampingthicknessfluctuationsandthusfavour
concentration c in Figure 10b, where data points are av- drainage-controlled rupture.
erages over 6 to 20 bubbles and errors bars show the Lhuissier & Villermaux have attributed the rupture
associated standard deviation. The average bubble life- of tap water bubbles to a marginal regeneration, which
time increases with surfactant concentration up to the produces thinner film patches in the vicinity of the bub-
cmc and seems to saturate around τ ∼80 s. The longer ble foot. However, careful observation of our TTAB-
8
stabilized bubbles reveals that marginal regeneration
cellsarealwayspresent,regardlessofthesurfactantcon-
centration (see the color image sequence in Figure 1b,
corresponding to c = 10 cmc). This suggests that both
rupturefactors–gravitationaldrainageandmarginalre-
generation–coexistforallsurfactantconcentrations,the
formerbeingdominantathighsurfactantconcentrations
and the latter governing the bursting at low concentra-
tions. Thedispersionofdatapointsobservedforc<cmc
may then be explained by the transition between those
rupture mechanisms.
The “fragility” of bubbles may also contribute to the
data dispersion at low surfactant concentrations. In-
deed, we already noted that drainage measurements at
c<0.8 cmc were impeded by the presence of the plastic Figure 11: The sensitivity of bubbles to an external per-
cylinder,whichcausedthebubblestoburstprematurely. turbation is tested by placing a small plastic obstacle
In order to demonstrate this effect in a more systematic next a bubble in continuous generation. Depending on
way, we place a small plastic obstacle (dipped into the the TTAB concentration, the bubble either (a) bursts
surfactant solution beforehand) in the way of inflating whenittouchestheobstacleor(b)goesthroughitwith-
bubbles of various concentrations. The image sequences out damage.
displayed in Figure 11 show that bubbles made from di-
lutesolutions(c=0.1et0.5cmc,Figure11a)burstupon
contacting the obstacle, while bubbles made from more
concentrated solutions (c = 1 et 10 cmc, Figure 11b)
penetrate it without damage. of hemispherical films coated onto a solid lens, could in-
For c < cmc, the bubbles are thus “fragile”, in the clude explicitely Marangoni and surface viscous stresses,
sense that they are more sensitive to external perturba- which are implicitely encompassed in the extrapolation
tions (the obstacle in Figure 11 but also dust particles length b in our approach.
forexample)thantheirmoreconcentratedcounterparts.
This fragility may be attributed to a lower value of sur- The position at which a hole nucleates in the bubble
face dilatational elasticity, leading to less efficient heal- cap was found to depend strongly on the surfactant con-
ingofsurfactantcoverageinhomogeneities,andprobably centration. Bubbles made from concentrated solutions
contributes to the dispersion of data points at low sur- (c > cmc) systematically burst from the apex, while
factant concentrations. the rupture nucleation point was observed to be quite
scattered at low concentrations (c < cmc), but getting
closerandclosertothebubblefootonaverage. Thislow-
5 Conclusion concentration behaviour is correlated with an increasing
fragility of the bubbles with respect to perturbations.
In this paper, we have studied how the drainage and Building on the ideas of Lhuissier & Villermaux [8], we
bursting of bubbles floating at the surface of a soapy suggestedthatbothgravitationaldrainageandmarginal
bath depend on the concentration of the cationic surfac- regeneration are involved in the bursting of bubbles, the
tant TTAB in water. We have shown that the decrease former prevailing at high surfactant concentration and
of the bubble cap thickness at the apex can neither be leading to a hole nucleation at the apex, and the latter
rationalisedbyadrainagemodelwithrigidliquid/airin- being responsible for bursting at the bubble foot at low
terfaces, nor by assuming that the interfaces are stress- concentrations. Besides, we observed that marginal re-
free. Weproposedasimplemodeldescribingthegravita- generation is present at all of the concentrations that we
tional drainage of a bubble with an intermediate bound- probed, butitseemstobecomeaweakercontributionto
ary condition at the interfaces (i.e. partially rigid inter- bursting as the concentration is increased. The fragility
faces),wherethedifferencewithrespecttotherigidcase of low-concentrated bubbles with respect to perturba-
isquantifiedbytheextrapolationlengthb. Itturnedout tionsmaybeacluetoexplainwhymarginalregeneration
thattheplugflowcontributiontothedrainagedynamics driven bursting occurs only at low surfactant concentra-
decreaseswhenthesurfactantconcentrationisincreased. tions. These results also open the question of the rela-
Thischangeislikelyrelatedtotheriseofasurfacestress tionbetweenthepropertiesofthemarginalregeneration
due to the increased presence of surfactants at the in- patches and the physicochemical properties of both the
terface. A more advanced model, in the spirit of the surfactant solutions and the surfactant molecules them-
one developed by Hermans et al. [27] for the drainage selves.
9
Acknowledgments
[11] E. Rio, W. Drenckhan, A. Salonen, and
D. Langevin. Unusually stable liquid foams.
TheauthorswouldliketothankEmelineAlvarezforher Advances in Colloid and Interface Science, 205:74–
help with the bubble bursting experiments. L.C. was 86, 2014.
supported by ANR F2F and this work also benefited
from the support of an ERC Starting Grant (Agreement [12] A. Vrij and J. T. G. Overbeek. Rupture of thin liq-
307280-POMCAPS). uid films due to spontaneous fluctuations in thick-
ness. Journal of the American Chemical Society,
90(12):3074–3078, 1968.
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10