Pbys.

J.

II

FFance(1997)

71331-1335

1997,

OCTOBER

Communication

Short

Instability

Kelvin-Helmholtz
Shigeyuki(~,*) andKomura

Iwayama (~)

Takahiro

Department of Mechanical System Engineering,Engineering, Kyushu Institute of Technology, Iizuka(~) Department of Control Engineering and Science,of Technology, Iizuka
Engineering, Kyushu
Institute(Received 21 April 1997,

revised

PACS.47.20.-k

Hydrodynamic

PACS.68.10.-m

Fluid

PACS.68.60.-p

Abstract.

We

fluid-fluid

andproperties

discuss

theScienceandSystems

FacultyScienceandSystems

Computerof820, Japan

July 1997)

24

stability

surfaces

PhysicalFaculty of Computer820, Japan

accepted

1997,

June

9

Monolayers

Langmuirof(~)

ofinterfaces

films,

thin

possibility ofnonelectric

instability

Kelvin-Helmholtz

ofapositive surface tension, this instability is expected to occur
value of the velocity of therelative
This is in
aircontrast
to thewater.
upper
which
for
transition
surfacetension.
negativeoccursalayer.

Even

for

a

Many amphiphilic molecules form
soluble in the bulk
are only slightly
and

1331

PAGE

has

attracted

botha

layer

Such

insoluble

water.

andacademicmonomolecularantechnologicalataninterests

monc-above

criticalabuckling

thetointerface

air-water

film is called

Langmuir

when

they

Langmuir monolayera[lj.

phenomtwo-dimensionalmonolayers exhibit various important class ofalso
attracted
fluctuation
third
have
intertheirandmorphology
dimension
into
great
ena [2j,al.
predicted
bucklingof
bothexperimentally.
MiInertheoreticallyandForinstance,etestsaLangmuir monolayers iri a fluid phase under a lateral compression which decreases the surface

Although Langmuir

tensionto 'ysurface

due
theunstable.

beII,'yepressurenegative,

even

to=

to'ye isthecompression

systemThe

surfacewhereintends

physical

atoprocess

[3j.

As

increaseits

inoftension

this

the

pure

surfaceinterfacial

instability

isand II is

watertensionarea

similar'yand
totwo-dimensionalabecomesthethatvery

interface
of

a

low
turns

plateand
out

under

compression.
observed in fluid monolayers and they collapse
system, however, buckling wasnever
multilayers even at positive tensions l'yo > II) upon increasing the surfacepressure.
Experimentally, the buckling transition has been found in other types of monolayers such as by
Bourdieu
monolayers using X-ray scattering, or by Saint-Jalmes
et al.
et al. [4] in polymerized
Formationof multilayers may be suppressed forin solid monolayers using light scattering [5].
these types of monolayers and the buckling takes place at positive 'y. Recently, Hu and Granek
head-tailof monolayers due to thehave
predicted the buckling
modulation
asymmetry [6j.
In

(*)

©

a

realform

toAuthor

Lesfor(ditions

correspondencede(e-mail:komurafliizuka.isc.kyutech.ac.jp)

Physique 1997

JOURNAL

1332zPHYSIQUE

DE

N°10

II~

AIR~P~(fI0TERPi

Fig. I.
velocity ofdenotes

theSchematictheairrepresentationof(with density p2)

displacementof

theLangmuirarelativeinterface

into

the

thetrough.
water

H is

thesurface(with density pi)

pressure

intheandU isz-direction-the((z)

z-direction.

hexagonal array of long-finger structure
induced byspontaneous
non-zeroanmonolayer.
So far, the problem of monolayer instability has been
withinInlimitedstaticarguments.
this article, we discuss the possibility of a dynamical instability of Langmuir monolayers, which
above.
different
physical origin compared to the static
instabilities
mentionedhas completely
which
when twoThis
instability is generally known as"Kelvin-Helmholtz
instability"
occurssuperposed fluids flow one over the other with a relative
horizontal
velocity [7, 8j. The physicalmechanismof this instability has been
described by Batchelor in terms of vorticity dynamics [9j;theaccumulationof vorticity atinterfaceand ofofrotationcertainpoints on theprocessneighbouring points occur together, leading to exponential growth of the
disturbance.
For
Langmuir manolayers, above a certain
criticalvalue of the velocity of the air
relativeto theKelvin-Helmholtz
instability will arise before the static bucklingtakestransitionwater, theplace. Our result shows that the critical velocity and the wavelength of the most
unstablemode are deeply affected by theof theadsorbed
amphiphilic molecules.
presence
shown
Kelvin-Helmholtz
It has beenthat theinstability is also relevant to lyotropic liquid
crystals in smectic phase (L~ phase) [10j. They discuss this instability in terms of "Reynolds
effect"triggered by shear flow in which the fluctuation
amplitude is amplified owing to theAs a result, mixing and strong
fluctuations
expected toincreasein lifetime.
concentration
are
They

anticipatedcurvatureoccur

nearof thetheinterface.

simplified Langmuir trough as depicted in Figure I where the surfacepressure
II is exerted
the monolayer along the x-axis.
We
that the Langmuir monolayer isonassumex~direction
relative to thein its fluid phase. Let us
denote by U the velocity of the air in theuniformand by ((x) the displacement of theWeinterface in thez-direction.
water
assumeadisplacement along the y-direction, and the depth of the water is taken as infinite. We describe
here both the air and theSince the flow is uniform,
incompressible
inviscid
fluids.
water
asboth of the velocity fields arefree,andbeconveniently
expressed in terms of therotation
canvelocity potential as v~
Hereafter
the lower
indices I
grad #~, with i7~#~2).
0 (I1,
refer to the waterand 2 to the air, respectively. Wethewinddoes
that
notcauseanyassume
flow of amphiphilicmolecules
thesurface.
In fact, this flow caninduce a backflow in theonbulk water [2j. As in reference Ii lj, we seek the velocity potential in the form
Consider

a==

#i
#2

"

=

Aie~~ cos(kxA2e~~~ cos(kx=cot),(1)uJt) + Ux.

(2)

INSTABILITYKELVIN-HELMHOLTZ

N°10TheNext

velocity of theof thez-component

wetheconsider

force balance

thisFormonolayer.

OFLANGMUIR

theairandwateratconditionatzMONOLAYERS

0=thearetheforce duebendingrigidity,

includinginterface1333

tourpose,

2j:
F

2

theIn

isdA
above,

andcurvatures,

-Pi

P2

andcurvature,re(piaccomit=-2~yHare the

0are

[llj.

Laplace's+~(2Hco)

(2H~

2K

-

=

satisfy

thec2)/2ci

(6)

the

The=zi7)BH,

issurface.

theatoundary+coH)-+-(cion

operator
Laplace-Beltrami

whichequationstheisgiven by [13jH

the~lement,spontaneous curvature. By usingpressions

for

mean

rvature,

Pi

when

0

thefter

inearization

respective

thefluctuation

the+ormulapressures,is

Bernoulli's=wherethei7(Bthebysurface

thehaveP~

wheretheiscodensitiesof thesurface

and g is theacceleration
owing to thetension

ofgravity.

fluctuation

Oneof thecanalso
surface

take

into

densitymolecules, namely, the compressibility of the monolayer [14j. This, however,ofhigher ordercontribution
equation (7) as long as the reference
tostateandtheinterface
is planer.
Moreover, K can also depend on the total area of the interface
hence on n [3j, but this effect will be neglected for simplicity. One finds from equation (7) thatmerely shifts (upward) the surface tension within the lowest orderthecurvature
spontaneous

amphiphilic
only gives riseoftocalculation.

procedure as in reference [llj, we look for the displacement ( in thesin(kx uJt). Notice that only terms of the first order in A2 are retained inperiodic form (aU~ in equation (7). After some calculation, we obtain the dispersion relation
evaluating v]
given byFollowing

thesame=~~~2~

withf(k)where fithe=part,
shall

onlypi p2 / (pi +p2)
becomesmotiondiscuss

thecaseand=~k4 +Apunstableof=~)~f~+

If uJ ispip2.

andthis isvanishing~~~

calledspontaneous~~~k2a>u2k + Apg,

complex(copositive imaginary
ainstability.
Hereafterwe0) just for the simplicity.
withnumberKelvin-Helmholtz
curvature(9)=

JOURNAL

1334

Results
staticspecialseveral

for

where

case

the'yeThemost=k.

vectorIIcis

becanrest,

atfromU

0.

-2(~Apg)~/~, f(k)=(8). First,

equationdeducedI.e.

N°10II

theIfconsider

wesurface

thesatisfies

tensionnegative for a certain
range of thewave(Apg/~)~" in this case. This situation
given by kc
exactly corresponds to the buckling instability predicted by MiIner et al. [3j and occurs onlyformechanicaldue to the lateralcompression of the Langmuir monolayer. This is notreasondynamicalinstability.
criticalsurface
Noticethattheexpected
tension is negative althoughait is extremely small, and the buckling instability is expected toat the vanishing surface
occur
tension in practice [3j.
Whenamphiphilic molecules, as originally considered by Kelvin, the interfatherearenocial property is solely determined by the surface
tension of water, 'ye. In this case, f(k) canbe
unstablemodenegative provided U > Uciis(4'yoApg/@~)~/~, and the most(pikci(Apgl'yo)~/~ [7-9, iii.
Typical values for air overI g cm~~,water
10~~ g cm~3,730 cm s~~ and kci
3.6 cm~~74 dyn cm~~) provide Uci
p2~yo(21r/kci
originally predicted by Kelvin.) Although thevalues
1.7 cm). (Roughly thesewerecompressibility and the viscosity of the air has been neglected in the calculation, the above
critical velocity gives a goodfor the onset of the instability [7j.
estimate
As the surface
II is increasedof adsorbed
amphiphilic
towards ~yo in thepressure
presencemolecules, it turns that the interface is mainly governed by the bending rigidity «. In principle,<~y'yecases

airPHYSIQUE

DE=modeunstable

becan

is="

=+~+~

~J+~+~~J

for any value of ~y, there exists
Here weconsider the case forplace.

takesandthetudeIn

thismodebefore,

used

as~y

criticalcertainwhich

0=

Uc abovestilliswhichthelarger than~y~criticalvelocity

Uc2(2/3~/~)~~/~>~~/~(Ap9)~/~,

thecase,unstable

mosta"

and~==10~~3
8 x 10~+J

103 cm~~whichbecomes

unstable.

staticbuckling

theis(Apg/3~)~"

is kc2interfaceaterg,

kc /3~".
findwe1).

(io)

With

thethatUc2

same~J

orders13

cmof magni-s~~and(27r/kc2
This length scale iswithaccessible
current
experimental techniques [4, 5j. Comparing with the pure water surface case, we see Uci » Uc2
and kci « kc2 indicating a
dramatic
effect of adsorbed
amphiphilic molecules on the surfaceofthesevaluescriticalby changing the surface pressure n.
property. One can expect acrossover

kc2

+~

7.6

The~J

surfacezero

emulsionsdensityThis

terface.

tensionAssuming[15].

thesincex

differencecaseamicroemulsious

also

be

may

under

theandalso

water.

Theby thecriticalvelocity
surfaceexertedtheforthe

relevantalsooil-waterinterfacesindestabilized

withsmallerUc2.

theair-waterin-oil-waterinterfacesinsaturateditbecanoil-watermonolayers
[16].

tois

to

take

deeplyplace above

affectedbytheathaninterface

shearing deformation
the possibility of
Kelvin-Helmholtz

monolayer.
toininterface,smallerhave discussed
summary,
weThis instability is expected

In

realized

oil-water(Ap) istype of instability

bicontinuous

is

flat

criticalpresenceat

micro-

instability of a Langmuir
velocity of the air relativeof amphiphilicmolecules

pressure.

Acknowledgments
One of theauthors

discussions.

This(Grant-in-Aid for(S.K.) would

like to thank Dr. H. Kodama and Dr. J.
Yamamoto
for helpful
supported by the Ministry of Education, Science and Culture, Japan
Scientific
ResearchNo. 08226233, No. 08740324).

work

is

KELVIN-HELMHOLTZ

N°10

INSTABILITYOF

LANGMUIRMONOLAYERS

1335

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