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Transverse and longitudinal waves induced and sustained by surfactant gradients at liquid-liquid interfaces
Transverse and Longitudinal Waves Induced and Sustained by Surfactant Gradients at Liquid-Liquid Interfaces X I A O - L I N C H U AND M A N U E L G. V E L A R D E Facultad de Ciencias, U.N.E.D., Apartado 60.141, Madrid 28.071, Spain
Received July 20, 1988; accepted October 10, 1988 Dispersion relations are derived for transverse and longitudinal modes of oscillation appearing as a result of transfer, adsorption and eventual accumulation of a surfactant at the interface of two liquids. Also provided here are the values of the Marangoni number sufficient to sustain these waves and a comparison between theoretical predictions and available experimental data. © 1989AcademicPress,Inc. 1. INTRODUCTION
, gitudinal wave has received m u c h less attention than the other since under most practical Capillary w a v e s - - o r r i p p l e s - - a n d gravity conditions, the motion is d a m p e d out m u c h waves have been well studied since Laplace, more rapidly than its transverse counterpart. Kelvin and Stokes ( 1 - 8 ) . Their properties Lucassen studied the longitudinal wave and were shown to be mainly determined by the proposed that in the elastic surface of a highly stress condition at liquid surfaces. The major viscous fluid longitudinal waves could be obinfluence of the boundary condition for nor- served easier than transverse waves (9). Lucassen showed that longitudinal waves real stress to the surface has been especially emphasized. While the ripples are rather are, to a major extent, related to the boundary transverse motions due to the deformation of condition for tangential stress with a frequency the surface there is yet another type of wave that depends on the viscosity and surface elasdiscovered years ago by Lucassen (6, 9-1 1 ). ticity modulus. Gravity-capillary waves, It refers to mostly longitudinal motion along however, have a frequency that depends on the surface, in the limit along a flat surface. gravity and on surface tension (Laplace overTheir existence is not surprising considering pressure) but not on viscosity. The latter only that a strong analogy is expected between a appears in the damping factor and frequencymonolayer-covered surface and a stretched deviation in the dispersion relation. Generally these waves, these oscillatory elastic membrane. The coverage with a surfactant monolayer--either by adsorption from motions, are damped albeit differently by vissolution or by spreading--gives indeed elastic cosity (5, 6, 9-13). However, if a nonequilibproperties to a surface so that it tends to resist rium distribution of surfactant/temperature the periodic surface expansion and compres- is imposed in the liquid that drives m a s s / e n sion which appears as wave motion. Normally, ergy transfer across the surface, the Marangoni any wave motion of the surface has both effect m a y transform the chemical energy into transverse and longitudinal components, and motion, overtaking the viscous dissipation and they are not separable. Only in the case of thus sustaining the wave motions. Sternling and Scriven (S&S, 14) were the small amplitude wave motion (corresponding to linear theory) transverse waves and longi- first to provide a mathematical description of tudinal waves can be considered as two gen- spontaneous convection at the interface of two uinely different modes of oscillation. The lon- fluids. In their now classic paper published in 471 0021-9797/89 $3.00 Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
472
C H U AND VELAR D E
1959, they clearly considered the Marangoni Sorensen et aL (26, 27), and Sorensen (28), effect as the "interface engine" that converts and by Sternling and Scriven (14) are longichemical energy into convection. Marangoni tudinal waves caused by different mechanisms stresses due to surfactant concentration or/ and appear in different wavelength regions. We and temperature gradients along the interface also study the transverse waves. generate convective motions that are transIn Section 2 we recall that the evolution mitted to the bulk. If there is high enough sur- equations are obeyed by disturbances upon the factant or thermal gradient in the bulk, con- quiescent state of the liquid-liquid system with vection may promote the surfactant or/and a fiat interface. We also derive in this section thermal distribution at the liquid-liquid in- the determinantal equation that provides the terface thus permitting disturbances to develop necessary and sufficient condition for the exin the form of steady cellular convection or istence of either a convective steady state or oscillatory convection ( 15-17). The latter case overstability. Section 3 is devoted to an acmay correspond to waves that appear as either count of numerical results obtained with the longitudinal or transverse capillary-gravity computer using the above mentioned deterwaves or a combination of both (4). minantal equation. In Section 4 we develop Although Sternling and Scriven were aware an asymptotic analysis that permits an indepth of transverse motions in experiments, and re- discussion of longitudinal and transverse ferred to them in their paper (14) as "localized waves appearing at different wavelength and stirring with rippling and twitching of the in- frequency ranges. Finally in Section 5 we terface" (17), they, however, treated the in- summarize the major results found and comterface as flat in order to simplify the mathe- pare our theoretical predictions with the scarce matical problem. This excluded the transverse available experimental data. waves in their analysis. Another approximation in S&S's model was the elimination of 2. DISTURBANCE EQUATIONS surface adsorption although they considered a surface-active solute. As pointed out by sevWe limit ourselves to consider a liquid-liqeral authors (11, 18-21 ), solute adsorption uid interface initially at rest and located at z and eventual accumulation at the surface takes = 0 (z is the vertical coordinate). The evoplace in liquids with surface-active solutes and lution equations and boundary conditions for drastically affects stability. The Sternling and Scriven theory has had disturbances upon the quiescent state are for subsequent developments by several authors both liquids and the interface (22-37). Sanfeld and collaborators studied a V - ~ + = O (q~= 1,2) [2.11 liquid-liquid interface model with due account of the deformable interface and solute adsorption at the interface. However, they only discussed one of the various possible modes, the long wavelength longitudinal waves. It was also shown in their results that the deformOt /3~#~= D~'2?o [2.3] ability of the interface has negligible influence on the stability of these waves. Here we consider the liquid-liquid model if z = f denotes the disturbed liquid-liquid developed by Sanfeld et aL and proceed to a interface we impose the following boundary complete analysis of all possible solutions. Dif- conditions (b.c.) ferent frequency regions are carefully exam0f ined. We show that the oscillations studied by ~-=~=~[~ (q~-- 1,2) [2.4] Ot Hennenberg et al. (23, 24), Sanfeld et al. (25), Journal of Colloid and Interface Science, Vol. 131,No. 2, September 1989
WAVES AT LIQUID-LIQUID
~,l~ = ~2[z = ~z
o~
(o~) o~
[2.5]
o~z
[ \Oy+-~-~
473
INTERFACES
face concentration: vl/l, 12/pl, v~pl/l 2, fill and Po. After changing the units, the dimensionless equations are (Note that the variables below without tildes have no dimension)
(q~= 1,2)
div g, = 0
=0
[2.11]
[2.6] --
= -Vpl
+ Vz~l
[2.121
Ot p~ ~ -
-
,~o a£-~ -
[ +[/~]-2[
= 0 0i]
= --Vp2 + NuV2~'2
Np at
OCl - - -- W 1 = S-1•2Cl
O3' + yo O~Z
o~
O~ _ [ D Oa]
-5-~--D~ at
~ =0
= k{b¢l~,
[2.13]
[2.71 [2.14]
Ot
[2.81
OC2
- - - w2 = S-1NDV2C2. Ot
[2.91
[2.151
Now the boundary conditions at z = ~-are where ~ = (~, k) is the disturbance velocity, /~ the disturbance pressure, ? the disturbance surfactant concentration, ~: the deviation of the interface from the motionless level initial position, P the excess accumulation of surfacrant at the interface, and ~ the disturbance on r. p accounts for liquid density, # = p ~ is the dynamic shear viscosity with u the kinematic viscosity, D the surfactant diffusivity, g the gravitational acceleration, /3 the surfactant gradient, p~ the surface excess density, #dil and #~h the surface dilational and shear viscosity, respectively. ~ is the surface tension and k I the Langmuir adsorption constant. The subscript ~bindicates the liquid: 4 --- 1 the liquid below and ~b = 2 the liquid above the interface. The subscript "0" accounts for the initial reference state and the bracket [f] the jumping property at the surface [ f ] = (f2 - A ) ~ . Now for universality of our analysis, we use suitable units to rescale the original dimensional quantities. Although this is not the only possible choice, we can choose the capillary length as the unit of length ~/~ l =
¢ro 1/91
p21
[2.10]
and the following scales for velocity, time, pressure, solute concentration and excess sur-
ul = u2 = uz
Owl Oz
Ow2 Oz
o~ Ot
--=
1
02~"
Bo
S C Ox 2
SC
[2.16]
Ow~ Oz
[2.17]
[2.18]
wz
~-- [p] OWz OW~ 1)-~-z =V~-- ~ -
+2(N,-
[2.191
/4E02~ (02 O2) H z S Ox2
Ox2
=
MS
O~5 (N.w2 - wl)
O O 2 ) Ow z Pt~-#Z~x 2 ~z
[2.20]
03" OUG 02"~ 1 - ~ ~- ~ x - S~I ~ x 2 )
Ocl Oz
0c2 Oz
+ . . . .
H~
3" = ~ - (cl - ~)
HzNz
3' = ~ - - -
(c2 - ~).
0
[2.21]
[2.22]
[2.23]
Journalof ColloidandInterfaceScience,Vol. 131, No. 2, September 1989
474
CHU AND VELARDE
We have used the following dimensionless groups N o = P~
pa
#2 N'` = - #1
density ratio,
k]31l 2 × - #aD1
viscosity ratio,
Po 3~l 2
surface excess solute number,
k;
Langmuir adsorption number, ratio of surface excess density to liquid density,
Hz = -
D2 ND = -D1
diffusivity ratio, ratio of Langmuir adsorption coefficients,
k12f12
N~ = kl1fll Ya
Schmidt number,
Sz = - Dz
surface Schmidt number,
#1D1 C = -[ao [p]gl2 Bo =
capillary number, bond number,
O"o
Bo+a 2 -
SC~
Ft ~
1 Pz -pal ~dil "~- ~ s h
~z =
S = -D1
-
elasticity (surfactant Marangoni) number,
#ll
ratio of surface viscosity to liquid viscosity.
Note that by choosing the capillary length as the unit of length Bo = 1. However we shall explicitly indicate the Bond number dependence in our results. Then after some standard and elementary albeit lengthy analysis, the necessary and sufficient condition for nontrival solution of the problem posed by Eqs. [2.11-2.15 ] with b.c. [2.16-2.23 ] is the following determinantal equation:
X(Np+ 1) +
a
+ PtX
a - aN, +
p2N'` - Pl
+ aN,, + a +Pa + p 2 N . - a + aN~, - p2N'` + PI
- r t X - ~ a 2 - aN,
Ea2/S
= 0,
[2.241
- a - Pl - p 2 N , _z(m+a I ~ q + rn]
m+a]
-L q-- a J + M s
X H z S ( X + a2/S~) + qlX + q2)t/Nz
where ml = ~ a 2 + X
[2.2.51
m2 = ~a 2 -b X N o / N u
[2.26]
ql = alfa~+ SX
[2.27]
q2 --= ga a + S X / N D
[2.281
Here a is the dimensionless Fourier waveJournal of Colloid and Interface Science, Vol, 131, No. 2, September 1989
number and ), the time constant whose real part determines the stability of the motionless state. Its imaginary part gives the frequency of the oscillation. Eq. [2.241 is the dispersion relation that in compact form is Ea2~_,~
~a +
S
O,
[2.29]
WAVES AT LIQUID-LIQUID INTERFACES
475
where 10~r.
~1 = HzS X+
~t
+ ql +
×{( B°+a2+SC~2(Np+l+aI't))a X (ml + a + N~(mz + a) + FiX + ~za 2)
101-'
10 ~
+ SCX(PtX + ~za2)(ml + a 10 -4
+ N~,(m2 + a)) + 4XSC × (ml + aN,)(m2N~ + a)}
[2.30]
.i
~2 = (BO "J~ a2 Jf- SC~2(N'° -~- e -jr a~t)
10'
100
102
' 60
I~)G. I. Typical overs[ability lines for transverse (solid line) and longitudinal (broken line) waves. No = 0.8, N, = 0.5, No = 2, Nz = 1, Bo = 1, C = 10-9, S = S:~ = 103, H = - 1 0 -9, az = 1 0 -5 a n d l~t = r/~ = 0 . T h e s e m o d e s o f oscillation appear for (vfrom/Vto) < 1 and (Dfrom/Dto) > 1.
×(HS-[m+a]/XI+HSZC~ [q + m J / ]
(b) IfND > 1 and N~ < 1 the marginal stability curves are those of Fig. 1.
× (ml + a + Nu(m2 + a)) + 2SC
(c) If No < 1 and ND > N, the marginal stability curves are those of Fig. 2.
× I(m2N, + a) ml + a [ ql + ml
(d) If No > 1 and N~ > 1 but not far from unity the marginal stability curves are those of Fig. 3. See also Ref. (14).
+ ( m l + Nua) q2rn2+m2 " [2.31] From now on, we concentrate on the analysis of this dispersion relation [2.29]. Only overstability will be investigated here, that is, we shall limit ourselves to the case Re (~) = 0 with I m ( ~ ) v~ 0.
(e) If No > 1 and N, > 1 but really away from unity the marginal stability curves are those of Fig. 4. See also Ref. (14).
It appears that the curves in case b (Fig. l ) and case c (Fig. 2) are rather similar to those found for air-liquid interfaces (30). This similarity reflects not only in the form of the curves, but also in the properties of those 3. NUMERICAL RESULTS curves. For instance, their dependence on the As expected from the large number of pa- parameters of the problem. Two independent rameters in the problem a discussion of nu- branches of marginal stability have minima merical results is far from simple. There is for the elasticity Marangoni number. Our calhowever a way out to simplify the issue: to list culation shows that one of these branches with the results found in figure form for cases of the minimum elasticity number in a region of rather general value. Thus Figs. 1-4 describe much lower frequency than the frequency stability thresholds in the (E, w) plane. For range of transverse waves does not depend on specific illustration let us choose Np = 0.8 and interface deformability but it is affected by the Nz = 1. Then recalling that N, = / 1 2 / / * 1 = N~,I surface adsorption, thus showing a feature of longitudinal waves. The other branch, with a No, m i n i m u m of elasticity number in the high fre(a) If N, > 1 and ND < 1, no oscillatory quency region, varies with C, i.e., with the insolution has been found. However, the system terfacial deformation and it is clearly a transverse wave. Note that the air-liquid interface is unstable to steady cellular disturbances. Journalof ColloidandInterfaceScience,Vol. 131, No. 2, September
1989
476
C H U A N D VELARDE
E
~,
10:~°
1 0 i6-
101ss
ql
1018 10 n
1014 -
i
t
/ ji
1012I 10 - a
10 2
10 o
102
~; 10 t /
FIG. 2. When the mass transfer direction is reversed, i.e., (Drrom/Dto) < 1 here, if the condition (Dfrom/Dto) > (V~om/vto) is satisfied the overstability lines are quite similar to the case depicted in Fig. 1. N~ = 0.8, N , = 0.3,
¢
10~
instability
convection
branch,
narrows
and narrows
region
but the until
/
z
/ / / / / /
to
10,
is a s p e c i a l c a s e o f i t e m b i n t h e l i m i t N D c a s e d ( F i g . 3 ) w e still s e e t h e t r a n s -
J
¢
TM
Go. F r o m
s
s¢
ND = 0 . 5 , N~ 1, B o = 1, C = 1 0 - 9 , S = Sz = 103, H = - - 1 0 -9, H~ = 10 -5 and rt = ~z = 0.
verse oscillatory
¢
FIG. 4. When we really move away from (Drrom/D,o) = 1 the transverse wave still existing in Fig. 3 here disappears but the SSLW remains. Np = 0.8, N , = 3, ND = 5,
Nz =
1, B o = 1, C = 10 -9, S = S~ = 103, H = - 1 0 -9,
Hz =
10 -5 and I?t = ~/z = 0.
E
it c a n n o t
be distinguished
f a c e as t h e p a r a m e t e r
with nonzero
t h e l i n e N , = 1, a n d f i n a l l y d i s a p p e a r s
1 0 x~
see Fig. 4). Transverse
waves
i n c a s e s a, b, a n d c. W e
x
sur-
goes further away from
have
( c a s e e,
are only found schematically
\ xx
i0 n
D from
x /'
10 s
/
s
s /
10 4
/ 4 /
10 -~
i
w
10 a
104
FIG. 3. AS we proceed to region ( P f r o m / / ~ ' t o ) > l and 1 the available region for transverse waves shrinks as we move away from (Dfrom/D~o) = 1. Longitudinal waves are much more affected as the overstability line does not show a minimum as discovered years ago by Scriven and Sternling (14). Np = 0.8, N~ = 1.2, Nr~ = 2, N~ = l, Bo = 1, C = 10 -9 , S = Sz = 103 , H = - 1 0 -9, Hz = 10 -s and Ft = ~Tz = 0.
(Dfrom/Dto) >
Journal of Colloid and Interface Science, V o l .
1 3 1 , N o . 2, S e p t e m b e r
1989
O0
1
t/from. t,,to
FIG. 5. The dashed area is the region where transverse waves are predicted by our theory (Sect. 4.2, Eq. 4.2.9 ). It corresponds to (Drrom[Dto) > ( ] A f r o m / # t o ) 2,
477
WAVES AT LIQUID-LIQUID INTERFACES
marked the shadowed area in ND -- N, space where we can find transverse waves (see Fig. 5 ). An explicit expression for the existence of transverse waves will be given later by an asymptotic expansion. The other branches in Figs. 3 and 4 are different from all the wave solutions discussed until now. The most striking feature is that these marginally stable curves have no minima, i.e., Ec --* 0 when --* 0 (or a --* 0). Furthermore, they are unaffected by the surface adsorption. However, the lack of influence upon these curves of the surface deformability indicates their longitudinal character. Remembering the oscillatory instability found by Sternling and Scriven (14) in an adsorption-free model, with the same critical behavior in low frequency (large wavelength) region, we consider this branch as the Sternling and Scriven wave.
TABLE I Parameter Values (Standard Ground Conditions) D v l
10-5 10 2 100 102 103 10-I 10 1o
S S~ C Bo H p~
103 103 10-9 10° 10_9 10-5 lO-lO
fl
10 1
?7z
10--8
kI ~sh, Tldil
10-5 10-10
g [p] Po,
pz
q-
He
)k3/2V-SHzTr3C5/2 } 1/2
_
~2 =(Bo+O(~2)) aTr2
~k3/2f"~ ~ q- 0 ( ~ 5/2)
4. ASYMPTOTIC EXPANSION
Following the data listed in Table I, we can analyze the problem by introducing a smallness parameter E = S -~ . Then we take
S= O(c-1), C= O(~3), H= O(~3), I~t = O ( ~ 3 ) ,
Bo = O(1)
H~--
and
~71"l }
[4.1.1]
-1/2
+O(E3),
[4.1.2]
where
~
~f~dND + 1
O(E3/2),
1 + 1/VS-
fNDD/N, -- 1
~/z = O(e3). ~2 - V ~ / N . + ~ff / N .
4.1. Long Wavelength Expansion-Long Wavelength Longitudinal Waves First we consider the long wavelength region. We assume that a ~ w and take w = O(e) and a = 0(~2). Thus, Eq. [2.29] and [2.30] become
~1 = (HzS)~e3/2 + ~ f ~
× {(Bo + o(~2))(~(1 + VN.~)~ 1/~ + a ( l + N.)e 2 + O(e7/2))}
= { x( 1 + ~ . N ~ ) ~ + a G ( 1 + U.)~ '/~
1
1+ 1 / V ~ - - - ~ - 1
~3 = (1 + N~N~)
/(
1)
1 + Nz----~
Substituting 2z and 22 into the Eq. [2.28] yields Ea 2rq
_]_ 3 / 2
{a( 1 + Nu)
G
+ ~Hz~3
_ aTr2(1 + fN~Np) l [4.1.3] V~71_ 1 J JournalofColloidandInterfaceScience,Vol. 131,No. 2, September1989
478
CHU AND VELARDE
At the marginal state, the 1.h.s. of Eq. [4.1.3] is real. Thus 1 E =-
a 2./1-1
Dfrom
[4.1.4] 1 + N.
a
7~2
H:r3o01/S
~rl
(1 + f'NuNp) = O.
[4.1.51 Using Eq. [4.1.5] in Eq. [4.1.4], we obtain the critical elasticity Marangoni number for oscillatory instability (overstability) E,=-{(~+
Nu)( 1/]/~-H:r,~3 1)} 2
× 7r 1
[4.1.6] It does take a finite value even when w --* 0. On the other hand, the capillary number does not appear in this expression. We have the longitudinal property of this wave. Thus we can safely take the flat interface approximation in the case of long wavelength longitudinal waves as already remarked by Sanfeld and collaborators (23, 24). From Eq. [4.1.4], we obtain:
Condition 1 sign(E) = - sign(zq ) = -sign(~
For the system to be oscillatory unstable (overstable) both conditions 1 and 2 must be simultaneously satisfied. Thus the conditions for long wavelength oscillatory instability are
- 1) = sign(N, - ND).
From Eq. [ 4.1.5 ], we have w
(}/-~D+ N~)(V1/N~ - 1)
a
f-SH:r~ ~r3
Now as the frequency and the wavenumber have to be positive, we also have:
Condition 2 sign(~rl) -- s i g n ( V 1 / N , - 1). Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
/)from
- - > - Dto /)to
and
/)from
--<
I.
[4.1.7]
/)to
Subscripts "from" and "to" give the direction of the mass transfer, e.g., surfactant transfer from the phase "from" to the phase "to". The physical meaning of this condition is that this oscillatory instability can be observed whether the mass transfer is from the high diffusivity fluid to the other or the other way around provided Dfrom/Dto > Pfrom/Vto-However, no oscillation is expected if mass transfer is from the higher viscosity liquid to the other. Our analytical results [4.1.7] agree quite satisfactorily with the numerical estimates for longitudinal waves in cases b and c. It is more restrictive than Sanfeld's condition (24) Dfrom/ Dto > Pfrom//)to thus showing that although necessary it is clearly not sufficient. Moreover Sanfeld's analysis does not provide a finite critical elasticity Marangoni number Ec (or else (Oc/OZ)c), below which all these waves, whatever long their wavelength may be, are damped. Note that this oscillatory motion is directly related to the surface adsorption as Hz appears in Eqs. [ 4.1.4-4.1.6 ]. Adsorption-free or saturated adsorption lead to an infinitely divergent critical elasticity Marangoni number thus illustrating the destabilizing effect of Hz. All these features are completely different from the properties of the waves discovered by Sternling and Scriven(14) which on the one hand are always unstable in the long wavelength limit, and on the other hand are not related to the surface adsorption.
4.2. High Frequency Expansion-Laplace Transverse Waves ( TW) Previous work by several authors (2, 3, 7, 30), indicates that transverse waves are easily found in the high frequency region. We take
WAVES AT LIQUID-LIQUID INTERFACES (..0 = O(~ -1) and a = O(1). Then Eqs. [2.292.30] become
which is a generalized Laplace equation for liquid-liquid interfaces (3, 6, 8 ). Substitution o f f ( B o ) = 0 in Eq. [4.2.5] yields
X (1 + VN~,No)E-3/2 + V~(f(Bo)a(1 + N~,)
+ 4~2SCf-NuNp)~ -1
479
+ EaZA = 0.
[4.2.8]
O(~-1/2)} [4.2.1]
+
This equation has a solution only when ~2-
" / F l U ( B ° ) E 3/2
sign(E) = sign(A) = s i g n ( N . - 1/~D). +
(a~r2f(Bo) X3/2V~
+ 2CAS~ + O(E5/2),
) E2
Thus we have the necessary condition for oscillatory instability
[4.2.2]
Dfrom (/£from] 2" > Dto \ #to ]
where
~/S + 1
~./ND
+ 1
- ]/~N, - ~
[4.2.3]
f ( B o ) = Bo + a 2 +
Relation [4.2.9] explains why in cases b, c and d one can find transverse waves while there is none in case e. This is illustrated in Fig. 5. The necessary condition for a minimum is dE(a, w(a))/da = 0. This yields the critical values aT = ~
X 2SC( 1 + Np) a
[4.2.10]
[4.2.4]
f
2Bo3/2
Replacing ~ and -2 ~ in Eq. [2.29] and setting the real and imaginary parts equal to zero leads to the following expressions: (1 + N ~ ) { f ( B o ) a ( 1
+ N~)
E [ = - C(1 + No)(V-~o - N,)
+ 4X 2SCVN,,Np } + Ea---~2
S X 2CA+ +
arc2f(Bo) ] ~w2
j=o
[4.2.5]
1
× (1 + 1/N.Np) + Ea2~rl} = O.
[4.2.61
J
We see that f(Bo) = 0
[4.2.9]
[4.2.71
[4.2.11]
[4.2.12]
Previous work (6) has shown how transverse waves can be damped at liquid-liquid interfaces. Now, we find that, as long as [4.2.9] is satisfied, transverse waves can be sustained if enough mass transfer crosses the interface, either from the liquid with lower diffusivity to the higher one or the other way around. The oscillatory motion is indeed a transverse wave because the wavenumber and frequency satisfy the Laplace relation, and moreover because the critical Marangoni elasticity number is inversely proportional to the capillary number C. Again we have found quite Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
480
CHU AND VELARDE
satisfactory agreement with the computer resuits given in Section 3.
Using Eqs. [4.3.1] and [4.3.2], from Eq. [2.29], we get Ea 2
4.3. Sternling and Scriven Waves (SSLW) In Section 4.1, we discussed the appearance of longitudinal waves by using the long wavelength approximation a ~ w. Let us explore now other wavelength ranges, for example a ~0. Our numerical results in Section 3 already gave indication (see Figs. 3 and 4) that, if the conditions Dfrom/Dto > 1 and Pfrom/Pto > 1 are satisfied, we can always find a marginal state for oscillatory instability. Its behavior is drastically different from L W L W as no minimum critical elasticity number can be found. It shares, however, a common property with LWLW: the marginal state is unaffected by the deformability of the interface. Consequently, we take it as a longitudinal wave. However, on the theoretical side, such a wavelength range brings substantial difficulties for an asymptotic expansion. Fortunately, computer search shows that these longitudinal waves change when we cross the line N, = 1. This suggests the possibility of performing a perturbation analysis around N, = 1 thus simplifying the problem. Assume N ; 1 = 1 + 6 with 6 = O(e 1/2) and oJ = O(1) as well as a = O( 1 ). Then
I+N.
S2w 2 1 +
.q-
1. +. N, .
.
(1 - ~ ) 2
.
I.
.
X(N, . . . +. f~D)
[2rnl(rnl + a ) ( 1 + N , ) 6
+ ml(1 - - N D ) ] ~-~~1/2. Marginal stability requires that Ec- 1---~
1+
w( 1/N~ - l )(N~ + ~ ) 2 R e { m l ( m l + a)}(1 + N,)
=
Re(m1)(1 - No) ~ ,
× {(ml + a)(1 + N~)~ -1/2 %.
q- 2m~ )~T" ~ -{- O(~ I/2)}
[4.3.1]
Bo + a 2 ~(ml + 1)(~r-~D - 1) ~1/2
--
[4.3.5]
where Re(o) denotes the real part of the complex quantity inside the bracket. Taking account of the signs in Eq. [4.3.4], we have sign(E) = sign(1 - 1/~D).
[4.3.6]
Eq. [4.3.5] can be written in the form F(w/a2, Np, N~,ND, . . . ) - 0 .
o~c/a 2 = 17( Np, N~, ND, • • ").
x
[4.3.4]
[4.3.7]
The solution of Eq. [4.3.7] gives the dispersion relation
Nz VND]
~2
[4.3.3]
[4.3.8]
Although we have been unable to obtain a compact analytical expression of this solution, we can, however, use Eq. [4.3.5] to obtain a necessary condition for the existence of this solution. This condition is
t
m l ( m l + 1)(ND-- 1) S~
sign ~ Jr,
E
+ 2 m 1 ~ 84 + O ( e 3/2) .
[4.3.21
Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
1 = sign(l - ~ ) .
[4.3.91
Now, assuming that we have a solution of Eq. [ 4.3.7 ] in the form [ 4.3.8 ], substituting it into Eq. [4.3.4] yields
481
WAVES AT LIQUID-LIQUID INTERFACES
e c = 1 _-
l+u--
× S2WcY(No, N,, ND, • • ").
[4.3.101
This equation shows that the marginal elasticity number tends to zero when w --~ 0 (or a --~ 0), a result that we expect from the numerical study. Combining [4.3.6] and [4.3.9], we obtain the necessary condition for the longitudinal wave Oftom /)from --> 1 and > 1. [4.3.11] )to vw Thus the oscillatory instability only occurs when the direction of mass transfer is from the fluid with higher diffusivity and higher kinematic viscosity to the other. This is analogous to the prediction given by Sternling and Scriven(14). In summary we find that for overstability to occur:
the necessary conditions for their existence. We have also shown the existence of two types of sustained longitudinal waves that we have called L W L W and SSLW, respectively, and delineated the regions in parameter space where they appear. Explicitly the results are (for a comprehensive account see Table II):
( 1 ) Laplace transverse waves exist for a critical elasticity number +
E~=-
with critical wavenumber: a Tc =
(3) The deformability of the interface does not play any influence on the critical elasticity number, which is consistent with the flat interface assumption made by Sternling and Scriven(14). (4) Surprisingly enough and contrary to the L W L W case, surface adsorption does not affect this new longitudinal wave which agrees with the result for Sternling and Scriven waves in the adsorption-free case.
B~O
and with critical frequency: X @ 2B°3/2 wo = SC(1 + N p ) "
( 1 ) Eq. [4.3.11 ] must be fullfilled. (2) The critical elasticity number found in the low frequency range is the same as that predicted by Sternling and Scriven (14) but different from the results obtained for LWLW.
C ( l + Np) ( I ~ D -- N u)
The necessary condition for the existence of this wave is -Dfrom - >
Dto
("from ] 2 \ #to /
(2) Long wavelength longitudinal waves exist for a critical elasticity number + N.)(l~-HzTrlTr3 1)} 2
E, = _{(~
× It is clear that the newly found waves are of the Sternling and Scriven oscillatory type. Finally as the surface adsorption plays no role, the conclusions in this section can be transferred verbatim to the case of energy transfer. 5. SUMMARY OF RESULTS AND COMPARISON WITH EXPERIMENTAL DATA
We have found sustained Laplace transverse waves at liquid-liquid interfaces and indicated
71"1
with a critical dispersion relation L
(I/~D + N.)(V1/N~ - 1)}a }
w c =
f~gzTrlTr3
"
The necessary conditions for the existence of this kind of instability are Ofrom gfrom > -and )to Pto
- -
Journal of Colloid and lnterfaee Science,
Pfrom
--
< 1
Pto
Vol.131,No. 2, September1989
482
CHU AND V E L A R D E TABLE II Summary of Necessary Conditions for Overstable Oscillations Dfrom
LW
Dto Yfrom //tO Of tom
Oto 0Jfrom //to
>
//from ~',o
- -
and
LWLW
<1
> I and
SSLW
>1
Ofr°m > (/*fr°"---'~m] 2 Dto \ #to /
TW
LAPLACE TW
with 7r1=
N D ~ - -
1
and 7r3=(1 +
)
1+
.
(3) S & S longitudinal waves exist for a critical elasticity number
I+N~ (1+ 1 × S2o~cY(Np, N., No,
• • • ),
where Y(Np, N,, ND, • • • ) is the solution of Eq. [4.3.7]. The dispersion relation is we = 7 ( N o, N,, No, • • -)a 2. The necessary conditions for the existence of this mode of instability are D from -
-
>
/)to
1
and
Vfrom
> 1.
/"to
Finally let us say a few words about the available experimental results (34-37). Linde and collaborators (31-35) have shown experimental evidence of Scriven and Sternling longitudinal waves as well as transverse waves in a gas-liquid case with heat or
Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
mass transfer from the gas to the liquid. Pertinent theory for the liquid layer open to air has already been provided in Ref. 30. On the other hand, Orell and Westwater (36, 37) have reported findings on the ethylene glycol-ethyl acetate system with acetic acid as surfactant. Typical parameter values as estimated by Sorensen (28) are listed in Table III. Phase 1 is the ethylene glycol and phase 2 the ethyl acetate. When acetic acid transfers from phase 1 to phase 2 all previous linear theories (14, 2328) as well as our results (Sect. 3, case a) predict instability in the form of steady cellular convection which is the experimental result reported by Orell and Westwater (36, 37). However, the latter authors also found an oscillatory instability that no available linear theory predicts. In agreement with Sorensen (28) we guess that this oscillatory motion is the result of a secondary instability at a higher Marangoni number than the steady instability threshold. Our guess is compatible with the numerical findings reported by H y m a n and Nicolaenko for the Kuramoto-Velarde nonlinear equation in Benard-Marangoni convection for thin liquid layers open to air (38). When the transfer of acetic acid is from phase 2 to phase 1, Sternling and Scriven (14) predicted stability while our theory (Sect. 3, case b) predicts instability for Marangoni numbers equal or higher than
E [ -- -
- - 1.3 * 10 9 C(1 + Np)(~-~D - N~,)
and
E~ = - { (f-N-~D+ Nj')(VI
l)} 2
× 7["1
- -4.8.1016
WAVES AT LIQUID-LIQUID INTERFACES TABLE Ili Typical Parameter Values According to Refs. (28) and (30, 31)
O¢]OP = -4.13.108 cm 2 s-2 a = 1.44 dyn c m - 1 k] = 1.27,10 7 cm /Zl = 0.103 poise 01 = 1.064 g c m -3 Dl = 0.21,10 -5 cm 2 s -I fl,a ,~ 0.05 g c m -4
Dz - 1.8.10 -5 cm 2 s l Fo = 2.67.10 -1° g cm -2 k~ = 1.93.10 -7 em t~2 = 6.36"10 -3 poise 02 = 0.9142 g cm -3 D2 = 3.4,10 -5 cm 2 s -t
a /~1 corresponds to a drop from the concentration 0.5 w% to zero over a distance of 1 mm.
for transverse and longitudinal modes of oscillatory convection, respectively. In terms of gradients these critical Marangoni numbers correspond to fiT = --565.2 g cm-2 and /3~ = -2.1 * 10 l° g cm -2. According to the parameter values listed in Table III, Orell and Westwater results (36, 37) should be in the stable region. Thus our theory is compatible with the available experimental evidence. However, further experimental work for higher values of the constraints is needed in order to validate or invalidate our theoretical predictions. Clearly these experimental results are not compatible with the predictions made by earlier authors (23-28) who predicted oscillatory instability albeit without a given threshold. Recently, the authors have rederived some of the results reported here by using a harmonic oscillator description of the interfacial motions (39-42). Moreover, they have extended the theory to the nonlinear region of limit cycle oscillation (43-46) and soliton behavior (47). ACKNOWLEDGMENTS This research has been sponsored in its earlier stage by the Stiftung Volkswagenwerk and latter by CAICYT (Spain) and by an EEC Grant. X.-L. Chu expresses his appreciation to the Spanish Science Policy General Directorate for a fellowship that enabled him to work at
483
UNED. M. G. Velarde acknowledges fruitful discussions with Dr. B. Nichols and the hospitality of CNLS, Los Alamos National Laboratory, where this work was completed. Both authors also acknowledge fruitful discussions with Professors H. Linde, A. Sanfeld and D. M. Hennenberg. REFERENCES 1. Lamb, H., "Hydrodynamics." Dover, New York, 1932. 2. Landau, L. D., and Lifshitz, E. M., "Fluid Mechanics." Pergamon, Oxford, 1959. 3. Levich, B. G., "Physicochemical Hydrodynamics." Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. 4. Miller, C. A., and Neogi, P., "Interfacial Phenomena." Dekker, Inc., New York, 1985. 5. Van den Tempel, M., and van de Riet, R. P., J. Chem. Phys. 42, 2769 (1965). 6. Hansen, R. S., and Ahmad, J., Prog. S u r f Membr. Sci. 4, 1 (1971). 7. Garcia-Ybarra, P. L., and Velarde, M. G., Phys. Fluids 30, 1649 (1987). 8. Velarde, M. G., Garcia-Ybarra, P. L., and Castillo, J. L., Physicochem. Hydrodyn. 9, 387 (1987). 9. Lucassen, J., Trans. Faraday Soc. 64, 2221, 2231 (1968). 10. Lucassen-Reynders, E. H., and Lucassen, J., Adv. Colloid Interface Sci. 2, 347 ( 1969 ). 11. Van den Tempel, M., and Lucassen-Reynders, E. H., Adv. Colloid Interface Sci. 18, 281 ( 1983 ). 12. De Voeght, F., and Joos, P., J. ColloM Interface Sci. 98, 20 (1984). 13. Cini, R:, Lombardini, P. P., Manfredi, C., and Cini, E., J. ColloM Interface Sci. 119, 74 (1987). 14. Sternling, C. V., and Seriven, L. E., A I C h E Journal 5, 514 (1959). 15. Velarde, M. G. and Normand, C., Sci. American 243, 78 (1980). 16. Velarde, M. G., and Castillo, J. L., in "Convective Transport and Instability Phenomena," (J. Zierep and Oertel, Jr., Eds.), pp. 235-246. Braun-Verlag, Karlsruhe, 1982. 17. Legros, J. C., Sanfeld, A., and Velarde, M. G., in "Fluid Sciences and Materials Science in Space," (H. U. Walter, Ed.) pp. 83-140. Springer-Verlag, N.Y., 1987. See also M. G. Velarde (Ed.) "Physicochemical Hydrodynamics. Interfacial Phenomena," Plenum Press, N.Y., 1988. 18. Lino S. P . , A I C h E J . 16, 375 (1970). 19. Palmer, H. J., and Berg, J. C., J. FluidMech. 51, 385 (1972). 20. Palmer, H. J., and Berg, J. C., A I C h E J. 19, 1082 (1973). 21. Chu, X.-L., and Velarde, M. G., J. Colloid Interface Sci. 127, 205 (1989),
Journalof Colloidand InterfaceScience, VoL 131,No. 2, September1989
484
CHU AND VELARDE
22. Gouda, J. H., and Joos, P., Chem. Eng. Sci. 30, 521 (1975). 23. Hennenberg, M., Sorensen, T. S., and Sanfeld, A., J. Chem. Soc. Trans. Faraday H 73, 48 (1977). 24. Hennenberg, M., Bisch, P. M., Vignes-Adler, M., and Sanfeld, A., in "Dynamics and Instability of Fluid Interfaces," (T. S. S~rensen, Ed.), pp. 227-259. Springer-Verlag, Berlin, 1978. 25. Sanfeld, A., Steinchen, A., Hennenberg, M., Bisch, P. M., Van Lamsweerde-Gallez, D., and Dalle-Vedove, W., in "Dynamics and Instability of Fluid Interfaces," (T. S. Sorensen, Ed.), pp. 168-204. Springer-Verlag, Berlin, 1978. 26. Sorensen, T. S., Hansen, F. Y., Nielsen, J., and Hennenberg, M., J. Chem. Soc. Trans. Faraday H 73, 1589 (1977). 27. Sorensen, T. S., Hennenberg, M., and Hansen, F. Y., J. Chem. Soc. Trans. Faraday II 74, 1005 (1978). 28. Sorensen, T. S., in "Dynamics and Instability of Fluids Interfaces," (T. S. Sorensen, Ed.), pp. 1-74. Springer-Verlag, Berlin, 1978. 29. Reichenbach, J. and Linde, H., J. Colloid Interf Sci. 84, 433 (1981). 30. Chu, X.-L., and Velarde, M. G., Physicochem. Hydrodyn. 10, in press. 31. Linde, H. and Loeschcke, K., Chem. lng. Tech. (GDR) 39, 65 (1966). 32. Linde, H. and Kunkel, E., Z. Wgirme-Stoffiibertr. 2, 60 (1969). 33. Linde, H. and Schwartz, P., Chem. Tech. (GDR) 26, 455 (1974). 34. Linde, H., in "Dynamics and Instability of Fluid In-
Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
35.
36. 37. 38.
39. 40. 41. 42. 43. 44. 45. 46.
47.
terfaces," (T. S. Sorensen, Ed.), pp. 75-119. Springer-Verlag, Berlin, 1978. Linde, H., in "Convective Transport and Instability Phenomena," (J. Zierep and H. Oertel, Eds.), pp. 256-296. Braun-Verlag, Karlsruhe, 1982. Orell, A., and Westwater, J. W., Chem. Eng. Sci. 16, 127 ( 1961 ). OreU, A., and Westwater, J. W., AIChE Z 8, 350 (1962). Hyman, J. M., and Nicolaenko, B., "Coherence and Chaos in the Kuramoto-Velarde Equation." LAUR-86-1388, Los Alamos, N.M., 1986. Velarde, M. G., and Chu, X.-L., Phys. Lett. A 131, 403 (1988). Velarde, M. G., and Chu, X.-L., ll Nuovo Cimento D, to appear (1989). Chu, X.-L. and Velarde, M. G., Il Nuovo Cimento D, to appear (1989). Chu, X.-L. and Velarde, M. G., II Nuovo Cimento D, to appear (1989). Chu, X.-L. and Velarde, M. G., Phys. Lett. A136, 126 (1989). Chu, X.-L. and Velarde, M. G., Phys. Lett. A, submitted for publication ( 1989 ). Velarde, M. G. and Chu, X.-L., Physica Scripta T 25, 231 (1989). Velarde, M. G. and Chu, X.-L., in "Phase Transitions in Soft Condensed Matter," (T. Riste and D. Sherrington, Eds.) Plenum Press, 1989, to appear. Chu, X.-L. and Velarde, M. G., Phys. Rev. Lett., submitted for publication ( 1989 ).
Received July 20, 1988; accepted October 10, 1988 Dispersion relations are derived for transverse and longitudinal modes of oscillation appearing as a result of transfer, adsorption and eventual accumulation of a surfactant at the interface of two liquids. Also provided here are the values of the Marangoni number sufficient to sustain these waves and a comparison between theoretical predictions and available experimental data. © 1989AcademicPress,Inc. 1. INTRODUCTION
, gitudinal wave has received m u c h less attention than the other since under most practical Capillary w a v e s - - o r r i p p l e s - - a n d gravity conditions, the motion is d a m p e d out m u c h waves have been well studied since Laplace, more rapidly than its transverse counterpart. Kelvin and Stokes ( 1 - 8 ) . Their properties Lucassen studied the longitudinal wave and were shown to be mainly determined by the proposed that in the elastic surface of a highly stress condition at liquid surfaces. The major viscous fluid longitudinal waves could be obinfluence of the boundary condition for nor- served easier than transverse waves (9). Lucassen showed that longitudinal waves real stress to the surface has been especially emphasized. While the ripples are rather are, to a major extent, related to the boundary transverse motions due to the deformation of condition for tangential stress with a frequency the surface there is yet another type of wave that depends on the viscosity and surface elasdiscovered years ago by Lucassen (6, 9-1 1 ). ticity modulus. Gravity-capillary waves, It refers to mostly longitudinal motion along however, have a frequency that depends on the surface, in the limit along a flat surface. gravity and on surface tension (Laplace overTheir existence is not surprising considering pressure) but not on viscosity. The latter only that a strong analogy is expected between a appears in the damping factor and frequencymonolayer-covered surface and a stretched deviation in the dispersion relation. Generally these waves, these oscillatory elastic membrane. The coverage with a surfactant monolayer--either by adsorption from motions, are damped albeit differently by vissolution or by spreading--gives indeed elastic cosity (5, 6, 9-13). However, if a nonequilibproperties to a surface so that it tends to resist rium distribution of surfactant/temperature the periodic surface expansion and compres- is imposed in the liquid that drives m a s s / e n sion which appears as wave motion. Normally, ergy transfer across the surface, the Marangoni any wave motion of the surface has both effect m a y transform the chemical energy into transverse and longitudinal components, and motion, overtaking the viscous dissipation and they are not separable. Only in the case of thus sustaining the wave motions. Sternling and Scriven (S&S, 14) were the small amplitude wave motion (corresponding to linear theory) transverse waves and longi- first to provide a mathematical description of tudinal waves can be considered as two gen- spontaneous convection at the interface of two uinely different modes of oscillation. The lon- fluids. In their now classic paper published in 471 0021-9797/89 $3.00 Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
472
C H U AND VELAR D E
1959, they clearly considered the Marangoni Sorensen et aL (26, 27), and Sorensen (28), effect as the "interface engine" that converts and by Sternling and Scriven (14) are longichemical energy into convection. Marangoni tudinal waves caused by different mechanisms stresses due to surfactant concentration or/ and appear in different wavelength regions. We and temperature gradients along the interface also study the transverse waves. generate convective motions that are transIn Section 2 we recall that the evolution mitted to the bulk. If there is high enough sur- equations are obeyed by disturbances upon the factant or thermal gradient in the bulk, con- quiescent state of the liquid-liquid system with vection may promote the surfactant or/and a fiat interface. We also derive in this section thermal distribution at the liquid-liquid in- the determinantal equation that provides the terface thus permitting disturbances to develop necessary and sufficient condition for the exin the form of steady cellular convection or istence of either a convective steady state or oscillatory convection ( 15-17). The latter case overstability. Section 3 is devoted to an acmay correspond to waves that appear as either count of numerical results obtained with the longitudinal or transverse capillary-gravity computer using the above mentioned deterwaves or a combination of both (4). minantal equation. In Section 4 we develop Although Sternling and Scriven were aware an asymptotic analysis that permits an indepth of transverse motions in experiments, and re- discussion of longitudinal and transverse ferred to them in their paper (14) as "localized waves appearing at different wavelength and stirring with rippling and twitching of the in- frequency ranges. Finally in Section 5 we terface" (17), they, however, treated the in- summarize the major results found and comterface as flat in order to simplify the mathe- pare our theoretical predictions with the scarce matical problem. This excluded the transverse available experimental data. waves in their analysis. Another approximation in S&S's model was the elimination of 2. DISTURBANCE EQUATIONS surface adsorption although they considered a surface-active solute. As pointed out by sevWe limit ourselves to consider a liquid-liqeral authors (11, 18-21 ), solute adsorption uid interface initially at rest and located at z and eventual accumulation at the surface takes = 0 (z is the vertical coordinate). The evoplace in liquids with surface-active solutes and lution equations and boundary conditions for drastically affects stability. The Sternling and Scriven theory has had disturbances upon the quiescent state are for subsequent developments by several authors both liquids and the interface (22-37). Sanfeld and collaborators studied a V - ~ + = O (q~= 1,2) [2.11 liquid-liquid interface model with due account of the deformable interface and solute adsorption at the interface. However, they only discussed one of the various possible modes, the long wavelength longitudinal waves. It was also shown in their results that the deformOt /3~#~= D~'2?o [2.3] ability of the interface has negligible influence on the stability of these waves. Here we consider the liquid-liquid model if z = f denotes the disturbed liquid-liquid developed by Sanfeld et aL and proceed to a interface we impose the following boundary complete analysis of all possible solutions. Dif- conditions (b.c.) ferent frequency regions are carefully exam0f ined. We show that the oscillations studied by ~-=~=~[~ (q~-- 1,2) [2.4] Ot Hennenberg et al. (23, 24), Sanfeld et al. (25), Journal of Colloid and Interface Science, Vol. 131,No. 2, September 1989
WAVES AT LIQUID-LIQUID
~,l~ = ~2[z = ~z
o~
(o~) o~
[2.5]
o~z
[ \Oy+-~-~
473
INTERFACES
face concentration: vl/l, 12/pl, v~pl/l 2, fill and Po. After changing the units, the dimensionless equations are (Note that the variables below without tildes have no dimension)
(q~= 1,2)
div g, = 0
=0
[2.11]
[2.6] --
= -Vpl
+ Vz~l
[2.121
Ot p~ ~ -
-
,~o a£-~ -
[ +[/~]-2[
= 0 0i]
= --Vp2 + NuV2~'2
Np at
OCl - - -- W 1 = S-1•2Cl
O3' + yo O~Z
o~
O~ _ [ D Oa]
-5-~--D~ at
~ =0
= k{b¢l~,
[2.13]
[2.71 [2.14]
Ot
[2.81
OC2
- - - w2 = S-1NDV2C2. Ot
[2.91
[2.151
Now the boundary conditions at z = ~-are where ~ = (~, k) is the disturbance velocity, /~ the disturbance pressure, ? the disturbance surfactant concentration, ~: the deviation of the interface from the motionless level initial position, P the excess accumulation of surfacrant at the interface, and ~ the disturbance on r. p accounts for liquid density, # = p ~ is the dynamic shear viscosity with u the kinematic viscosity, D the surfactant diffusivity, g the gravitational acceleration, /3 the surfactant gradient, p~ the surface excess density, #dil and #~h the surface dilational and shear viscosity, respectively. ~ is the surface tension and k I the Langmuir adsorption constant. The subscript ~bindicates the liquid: 4 --- 1 the liquid below and ~b = 2 the liquid above the interface. The subscript "0" accounts for the initial reference state and the bracket [f] the jumping property at the surface [ f ] = (f2 - A ) ~ . Now for universality of our analysis, we use suitable units to rescale the original dimensional quantities. Although this is not the only possible choice, we can choose the capillary length as the unit of length ~/~ l =
¢ro 1/91
p21
[2.10]
and the following scales for velocity, time, pressure, solute concentration and excess sur-
ul = u2 = uz
Owl Oz
Ow2 Oz
o~ Ot
--=
1
02~"
Bo
S C Ox 2
SC
[2.16]
Ow~ Oz
[2.17]
[2.18]
wz
~-- [p] OWz OW~ 1)-~-z =V~-- ~ -
+2(N,-
[2.191
/4E02~ (02 O2) H z S Ox2
Ox2
=
MS
O~5 (N.w2 - wl)
O O 2 ) Ow z Pt~-#Z~x 2 ~z
[2.20]
03" OUG 02"~ 1 - ~ ~- ~ x - S~I ~ x 2 )
Ocl Oz
0c2 Oz
+ . . . .
H~
3" = ~ - (cl - ~)
HzNz
3' = ~ - - -
(c2 - ~).
0
[2.21]
[2.22]
[2.23]
Journalof ColloidandInterfaceScience,Vol. 131, No. 2, September 1989
474
CHU AND VELARDE
We have used the following dimensionless groups N o = P~
pa
#2 N'` = - #1
density ratio,
k]31l 2 × - #aD1
viscosity ratio,
Po 3~l 2
surface excess solute number,
k;
Langmuir adsorption number, ratio of surface excess density to liquid density,
Hz = -
D2 ND = -D1
diffusivity ratio, ratio of Langmuir adsorption coefficients,
k12f12
N~ = kl1fll Ya
Schmidt number,
Sz = - Dz
surface Schmidt number,
#1D1 C = -[ao [p]gl2 Bo =
capillary number, bond number,
O"o
Bo+a 2 -
SC~
Ft ~
1 Pz -pal ~dil "~- ~ s h
~z =
S = -D1
-
elasticity (surfactant Marangoni) number,
#ll
ratio of surface viscosity to liquid viscosity.
Note that by choosing the capillary length as the unit of length Bo = 1. However we shall explicitly indicate the Bond number dependence in our results. Then after some standard and elementary albeit lengthy analysis, the necessary and sufficient condition for nontrival solution of the problem posed by Eqs. [2.11-2.15 ] with b.c. [2.16-2.23 ] is the following determinantal equation:
X(Np+ 1) +
a
+ PtX
a - aN, +
p2N'` - Pl
+ aN,, + a +Pa + p 2 N . - a + aN~, - p2N'` + PI
- r t X - ~ a 2 - aN,
Ea2/S
= 0,
[2.241
- a - Pl - p 2 N , _z(m+a I ~ q + rn]
m+a]
-L q-- a J + M s
X H z S ( X + a2/S~) + qlX + q2)t/Nz
where ml = ~ a 2 + X
[2.2.51
m2 = ~a 2 -b X N o / N u
[2.26]
ql = alfa~+ SX
[2.27]
q2 --= ga a + S X / N D
[2.281
Here a is the dimensionless Fourier waveJournal of Colloid and Interface Science, Vol, 131, No. 2, September 1989
number and ), the time constant whose real part determines the stability of the motionless state. Its imaginary part gives the frequency of the oscillation. Eq. [2.241 is the dispersion relation that in compact form is Ea2~_,~
~a +
S
O,
[2.29]
WAVES AT LIQUID-LIQUID INTERFACES
475
where 10~r.
~1 = HzS X+
~t
+ ql +
×{( B°+a2+SC~2(Np+l+aI't))a X (ml + a + N~(mz + a) + FiX + ~za 2)
101-'
10 ~
+ SCX(PtX + ~za2)(ml + a 10 -4
+ N~,(m2 + a)) + 4XSC × (ml + aN,)(m2N~ + a)}
[2.30]
.i
~2 = (BO "J~ a2 Jf- SC~2(N'° -~- e -jr a~t)
10'
100
102
' 60
I~)G. I. Typical overs[ability lines for transverse (solid line) and longitudinal (broken line) waves. No = 0.8, N, = 0.5, No = 2, Nz = 1, Bo = 1, C = 10-9, S = S:~ = 103, H = - 1 0 -9, az = 1 0 -5 a n d l~t = r/~ = 0 . T h e s e m o d e s o f oscillation appear for (vfrom/Vto) < 1 and (Dfrom/Dto) > 1.
×(HS-[m+a]/XI+HSZC~ [q + m J / ]
(b) IfND > 1 and N~ < 1 the marginal stability curves are those of Fig. 1.
× (ml + a + Nu(m2 + a)) + 2SC
(c) If No < 1 and ND > N, the marginal stability curves are those of Fig. 2.
× I(m2N, + a) ml + a [ ql + ml
(d) If No > 1 and N~ > 1 but not far from unity the marginal stability curves are those of Fig. 3. See also Ref. (14).
+ ( m l + Nua) q2rn2+m2 " [2.31] From now on, we concentrate on the analysis of this dispersion relation [2.29]. Only overstability will be investigated here, that is, we shall limit ourselves to the case Re (~) = 0 with I m ( ~ ) v~ 0.
(e) If No > 1 and N, > 1 but really away from unity the marginal stability curves are those of Fig. 4. See also Ref. (14).
It appears that the curves in case b (Fig. l ) and case c (Fig. 2) are rather similar to those found for air-liquid interfaces (30). This similarity reflects not only in the form of the curves, but also in the properties of those 3. NUMERICAL RESULTS curves. For instance, their dependence on the As expected from the large number of pa- parameters of the problem. Two independent rameters in the problem a discussion of nu- branches of marginal stability have minima merical results is far from simple. There is for the elasticity Marangoni number. Our calhowever a way out to simplify the issue: to list culation shows that one of these branches with the results found in figure form for cases of the minimum elasticity number in a region of rather general value. Thus Figs. 1-4 describe much lower frequency than the frequency stability thresholds in the (E, w) plane. For range of transverse waves does not depend on specific illustration let us choose Np = 0.8 and interface deformability but it is affected by the Nz = 1. Then recalling that N, = / 1 2 / / * 1 = N~,I surface adsorption, thus showing a feature of longitudinal waves. The other branch, with a No, m i n i m u m of elasticity number in the high fre(a) If N, > 1 and ND < 1, no oscillatory quency region, varies with C, i.e., with the insolution has been found. However, the system terfacial deformation and it is clearly a transverse wave. Note that the air-liquid interface is unstable to steady cellular disturbances. Journalof ColloidandInterfaceScience,Vol. 131, No. 2, September
1989
476
C H U A N D VELARDE
E
~,
10:~°
1 0 i6-
101ss
ql
1018 10 n
1014 -
i
t
/ ji
1012I 10 - a
10 2
10 o
102
~; 10 t /
FIG. 2. When the mass transfer direction is reversed, i.e., (Drrom/Dto) < 1 here, if the condition (Dfrom/Dto) > (V~om/vto) is satisfied the overstability lines are quite similar to the case depicted in Fig. 1. N~ = 0.8, N , = 0.3,
¢
10~
instability
convection
branch,
narrows
and narrows
region
but the until
/
z
/ / / / / /
to
10,
is a s p e c i a l c a s e o f i t e m b i n t h e l i m i t N D c a s e d ( F i g . 3 ) w e still s e e t h e t r a n s -
J
¢
TM
Go. F r o m
s
s¢
ND = 0 . 5 , N~ 1, B o = 1, C = 1 0 - 9 , S = Sz = 103, H = - - 1 0 -9, H~ = 10 -5 and rt = ~z = 0.
verse oscillatory
¢
FIG. 4. When we really move away from (Drrom/D,o) = 1 the transverse wave still existing in Fig. 3 here disappears but the SSLW remains. Np = 0.8, N , = 3, ND = 5,
Nz =
1, B o = 1, C = 10 -9, S = S~ = 103, H = - 1 0 -9,
Hz =
10 -5 and I?t = ~/z = 0.
E
it c a n n o t
be distinguished
f a c e as t h e p a r a m e t e r
with nonzero
t h e l i n e N , = 1, a n d f i n a l l y d i s a p p e a r s
1 0 x~
see Fig. 4). Transverse
waves
i n c a s e s a, b, a n d c. W e
x
sur-
goes further away from
have
( c a s e e,
are only found schematically
\ xx
i0 n
D from
x /'
10 s
/
s
s /
10 4
/ 4 /
10 -~
i
w
10 a
104
FIG. 3. AS we proceed to region ( P f r o m / / ~ ' t o ) > l and 1 the available region for transverse waves shrinks as we move away from (Dfrom/D~o) = 1. Longitudinal waves are much more affected as the overstability line does not show a minimum as discovered years ago by Scriven and Sternling (14). Np = 0.8, N~ = 1.2, Nr~ = 2, N~ = l, Bo = 1, C = 10 -9 , S = Sz = 103 , H = - 1 0 -9, Hz = 10 -s and Ft = ~Tz = 0.
(Dfrom/Dto) >
Journal of Colloid and Interface Science, V o l .
1 3 1 , N o . 2, S e p t e m b e r
1989
O0
1
t/from. t,,to
FIG. 5. The dashed area is the region where transverse waves are predicted by our theory (Sect. 4.2, Eq. 4.2.9 ). It corresponds to (Drrom[Dto) > ( ] A f r o m / # t o ) 2,
477
WAVES AT LIQUID-LIQUID INTERFACES
marked the shadowed area in ND -- N, space where we can find transverse waves (see Fig. 5 ). An explicit expression for the existence of transverse waves will be given later by an asymptotic expansion. The other branches in Figs. 3 and 4 are different from all the wave solutions discussed until now. The most striking feature is that these marginally stable curves have no minima, i.e., Ec --* 0 when --* 0 (or a --* 0). Furthermore, they are unaffected by the surface adsorption. However, the lack of influence upon these curves of the surface deformability indicates their longitudinal character. Remembering the oscillatory instability found by Sternling and Scriven (14) in an adsorption-free model, with the same critical behavior in low frequency (large wavelength) region, we consider this branch as the Sternling and Scriven wave.
TABLE I Parameter Values (Standard Ground Conditions) D v l
10-5 10 2 100 102 103 10-I 10 1o
S S~ C Bo H p~
103 103 10-9 10° 10_9 10-5 lO-lO
fl
10 1
?7z
10--8
kI ~sh, Tldil
10-5 10-10
g [p] Po,
pz
q-
He
)k3/2V-SHzTr3C5/2 } 1/2
_
~2 =(Bo+O(~2)) aTr2
~k3/2f"~ ~ q- 0 ( ~ 5/2)
4. ASYMPTOTIC EXPANSION
Following the data listed in Table I, we can analyze the problem by introducing a smallness parameter E = S -~ . Then we take
S= O(c-1), C= O(~3), H= O(~3), I~t = O ( ~ 3 ) ,
Bo = O(1)
H~--
and
~71"l }
[4.1.1]
-1/2
+O(E3),
[4.1.2]
where
~
~f~dND + 1
O(E3/2),
1 + 1/VS-
fNDD/N, -- 1
~/z = O(e3). ~2 - V ~ / N . + ~ff / N .
4.1. Long Wavelength Expansion-Long Wavelength Longitudinal Waves First we consider the long wavelength region. We assume that a ~ w and take w = O(e) and a = 0(~2). Thus, Eq. [2.29] and [2.30] become
~1 = (HzS)~e3/2 + ~ f ~
× {(Bo + o(~2))(~(1 + VN.~)~ 1/~ + a ( l + N.)e 2 + O(e7/2))}
= { x( 1 + ~ . N ~ ) ~ + a G ( 1 + U.)~ '/~
1
1+ 1 / V ~ - - - ~ - 1
~3 = (1 + N~N~)
/(
1)
1 + Nz----~
Substituting 2z and 22 into the Eq. [2.28] yields Ea 2rq
_]_ 3 / 2
{a( 1 + Nu)
G
+ ~Hz~3
_ aTr2(1 + fN~Np) l [4.1.3] V~71_ 1 J JournalofColloidandInterfaceScience,Vol. 131,No. 2, September1989
478
CHU AND VELARDE
At the marginal state, the 1.h.s. of Eq. [4.1.3] is real. Thus 1 E =-
a 2./1-1
Dfrom
[4.1.4] 1 + N.
a
7~2
H:r3o01/S
~rl
(1 + f'NuNp) = O.
[4.1.51 Using Eq. [4.1.5] in Eq. [4.1.4], we obtain the critical elasticity Marangoni number for oscillatory instability (overstability) E,=-{(~+
Nu)( 1/]/~-H:r,~3 1)} 2
× 7r 1
[4.1.6] It does take a finite value even when w --* 0. On the other hand, the capillary number does not appear in this expression. We have the longitudinal property of this wave. Thus we can safely take the flat interface approximation in the case of long wavelength longitudinal waves as already remarked by Sanfeld and collaborators (23, 24). From Eq. [4.1.4], we obtain:
Condition 1 sign(E) = - sign(zq ) = -sign(~
For the system to be oscillatory unstable (overstable) both conditions 1 and 2 must be simultaneously satisfied. Thus the conditions for long wavelength oscillatory instability are
- 1) = sign(N, - ND).
From Eq. [ 4.1.5 ], we have w
(}/-~D+ N~)(V1/N~ - 1)
a
f-SH:r~ ~r3
Now as the frequency and the wavenumber have to be positive, we also have:
Condition 2 sign(~rl) -- s i g n ( V 1 / N , - 1). Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
/)from
- - > - Dto /)to
and
/)from
--<
I.
[4.1.7]
/)to
Subscripts "from" and "to" give the direction of the mass transfer, e.g., surfactant transfer from the phase "from" to the phase "to". The physical meaning of this condition is that this oscillatory instability can be observed whether the mass transfer is from the high diffusivity fluid to the other or the other way around provided Dfrom/Dto > Pfrom/Vto-However, no oscillation is expected if mass transfer is from the higher viscosity liquid to the other. Our analytical results [4.1.7] agree quite satisfactorily with the numerical estimates for longitudinal waves in cases b and c. It is more restrictive than Sanfeld's condition (24) Dfrom/ Dto > Pfrom//)to thus showing that although necessary it is clearly not sufficient. Moreover Sanfeld's analysis does not provide a finite critical elasticity Marangoni number Ec (or else (Oc/OZ)c), below which all these waves, whatever long their wavelength may be, are damped. Note that this oscillatory motion is directly related to the surface adsorption as Hz appears in Eqs. [ 4.1.4-4.1.6 ]. Adsorption-free or saturated adsorption lead to an infinitely divergent critical elasticity Marangoni number thus illustrating the destabilizing effect of Hz. All these features are completely different from the properties of the waves discovered by Sternling and Scriven(14) which on the one hand are always unstable in the long wavelength limit, and on the other hand are not related to the surface adsorption.
4.2. High Frequency Expansion-Laplace Transverse Waves ( TW) Previous work by several authors (2, 3, 7, 30), indicates that transverse waves are easily found in the high frequency region. We take
WAVES AT LIQUID-LIQUID INTERFACES (..0 = O(~ -1) and a = O(1). Then Eqs. [2.292.30] become
which is a generalized Laplace equation for liquid-liquid interfaces (3, 6, 8 ). Substitution o f f ( B o ) = 0 in Eq. [4.2.5] yields
X (1 + VN~,No)E-3/2 + V~(f(Bo)a(1 + N~,)
+ 4~2SCf-NuNp)~ -1
479
+ EaZA = 0.
[4.2.8]
O(~-1/2)} [4.2.1]
+
This equation has a solution only when ~2-
" / F l U ( B ° ) E 3/2
sign(E) = sign(A) = s i g n ( N . - 1/~D). +
(a~r2f(Bo) X3/2V~
+ 2CAS~ + O(E5/2),
) E2
Thus we have the necessary condition for oscillatory instability
[4.2.2]
Dfrom (/£from] 2" > Dto \ #to ]
where
~/S + 1
~./ND
+ 1
- ]/~N, - ~
[4.2.3]
f ( B o ) = Bo + a 2 +
Relation [4.2.9] explains why in cases b, c and d one can find transverse waves while there is none in case e. This is illustrated in Fig. 5. The necessary condition for a minimum is dE(a, w(a))/da = 0. This yields the critical values aT = ~
X 2SC( 1 + Np) a
[4.2.10]
[4.2.4]
f
2Bo3/2
Replacing ~ and -2 ~ in Eq. [2.29] and setting the real and imaginary parts equal to zero leads to the following expressions: (1 + N ~ ) { f ( B o ) a ( 1
+ N~)
E [ = - C(1 + No)(V-~o - N,)
+ 4X 2SCVN,,Np } + Ea---~2
S X 2CA+ +
arc2f(Bo) ] ~w2
j=o
[4.2.5]
1
× (1 + 1/N.Np) + Ea2~rl} = O.
[4.2.61
J
We see that f(Bo) = 0
[4.2.9]
[4.2.71
[4.2.11]
[4.2.12]
Previous work (6) has shown how transverse waves can be damped at liquid-liquid interfaces. Now, we find that, as long as [4.2.9] is satisfied, transverse waves can be sustained if enough mass transfer crosses the interface, either from the liquid with lower diffusivity to the higher one or the other way around. The oscillatory motion is indeed a transverse wave because the wavenumber and frequency satisfy the Laplace relation, and moreover because the critical Marangoni elasticity number is inversely proportional to the capillary number C. Again we have found quite Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
480
CHU AND VELARDE
satisfactory agreement with the computer resuits given in Section 3.
Using Eqs. [4.3.1] and [4.3.2], from Eq. [2.29], we get Ea 2
4.3. Sternling and Scriven Waves (SSLW) In Section 4.1, we discussed the appearance of longitudinal waves by using the long wavelength approximation a ~ w. Let us explore now other wavelength ranges, for example a ~0. Our numerical results in Section 3 already gave indication (see Figs. 3 and 4) that, if the conditions Dfrom/Dto > 1 and Pfrom/Pto > 1 are satisfied, we can always find a marginal state for oscillatory instability. Its behavior is drastically different from L W L W as no minimum critical elasticity number can be found. It shares, however, a common property with LWLW: the marginal state is unaffected by the deformability of the interface. Consequently, we take it as a longitudinal wave. However, on the theoretical side, such a wavelength range brings substantial difficulties for an asymptotic expansion. Fortunately, computer search shows that these longitudinal waves change when we cross the line N, = 1. This suggests the possibility of performing a perturbation analysis around N, = 1 thus simplifying the problem. Assume N ; 1 = 1 + 6 with 6 = O(e 1/2) and oJ = O(1) as well as a = O( 1 ). Then
I+N.
S2w 2 1 +
.q-
1. +. N, .
.
(1 - ~ ) 2
.
I.
.
X(N, . . . +. f~D)
[2rnl(rnl + a ) ( 1 + N , ) 6
+ ml(1 - - N D ) ] ~-~~1/2. Marginal stability requires that Ec- 1---~
1+
w( 1/N~ - l )(N~ + ~ ) 2 R e { m l ( m l + a)}(1 + N,)
=
Re(m1)(1 - No) ~ ,
× {(ml + a)(1 + N~)~ -1/2 %.
q- 2m~ )~T" ~ -{- O(~ I/2)}
[4.3.1]
Bo + a 2 ~(ml + 1)(~r-~D - 1) ~1/2
--
[4.3.5]
where Re(o) denotes the real part of the complex quantity inside the bracket. Taking account of the signs in Eq. [4.3.4], we have sign(E) = sign(1 - 1/~D).
[4.3.6]
Eq. [4.3.5] can be written in the form F(w/a2, Np, N~,ND, . . . ) - 0 .
o~c/a 2 = 17( Np, N~, ND, • • ").
x
[4.3.4]
[4.3.7]
The solution of Eq. [4.3.7] gives the dispersion relation
Nz VND]
~2
[4.3.3]
[4.3.8]
Although we have been unable to obtain a compact analytical expression of this solution, we can, however, use Eq. [4.3.5] to obtain a necessary condition for the existence of this solution. This condition is
t
m l ( m l + 1)(ND-- 1) S~
sign ~ Jr,
E
+ 2 m 1 ~ 84 + O ( e 3/2) .
[4.3.21
Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
1 = sign(l - ~ ) .
[4.3.91
Now, assuming that we have a solution of Eq. [ 4.3.7 ] in the form [ 4.3.8 ], substituting it into Eq. [4.3.4] yields
481
WAVES AT LIQUID-LIQUID INTERFACES
e c = 1 _-
l+u--
× S2WcY(No, N,, ND, • • ").
[4.3.101
This equation shows that the marginal elasticity number tends to zero when w --~ 0 (or a --~ 0), a result that we expect from the numerical study. Combining [4.3.6] and [4.3.9], we obtain the necessary condition for the longitudinal wave Oftom /)from --> 1 and > 1. [4.3.11] )to vw Thus the oscillatory instability only occurs when the direction of mass transfer is from the fluid with higher diffusivity and higher kinematic viscosity to the other. This is analogous to the prediction given by Sternling and Scriven(14). In summary we find that for overstability to occur:
the necessary conditions for their existence. We have also shown the existence of two types of sustained longitudinal waves that we have called L W L W and SSLW, respectively, and delineated the regions in parameter space where they appear. Explicitly the results are (for a comprehensive account see Table II):
( 1 ) Laplace transverse waves exist for a critical elasticity number +
E~=-
with critical wavenumber: a Tc =
(3) The deformability of the interface does not play any influence on the critical elasticity number, which is consistent with the flat interface assumption made by Sternling and Scriven(14). (4) Surprisingly enough and contrary to the L W L W case, surface adsorption does not affect this new longitudinal wave which agrees with the result for Sternling and Scriven waves in the adsorption-free case.
B~O
and with critical frequency: X @ 2B°3/2 wo = SC(1 + N p ) "
( 1 ) Eq. [4.3.11 ] must be fullfilled. (2) The critical elasticity number found in the low frequency range is the same as that predicted by Sternling and Scriven (14) but different from the results obtained for LWLW.
C ( l + Np) ( I ~ D -- N u)
The necessary condition for the existence of this wave is -Dfrom - >
Dto
("from ] 2 \ #to /
(2) Long wavelength longitudinal waves exist for a critical elasticity number + N.)(l~-HzTrlTr3 1)} 2
E, = _{(~
× It is clear that the newly found waves are of the Sternling and Scriven oscillatory type. Finally as the surface adsorption plays no role, the conclusions in this section can be transferred verbatim to the case of energy transfer. 5. SUMMARY OF RESULTS AND COMPARISON WITH EXPERIMENTAL DATA
We have found sustained Laplace transverse waves at liquid-liquid interfaces and indicated
71"1
with a critical dispersion relation L
(I/~D + N.)(V1/N~ - 1)}a }
w c =
f~gzTrlTr3
"
The necessary conditions for the existence of this kind of instability are Ofrom gfrom > -and )to Pto
- -
Journal of Colloid and lnterfaee Science,
Pfrom
--
< 1
Pto
Vol.131,No. 2, September1989
482
CHU AND V E L A R D E TABLE II Summary of Necessary Conditions for Overstable Oscillations Dfrom
LW
Dto Yfrom //tO Of tom
Oto 0Jfrom //to
>
//from ~',o
- -
and
LWLW
<1
> I and
SSLW
>1
Ofr°m > (/*fr°"---'~m] 2 Dto \ #to /
TW
LAPLACE TW
with 7r1=
N D ~ - -
1
and 7r3=(1 +
)
1+
.
(3) S & S longitudinal waves exist for a critical elasticity number
I+N~ (1+ 1 × S2o~cY(Np, N., No,
• • • ),
where Y(Np, N,, ND, • • • ) is the solution of Eq. [4.3.7]. The dispersion relation is we = 7 ( N o, N,, No, • • -)a 2. The necessary conditions for the existence of this mode of instability are D from -
-
>
/)to
1
and
Vfrom
> 1.
/"to
Finally let us say a few words about the available experimental results (34-37). Linde and collaborators (31-35) have shown experimental evidence of Scriven and Sternling longitudinal waves as well as transverse waves in a gas-liquid case with heat or
Journal of Colloid and Interface Science, Vol. 131, No. 2, September 1989
mass transfer from the gas to the liquid. Pertinent theory for the liquid layer open to air has already been provided in Ref. 30. On the other hand, Orell and Westwater (36, 37) have reported findings on the ethylene glycol-ethyl acetate system with acetic acid as surfactant. Typical parameter values as estimated by Sorensen (28) are listed in Table III. Phase 1 is the ethylene glycol and phase 2 the ethyl acetate. When acetic acid transfers from phase 1 to phase 2 all previous linear theories (14, 2328) as well as our results (Sect. 3, case a) predict instability in the form of steady cellular convection which is the experimental result reported by Orell and Westwater (36, 37). However, the latter authors also found an oscillatory instability that no available linear theory predicts. In agreement with Sorensen (28) we guess that this oscillatory motion is the result of a secondary instability at a higher Marangoni number than the steady instability threshold. Our guess is compatible with the numerical findings reported by H y m a n and Nicolaenko for the Kuramoto-Velarde nonlinear equation in Benard-Marangoni convection for thin liquid layers open to air (38). When the transfer of acetic acid is from phase 2 to phase 1, Sternling and Scriven (14) predicted stability while our theory (Sect. 3, case b) predicts instability for Marangoni numbers equal or higher than
E [ -- -
- - 1.3 * 10 9 C(1 + Np)(~-~D - N~,)
and
E~ = - { (f-N-~D+ Nj')(VI
l)} 2
× 7["1
- -4.8.1016
WAVES AT LIQUID-LIQUID INTERFACES TABLE Ili Typical Parameter Values According to Refs. (28) and (30, 31)
O¢]OP = -4.13.108 cm 2 s-2 a = 1.44 dyn c m - 1 k] = 1.27,10 7 cm /Zl = 0.103 poise 01 = 1.064 g c m -3 Dl = 0.21,10 -5 cm 2 s -I fl,a ,~ 0.05 g c m -4
Dz - 1.8.10 -5 cm 2 s l Fo = 2.67.10 -1° g cm -2 k~ = 1.93.10 -7 em t~2 = 6.36"10 -3 poise 02 = 0.9142 g cm -3 D2 = 3.4,10 -5 cm 2 s -t
a /~1 corresponds to a drop from the concentration 0.5 w% to zero over a distance of 1 mm.
for transverse and longitudinal modes of oscillatory convection, respectively. In terms of gradients these critical Marangoni numbers correspond to fiT = --565.2 g cm-2 and /3~ = -2.1 * 10 l° g cm -2. According to the parameter values listed in Table III, Orell and Westwater results (36, 37) should be in the stable region. Thus our theory is compatible with the available experimental evidence. However, further experimental work for higher values of the constraints is needed in order to validate or invalidate our theoretical predictions. Clearly these experimental results are not compatible with the predictions made by earlier authors (23-28) who predicted oscillatory instability albeit without a given threshold. Recently, the authors have rederived some of the results reported here by using a harmonic oscillator description of the interfacial motions (39-42). Moreover, they have extended the theory to the nonlinear region of limit cycle oscillation (43-46) and soliton behavior (47). ACKNOWLEDGMENTS This research has been sponsored in its earlier stage by the Stiftung Volkswagenwerk and latter by CAICYT (Spain) and by an EEC Grant. X.-L. Chu expresses his appreciation to the Spanish Science Policy General Directorate for a fellowship that enabled him to work at
483
UNED. M. G. Velarde acknowledges fruitful discussions with Dr. B. Nichols and the hospitality of CNLS, Los Alamos National Laboratory, where this work was completed. Both authors also acknowledge fruitful discussions with Professors H. Linde, A. Sanfeld and D. M. Hennenberg. REFERENCES 1. Lamb, H., "Hydrodynamics." Dover, New York, 1932. 2. Landau, L. D., and Lifshitz, E. M., "Fluid Mechanics." Pergamon, Oxford, 1959. 3. Levich, B. G., "Physicochemical Hydrodynamics." Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. 4. Miller, C. A., and Neogi, P., "Interfacial Phenomena." Dekker, Inc., New York, 1985. 5. Van den Tempel, M., and van de Riet, R. P., J. Chem. Phys. 42, 2769 (1965). 6. Hansen, R. S., and Ahmad, J., Prog. S u r f Membr. Sci. 4, 1 (1971). 7. Garcia-Ybarra, P. L., and Velarde, M. G., Phys. Fluids 30, 1649 (1987). 8. Velarde, M. G., Garcia-Ybarra, P. L., and Castillo, J. L., Physicochem. Hydrodyn. 9, 387 (1987). 9. Lucassen, J., Trans. Faraday Soc. 64, 2221, 2231 (1968). 10. Lucassen-Reynders, E. H., and Lucassen, J., Adv. Colloid Interface Sci. 2, 347 ( 1969 ). 11. Van den Tempel, M., and Lucassen-Reynders, E. H., Adv. Colloid Interface Sci. 18, 281 ( 1983 ). 12. De Voeght, F., and Joos, P., J. ColloM Interface Sci. 98, 20 (1984). 13. Cini, R:, Lombardini, P. P., Manfredi, C., and Cini, E., J. ColloM Interface Sci. 119, 74 (1987). 14. Sternling, C. V., and Seriven, L. E., A I C h E Journal 5, 514 (1959). 15. Velarde, M. G. and Normand, C., Sci. American 243, 78 (1980). 16. Velarde, M. G., and Castillo, J. L., in "Convective Transport and Instability Phenomena," (J. Zierep and Oertel, Jr., Eds.), pp. 235-246. Braun-Verlag, Karlsruhe, 1982. 17. Legros, J. C., Sanfeld, A., and Velarde, M. G., in "Fluid Sciences and Materials Science in Space," (H. U. Walter, Ed.) pp. 83-140. Springer-Verlag, N.Y., 1987. See also M. G. Velarde (Ed.) "Physicochemical Hydrodynamics. Interfacial Phenomena," Plenum Press, N.Y., 1988. 18. Lino S. P . , A I C h E J . 16, 375 (1970). 19. Palmer, H. J., and Berg, J. C., J. FluidMech. 51, 385 (1972). 20. Palmer, H. J., and Berg, J. C., A I C h E J. 19, 1082 (1973). 21. Chu, X.-L., and Velarde, M. G., J. Colloid Interface Sci. 127, 205 (1989),
Journalof Colloidand InterfaceScience, VoL 131,No. 2, September1989
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