3682
Langmuir 1993, 9, 3682-3690
Liquid/Liquid Dewetting FranGoise Brochard Wyart,' Pascal Martin, and Claude Redon Institut Curie, Section de Physique et Chimie, 11, rue Pierre et Marie Curie, 75231 Paris Cedex 05, France Received July 6,1993. I n Final Form: September 17,199340 We study theoretically the mechanisms of dewetting of liquid A deposited (by solvent evaporation) on an immiscible, not wettable, liquid B (S= YB - (YA + ym) < 0, where yi, are the interfacial tensions). The A film is unstable below a critical thickness e, (-mm) controlled by gravity effects. Macroscopic films (e > mm) are metastable and evolve by nucleation and growth of holes (radius R(t)) surrounded by a rim, which moves at velocity V = dR/dt. Depending on the viscosities (?)A, m),the densities (PA, p ~ of) the two liquids, and the A film thickness, e, we expect four regimes: (i) inertial (V = (151/pAe)1/2;(ii) viscous with "A" friction dominant (V = ISIedqA); (iii) Viscous with "B"friction dominant (V = lSl/q~);(iv) viscous where l is the width of the rim and eE the equilibrium contact angle (assumed inertial (V = lS12/3/(qBPBl)1/3); 0) and evolve by nucleation and growth of a hole in the A film. The radius of the hole must be larger than R, e/sin d E to initiate the growth." As the hole of radius R ( t ) grows, it is surrounded by a liquid rim (Figure 5 ) which collects the liquid. Our aim is to study the velocity of the moving rim V = d R / d t . The growth of the hole is very dependent upon the viscosity of the B liquid substrate. Two cases will be discussed (a) the substrate B is much more viscous than the spread liquid A and behaves like a solid substrate; (b) the spread liquid A is more viscous than the B substrate, "liquid substrate behavior". We shall work mainly at the level of scaling arguments. For case a, ref 7 describes the basic dynamic equation of the three phase contact line and refs 8 and 9 contain some relevant experimental observations of the wetting of a liquid substrate. a. Solid Substrate Behavior. If TB > ?A/&, where 0~ is the angle pictured in (Figure 3) (-5' = ' / 2ydE2) , the substrate behaves like a solid. This situation has already been studied in great detail.' Two regimes are expected (1) a uiscous regime for a viscous A liquid with a small
-
(10)Langmuir, I. J. Chem. Phys. 1933, I , 756. (11) Sykes, C. C. R. Acad Sci. 1991,313,607. ( 1 2 ) Redon, C.; Brochard, F. Macromolecules, in press.
Brochard Wyart et al.
3684 Langmuir, Vol. 9,No. 12, 1993 RIM
DRY
1
I
RIM
HOLE
I
I
WET
G
v
l
NATIVE
conservation of A volume, leading to
-
-
l ( t ) (Re/eE)1'2 (11) R increases linearly with time and l(t) a2. Inertial Regime. For a very low viscosity liquid, the dissipation becomesvery small and we reach the inertial regime. This is somewhat similar to the bursting of soap films14J6 or to the dewetting of superfluid helium.14 The driving force on the rim is constant and equal to -S, and the rim velocity is constant! This result does not violate the fundamental laws of mechanics because the mass M of the rim increases with time. Let us derive the motion of the rim for a one-dimensional geometry
with momentum density p = M ( t ) V . Assuming constant velocity, this is equivalent to
V
where M is the mass per unit length of the rim a t time t. With M = for the 1-D geometry, eq 13 leads to
v = (IsI/PAe)1/2
e + + + + + + + + + + + + + + + Figure 5. Growth of holes in a film deposited (a) on a solid substrate, hydrodynamic flows in the moving rim (Poiseuille flows);(b)on a bulk liquid substrate, flowsin the rim (plugflows) and induced in the substrate (penetrationlength d); (c)on a thin
liquid substrate
(14) Equation 16can also be derived from the transfer of kinetic energy
into surface energy. If A is the surface area per unit length of the hole, it leads to
M V = A IS1 (16) with M = RepA and A = R for the ribbon geometry and M = nR2epAand A = nR2 for the circular hole. In both cases, eq 16leads to eq 14because curvature is negligiable. This differed by a factor of 2 from the results in refs 14 and 16 because they assumed E~in= llZMv2. Conclusion. The location of the crossover between inertial and viscous regimes is found by equating eq 10 and eq 14. This leads to VA* 8E2(pAeyA)1/2 a few C P for 30°, e = 10 pm and y 40 mN m-l. If C VA*, inertial effects are dominant. The Reynolds number Re at the crossover is Re = PA(Ve/TA*)= l / h . (The h-lfactor arises from the increase of dissipation in liquid wedge.) In usual conditions we expect Reynolds numbers which are large, but not huge; turbulent instabilities are probably absent. b. Substrate "Liquid"Behavior. If the viscosity of the substrate TJBis smaller than V d e E , the flow in the rim is a plug flow,' and the viscous dissipation is dominated by the dissipation in the substrate. We expect three regimes: (1)viscous regime; (2) viscoinertial regime, if the & is an adiabatic contact angle defined by IS1 = ' / ~ Y A ~ E ~penetration . length of the flow in the substrate induced by The left-hand side of eqs 8a and 8b represents the the motion of the rim is smaller than ita size 1; (3) pure uncompensated Young force, while the right-hand side inertial regime if the substrate viscosity is very small. describes the viscous force due to the flow in a wedge of bl. Viscous Regime (Figure 5b,c). Joanny7has derived angle ed. In is a logarithmic factor of order 10 and is due the dynamics of a contact line for both (a) thin and (b) to a divergence of the dissipation in a wedge. thick liquid substrates. Let us start with the case of a thin The width of the rim l(t)is much smaller than R ( t ) ,and B substrate, where the lubrication approximation sim,thus V M Vp = V . Equations 8a and 8b then lead to plifies the description. contact angle; (2) an inertial regime for a very low viscosity liquid and large contact angle. al. Viscous Regime. The growth of a hole in the A film is shown in Figure 5a. The viscous B liquid does not move during the fast growth of the hole of radius R ( t ) . The motion of the rim which appears ahead of the dry patch is controlled by a competition between capillary forces and viscous flow.13 We briefly summarize the results here, following ref 1. The rim cross section (Figure 5a) is a portion of a circle (with a dynamic contact angle because the pressure is expected to equilibrate rapidly. The dynamical contact angle ed is related to the velocities V Mand Vp of the inner (M) and outer (P) contact lines which limit the rim. For small contact angles (eE, ed > L), the viscous stress at the AIB interface is = kV (17) where k = 4 d L (the factor 4 arises from the condition of zero B flux7). The total friction force F, on the rim is then uXZ
(18)
The total driving force Fd, assuming e
l *B where VB = d p ~ Using . V = S/m, we expect a crossover between viscous and viscoinertial regime for 1 = 1, = m2/ PBS. For a rim larger than I,, the flows of the substrate are confined to a diffusion layer of thickness d = (Qtb)'/2, where tb = 11V, is the passage time of the rim (Figure 5b). We can estimate the friction force from the dissipation per unit length of the rim moving at velocity V
T$ = qB($?d = F,V Balancing these forces leads to the rim's motion: 4VB
v (1/L) = -s
and (20) Fv
Equation 11 (A conservation) and eq 20 yield
1 = VB(2)
The balance F, = Fd (eqs 19 and 25) leads to The velocity of dewetting (eq 20) decreases with time, because the friction on the moving rim increases linearly with ita size 1. Equation 21 describes also the growth of holes in entangled polymer melt deposited on extremely smooth surfaces, when the polymer slips at the solid surface.12 ii. General Case: Dewetting on a Bulk Substrate (Figure 5b). We follow the scaling description of the contact line motion of ref 7. Taking for the viscous stress uxz at the AIB interface near a contact line moving at velocity V, uxz m(V/x), Joanny finds that the friction force Fv = J uxz dx acting on the contact line is
-
Fv
?Bv
(22)
We can justify this result by a dissipation argument. As the rim (Figure 5b) moves, it induces flows in B in a region of size 1. The viscous dissipation per unit length of rim can be written as
which leads exactly to eq 22. The balance between F, (eq 22) and Fd (eq 19) leads to
where VB*= 7 1 7 ~ For . this B viscous regime, the velocity of dewetting on a bulk substrate is also constant in time, independent upon the A viscosity and thickness e as soon
as e F,, the inertia of the rim is dominant and dpldt ISl, which leads to V = (ISVpAe)1/2,as discussed in section a2. The crossover between viscous (or viscoinertial) and inertial regimes is given by
-
Fv(V = (ISI/pAe)'/2) = S (29) if 1 C I,, Fv = m V (eq 22), and eq 29 leads to m*2 = pASe. The regime is inertial for m < TB* = (pdSJe)1/2, or for e > e, = m2/p~ISI.With PA = 1 g cm4, S = 10 mN m-l, and e = 10 pm, m*= 10 cP. If 1 > I,, I;, = (tlBPB1)1/2v/2 (eq 251, and eq 29 leads to 1 = 1,' = s'/2pA3/28/2/qBpB. The condition 1,' > 1, is satisfied if T)B2 C p@Ie. Therefore, for 1 > l,', we expect a viscoinertial regime and for 1 < l;, an inertial regime. The different regimes are shown in a diagram 1 versus VB (Figure 6): If 9 ~ < 2 pAlSle, we expect a pure inertial regime if 1 C 1,' and a viscoinertial regime for 1 > 1;. If V B > ~ pBlSle, we expect the viscous regime, controlled by the viscosity of the substrate if 1 < I,, and a viscoinertial regime for 1 > 1,. For m = VB*, 1,' = 1, = (pA/pB)e. It shows that the domains of existence of the three regimes pictured in Figure 6 are large and they can easily be observed. Conclusion. Liquidlliquid dewetting of macroscopic films ( e > pm) may have important applications in polymer coextrusion,float glass industry, and more generally in all
Brochard Wyart et al.
3686 Langmuir, Vol. 9,No. 12,1993
et
FWf
I
INERTIAL
e pa/pe- -
----+
e
I
*
1
I% pI.s
Figure 6. Diagram in coordinate 1 (width of the rim), m2(m = substrate viscosity) showing the three regimes of liquid/liquid dewetting. processes involving stratified liquids. The liquid LAfilm may be deposited by solvent evaporation. This is the usual way to deposit monolayers of unsoluble surfactant at the water surface. The choice of the substrate is very broad. Fluorinated species have low surface energies and can be used as nonwettable substrate for polymers such as poly(dimethylsiloxane) (PDMS) which is liquid at ambient temperature. Experiments using this system will be published in a following paper. Deposited on LB liquid substrates LA liquid films are metastable below a critical thickness e, if the spreading coefficient is negative. They evolve by nucleation and growth of holes in the A film. (a) if the substrate is more viscous (such as a polymer melt of high molecular weight) than liquid A (VB > V d e E ) , it behaves like a solid and we expect two regimes: (i) a viscous regime with A friction dominant, observed experimentally in ref 3 on silanized wafers; (ii) an inertial regime if VA is not too large, VA < VA* = (PAe1sldE)'/2 a 10 pm which has been reported only for few CPfor e large contact angle (C.Andrieu, private communication). This regime controls also the rupture of soap films and is described experimentally in ref 15. (b) If the substrate is less viscous than the deposited A liquid, the flows in A are plug flows and they induce flows in the substrate, where the viscous dissipation becomes dominant. We expect three regimes (Figure 6): (i) a viscous regime with B friction dominant (very similar to the viscous regime controlled by A friction (case a)), but here no dissipation arises in a wedge and the friction force on the moving rim is simply VBV instead of TAV/eE. (Actually,we observed this regime using PDMS in a broad range of polymer weight deposited on poly(methy1trifluoropropylsiloxane) of viscosity Y = 300 cS. The velocity of the order of mm/s is independent upon the PDMS viscosity.); (ii) an inertial regime if VB is small (118 < VB* = VA*/BE) identical to the inertial regime of case a; (iii) a viscoinertial regime intermediate between viscous and inertial regime, where both viscosity and inertia of the substrate are involved. This regime arises for width of the rim larger than the threshold values I,, I,' as shown in Figure 6. For VB, = VB*, I, = I,' = pA/pB)e is small. This means that this regime can easily been observed. This is the most original process, only expected for liquid/liquid dewetting. There is nothing analogous in the physics of soap films, suspended in air, either in the rupture of soap films (inertialmode) or in the growth of black films (viscous mode) but viscoinertial regimes have also been predicted for the bursting of soap films in a viscous environment.16 In our case of liquid/liquid dewetting, capillary waves can also be generated, if the rim moves faster than a critical velocity V* = (4(?qj/pB)g)'/4.'s Thisaspect will be discussed in more detail in a separate note.
-
-
Figuw 7. Free energy per unit area F vs thickness e in the case of a substrate wetted by a liquid f i b of thickness ranging from molecular sizes to micrometers. Vs decreasing thickness, f i are metastable (ei < e < e, gravity regime) and unstable (e < e, van der Waals regime).
Our case is simplerthan the rupture of soap films because interfacial tensions remain constant during the dewetting process. In soap film rupture, the density of surfactant molecules increases in the rim, leading to Marangoni flows and longitudinal waves emitted from the rim, which is surrounded by an "aureole". However, the velocity of the rim is not strongly modified by this dissipative process.
11. Dewetting of Microscopic Films A. Introduction. While thick films (greater than micrometers) are controlled by gravity, for thinner films long range forces are dominant.s Surfaces tensions YA and YB and interfacial tension y m are defined for semiinfmite media. For the stratified liquid pictured in Figure 2, long range forces give corrections to capillarity energies if the thickness e becomes very small and the free energy (per unit area) has to be written as F(e) = YB+ YA
1
+ R e ) +p
e 2
(30)
At large e, P ( e ) tends to zero. When e is larger than the molecular size ao, P(e)is controlled by long range forces. For van der Vaals liquids considered here
AH P(e) = 129re2
a,
< e < X (ultraviolet wavelength)
(31) AH= A m - AM is the differenceof two Hamaker constants and may be of either sign. We assume here AH < 0, i.e. A more polarizable than B. In the following, we shall write p~V67r= ya2, which defines a molecular length a. For small thicknesses P(e-0) = S = YB + YA) (32) The stability of the A film is derived directly from the curve F(e) pictured in Figure 7. The thickness ei ( - ( ~ - l a ) l / ~ pm) defined by I"'(ei) = 0 separates two regimes: (i) If ei < e < e,, the films are metastable (F'(e)> 0) and evolve by nucleation and growth. Gravity is dominant. This corresponds to the macroscopic regime discuseed in part I. (ii) If e < ei, films are unstable (P'(e)< 0): the thermal fluctuations of the film thickness are amplified and the film breaks up spontaneously into a myriad of droplets.
-
LiquidlLiquid Dewetting
Langmuir, Vol. 9,No. 12,1993 3687
U.
Figure 8. Cellular patternsafter dewettingof nanoscopic films, holes appear spontaneously, and grow to form (a) transient 2d foams and (b)chaplets of droplets arranged in hexagonal patterns.
Figure 10. Collectivemode of a stratifiedliquid (a) transverse bending mode; (b) longitudinal peristaltic mode (shown here for ?‘AB
< ?‘A).
gradient in the film and then a liquid flow. In the lubrication approximation, we have a Poiseuille f l ~ w . ~ J ~ JA
+
Solid Substrats
Figure 9. Amplification of capillary waves of a liquid film
-
z =e
+ UAeiq~e-t17
(Figure 9)
This modulation induces a Laplace and disjoiningpressure (18)Vrij, A. Diacues. Faraday SOC.1966,42.
(33)
writing the volume conservation equation az
deposited on a solid (or very viscous) substrate.
This is the “spinodal decomposition” regime, by analogy with phase transitions also observed in the rupture of soap films.’8 The film instability is driven by van der Waals forces and correspondsto the microscopicregime discussed here. Unstable thin polymer films deposited on solid substrates have been studied recently, both experimentally and the~retically.~s The picture in the f i a l stage of the dewetting process is represented in Figure 8; droplets are arranged in a polygonal pattern. The size of the polygons increases with thickness as e2 and the size of the droplets as el.&! This has been checked using polystyrene on wafers with surface of unknown chemical composition6and also observed with PDMS deposited on a smooth silanized silicon wafer.12 The interpretation is the following: Capillary waves are amplified. The wavelength of the fastest mode is 2~qm-l e2/aand the corresponding rise time is 7 m - l ~V ~ * a ~ J eAfter 6 . a certain growth time, a dry spot appears in each trough of the wave. This generates a cellular pattern of dry patches. Each patch grows and pushed out a rim. When two rims meet, they stop and break into droplets of size L = (e3Ja)ll2. The dewetting of a unstable A film deposited on a highly Viscous substrate (7]B > qAeJa as shown in section B3) is identical to the dewetting of a solid discussed in detail in ref 4, if the AJB interface cannot follow adiabatically the deformation (Figure 9). Only the A free surface is modulated. Let us remind briefly the time evolution of a sinusoidal modulation of the film thickness:
-P ( e ) z ’ ]
J = #yAd’’ e3
+ + + + + + + + +
d i v J + -at= o
(34)
one finds the dispersion relation
(35) The fastest mode among the unstable modes (1Jr < 0) is
(36)
-
-
-
(37)
with VA* 103 ms-l, eJa lo2, one expects rm 1h. qmprescribes the scale length of the fiial dewetting picture. Our aim now is to study unstable A film exposed to air, and floating on low viscosity liquid. We first introduced the collective modes of a stratifiedsystem, where now both free A surface and AJB interfaces are deformed. We then study the statics properties of thermal fluctuations and the dynamics for unstable film deposited fist on thin and then on bulk substrates. B. Collective Modes of a Stratified System. We discuss here the surface mode of a BJNair system in the limit where the wavelength 2 ~ Jisq much larger than the film thickness e. As for soap film^'^.^' one expects two fundamental modes shown in Figure 10 (1)a transuerse bending mode (associated with vertical displacement) governed by the total surface tension; (2) a longitudinal peristaltic mode (accompained by thickness fluctuations) associated with horizontal displacements. This will turn out to be the crucial mode for dewetting. These two modes have been studied in detail both theoretically and experimentally for a liquid covered by (19)de Gennee, P.G.C.R. Acad. Sci. 1969, W , 1207.
Brochard Wyart et al.
3688 Langmuir, Vol. 9, No. 12, 1993 a monolayer, or few fluid layers.w.26 Here we extend these discussions to a liquid substrate covered by a mesoscopic f i b . We first discuss the static properties of the deformations of a mesoscopic film. Then, for dynamics, we find it convenient to consider first the case of a thin B substrate, where the backflows in the B region are particularly simple. Finally we extend the result to the more complex case of a bulk B liquid, using the flow analysis of Lucassen.22 1. Elastic Energy of a Deformed A Film. The elastic energy associated with deformations u1 of the free surface and u2 of the A/B interface is given by
(e + u1- u2)P0)dx (38) where P(e) is the long range van der Waals energy. P(e) = -(AJ/127re2and Po a Lagrange multiplier introduced to ensure the conservation of the A volume. In order to separate the transverse and the longitudinal modes, it is useful to define new variables
u = u1 - u2 u1=
TAB + 7 A u2 = -
u+u
yA u + u TAB + YA
u is associated with the longitudinal mode (if u = 0, the Laplace pressure YAU”1+ y ~ ~ u = ” 20)and u is associated with the transverse mode. The free energy becomes
or in Fourier space Fq
=1
+ yAB)q uq
+
+
I
+
+ + + + + Figure 11. Peristaltic mode of a liquid f i i deposited on a thin liquid substrate. E, = e2(yq2 P’)
+
and we propose to call it the Lucassen modulus. This shows up as follows: The A mass conservation is aV/at + div(e V z )= 0;setting V , = a{/& leads to uq + ieqf, = 0, and uXz= By/dx = -e2q2(yq2 P){, = Ea2f/dx2by definition of an elastic modulus E.22 The A film elastic modulus, associated with A surface fluctuations is
+
E, = e2(yq2+ P”) (44) For P‘ C 0, E, C 0 if q < qc = (pvy)1/2 ale2. 2. Dynamics of Collective Modes. (a) Film on a Thin Liquid Substrate. The A liquid f i i is deposited on a liquid substrate, which has a thickness L above a solid plane. If both L and e are small compared to the wavelength of the capillary mode, the Naviel-Stokes equations are simplified and one can use the lubrication approximation (Figure 11). A deformation of the surface film, specified by u1 and u2 leads to a Laplace pressure P, eq 41, which induces a Poiseuille flow and a surface pressure gradient uxz(eq 43), which induces a shear flow in the liquid substrate. The velocity u,(z) is then
-
The flow of the B liquid is
+ l2( y q 2 + P’(e))ut
(40)
where l l y = (UTA) + (l/ym) is the effective surface tension. We set as before (AHIIGu= ya2 and P’= -3y/e4. From eq 39, we can derive: (i) the Laplace pressure across the A film
P = -(TA + yAB)u
I‘
(41)
The flow A is
The conservation of A and B leads to au -+ at
(ii) the pressure inside the A film (or the A chemical potential pA/uo = p, where uo is the A molecular volume)
divJA = 0
(48)
and to & at+ d i v J , = O This pressure give rise to a tangential constraint
-
(43) dx = e(yu//t ~ y ’ u ’ ) Remark: we found it useful to introducethe (wave vector dependent) elastic modulus uxz=: -e
(20) Lucaesen Van der Tempel, J.; Vrij, A.; Heeeeline,F.Ned. Akad. Wet. Roc., Ser. B 1970, 73. (21) Haskell,R. C.; Petersen, D. C.; Johnson, M.W. Phys.Reo. E 1999, 47, 439. (22) Lucaeeen, J. Trans. Faraday SOC.1968,642221. (23) Kramer, L. J. Chem. Phys. 1971,55,2097. (24) Langevin, D. Thesis Parie (1975). (25) Brochard,F.; Joanny, J. F.;Andelman, D. Physics of Amphiphilic Layers; Meunier, J., Langevin, D., Boccara, N.,Ede.; 1987. (28) Sekimoto, K.; Oguma, R.;Kawasaki, K.Ann. Phye. 1987, 176, 259.
-
(49)
where we have assumed JA (e/L)JB > 1 (given by d2+ Q, 0)and overdamped if Q, > 1. If QEis large and positive, the roots of eq 54 are
or
(55)
The modes are propagative and involve both inertia ( P B ) and viscosity ( 1 ~ )One . can check that the elastic energy is balanced by the kinetic energy of a slab of liquid of thickness l,, the penetration length of the longitudinal mode (l/h = (42 + iop/a)l/2). If QE is negative, i.e. q < qc, one has only one root X’
=) Q E ~ ~ / ~
or a negative relaxation time
i.e. (62)
For q < qc,the thickness modulation is amplified and the fastest mode correspondsto q = 0. For the thin substrate, eq 52 is valid if ?A < qB(q2Le)-l. For the bulk substrate, eq 58 is valid if ‘IA < tlB(qe)-l. 3. Rupture of Unstable Film. We have calculated the fluctuations of stratified liquids in the limit qe
