Langmuir 2002, 18, 4795-4798
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Dewetting versus Rayleigh Instability inside Capillaries G. Callegari,† A. Calvo,† J.-P. Hulin,‡ and F. Brochard-Wyart*,§ Grupo de Medios Porosos, Facultad de Ingeniera, Universidad de Buenos-Aires, Paseo Colon 850, 1063, Buenos Aires, Argentina, Laboratoire FAST (UMR 7608), Baˆ timent 502, Campus Paris-Sud, 91405 Orsay, France, and Institut Curie, UMR 168, Laboratoire Physico-Chimie Curie, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France Received December 28, 2001. In Final Form: March 15, 2002 We study the dewetting of liquid films deposited inside nonwettable and wettable capillaries. Two processes compete: (i) Rayleigh instability (i.e., amplification of thickness fluctuations) and (ii) dewetting by nucleation and growth of a dry zone limited by a rim collecting the liquid. At times shorter than the characteristic time τM of the growth of the Rayleigh instability, we expect two regimes: (i) annular rims and drying at constant velocities and (ii) columnar rims with drying velocities decreasing versus time. For wettable capillaries, in a certain regime of thin thicknesses, the Rayleigh instability is absent and dewetting is the only process to remove the film.
I. Introduction The dewetting of liquid films deposited on planar nonwettable substrates has been intensively studied during the past 10 years.1-5 Films dewet if the spreading coefficient S (S ) γSO - (γSL + γ), where γij are the S/air, S/L, and L/air interfacial tensions) is negative and for thicknesses e below a critical value ec. ec is also the height of large flat drops, which results from a competition between gravitational and surface energies,
1 Fge 2 ) |S|()γ(1 - cos θE)) 2 c
(1)
where F is the liquid density, g is the gravitational acceleration and θE the contact angle. Below ec, the film is metastable. It dewets by nucleation and growth of a dry patch [radius R(t)], surrounded by a liquid rim, which collects the liquid.1-3 The velocity of dewetting Vd ) dR/dt is constant in time because the driving force on the rim,
1 1 FM ) γ + γSL - γSO - Fge2 ) Fg(ec2 - e2) 2 2
(2)
is constant, and the friction, assumed to be localized in the two wedges bounding the rim, is also constant,
Vd FV ) η ln θE
(3)
where ln is a logarithmic factor, which describes the divergence of the viscous dissipation in the two liquid wedges and which also includes numerical coefficients. † Universidad de Buenos-Aires. E-mail:
[email protected],
[email protected]. ‡ Laboratoire FAST. E-mail:
[email protected]. § Institut Curie. E-mail:
[email protected].
(1) Sharma, A.; Ruckenstein, E. J. Colloid Interface Sci. 1985, 106, 12. (2) Redon, C.; Brochard-Wyart, F.; Rondelez, F. Phys. Rev. Lett. 1991, 66, 715. (3) Andrieu, C.; Sykes, C.; Brochard-Wyart, F. Langmuir 1994, 10, 2077. (4) Reiter, C. Science 1998, 282, 888. (5) Seemann, R.; Herminghaus, S.; Jacobs, K. Phys. Rev. Lett. 2001, 86, 55, 34.
For viscous liquids of interest here, inertia is negligible and the balance of forces FM ) FV leads, in the limit of thin films (e , ec) and small contact angles θE, to
Vd )
(| |
θE 1 S - Fge2 ≈ V* θE3 ηln 2
)
(4)
where V* ≈ γ/η is a characteristic liquid velocity. Ultrathin films are unstable4,5 (because of long-range VW forces) and dewet by amplification of capillary waves. This regime is called “spinodal dewetting” by analogy with the dynamics of phase separation in binary mixtures. We want to extend this picture of dewetting to confined geometries. Despite numerous applications of dewetting in thin capillaries, from the fast drying of porous materials to petroleum engineering, relevant experiments have been performed only recently.6 This study was interpreted, assuming that the “flat” picture is nearly valid. Our aim here is to study how the laws of dewetting are modified by curvature and confinement. We first compare the stability of films deposited on a plane and a curved cylindrical substrate. A cylindrical fluid interface is unstable, as first analyzed by Lord Rayleigh. Here, we analyze the dynamics of the modified Rayleigh instability, when we are dealing with a film and not with a full cylinder. Then, we derive the characteristic time τM for the growth of the instability. We describe finally the dewetting at times t < τM in the two regimes: annular rims and columnar rims. II. Stability of Films inside Capillaries A. Stability of Films. A film of thickness e is deposited inside a capillary of radius b. We assume that b is always small compared to the capillary length κ-1 ) xγ/Fg (≈mm); gravity effects may then be ignored. The free energy f(e) is composed of interfacial energies and longrange forces, which play an important part only for thin films (e < λ (a typical wavelength) ≈ 1000 Å). It can be written, per unit length of the capillary,
f(e) ) 2πbγSL + 2π(b - e)γ + 2πb P(e) (6) Callegari, G.; Calvo, A.; Hulin, J. P. Preprint.
10.1021/la011862w CCC: $22.00 © 2002 American Chemical Society Published on Web 05/16/2002
(5)
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This modulation of the pressure induces a flux Q(x),
Q(x) ) Figure 1. Rayleigh instability inside a capillary.
[
]
1 -γ 1 b P′′(e) + + P′(e) 2π(b - e) (b - e)2 b - e (b - e)2 (6)
In the limit e . b1/3a2/3, f ′′u = [3γa2/e4 - γ/b2](1/2πb). For a wetting liquid ( > 0), eq 6 shows that f ′′(u) > 0 if e < ei ) a1/2b1/231/4 and f ′′(u) < 0 if e > ei. For a nonwetting liquid ( < 0), f′′(u) < 0 at all thicknesses e. This shows that inside a capillary, films are generally unstable because of the sign of the curvature imposed by the geometry (f′′ < 0): this corresponds to the Rayleigh instability.8 However, as pointed out earlier,9 wetting films can be stabilized by van der Waals forces (A > 0) at very small thicknesses (e < ei). B. Equilibrium of a Film with a Drop. If a columnar drop is injected inside the capillary, what is now the thickness of the film in equilibrium with the drop? The chemical potential µ inside the film (µ/v ) f ′(e)de/du), where v is the molecular volume) and in the droplet (µ/v ) -2γ/(b - e)) must be equal at equilibrium. This leads to
2γ γa2 γ - =3 b b e
-
(7)
If > 0, a wetting film of thickness ew ≈ a2/3b1/3 coexists with the drop. More detailed analysis can be found in refs 10 and 11. On the other hand, if < 0, the capillary remains dry (e ) 0). Typically, for b ) 100 µm, a ) 1 Å, ew = 10 nm, and ei ≈ 100 nm. To summarize, liquid films inside capillaries are generally unstable and break spontaneously into droplets to decrease the liquid/air interface area. Only microscopic films of wetting liquids can be stabilized by van der Waals interactions. However, the development of the Rayleigh instability can be very slow for thin films and dewetting by nucleation and growth of dry cylindrical patches can be observed, if it is faster. We shall discuss now the dynamics of the Rayleigh instability. III. Rayleigh Instability inside a Capillary We assume a modulation of the film thickness e ) e0 + δe(x) (Figure 1). In the limit of small amplitudes δe, the pressure p(x) generated by this fluctuation is
γ γa2 - γe′′ b-e e3
p ) -
(
)
(9)
The volume conservation imposes
where P(0) ) S, P(∞) ) 0, and P(e) ) A/12πe2, for a simple van der Waals liquid (and e < λ). The Hamaker constant A ) ASL - ALL can be positive or negative, so that we set A/6π ) γa2, where ) (1. We shall consider both wetting (A > 0, S > 0) and nonwetting (A < 0, S < 0) liquids. For a film inside a cylinder, the volume per unit length is u ) π[b2 - (b - e)2] ) πe(2b - e). The stability of the film is determined by the sign of the curvature of f(u):7
f ′′u )
2πb e3 dp 3 η dx
(8)
∂Q ∂e ) 2πb ∂x ∂t
( )
(10)
Looking for a solution δe ) δe0 eiqxe-t/τq, neglecting secondorder terms in ∂e/∂x, and assuming e , b, we find the relaxation time τ(q),
-
[(
)
]
a2b2 1 γe3q2 ) 1 3 - b2q2 4 τq 3ηb2 e
(11)
In the complete wetting case > 0, the fluctuation is damped if e < ei and amplified for e > ei. This means that the Rayleigh instability is absent for films of thickness ew < e < ei and dewetting is the only process to remove the film. For partial wetting, < 0, the mode is always amplified at small q values (qb < 1). Neglecting the 1/e4 term, the fastest mode corresponds to qM ) 1/bx2. The corresponding characteristic time τM is given by
γe3 1 ) τM 12ηb4
(12)
For thin films (e , b), assuming that the velocity of dewetting is still Vd ) cte(γ/η)θE3, we can define a characteristic maximal length LM by
Vd τ M ) L M
(13)
b4 3 θE e3
(14)
It leads to
LM ≈
Dewetting may be observed only over lengths of film L < LM for which the Rayleigh instability is not yet developed. Above LM, droplets build up all along the capillary. For b ) 1 mm, b/e ) 102, θE ≈ 10-1, and LM ≈ 1 m. IV. Dewetting of Nonwettable Capillaries (S < 0) The original liquid film is assumed to be left behind on the capillary walls by a columnar drop moving at a high velocity U (Figure 2a). This corresponds to the experimental procedure used in ref 6. The thickness e(U) of the film, studied in refs 12 and 13, follows the Landau Levich law e(U) ≈ b(U/V*)2/3, if U > Uc (≈V* θE3), where V* ) γ/η. At time t ) 0, we stop the drop. The film exposed at one end to the dry capillary dewets (Figure 2b), and the drop moves back (Figure 2c). This retraction may be slow (7) Brochard-Wyart, F.; di Meglio, J. M.; Que´re´, D. C. R. Acad. Sci. Paris 1987, 304, 553. Brochard-Wyart, F. J. Chem. Phys. 1986, 84, 8. (8) Lord Rayleigh On the stability of jets; Cambridge, England, 1899; Scientific papers (361-371). (9) di Meglio, J. M.; Que´re´, D.; Brochard-Wyart, F. C. R. Acad. Sci. 1989, 309, 19. (10) Churaev, N. V.; Starov, V. M.; Derjaguin, B. V. J. Colloid Interface Sci. 1982, 89, 16. (11) Churae, N. V. Liquid and vapor flows in porous bodies: surface phenomena; Topics in Chemical Engineering, Vol. 13; Gordon and Breach: Amsterdam, 2000; p 44. (12) Que´re´, D. C. R. Acad. Sci. 1992, 313, 213. (13) Cachile, M.; Chertcoff, R.; Calvo, A.; Rosen, M.; Hulin, J. P.; Cazabat, A. M. J. Colloid Interface Sci. 1996, 182, 483.
Dewetting versus Rayleigh Instability
Langmuir, Vol. 18, No. 12, 2002 4797
respect to L taken at a constant volume Ω ) Lu with
2πbγ˜ SO )
]
∂F ∂L
) Ω
dF de du F +L L de du dL
It leads to 2πbγ˜ SO ) 2πbγSL + 2π(b - e)γ + γu/(b - e). The driving force fd ) 2πb(γ˜ SO - γSO) is then fd ) 2πb[(γSL + γ) - γSO] + π[e2γ/(b - e)] in agreement with eq 16. Instead of decreasing with e (as it does for flat substrates), the total driving force increases! The velocity becomes then
Vd )
Figure 2. Dewetting of films deposited inside a capillary of radius b. (a) Deposition of a film by a drop pushed at velocity U. (b) Dewetting by nucleation and growth of a dry patch in the early stage: the rim is annular. (c) Dewetting in the late stage: the rim is columnar. (d) Dewetting of a wettable capillary: a film (thickness e1) is left behind the drop.
compared to dewetting by the film border, if the columnar drop is large, and was not observed in the Buenos-Aires experiments.6 We discuss both the retraction of the columnar drop and the dewetting of the film at its contact line with the dry surface. We focus on viscous liquids. The motion of the rim results from a balance between capillary forces and viscous forces. We can alternatively write an energy balance: the gain of surface energy is converted into viscous dissipation. A. Annular Rims. We study the dewetting of a film exposed at one end to the dry capillary. At short times, the rim collecting the liquid is annular. In the limit e/b , 1, we expect a constant dewetting velocity,2,3 given by eq 4,
|S| θE Vd ) η ln
(15)
which expresses that the driving force 2πb|S| on the rim is balanced by the friction force, ≈2πbη(V/θE)ln. For thicker films, what is the driving force on the rim? Naı¨vely, we would write fcapillary ) 2πb(γSL) + 2π(b - e)γ - 2πbγSO ) 2πb[|S| - (e/b)γ]. But, in fact, the film is under a negative Laplace pressure P0 - [γ/(b - e)]. There results a force component fLaplace ) [γ/(b - e)]π[b2 - (b - e)2] parallel to the tube axis and obtained by integrating pressure over a section of the film perpendicular to the axis in the constant thickness region. The overall force is fd ) fc + fL.
e2 γ fd ) 2πb|S| + π b-e
(16)
We can also derive fd from the surface energy of the film coating the cylinder. Per length L of the cylinder, F ) [2πbγSL + 2π(b - e)γ]L. We name γ˜ SO the surface energy of the wet capillary obtained from the derivative of F with
(| |
θE e2 γ S + ηln 2b(b - e)
)
(17)
Equation 17 shows that the effect of the curvature becomes important for thicknesses e ≈ bθE. The volume of the rim (≈2πbθEl2) is equal to 2πbL1e, where L1 is the length of the dried capillary); it increases up to a volume of the order of b3, where the annular shape becomes unstable and a liquid bridge is formed. This transition at L1 ≈ b2/e should be observed if L1 < Lc, that is, at times t1 ) L1/Vd shorter than the characteristic time τM for the development of the Rayleigh instability. We describe now the regime expected when the rim becomes a columnar drop (Figure 2c). This regime should also apply to the retraction of a drop, first moving at a fast velocity U and suddenly stopped at t ) 0 (Figure 2a). B. Columnar Rims. We describe now the motion of the columnar drop (Figure 2c), assuming thick films (the van der Waals interactions are negligible). The capillary energy W can then be written:
W ) bγSOL1 + bγSL(Lg + L2) + γ(b - e)L2 (18) 2π where L1 is the length of the dry capillary, Lg is the length of the cylindrical droplet, and L2 is the length of the wet capillary. We can neglect the liquid/air surface energies of the drop, because they are constant in time. L1, Lg, and L2 are related by the two equations
L1 + Lg + L2 ) L and Lgb2 + L2[b2 - (b - e)2] ) Ω/π ) cte (19) expressing the conservation of the total length and of the liquid volume, respectively. As the droplet moves at velocity V ) dL1/dt, the production of surface energy W ˙ ) dW/dt is
dL2 dL1 W ˙ ) |b(γSO - γSL) + γ(b - e) | 2π dL1 dt
(20)
With dL2/dL1 ) -b2/(b - e)2, it leads to
e -W ˙ ) -S + γ V 2πb b-e
[
]
(21)
The corresponding driving force fD/2πb ) |S| + γe/(b e) can also be computed from the drop of the Laplace pressure between the two ends of the columnar drop:14
∆P ) -
2γ cos θE γ γ +2 ) 2 (1 - cos θE) + b b-e b fd e 2γ ) 2 (22) b(b - e) πb
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The equation of motion of the drop is given by W ˙ + TS˙ ) 0, where the dominant contribution to TS˙ is assumed to be the viscous dissipation in the drop. For a Poiseuille flow, TS˙ ) 8πηLgV2 defines the viscous force fv ) 8πηLgV on the drop. The equality fd ) fv gives the velocity V of the drop,
{| |
e b S +γ b-e V) 4ηLg
}
Vd = (23)
Velocity increases with the film thickness and decreases with the length of the drop. Equation 23 is valid if dissipation at the contact line is negligible. It can be taken ˜ g ) Lg + b ln/4θE in eq into account by replacing Lg by L 23. This type of dewetting was not observed in the experiments of ref 6: this may be due to hysteresis effects associated with differences between the advancing and receding contact angles at the front meniscus. V. Dewetting of Wettable Capillaries (S > 0) In the condition of complete wetting, a drop moving at any velocity leaves behind a film of thickness e(U).12 The film is metastable for e < xab and becomes unstable at larger thicknesses. At time t ) 0, we stop the droplet and we study its backward motion (Figure 2d). We consider the motion of a columnar droplet (length Lg) separating two films of thickness e1 and e2 (>e1) and length L1 and L2, respectively. If V is the droplet velocity, the front velocities Vs1 and Vs2 are related to V by Vs1 ) V[b2/(b - e1)2] and Vs2 )V[b2/(b - e2)2], respectively. The production of surface energy W ˙ is W ˙ ) 2πγ[(b - e1)Vs1 - (b - e2)Vs2]. Using the relations between Vs1,2 and V1 allows computation of W ˙ and the driving force
(
˙ /V ) 2πγb2 fd ) -W
)
1 1 b - e2 b - e1
fd is balanced by the friction force fV ) 8πηLgV, so that the columnar droplets move in the direction of the thicker film (e2) with a velocity V:
V)
γ e2 - e1 b2 4η Lg (b - e1)(b - e2)
of the film left behind the retracting droplet. Because the velocity of the retracting droplet is very small, one can assume that e1 = a2/3b1/3 (the equilibrium thickness of a film in coexistence with a drop described in the first section). If e1 , e, the dewetting velocity Vd for a wetting capillary is
(24)
For our case, e2 is the initial thickness of the film deposited in the capillary (e2 ) e) and e1 is the thickness
γ e 4η Lg
(25)
If e < xab, this process is the only one to remove the liquid film. If e > xab, this process can be observed only at times t < τM (eq 13) before the Rayleigh instability has developed. This condition defines a maximal length L′M ) b4/e2Lg (≈m for b ) 1 mm, b/e ) 102, and Lg ) 1 cm). VI. Concluding Remarks The dewetting of a liquid film inside a circular capillary displays two novel features: The curvature gives rise to a driving force increasing with the film thickness. An enhancement of the dewetting velocity with film thickness has indeed been observed experimentally.6 This result was unexpected since, for planar substrates, dewetting is slower at large film thicknesses. More quantitative analyses are under way. The swelling rim ultimately becomes a columnar drop. The resulting gain of liquid/air surface energies increases the driving force for dewetting, whereas the viscous dissipation increases with the length of the cylindrical drop. We expect to stimulate experiments testing these predictions. The main advantage of drying by dewetting is the complete removal of the liquid film. In the Rayleigh instability process, droplets of liquids are left all along the capillary. We show here that we can adjust the thickness e of the film to favor the dewetting process, as shown by eq 14, and to improve the drying of capillaries. Acknowledgment. This work has been partly supported by the ECOS A97-E03 and PICS No. 561 cooperation programs between France and Argentina. We also thank A. Buguin, O. Rossier, and P. G. de Gennes for comments and a critical lecture of the manuscript. LA011862W (14) Notice that we assume that the dynamical contact angle θD is equal to the static equilibrium contact angle θE, which is valid when the viscous dissipation at the contact line is small compared to the dissipation in the columnar drop.
