Langmuir 1992,8, 2324-2329
2324
Dynamics of Liquid Rim Instabilities F. Brochard-Wyart' and C. Redon Laboratoire de Physico-chimie des Surfaces et Interfaces,t Imtitut Curie, 75231 Paris Cedex 05, France Received February 18,1992. In Final Form: April 17,1992 We estimate the relaxation times of the varicose modes of a liquid rim deposited on a flat solid surface, which have been found to be unstable below a wave vector &. We fiid (a) a purely viscous regime (for highly viscous liquids and small contact angles) and (b) a viscoinertial regime in the opposit limit. In our dewettingexperiments, we do observe instabilitiesof moving rims, especiallywhen they are thin. Polymer liquids usually give broad rims which are nearly stable.
I. Introduction Following the pioneering work of Plateau' and Rayleigh2on the breakup of liquid jets, Sekimotoet aL3 have shown recently that liquid ribbons deposited on a flat solid surface display a varicose instability for wave vector q below a critical value q,. The driving force (for liquid cylinders, or portions of cylinders) is provided by a gain of surfaceenergy. A general, statictheory of the instability (with contact lines obeying the same Young condition for partial wetting) was constructed in ref 3, in particular for the ridge geometry. Our aim here is to discuss the dynamics of these instabilities. To begin, we consider a strip of wetted region extending along the y axis, in conditions of partial wetting (that is finite contact angle de). If the distance L between the two contact lines is less than the capillary length K-' = (r/pg)'i2, where 7 is the liquid surface tension, p the mass per unit volume, and g the gravitational constant, the equilibrium shape is a portion of a cylinder with a contact angle 8,. If L > K-', the rim is flattened by gravity with a height e = 2x-I sin (e$2) in the flat region. Our aim is to study the deformations of the rim's contour, characterized by the displacement of the two contact lines around the equilibrium positions at x = *L/2 (Figure 1). The elasticity of one single contact line was first constructed in ref 4 and is quite anomalous. The energy per unit length of a weakly distorted line with amplitude uq and wave vector q, larger than K, is
and cannot be described in terms of a line tension, which would give an energy proportional to q2. The JqJ factor in eq 1 comes from distorsions of the liquid-gas surface extending up to distances q-'. On the other hand, for a small wave vector (q < K), the distorsions are screened out beyond the capillary length and the elastic energy returns to the classical q2 dependence. The correct interpolation between these two limits is3 The deformationsof two coupled lines, commonly called the Yridge"geometry, has been discussed in ref 3 in terms of two types of eigen modes: (1)zigzag modes where the t Laborabire associbau CNRS (URA 1379) et A l'Universit4 Pierre
et Marie Curie. (1) Plateau,J. Statique exp6rimentale et tMoriquedes liquidessoumis aux seules forces molheulaires; Gauthier-Villare: Paris, 1873. ( 2 ) Lord Raleigh Scientific Papers; Cambridge University Press: Cambridge, 1899, PMOS.Mag. 1892, 34,145. (3) Sekimoto, K.; Oguma, R.; Kawasaki, K. Ann. Phys. 1987, 176. (4) Joanny, J. F.; de Gennes, P. G. J. Chem. Phys. 1984,81,552.
-Li2
0
L12
X
Figure 1. Strip of wetted region. The liquid cylinder has a width L and extends infinetely along the y-axis. deformationsof the two lines are in phase and (2) "varicose modes" (or 'peristaltic modes"), where the deformations are out of phase (Figure 2). In this case, the modulations of the distance between the contact lines are associated with modulations of the thickness in the y direction and of the Laplace pressure. If the pressure becomes smaller in the thicker part, the fluctuation is unstable, and the rim breaks into droplets. If uqis the outward displacement of the two contact lines, the energy per unit length for a varicose mode of wave vector q, in the limit KL G< 1, can be written as3
(3) (i) At a large wave vector (Lq >> 1)) Fq = 2f1q, and consequently, the energy is twice the energy of one single contact line. (ii) Fqbecomes negative for q < q, (Lq, = 2.408). This means that the surface energy of the deformed rim is smaller, and the varicose modes are u t a b l e for q < 2.408/L. (iii) Equation 3 does not apply for KL >> 1. In this limit, Sekimoto et ale3found (4)
where q', = 2~exp(-d/2). As L increases, the wavelength 2?r/q', of unstable modes increases exponentially and the two lines are nearly decoupled. Our aim here is to discuss the dynamics of the unstable modes at small wave vectors. In section 11, we review briefly the dynamics of one single line which has been studied both experimentally6and theoretically? In section 111, we analyze the motion of two coupled lines. We calculatethe growth times and the wavelengthof the fastest unstable mode. In section IV, we describeour observations on the instabilities of rims, formed in the course of dewetting experiments' after nucleation and growth of a dry spot. ~~
(5) OndarMu, T.; Veysei6, M. Nature 1991,362, 418. (6) Brochard-Wyart, F.;de Gennea, P. G. Langmuir 1991, 7, 3216. (7) Redon,C.;Brochard-Wyart,F.;Rondelez,F.Phys.Rev.Lett. 1991, 66,715.
0743-7463/92/2408-2324$03.00/0 0 1992 American Chemical Society
Langmuir, Vol. 8, No. 9,1992 2325
Dynamics of Liquid Rim Instabilities a
sinusoidaly distorted contact line was first derivedQby writing that the elastic energy is burnt in viscous dissipation, T$:
Y
+
(d/dt)(f,) = y8;((q2 K')@ - ~ ) ~= l-T$ i (5) When a contact line moves at velocity li (normal to the line), the hydrodynamic dissipation per unit length of the line inside the wedge of angle 0 isa
TSI = -31(q/B)li2 (6) where q is the liquid viscosity and I a logarithmic factor reflecting the singularity of the flow near the contact line. 1 is of order 10 (ref 7) and may be treated as a constant. From eqs 5 and 6, the relaxation is exponential: li, = -(l/Tq)Uq
(7)
where
+
1 / ~ , ( V*/6Z)e,3((q2
K')~/'
-K)
(8)
V* = y/q is a typical velocity of wetting.
0
LIZ X
(b) Viscoinertial Modes. The derivation of eq 4 assumed that the velocity gradient in the liquid wedge was spread over all the wedge thickness z = Ox. However, the velocity field induced by longitudinal waves of frequency w is screened out within a penetration length X:10,11
b
X = (q/wp)'/2 (9) Here we study the case A < fIq-l/i, when the dissipation is confined inside a thin layer of thickness X near the free surface. Integrating the velocity gradient (V/XI2spread over a distance q-l (alongx ) and a distance X (along z ) and omitting all coefficients, the dissipation becomes T$ = q(li2/X)q-l. If q < K , the deformation is screened by K - ~and the dissipation is limited to a re ion of size K-' near the contact line. An expression of T extrapolating to these two limits can be written as
8
2n
T$ = -7 -r; li2 (q2 +1K2)1/2
(10)
Equations 5, 9, and 10 lead to
~.
-LIZ
LIZ X
Figure 2. Schematic representation of the sinusoidal deformations of the two contact lines with wave vector q: (a) when the amplitudesd are in phase, zigzag modes, (b) when the
Note that the prefactor in eq 11is a complex number: the imaginary and real parts are comparable. The deformations relax with an oscillation which is a signatureof inertial effects. The imaginary part can also be calculatedthrough the balance of elastic and inertial energya6A remarkable feature of the dispersion eq 11 is that K cancels. (c)Discussion. The viscoinertialmode shows up only if the hydrodynamic screening length is smaller than the wedge thickness:
amplitudes are out of phase, varicose modes.
11. Collective Modes of One Single Line The relaxation times of a contact line have been measured recently by T. OndarGuhu and M. Veyssi6.6The periodic distortion of the contact line is obtained by preparing a sequence of equidistant droplets, and causing them to coalesce with a macroscopic wedge of liquid. The liquids are deposited on a model surface, with almost no hysteresis. A theoretical picture has been given in parallel with these experiments.6 Two regimes have been predicted. (a) Viscous Regime. The relaxation time 7, of the
i.e.
-
where q* ypOe6/q2. The constant is not known but has to be a small number because the i factor enters in the denominator with a power of 3, but we ignore the exact (8) de Gennes, P. G. Rev. Mod. Phys. 1985,57,827. (9) de Gennes, P. G. C. R. Acad. Sci., Ser. 2 1986,302,731. (IO) Landau, E.;Lifschitz, E. Fluid Mechanics; Mir, Ed.; 1971; p 111. (11) Levich, V. Physical Hydrodynamics; Prentice Halk Englewood Cliffs, NJ, 1962.
Brochurd- Wyart a d Redon
2326 Langmuir, Vol. 8, No.9,1992 Table I. Characteristic Parameters for the Liquid-Solid P*S* liauid solid a* (cm-1) K (cm-9 octane Si-F 106 Si-H 440 5.51 nonane Si-F 7 X 1o.L Si-H 770 5.46 decane Si-F 5 X 1o.L Si-H 860 5.37 dodecane Si-F 4.25 X lo" hexadecane Si-H 1750 5.23 Si-F 1.3 X lo" 15 6.6 PDMS (M,= 5970) Si-F PDMS (M,= 17 250) Si-F 0.6 6.6 4 q* = yp~?~5/q and the inverse capillary length K = (pg/y)1/2. AU physical variables are handbook values except for 0, which is the measured contact angle, with O h 0 accuracy. Octane, nonane, and decane are volatile, and Marangoni effects are superimposedto the Rayleigh instability. It leads to a highly distorted rim shown in Figure 7. Dodecane and hexadecane form etable rims on Si-H and dropletson Si-F (Figures5 and 6). A typical experimentwith PDMS is shown in Figure 4. According to our model, viscoinertial regimes can be expected for the largest q* values.
I
10
~~
numerical coefficient. q* values are given in Table I for alkanes and silicone oils. If q < K , eq 13 leads to q > (K/q*)ll2K. If q > K, eq 13 leads to q < q*. Thus, if q* < K, eq 13 has no solutions. If q* > K, the viscoinertial regime extends from a wave vector (K/q*)1/2Kup to q*. In conclusion, the relaxation of the contact line of very viscous liquids is always governed by the purely viscous regime. The viscoinertial regime shows up only for lowviscosity liquids and large contact angles.
111. Growth Times of Liquid Rim Instabilities The relaxation spectrum of the two coupled lines can also be deduced by the transfer of elastic energy into viscous
dissipation: (d/dt)(F,) = -TS
(14)
where F, is the elastic energy of a varicose mode of wave vector q given by eqs 3 and 4. The main difference with the deformation of one single line is the flow in the y direction of the rim induced by the displacement of the two contact lines. The mean velocity V, along they axis is related to the displacement u of the line by the incompressibility condition (div V = 0) 2UlL
+ qv, = 0
(15)
The velocity V, is dominant for long wavelength deformations. (a) Viscous Regimes. (1) Circular Rims (KL > 1: the two lines are decoupled, and T$ is twice the dissipation of one single line:
TS = -(61/8)qIL2 (16) If qL > 1, we recover the dispersion equation for the fluctuations of one single contact line. (ii) Below qc, the sign of 1/7, is negative. Capillary excitations are attenuated for q > qc, and are amplified below. The rim breaks into droplets. The distance between droplets is the wavelength of the fastest unstable mode. A plot of 117, versus q is shown in Figure 3a. The fastest rise time corresponds to ~1 L/2 = 314. (2) Flat Rims (KL>> 1). If L > K - ~ , the rim is flattened by gravity, and the thickness of the flat region is e = K-lO,, (e, 1and qL (a + 1 / w 3
(27)
with q* = ypBe5/q2. If q > 1/L, the viscoinertial regime shows up for q < q*. If q < 1/L, it appears for q > (l/L)(l/(Lq*)1/2).Thus inequality 27 is never satisfied if q* < 1/L. If q* > 1/L, the viscoinertial regime should appear for q vectors ranging from (l/L)(l/(Lq*)1/2)up to q*.
Conclusion. Viscoinertial regimes should appear only for liquid rims of width L larger than a threshold value q*-l, i.e., for low-viscosity liquids and high contact angles (q*-l can be very small (Table I)). On the other hand, for viscous liquids, q*-' becomes larger than the capillary length, where the varicose modes are quasistable. For both regimes, the fastest unstable mode corresponds to q&/2 = 3/4, i.e., a distance between droplets equal to 4L..
Brochard- Wyart and Redon
2328 Langmuir, Vol. 8, No. 9,1992
3 Mln
f min 30
2 rnm 30 5 mrn 30 Figure 4. Snapshot of an expanding dry patch in a PDMSfilm deposited on a fluorinated wafer of 5-cm diameter. The rim surrounding the hole is circular and has a uniform width of 0.1 mm in the first image and grows to 1mm a t the last stage of the experiment. Full view size: 4 cm. In the experimental time, one observes the first stage of the instability. Notice that the rim is modulated with a wavelength roughly equal to 4L = 3 mm (in agreement with our theoretical predictions).
Figure 5. Time evolution of a dodecane rim on a fluorinated wafer (4 = 5cm). The rim outline has been drawn a t a fixed time interval (*/Ms); the regular spacingbetween each contour therefore evidences a constant dewetting velocity. The rim is unstable and droplets appear. Magnifing power: X6.
droplets which grow and move with the rim. It is worth pointing out that the number of droplets obviouslydepends on initial film thickness and stays constant during the
Figure 6. Time evolution of a hexadecane rim on a fluorinated water (4 = 5 cm). The rim breaks into droplets. The droplets are growing through the rim draining, and their number is a constant, depending on the initial film thickness.
whole dewetting experiment, as evidenced in Figure 6 where the liquid is hexadecane on wafer Si-F. We conclude, that, as soon as the droplets are formed, they drain the liquid in the rim and grow. That can be compared to the observations made by J. Melo et al. on spinning drop instabilities.12 In their case, a liquid drop is deposited
Dynamics of Liquid Rim Instabilities
Langmuir, Vol. 8, No. 9, 1992 2329
hysteresis, or more simply dusts on the surface, and the droplets formed in the rim can be pinned on the defects and therefore be let on the substrate. In this case, the solid is covered by macroscopicdroplets in the final stage of the experiment.
Figure 7. Dewetting rim for a slightly volatile alkane. The rim is very irregular and drives liquid trails (hatched zone) which evaporate.
on a rotating wetting substrate. The droplet in the centrifugal force is deformed, and a bump appears near the contact line, which is unstable and breaks into small droplets which develope well-defined fingers whose number is fixed at the onset of the unstability. In our case, the strong perturbation due to the rim instability does not seem to affect the hydrodynamics of the whole system since the velocity of the rim fragmented in droplets is equal to the dewetting velocity v d measured for the stable rim; that is, v d = (r/7l)Oe3. Another pattern is shown in Figure 7. In this case, the liquid is a small chain alkane (octane) and is therefore slightlyvolatile. We have represented two outlines of the rim at the same time. These two borders correspond (i) to the three-phasecontact line at the liquid-solid boundary and (ii)to the beginning of the macroscopicrim, evidenced as a black line on the video frame because of a sharp change in the contact angle of the drop profile a t this point. On the contrary, the first line is very difficult to localize and there seems to be a continuous transition from the dry substrate to the macroscopicliquid film (after the second line). When the rim recedes, it takes with it some small and thinner liquid tongues. These comovingliquid trails evaporate as they move. As in the previous cases, the solid substrate is perfectly dry in the final stage of the dewetting experiment. Finally,the presence on the solid of macroscopicdefects can also alter the rim evolution. If the solid is chemically homogeneous, we have observed that the liquid rim is formed of regularly disposed droplets moving together with it and growing through the drainage of the rim. For a nonideal substrate, there can be strong contact angle
V. Conclusion Narrow liquid rims as opposed to thick rims are always unstable and break spontaneously into droplets. We have estimated the rise time 7 of the fluctuations of the contour of the rims, which are unstable below a wave vector q,. In the linear approximation of small deformations, we expect two regimes: First, a viscous regime where q*L< 1. The fastest mode is characterized by a wave vector ~m = 1.5/L and a time constant 7v = l L/V*Oe3. Second, an inertial regime where q*L> 1. The fastest mode is described by qm= 1.5/L and 7 i = ( ~ * L ) l / ~For 7 ~ both . regimes, the distance between the droplets is equal to 4L. To distinguish between the two regimes, one would need careful measurements of the rise times 7 = L*,where x = 1for the viscous regime and x = 4/3 for the inertial regime. Experimentally, we have observed the instabilities of movingrims which are formed spontaneously in dewetting experiments. We expect in this case that the motion of the rim is not coupled to the varicose mode. It is however very difficult to observe the onset of the instabilities; i.e., the linear regime which has been calculated here and the experimental observations are only qualitative. Fortunately, the number of droplets is constant, and it will be possible in principle to check the relationship between ~m and L (or e ) . Delicate optical measurements of the film thickness, dewetting in a few seconds, are currently under way to relate quantitatively the number of droplets to the film thicknesses. We have focused our attention on the development of the instability: we have found that (i) the number of droplets is constant, (ii) the formed droplets continue to grow by draining liquid from the rim, (iii) they are dragged along by the moving rim and deposited on the solid surface only if defects or dust are present, (iv) for slightly volatile liquids, a thin film forms behind the macroscopic wedge in motion, which must be due to the Marangoni effect associated with the thermal gradient produced by the liquid evaporation, and (v) we do not think that there is an upper critical thickness for the instability. As the film becomes too thick, we enter the gravityregime, where the rise time increasesexponentially. We only see very small rim modulations, because the droplets cannot develop in the experimental time.
Acknowledgment. We thank Hubert Hervet, PierreGilles de Gennes, and Bradford Factor for stimulating discussions and for reading the paper. Registry No. Si,7440-21-3;nonane, 111-84-2;hexadecane, 544-76-3;octane, 111-65-9;decane, 124-18-5;dodecane, 112-403.
