Langmuir 1993,9, 2220-2224
2220
Motions of Droplets on Hydrophobic Model Surfaces Induced by Thermal Gradients J. B. Brzoska, F. Brochard-Wyart,’ and F. Rondelez Laboratoire de Physico-chimie des Surfaces et des Interfaces,t Institut Curie, Section de Physique et Chimie, 11 rue Pierre et Marie Curie, 75005 Paris, France Received November 23,1992. I n Final Form: May 11,1993 We have studied experimentallythe motions of liquiddroplets depositedon nonwettablesurfaces (silanized silicon wafers) submitted to horizontal thermal gradients VT. Systematic measurements of the drift velocity V for different droplet radii R show the following: (i) the droplets move only above a critical radius R, which depends on contact angle hysteresis and is inverselyproportional to VT; (ii)above R,, V increases linearly with R and VT; (iii) for large radii R (droplets flattened by gravity), V reaches a saturation value. The droplet contour is no longer circular, but presents two straight-line segments in the direction of VT. The flow patterns at the free surface of the dropletshave been monitoredby video recording the displacement of floating particles.
Introduction The displacement of liquid droplets under the effect of heat can be observed in everyday life, when heating a wet saucepan or when soldering copper tubes (the tin droplets move away from the flame). Although this phenomenon has been reported long ago,1 it is only very recently2that one of us has studied theoretically the motions of droplets on “model” surfaces (no hysteresis) induced by either chemical or thermal gradients. In the case of chemically heterogeneous surfaces, the droplets are predicted to move toward the regions of lower surface energies? This has been checked experimentally on two types of surfaces, showing either a discontinuity in wettability (obtained by partial silanization)3 or a continuous gradient of wettability generated by diffusion of silanes in the vapor p h a ~ e . ~ ~ ~ In the case of a thermal gradient, the temperature dependence of the contact angle 0, also leads to a motion in the direction of the lowest surface energies. However, the situation is more complex than for chemical gradients, because the liquid/air surface tension y is also temperature dependent: therefore V T induces a Vy, and this gives rise toa Marangoni flow (see Figure l),which drives the droplet toward the cold region. For weak thermalgradients, these two effects are additive, and the sign of the overall motion is not obvious: droplets can be driven toward the regions of higher surface energies! This is in fact the most common case. This ambiguity in the sign is reminiscent of the Soret effect in binary mixtures: a solute can move toward either the cold or the hot region. We present here a quantitative study of the motions of liquid droplets deposited on smooth hydrophobic surfaces and submitted to a horizontal thermal gradient. The surfaces which have been used are characterized by a low contact angle hysteresis and have been prepared by silanization of silicon wafers a t controlled temperature:6 typical contact angle hysteresis was 6 = cos 0, - cos 0, = 1.5 X for silicone oils, 6 = 10-2 for n-alkanes. The Laboratoireassoci6 au CNRS (URA 1379),et al’Univerait.4 Pierre et Marie Curie (Paris VI). (1) Marangoni, G. G. M. Ann. Phyu. Poggendorf 1871,143. (2) Brochard-Wyart, F. Langmuir 1989,5, 432. (3) Greenspan, H. P. J. Fluid Mech. 1978,84,125. (4) Ondequhu, T.; Raphael, E. C. R. Acad. Sci. 1992,314 (2), 453. (5) Chaudhury, M.; Whitesides, G. M. Science 1992, 256, 1539. (6) Maoz, R.; Sagiv, J. J. Colloid. Interface Sci. 1984, 100 (2), 465. Brzoeka, J. B.; Shahidzadeh, N.; Rondelez, F. Nature 1992,360.
experiments have been performed on a series of different molecular weight Mw silicone oils (PDMS), which have nearly identical surface tension, but widely different viscosities q. For each compound and each VT, droplets of several radii R were deposited on the substrate and this velocity was monitored by video recording. Direct measurements of the surface flow pattern were also performed by using a decoration technique. The existing theoretical predictions in ref 2 for the motions of droplets induced by a thermal gradient are restricted to perfect surfaces with zero contact angle hysteresis. In our experiments, we have observed that even a slight hysteresis hinders the motion of small droplets. For each thermal gradient, there exists a critical radius Rc, below which the droplets are immobile (no motion of the center of mass), even though Marangoni flows are clearly visible a t the free liquid surface. Above R,, droplets start to move. Our aim here is to study quantitatively the dependence of R, upon 6 and strength of the thermal gradient, the center of mass velocity V of the droplets above R,, the contour of the moving droplets, and the flow pattern at the free surface. Section I gives a reminder on the theoretical motions of liquid droplets on perfect surfaces (6 = 0) under a thermal gradient, and section I1extends these predictions to finite contact angle hysteresis. Section I11 describes the experimental procedure and setup. In section IV, we present the experimental data and compare them to the theoretical predictions for real surfaces with 6 Z 0.
I. Droplet Motion on Model Surfaces We.start by recalling the theoretical predictions2 for the motions of liquid droplets deposited on a perfect nonwettable surface. The spreading coefficient S = yso - ( y s ~+ y) is negative, where the yij are the solid/air, solid/liquid, and liquid/air interfacial tensions, respectively. The contact angle at the triple line 0, is finite. Furthermore, we assume no contact angle hysteresis (6 = 0). In the limit of small angles 0, considered here, S = -‘/zyBe2. Under an uniform thermal gradient, 0,(T) and y ( T ) become spatially dependent. This has two effects: (a) capillary forces which drive the droplets toward the regions of large S, in order to decrease the surface energy; (b) Marangoni flows induced by the surface tension gradients. We will only discuss here the one-dimensional geometry shown in Figure 1. Physically, this corresponds to a strip of liquid (along y ) moving under a gradient
0743-7463/93/2409-2220$04.0O/Q0 1993 American Chemical Society
Langmuir, Vol. 9, No.8, 1993 2221
Droplet Motion by Thermal Gradients
where 1, = In (Lla) is a logarithmic factor describing the divergence of the dissipation in a liquid wedges (a is a 10). For a flat ridge, Z eo and molecular size and 1,
+
TFigure 1. Side view of a 1-Dliquid ridge advancing from the side of a hydrophobic solid substrate hot (T+)to the cold (T) and velocity profile u(z,z) inside the liquid. The local velocity reaches ita maximum value US at the interface, changes ita sign due to the backflow of the deep regions of the ridge, and is zero on the solid substrate to satisfy the no-slip condition of simple liquids. The resulting mean advancing velocity for the ridge is V.
(parallel to x ) , which occupies an interval X A < x < XB. If L = XB - X A is smaller than the capillary length r1= d ( y / p g ) ’ I 2 ,the ridge has a cylindrical cross section. For L > K-1, the ridge is flat (“mesa” type), with a thickness eo = 2K-1 sin(ee/2). It should be noted that the onedimensional geometry is not physical since liquid ridges are unstable.7 However, it should give the main features of the 2-D case (spherical droplets), at least within geometrical prefactors. The equation of motion of the rim under a thermal gradient is deduced from a balance between the viscous and the capillary forces: The driving force acting on the liquid ridge is the unbalanced Young force F d
F,j = (7~0 - YSL)B - ( Y S -~ YSL)A
(1)
The viscous force is the integral of the viscous stress uxz at the solid/liquid interface
In the lubrication approximation, valid for 8, 6 mm), we observe a saturation of the velocity and a deformation of the droplet contour characterized by the apparition of two straight-line segments in the direction of VT, as predicted by Dussan for similar pr0blems.~J6J7 Most of these data are well-described by an extension of the hydrodynamic theory,2 initially developed for the motion of droplets on ideal surfaces and now extended to incorporate finite contact angle hysteresis. (16)Dussan, E.B.,V. J. Fluid Mech. 1983,151,l. (17)Du", E.B.,V. J. Fluid Mech. 1983,174,381.
