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Langmuir 1999, 15, 2674-2683
Thermocapillary Deposition of a Fluid Droplet Normal to a Planar Surface Shih H. Chen Department of Chemical Engineering, Hwa Hsia College of Technology and Commerce, Taipei 235, Taiwan Received July 7, 1998. In Final Form: January 13, 1999
An analytical study is presented for the thermocapillary migration of a fluid sphere in a constant applied temperature gradient perpendicular to a planar surface. The Peclet and Reynolds numbers are assumed to be small, so that the appropriate energy and momentum equations of the fluids inside and outside the droplet are governed by the Laplace and Stokes equations, respectively. The asymptotic formulas for the temperature and velocity fields in the quasisteady situation are obtained by using a method of reflections. The plane surface may be a solid wall or a free surface. When the droplet is migrating normal to a solid plane, the boundary effect of the planar surface retards the droplet motion, reducing the thermocapillary velocity of the droplet. In the situation of droplet migration toward a free surface due to thermocapillarity, the droplet velocity can be either greater or smaller than that which would exist in the absence of the plane surface. In general, the boundary effect on the thermocapillary migration is found to be weaker than that on the motion driven by a gravitational force. However, the interaction between the plane and the droplet can be very strong when the gap thickness approaches zero. Considering thermocapillary mobility, the deposition time for a droplet translating across the thermocapillary boundary layer is integrated. Also, it is predicted that the deposition time will be postponed if the fluid sphere is migrating normal to a solid wall. However, the deposition time for a droplet moving normal to a free surface may be shorter than predicted if there is no boundary influence. Generally speaking, a free surface exerts less influence on the droplet movement than does a solid surface.
Introduction When a droplet of one fluid is suspended in another fluid in which it is immiscible, it will migrate toward the warmer side if the surrounding fluid has a temperature gradient. This is due to the temperature-induced surface tension gradient at the droplet-fluid interface. Thermocapillary phenomena have become a subject of current interest following the development of the orbiting spacecraft and opportunities for experimenting and manufacturing under near-weightless conditions. For example, the removal of unwanted gas bubbles or liquid drops in a continuous phase by thermocapillary forces is a possibility during the processing of materials in a reduced gravity environment provided by an orbiting laboratory. Thermocapillary motion was first demonstrated by Young et al.1 who observed the movement of bubbles in a vertical liquid bridge in the gap between the anvils of a micrometer. They also obtained a theoretical prediction for the migration velocity of a spherical droplet of radius a, present in an immersion fluid of viscosity η with an imposed linear temperature distribution T∞(x). With a limit of zero for the Reynolds and Peclet numbers, the droplet velocity U(0) is related to the uniform temperature gradient ∇T∞ by
U (0) )
2 a ∂γ ∇T∞ ∂T (2 + 3η*)(2 + k*) η
(
)
(1)
Here, η* and k* are the ratios of viscosities and thermal conductivities, respectively, between the internal and the surrounding fluids, and ∂γ/∂T is the gradient of the interfacial tension γ with the local temperature T. In eq 1, all physical properties are assumed to be constant except (1) Young, N. O.; Goldstein, J. S.; Brock, M. J. J. Fluid Mech. 1959, 6, 350.
for the interfacial tension, which is assumed to vary linearly with temperature. Equation 1 is valid only for a single droplet in continuous phase that extends to infinity in all directions. However, in practical applications of thermocapillary motion, droplets usually are not isolated and might interact with nearby droplets2-7 and/or boundaries.8-17 During the past 2 decades, much progress has been made in the theoretical analysis concerning the applicability of eq 1 for a fluid droplet in a variety of bounded systems.18-21 Several important conclusions result from these investigations of droplet interactions in thermocapillary migration. First, the droplet interaction effects on thermocapillary migra(2) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1981, 83, 199. (3) Meyyappan, M.; Wilcox, W. R.; Subramanian, R. S. J. Colloid Interface Sci. 1983, 94, 243. (4) Feuillebois, F. J. Colloid Interface Sci. 1989, 131, 267. (5) Keh, H. J.; Chen, S. H. Int. J. Multiphase Flow 1990, 16, 515. (6) Keh, H. J.; Chen, L. S. J. Colloid Interface Sci. 1992, 151, 1. (7) Keh, H. J.; Chen, L. S. Chem. Eng. Sci. 1993, 48, 3565. (8) Sadhal, S. S. J. Colloid Interface Sci. 1983, 95, 283. (9) Mayyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1984, 97, 291. (10) Mayyappan, M.; Subramanian, R. S. J. Colloid Interface Sci. 1987, 115, 206. (11) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1990, 137, 170. (12) Chen, S. H.; Keh, H. J. J. Colloid Interface Sci. 1990, 137, 550. (13) Ascoli, E. P.; Leal, L. G. J. Colloid Interface Sci. 1990, 138, 220. (14) Barton, K. D.; Subramanian, R. S. J. Colloid Interface Sci. 1991, 141, 146. (15) Chen, J.; Dagan, Z.; Maldarelli, C. J. Fluid Mech. 1991, 233, 405. (16) Loewenberg, M.; Davis, R. H. J. Colloid Interface Sci. 1993, 160, 265. (17) Wang, Y.; Mauri, R.; Acrivos, A. J. Fluid Mech. 1994, 261, 47. (18) Anderson, J. L. Int. J. Multiphase Flow 1985, 11, 813. (19) Acrivos, A.; Jeffrey, D. J.; Saville, D. A. J. Fluid Mech. 1990, 212, 95. (20) Zhang, X.; Davis, R. H. J. Colloid Interface Sci. 1992, 152, 548. (21) Satrape, J. V. Phys. Fluid A 1992, 4, 1883.
10.1021/la980833r CCC: $18.00 © 1999 American Chemical Society Published on Web 03/19/1999
Deposition of a Fluid Droplet
Figure 1. Geometric sketch for a droplet in the proximity of a plane.
tion are in general much weaker than those on sedimentation, because the disturbance to the fluid velocity field caused by a thermocapillary migrating droplet decays faster (as r-3, where r is the distance from the particle center) than that caused by a settling droplet (as r-1). In sedimentation, there is a net gravitational force exerted on the droplet and this force is balanced by a nonzero hydrodynamic force. However, there is no hydrodynamic force exerted on the droplet in thermocapillary migration. As a consequence, the disturbance velocity fields in the surrounding fluid for the two situations decay at different rates with r. Second, for the two identical liquid droplets aligned parallel to the prescribed temperature gradient, the interaction effects make each droplet move faster than it would if isolated; however, for the two identical droplets undergoing thermocapillary motion normal to their line of centers, each droplet migrates slower than when isolated. Third, the migrating velocity of each bubble in a collection of identical, which can be arbitrarily oriented, is unaffected by the presence of the others. Fourth, the influence of interactions between droplets is in general far greater on the smaller one than on the larger one. The objective of this work is to obtain an analytical solution to the problem of the thermocapillary motion of a fluid sphere perpendicular to a nearby surface, which has been formulated by using a method of spherical bipolar coordinates.11,12 The infinite plane boundary can be a solid wall and/or a free surface. The quasi-steady energy and momentum equations applicable to the system are solved by using a method of reflections. The analytical results to correct eq 1 for various cases of flow are presented. Though the asymptotic solution obtained in the present article is not exact, it is more convenient in application than those obtained in the previous works.11,12 Our results for various values of k* and η* are in good agreement with those of the exact solution if the particle and planar surface are not too close. It is found that the effect of the presence of the plane surface on the droplet velocity can be significant when the distance between the droplet and plane become small. Moreover, the deposition times for the thermocapillary droplet normal to the plane wall are integrated by considering its thermocapillary mobility. Problem of a Droplet Moving toward a Plane. As shown in Figure 1, we consider the thermocapillary motion of a spherical droplet of radius a in the direction normal to an infinite surface of constant temperature located at a distance h from the center of the sphere. A linear temperature field T∞(x) with a uniform thermal gradient E∞ez (equal to ∇T∞) is imposed on the surrounding fluid medium far away from the droplet; ez is the axial unit vector in the circular cylindrical coordinate system (F,φ,z). For convenience, the center of the fluid sphere is chosen to be the origin of the coordinate frame. In addition to the circular cylindrical coordinates, the spherical coordinate
Langmuir, Vol. 15, No. 8, 1999 2675
system (r,θ,φ) is also employed. The fluids of the droplet and of the external fluid are assumed to be Newtonian and incompressible. Gravitational effects are ignored. Our purposed here is to determine the correction to eq 1 for the droplet velocity due to the presence of a solid plane and/or a free surface. The transport of momentum and energy is inherently unsteady in this problem of thermocapillary migration of a droplet normal to a plane wall. However, the problem can be considered quasisteady if the Peclet and Reynolds numbers are small (the effects of convection are neglected). The energy equation governing the temperature distributions for the external and internal fluids are
∇2T ) 0
(2a)
∇2Ti ) 0
(2b)
respectively. The boundary condition at the droplet interface requires that the temperature and normal component of heat flux be continuous. Also, the temperature of the surrounding fluid must approach the linear prescribed field far away from the droplet and the temperature inside the fluid sphere is finite everywhere. Thus, one has
r)a
T ) Ti k
r
