Thermocapillary Motion of Drops on a Solid Surface

Normally, when a liquid drop is placed on a solid surface, it will stay where it is introduced and will assume a certain shape dictated by the contact angle and the relative importance of the gravitational force and the surface tension force on the fluid.  Numerous examples can be found in our daily lives where we see this phenomenon, as for example in the kitchen or bathroom sink, or when we notice raindrops on a window or the windshield of a car.  An example of a typical drop lying on a surface is shown in the figure.

An important material property of a given liquid, solid, and surrounding gas, is the equilibrium contact angle.  It is the angle between a tangent to the liquid surface at the contact line and the solid surface, measured in the liquid phase.   When the equilibrium contact angle is large, we say that the liquid does not wet the solid well.  So, for a given liquid, a "wettable" surface implies  zero or a relatively small contact angle.

In certain applications such as condensation heat transfer, a laboratory on a chip, and removing debris in ink jet printing, it is useful to be able to move a drop that is located on a solid surface.  We can imagine blowing a gas over the liquid to make it move, but this is a rather inefficient process, and cannot always be used.    One way to cause a drop to move spontaneously on a solid surface is to form a wettability gradient on the surface by suitable chemical treatment.  In a companion description, we describe our past research efforts in that area.   We are currently studying the motion of a drop on a solid surface that is caused by applying a temperature gradient on the surface, which leads in turn to a corresponding variation of temperature along the free surface of the drop.  Because the surface tension of a liquid typically decreases with increasing temperature, the result is a gradient of surface tension along the free surface.  The tangential stress associated with this gradient causes the liquid to move from the warm to the cool side along the free surface, leading to a return flow from the cool side to the warm side along the bottom of the drop.  This motion of the liquid leads to a shear stress exerted on the solid surface by the drop, which upon integration, leads to a hydrodynamic force exerted by the drop on the solid.  Because the solid is fixed in place, the resulting reaction on the drop propels the drop in the direction opposite to that of this force, namely from warm regions to cool regions on the solid surface.  We note that this direction of motion along a solid surface is the precise opposite of that for a drop or bubble that is freely suspended in a second fluid.  Such freely suspended objects move toward warm regions in the fluid. 

The motion of a drop that arises from the action of surface tension gradients is called "thermocapillary motion."  H. Bouasse (Capillarite et phenomenes superficiels, Delagrave: Paris, 1924)  appears to have been the first person to report that drops of liquid can be made to move on a solid by using a temperature difference.   Bouasse heated the lower end of a metal wire and caused drops attached to the wire to move upward against the pull of gravity and toward cooler regions on the wire.  The subject remained unstudied for many decades after the appearance of the report by Bouasse.  Results from two recent studies other than our own can be found in Brzoska, J.B., Brochard-Wyart, F., and Rondelez, F. Langmuir 9, 2220 (1993), and  Chen, J.Z., Troian, S.M., Darhuber, A.A. and Wagner, S., J. Appl. Phys. 97, 014906 (2005). 

To view a brief video of a drop of decane of approximate footprint radius 1.6 mm moving under the action of a temperature gradient of 1.05 K/mm on a Polydimethylsiloxane (PDMS)-coated glass slide, please click on the link below.

video of  drop motion

The actual speed of the drop is one-fifth that observed in the video, which has been speeded up for convenience.  In the video, you are seeing a top view of the drop; you can see that it is almost a circle, but not quite.  In fact, drops moving in relatively large temperature gradients deform visibly in this view.  Here is a top view of a drop of a drop of nominal footprint radius 1.45 mm that is moving  in a temperature gradient of 2.77 K/mm.  As you can see, the diameter in the direction of motion is significantly smaller than that in the direction normal to it, the ratio being 0.88.

We have performed experiments on the motion of isolated drops on  PDMS-coated glass slides.  We used video techniques to capture images of the moving drops, and subsequently analyzed the images frame-by-frame to determine drop shapes and velocities.   We  also have developed an approximate theoretical description of the motion of a drop driven by a temperature gradient.  A general theoretical formulation involves writing the applicable unsteady Navier-Stokes and continuity equations, subject to simplifying assumptions, along with the relevant boundary conditions and solving them.  This is a formidable task, but the following  experimental observations permit us to simplify the model.  A Reynolds number based on the height of the drop as the length scale and the speed of the drop as the velocity scale is found to be no larger than 0.1 in the experiments, so that the flow within the drop can be approximated as being inertia-free, or Stokes flow.  Likewise, a Peclet number defined using the drop height as the length scale is found to be no larger than 0.9, so that conduction is the dominant mechanism for heat transport within the drop.  The capillary number for moving drops is sufficiently small to treat the shape as being closely approximated by the static shape.  Furthermore, the Bond number is also small, so that gravitational distortion of the shape can be neglected as a first approximation.  This means that the moving drops should assume the shape of a spherical cap, which we find to be consistent with observation.

In calculating a theoretical prediction for the velocity, we employ the quasi-steady approximation.  This means that the hydrodynamic force acting on the drop is set equal to zero, because there is no other force acting on the drop in the direction of motion.  If you are wondering how a drop can move when the hydrodynamic force is zero, recall that initially the drop is stationary, and it experiences a reaction to the hydrodynamic force it exerts on the solid surface because of the thermocapillary flow within the drop.  It is that reaction force which initially accelerates the drop from rest; however, as the drop begins to move, the contribution to the hydrodynamic force exerted on the solid surface from that movement actually opposes the force exerted by the drop on the solid that arises from the thermocapilary flow contribution.  Thus, the net reaction force on the drop is reduced once the drop begins to move.  The faster the drop moves, the less is the reaction force exerted by the solid on the drop, until the drop moves at a steady speed when the reaction force exerted by the solid on the drop is zero.

The validity of the quasi-steady approximation depends on the magnitudes of certain important time scales.  If the drop is at a given location on the surface, and motion is suddenly initiated, it will take a certain amount of time for the flow within the drop to achieve steady state; this is the viscous relaxation time scale, and in the experiments, the maximum value of this time scale is 70 ms.   Likewise, the time scale for conduction over the height of the drop to lead to a steady temperature gradient field is less than one second.   So long as the physical properties of the drop do not change substantially in that one second period as the drop moves to a region of different temperature on the solid surface, the quasi-steady assumption should be good.   In the experiments, the distance moved by the fastest-moving drop in one second was  about 0.8 mm.  Over such a short distance, the viscosity and density of the drop do not change by much, even in the steepest temperature gradient that was used.  Therefore,  after a brief initial transient, acceleration effects can be neglected in developing a theoretical model.

The theoretical approach involves using the lubrication approximation, but retaining the spherical cap shape of the drop.  The shear stress is evaluated at the solid surface and integrated over the area beneath the drop. In performing this integration, we cannot go all the way to the contact line, because the shear stress scales as the inverse of the distance from the contact line, and becomes singular at the contact line.  The appearance of this singularity is well-known, and various ideas have been explored in the past for relieving it.  The most common idea is to assume that the no-slip boundary condition is not valid in the immediate vicinity of the contact line.  This hypothesis is supported by molecular dynamics simulations of model systems.   The resulting hydrodynamic force is set equal to zero to determine the quasi-steady velocity of the drop.  The experimental results along with the theoretical analysis  and a comparison are presented in our recent article:

V. Pratap, N. Moumen,  and R.S. Subramanian, Thermocapillary Motion of a Liquid Drop on a Horizontal Solid Surface, Langmuir,  24, 5185--5193 (2008).

The PowerPoint slides used in the Master's thesis defense presented to the Department of Chemical and Biomolecular Engineering at Clarkson by Mr. Vikram Pratap, who worked on this topic for his M.S. research,  can be viewed by clicking on the link below.

  V. Pratap defense

Currently,  there are no active ongoing experimental projects in this area.

Revised, April 2013