Motion of Drops on a Solid Surface due to a Wettability Gradient

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.  In this research program,  we studied the spontaneous motion of a drop on a solid surface.  Chaudhury and Whitesides ("How to make water run uphill" Science, 256, 1539-1541, 1992) showed that by treating a silicon surface with an alkylsilane, a gradient in "wettability" can be formed on that surface.  This means that the silicon surface, which is ordinarily quite wettable, can be made less wettable on one end, gradually changing to a more wettable surface at the other end.  Chaudhury and Whitesides placed drops of different liquids on the surface and showed that they moved from the less wetted end to the more wetted portion of the surface, even against the pull of gravity.  

To view a brief video of a drop of tetraethylene glycol moving on a treated surface, please click on the link below.

video of  drop motion

The drop is initially present at a location on the surface where the back portion is on a homogeneous surface and the front portion is on a surface with a gentle contact angle gradient.   As the drop begins to advance, it enters a region of steep contact angle variation, and therefore moves more rapidly, and near the latter part of its traverse, the gradient becomes gentle again, leading to slower motion of the drop.  The initial footprint of the drop in this video is approximately 0.9 mm in radius, and is practically a circle.  You are seeing a side view of the drop, and the shape of this profile is that of a spherical cap.

It is also possible the cause the motion of a drop on a horizontal solid surface by applying a temperature gradient on the surface.  Our research in this area is described in a companion write-up.

The write-up here describes our past research effort in this area. 

We have performed experiments on the motion of isolated drops on treated surfaces.  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 approximate theoretical descriptions of the motion of a drop driven by a gradient in contact angle.  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 drop footprint radius as the length scale and the speed of the drop as the velocity scale is found to be no larger than 0.01 in the experiments, so that the flow within the drop can be well-approximated as being inertia-free, or Stokes flow.  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 at a given instant, the driving force acting on the drop is equal in magnitude to the steady hydrodynamic resistance offered by the solid.  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 about 65 ms.  In addition, if a fixed driving force is applied to initiate motion of the drop from rest, and the hydrodynamic resistance scales linearly with the velocity of the drop with a constant resistance coefficient, the drop will approach a "steady" velocity exponentially.  The time scale associated with this exponential rise is the ratio of the mass of the drop to the resistance coefficient.  In the experiments, the maximum value of this time scale is found to be 0.2 ms.  So, it is evident that acceleration effects are entirely negligble, and the quasi-steady assumption would be good if the drop moves only a tiny distance in 65 ms so that the conditions on the gradient surface that determine the driving force and the hydrodynamic resistance are sensibly constant.  This is indeed true in the experiments, because the maximum velocity observed in the experiments is of the order of  2 mm/s, which means that a drop would move at most a distance of 130 micrometers in the time it takes to achieve viscous relaxation.   Over such a small distance, both the driving force and the hydrodynamic resistance can be approximated as being constant.

The driving force is evaluated from the variation of the contact angle around the periphery of the drop.   We estimate the quasi-steady hydrodynamic resistance to the motion of the drop using two approaches.  The first involves approximating the drop as a series of differential wedges.  Within each wedge, the Stokes flow can be described using a solution reported in  Cox, R.G., "The dynamics of spreading of liquids on a solid surface. Part 1. Viscous flow," J. Fluid Mech. 168, 169-194, 1986.  From this result, the shear stress exerted by the solid on the liquid can be evaluated and integrated in the area underneath the drop to obtain the drag.   The second approach involves using the lubrication approximation, but retaining the spherical cap shape of the drop.  Once again, 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 theoretical analysis as well as sample predictions are reported in the following article.

R.S. Subramanian, N. Moumen, and J.B. McLaughlin, Motion of a Drop on a Solid Surface Due to a Wettability Gradient, Langmuir,  21, 11844-11849 (2005).

The experimental data are compared with predictions from the quasi-steady theory in the following article. 

N. Moumen, R.S. Subramanian, and J.B. McLaughlin,  Experiments on the Motion of Drops on a Horizontal Solid Surface Due to a Wettability Gradient, Langmuir,  22, 2682-2690 (2006).

The principal findings from the experiments are summarized here.   The drops, of typical footprint radius 0.33 - 1.33 mm at the beginning of the traverse, move at a speed that varies over the gradient surface, because both the driving force and the hydrodynamic resistance change with position.  Typically, the speed increases rapidly to a maximum over the first few mm of traverse, and then decreases by as much as an order of magnitude over the next several mm of traverse.   The  prediction from the wedge model does better than that from  lubrication theory, but there is still a gap between the data and the prediction.  This gap is mostly closed over most of the traverse by accommodating contact angle hysteresis.   For more details, please consult the manuscript.

The PowerPoint slides used in a seminar presented to the Department of Chemical and Biomolecular Engineering at Clarkson by Dr. Nadjoua Moumen, who worked on this topic for her Ph.D. research,  can be viewed by clicking on the link below.

N. Moumen seminar

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

Revised, April 2013