F16LateralRetentionForce

The Effect of Lateral Retention Forces on Sessile and Pendant Drops of Water on a Plexiglass Surface

Jack Goodsell & Prashant Dhakal

University of Minnesota - Twin Cities

School of Physics and Astronomy

Minneapolis, MN 55455

Abstract

The lateral retention force on sessile (top) and pendant (bottom) drops of distilled water on a rotating Plexiglass surface was experimentally determined. The maximum retention force of the drops was determined from the measured rotational speed of the Plexiglass when a drop began to slide. It was found that the experimentally determined ratio of Fbottom/Ftop = 1.10 ± 0.12 was off by 0.38 standard deviations from the predicted value of our model: Fbottom/Ftop = 1.14.

Introduction & Theory

The adhesive force of a liquid drop in direct contact with a solid surface is known as lateral adhesion. This retention force is a result due to several effects such as the surface tension between the drop and the surface, pressure differences in the inside and outside of the drop, and the deformability of the droplets on the surface. The research was stimulated by the counterintuitive report that the pendant drop requires a larger force than the sessile drop to begin sliding.

When the fluid mechanics behind a water drop resting on a surface are studied, it is seen that at rest the sum of tension forces at each point of contact in the drop is zero. The equation of equilibrium is given by Young’s equation [1]

where γsv, γsl, and γlv represent the surface tensions between solid-vapor, solid-liquid and liquid- vapor phases, and θ is the angle of contact between the liquid-vapor and liquid-solid phases as shown in Figure 1 [1].

Although there has been no generally accepted model of the maximum retention force on the pendant and sessile drops, one model we have explored derives an expression for this maximum force. The model is derived from investigating a liquid drop resting on an inclined surface as seen in figure 2.

The drop resting on an inclined plane can be analyzed by applying equation 1 to both the advancing edge and receding edge in a force balance to obtain equation 2 [2].

By considering the drop at the moment just before sliding, equation 2 may be expanded into a final equation for the maximum retention force of the drop. In the case of the drop on the inclined plane, the maximum force before sliding (Fmax) is given by

where k is a dimensionless quantity that depends on the shape of the drop, w is the width of the drop, γlv is the surface tension between the drop and the atmospheric air, θr is the receding contact angle, and θa the advancing contact angle. See figure 2 for illustration of angles.

This inclined drop model seems drastically different than the experimental setup; however, the similarity can be explained by replacing the gravitational force deforming the drop in the model and replacing it with the centripetal force caused by the rotation seen in the experiment. The centripetal force required to maintain circular motion is given by

where m is the mass of the rotation object, r is the distance from the axis of rotation, and ω is the rotational velocity in radians/second of the plexiglass.

To compare the retention force of the pendant drop to the sessile drop, a force ratio can be used as seen in equation 5.

It is assumed that the width of the top and bottom drop will be equal in the case of equal volume drops in the microliter range. Due to this assumption, the only variables that should affect the force ratio of the bottom drop to the top drop are the contact angles of each drop. Experimental data was referenced to determine what the expected contact angles would be for the experiment. For a 3.3 μL drop about to slide the contact angles are ƟA,top=35.5°, ƟR,top=30.3°, ƟA,bottom=40.0, ƟR,bottom=34.7° [3]. An assumption was made that these contact angles would remain constant as the volume of the drops increased up to 25 μL in the experiment. When entering these contact angles in to equation 5, a theoretical force ratio is determined to be:

Experimental Technique

The experimental setup for observing lateral retention in between water and plexiglass consisted of a rotating plexiglass disk. A stepper motor was used for the rotational control. The plexiglass disk was secured to a block mounted on the stepper motor's axle. The stepper motor controller was used to control the angular velocity of the motor, which in turn was controlled with a function generator capable of inputting different frequency signals into the stepper motor. A camera was mounted above the rotating Plexiglass disk to have a top down view of the drop. The camera frame rate was 120 fps (frames per second) which was sufficient for data accumulation when the drops could not be observed by eye.

The placement of the drops onto the plexiglass disk was carried out with a P20 micropipette which has the ability to place drops as small as 2 μL. To obtain a pendant drop, the water was placed on the top surface of the disk and then the disk was quickly flipped by hand ensuring the drop did not slide since the drops would typically split into multiple drops when sliding. In initial trials it was determined that it was too difficult to observe the water droplets so blue dye was applied to the water used for the sessile drops and red dye was applied to the water used for the pendant drops. It was noted that the dyes could possibly affect the data so trials were run with the dye colors reversed to ensure similar results. It was also important that the pendant and sessile drops were placed equidistant from the axis of rotation to ensure they experienced identical centripetal forces.

Results & Data Analysis

As a result of the data being analyzed as a force ratio, only the velocities of the two drops are required to obtain the final ratio as seen in equation 7.

Despite the simplicity of the force ratio itself, understanding the errors present in the conducted experiment is the crucial step in determining if the model used accurately represents the experimental data. In order to determine this experimental error, the uncertainty in all the measured quantities has to be considered in the final ratio. The propagation of uncertainty for each retention force is shown in equation 8.

In equation 8, all the values of σ are the uncertainty values of the respective measurements. σfrequency = 100 Hz which is the resolution of the function generator used since it was increased in 100 Hz increments, σradius= 0.1 cm, and σvolume = 0.1 μL. From this data, it is clear that the largest source of uncertainty in our measurements is in the frequency input into the stepper motor.

The final uncertainty of the force ratio can be determined by propagating the uncertainty through the division of the two Forces as shown in equation 9.

Discussion and Conclusion

In conclusion, the experimentally determined ratio of Fbottom/Ftop = 1.10 ± 0.12 was off by 0.38 standard deviations from the predicted value of Fbottom/Ftop = 1.14. This allows us to conclude that the theoretical model describing the drop on an incline accurately predicts the behavior of the water droplets on the rotating plexiglass surface in drops ranging from 5 μL to 25 μL. It should be noted that the largest source of uncertainty was the frequency resolution and more precise results could be obtained if the frequency was to be increased by a value smaller than 100 Hz at a time.

Opportunities for Expansion on Current Experiment

There are several ways in which the accuracy and scope of the experiment at hand could be expanded. Counducting trials with an increase in signal frequency less than 100 Hz is highly encouraged to help minimize the error and lead to more precise measurements. Similarly, performing trials with different liquids such as vinegar, glycerin, or an alkaline solution would provide an opportunity to see whether the experimental result agrees with the same theoretical model or if it is dependent on the high adhesive qualities of water.

Acknowledgements

We would like to thank our advisor Dr. Dan Dahlberg for his guidance throughout the experimental process. Our immense gratitude is extended to Kevin Booth and Kurt Wick for their assistance and counsel throughout this research.

References

1. R. Madrid, T. Whitehead, G. Irwin, “Comparison of the lateral retention forces on sessile and pendant water drops on a solid surface”, A.J. Phys. 83, 531 (2015)

2. C. W. Extrand and A. N. Gent, “Retention of liquid drops by solid surfaces,” J. Colloid Interface Sci. 138, 431–442 (1990).

3. R. Tadmor, P. Bahadur, A. Leh, H. E. N’guessan, R. Jaini, and L. Dang, “Measurement of lateral adhesion forces at the interface between a liquid drop and a substrate,” Phys. Rev. Lett. 103, 266101 (2009).