Using drop shape analysis to obtain surface tension

Wait a minute - who said we had to use drop shape analysis to measure surface tension?

There are many different methods to measure surface tension. Why use something as complex as drop shape analysis?

It turns out that drop shape analysis (DSA) has the widest measurement range of all techniques. DSA allows surface or interfacial tension to be measured on all sorts of systems, such as molten metals and glasses, liquefied gases under pressure, or tiny drops of emulsions where surface tension is 0.0001 mN/m. Versatility of the technique is based on the ability to look at a drop and measure it, rather than having to come in direct contact with it.
When a drop of liquid forms at the end of a needle, it takes a shape which was first described by French mathematician Pierre-Simon Laplace and English scientist Thomas Young around 1802. This shape is the balance between two forces acting on the drop-gravity and surface tension.
Gravity elongates the drop (based on its density), while surface tension opposes deformation, keeping the drop closer to spherical on the needle. A form of the Young-Laplace equation can be used to describe the balance of forces at every point around the drop boundary.
Thus, the Young-Laplace equation can be used to plot the theoretical shape a drop should have if surface tension and density of the drop are known along with the acceleration due to gravity when measured. Gravity and density values are easily obtained experimentally and careful measurement from a drop photograph or other image will provide a good set of x,y coordinates for a drop boundary.
It may therefore be possible to use shape information on a drop with unknown surface tension along with the Young-Laplace equation to solve surface tension. Unfortunately, there is no analytical solution to the Young-Laplace equation. The equation can only be solved for surface tension by successive approximation.

Mechanics of the TRACKER are used to produce a liquid drop [C] at the end of a needle [A] or to deposit a drop on a substrate [B] for contact angle measurement. A drop can be pendant (hanging down) as shown in [A], or it can rise beneath the surface of a more dense liquid (rising drop). A bubble of gas takes the shape of a drop if it is in the rising configuration. Thus it is possible to measure "drops" on gas/liquid or liquid/liquid systems.
Optical system of the TRACKER is shown schematically on the left. A light source [D] illuminates the needle and drop [A and C]. A drop image is captured by a video camera [E] fitted with a telecentric lens [F]. The boundary of the drop edge is then digitized to produce a series of x,y values for analysis.

Before measurement, the needle holding the drop is adjusted to be as vertical as possible. This makes the drop image symmetrical, so only one side needs to be analyzed. A boundary [A] is set in the software so only useable data are acquired. This limit corresponds to the needle tip. During measurement, software generates an initial value for surface tension and drop radius (at top or bottom of the drop) to produce a set of theoretical coordinates for the drop based on the Young-Laplace equation [B]. Values for each experimental and theoretical coordinate are compared to produce an overall error margin. The initial surface tension and drop radius values are then varied slightly and the process is repeated. After a number of iterations, a minimum error between theoretical and experimental values is obtained, and the surface tension and drop radius values yielding this lowest error are reported.

Faster data analysis by TRACKER software

Development of software for the TRACKER has allowed it to measure faster and more precisely than had been thought possible just a few years ago. For example, depending on how many data points (coordinates) are chosen and how low the error limit is set, fitting calculations for the Young-Laplace equation can require millions of iterations. Many powerful mathematical tools have been developed and tested to accelerate these calculations without sacrificing accuracy.TRACKER software can now measure and report up to 25 measurements per second.
This speed allows real time analysis of the variation in interfacial properties and enables many other dynamic measurements:

• Kinetics of surface tension change
• Measurement of Gibbs elasticity
• Drop oscillation for measurement of interfacial rheology