Introduction

The opposing forces of gravity and surface tension determine a droplet's shape. Thus, we can work backwards from a droplet's shape and the known force of gravity to its surface tension. This article investigates the shapes of drops with some solid support. Drops may hang from a support (pendant) or rest on top of it (sessile):

We will concentrate on pendant drops and ignore some of the issues peculiar to sessile drops, such as contact angle. (Lahooti et al., 1996, binder)

The Laplace Equation of Capillarity

The Laplace equation of capillarity is also known as the Laplace-Young equation. Its basic form is:

where is the difference in pressure between the droplet and the surrounding medium, is the surface tension, and R₁ and R₂ are radii of curvature. Note that the sum 1/R₁ + 1/R₂ will be the same no matter which pair of radii we choose; R₁ need not be the radius of maximum curvature. (Adamson, 1990, Measurement binder)

If gravity is the only outside force acting on a droplet, we can write the pressure difference as:

where is the pressure at a fixed reference point, is the difference in density between the droplet and the substance surrounding it, g is the local acceleration due to gravity, and z is the vertical distance from the reference point.

Let us consider an axisymmetric drop with its apex at the origin of our coordinate system:

Since the drop is axisymmetric, at the apex we have

1/R₁ = 1/R₂ = b

where b is also known as the curvature, so

= 2 b

Thus, the shape of the drop is determined by b, , g, and . ADSA, a popular program for drop shape analysis, takes g, , and the coordinates of several points on a droplet's edge as input, and returns b and as the output. The program also provides information about the drop's volume and surface area. (Lahooti et al., 1996, binder)

ADSA as a Film Balance

ADSA has been used to measure surface tension as a function of changing surface area. This application was conceived as an improvement to the Langmuir-Wilhelmy trough, and it was first used for non-biological surfactants. In the ADSA film balance, a droplet hangs from a capillary:

The drop volume is varied with time using a syringe or microinjector. (c.f. Wege et al., 1999, online and Prokop et al., 1998, binder) Surfactant is usually mixed with a solvent, then spread at the surface of the drop. There are two ways to spread the surfactant. First, surfactant may be spread directly at the surface of the drop:

With this method, some surfactant particles will enter the bulk, rather than remaining at the drop surface. This may be a problem if accurate measurement of the number of molecules per surface area is desired.

Surfactant may also be spread on top of the capillary and allowed to drip onto the bubble:

This ensures that no surfactant enters the bulk. However, some solution may remain on the capillary, and the method may increase the probability of surfactant leaking onto the capillary at low surface tensions. (Wege et al., 1999, online)

Dissolving surfactant in the bulk solution is not impossible. This method is not popular because it is more difficult to control the amount of surfactant at the interface, especially when the volume of the drop is changing. (c.f. Li et al., 2, 1996, binder, Cabrerizo-Vílchez et al., 1999, online)

Once surfactant has had time to spread, the drop's volume is varied with time and the results are filmed. A profile is extracted from each frame. The profiles are analyzed under the assumption that the drop is axisymmetric and Laplacian, and that its surface tension is the same as the surface tension of a static droplet with the same shape. (Wege et al., 1999, online)

Comparison with Other Measurement Devices

Orientation

Because the ADSA film balance uses a pendant drop, under the force of gravity, buoyant surfactant (such as some forms of pulmonary surfactant replacement) will float away from the interface, while dense surfactant will fall toward it. This reverses the pattern seen in bubble surfactometers and the Langmuir-Wilhelmy trough. (Park et al., 1999, online) Gravity also affects the choice of solvent. (Wüstneck, N., et al., 2000, online)

Limited Fluid Volume

Compared to the Langmuir-Wilhelmy trough or the bubble surfactometers, the ADSA film balance has relatively little bulk fluid available for surfactant to desorb into or adsorb from. Thus, equilibration time is slow compared to the captive bubble surfactometer. (Prokop et al., 1998, binder) This might make it an ideal tool for detecting differences in surfactant behavior at high concentrations.

Advantages of the ADSA Film Balance

• Small volume of surfactant required
• Easy to spread surfactant at interface
• High range of compression rates. For example, one study used rates of 1.6 - 371 A²/ molecule DPPC/ min, compared to 3.8 - 31.8 A²/ molecule DPPC/ min for the Langmuir-Wilhelmy trough. (Jyoti et al., 1996, binder)
• ADSA program is precise and provides convenient error estimates

Disadvantages of the ADSA Film Balance

• Drop may fall off
• Leakage
• Axisymmetry requirement
• Measurements with surfactant dissolved in the bulk may be more complicated
• Increase in error rates at very high or very low surface tension

(Prokop et al., 1998, binder; Wege et al., 1999, online)

Why Should We Use the ADSA Film Balance?

The captive bubble surfactometer (CBS) has very few difficulties with leakage, and the ADSA program may be combined with the CBS for extremely accurate analysis. (c.f. Prokop et al., 1998, binder) Why should we prefer the ADSA droplet-based film balance?

• Spreading surfactant at the surface is simpler, and evaporated solvent dissipates more easily (Wüstneck, N., et al., 2000, online)
• We may investigate high surfactant concentration without visibility issues (Putz et al., 1994, Measurement binder)
• We may investigate the effects of the limited availability of bulk solution