Spinning drop method

An approximate theory was developed by Bernard Vonnegut[3] in 1942 to measure the surface tension of the fluids, which is based on the principle that the interfacial tension and centrifugal forces are balanced at mechanical equilibrium. This theory assumes that the droplet's length L is much greater than its radius R, so that it may be approximated as a straight circular cylinder.

The relation between the surface tension and angular velocity of a droplet can be obtained in different ways. One of them involves considering the total mechanical energy of the droplet as the summation of its kinetic energy and its surface energy:

${\displaystyle E=E_{k}+\gamma _{s}}$

The kinetic energy of a cylinder of length L and radius R rotating about its central axis is given by

${\displaystyle E_{k}={\frac {1}{2}}I\omega ^{2}={\frac {1}{4}}mR^{2}\omega ^{2}}$

in which

${\displaystyle I={\frac {1}{2}}mR^{2}}$

is the moment of inertia of a cylinder rotating about its central axis and ω is its angular velocity. The surface energy of the droplet is given by

${\displaystyle \gamma _{s}=2\pi LR\sigma ={\frac {2V}{R}}\sigma }$

in which V is the constant volume of the droplet and σ is the interfacial tension. Then the total mechanical energy of the droplet is

${\displaystyle E=E_{k}+\gamma _{s}={\frac {1}{4}}\Delta \rho VR^{2}\omega ^{2}+{\frac {2V}{R}}\sigma }$

in which Δρ is the difference between the densities of the droplet and of the surrounding fluid. At mechanical equilibrium, the mechanical energy is minimized, and thus

${\displaystyle {\frac {dE}{dR}}=0={\frac {1}{2}}\Delta \rho VR\omega ^{2}-{\frac {2V}{R^{2}}}\sigma }$

Substituting in

${\displaystyle V=\pi LR^{2}}$

for a cylinder and then solving this relation for interfacial tension yields

${\displaystyle \sigma ={\frac {\Delta \rho \omega ^{2}}{4}}R^{3}}$

This equation is known as Vonnegut’s expression. Interfacial tension of any liquid that gives a shape very close to a cylinder at steady state, can be estimated using this equation. The straight cylindrical shape will always develop for sufficiently high ω; this typically happens for L/R > 4.[1] Once this shape has developed, further increasing ω will decrease R while increasing L keeping LR² fixed to meet conservation of volume.

The full mathematical analysis on the shape of spinning drops was done by Princen and others.[4] Progress in numerical algorithms and available computing resources turned solving the non linear implicit parameter equations to a pretty much 'common' task, which has been tackled by various authors and companies. The results are proving the Vonnegut restriction is no longer valid for the spinning drop method.

The spinning drop method is convenient compared to other widely used methods for obtaining interfacial tension, because contact angle measurement is not required. Another advantage of the spinning drop method is that it is not necessary to estimate the curvature at the interface, which entails complexities associated with shape of the fluid drop.

On the other hand, this theory suggested by Vonnegut, is restricted with the rotational velocity. The spinning drop method is not expected to give accurate results for high surface tension measurements, since the centrifugal force that is required to maintain the drop in a cylindrical shape is much higher in the case of liquids that have high interfacial tensions.