The spinning drop method or rotating drop method is one of the methods used to measure interfacial tension. Measurements are carried out in a rotating horizontal tube which contains a dense fluid. A drop of a less dense liquid or a gas bubble is placed inside the fluid. Since the rotation of the horizontal tube creates a centrifugal force towards the tube walls, the liquid drop will start to deform into an elongated shape; this elongation stops when the interfacial tension and centrifugal forces are balanced. The surface tension between the two liquids (for bubbles: between the fluid and the gas) can then be derived from the shape of the drop at this equilibrium point. A device used for such measurements is called a “spinning drop tensiometer”. The spinning drop method is usually preferred for the accurate measurements of surface tensions below 10−2 mN/m. It refers to either using the fluids with low interfacial tension or working at very high angular velocities. This method is widely used in many different applications such as measuring the interfacial tension of polymer blends and copolymers.
Theory An approximate theory was developed by Bernard Vonnegut 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:
E = E k + γ s {\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
E k = 1 2 I ω 2 = 1 4 m R 2 ω 2 {\displaystyle E_{k}={\frac {1}{2}}I\omega ^{2}={\frac {1}{4}}mR^{2}\omega ^{2}}
in which
I = 1 2 m R 2 {\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
γ s = 2 π L R σ = 2 V R σ {\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
E = E k + γ s = 1 4 Δ ρ V R 2 ω 2 + 2 V R σ {\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
d E d R = 0 = 1 2 Δ ρ V R ω 2 − 2 V R 2 σ {\displaystyle {\frac {dE}{dR}}=0={\frac {1}{2}}\Delta \rho VR\omega ^{2}-{\frac {2V}{R^{2}}}\sigma }
Substituting in
V = π L R 2 {\displaystyle V=\pi LR^{2}}
for a cylinder and then solving this relation for interfacial tension yields
σ = Δ ρ ω 2 4 R 3 {\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. Once this shape has developed, further increasing ω will decrease R while increasing L keeping LR2 fixed to meet conservation of volume.
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