Young's Equation Calculator: Contact Angle, Spreading Coefficient, Work of Adhesion and Rayleigh-Plateau Jet Breakup

How this calculator works

For a drop, enter the solid-vapour, solid-liquid and liquid-vapour tensions. The calculator gives cos theta = (gamma SV - gamma SL) / gamma LV, the contact angle, the spreading coefficient and the work of adhesion.

For a jet, enter the radius, the surface tension and the density. It gives the shortest unstable wavelength 2 pi R, the fastest growing wavelength of about 9 R, the droplet radius and the time to break.

Formulas used

• \gamma_{SV} = \gamma_{SL} + \gamma_{LV}\cos\theta
• S = \gamma_{SV} - \gamma_{SL} - \gamma_{LV}
• W_a = \gamma_{LV}(1 + \cos\theta)
• \lambda > 2\pi R
• \tau = \sqrt{\rho R^3/\gamma}

Worked example

With the default numbers in the calculator above:

1. Cosine of the contact angle (Young's equation): 0.56
2. Contact angle, measured through the liquid: 56.25 degrees (the liquid wets the surface)
3. Spreading coefficient: -32.00 mN/m (negative: the liquid stays as a drop)
4. Work of adhesion between the solid and the liquid (Young-Dupre equation): 112.00 mN/m (112.00 mJ per square metre)
5. Work of cohesion of the liquid, for comparison: 144.00 mN/m (a liquid wets the surface when adhesion is stronger than half the cohesion)

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: What do you want? Breakup of a liquid jet (Rayleigh-Plateau).

1. Shortest wavelength that can grow: a disturbance longer than the circumference makes the jet unstable: 6.28 mm
2. Wavelength that grows fastest, and so is the one that breaks the jet (inviscid liquid): 9.02 mm
3. Radius of the droplets (the volume of one wavelength of jet becomes one drop): 1.89 mm (1.89 times the jet radius)
4. Capillary time scale: 3.73 ms
5. Time for the disturbance to grow and break the jet (about 2.9 time scales): 10.84 ms
6. Number of drops formed each second by a jet at 1 m/s: 110.86 per second for each metre per second of jet speed

Common questions

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Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Young (1805), An Essay on the Cohesion of Fluids; Rayleigh (1878), On the instability of jets; de Gennes, Capillarity and Wetting Phenomena.

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