Young Laplace Equation Calculator

Surface tension gamma has units of N/m, which are identical to J/m^2, so it can be read either as a force per unit length along the interface or as an energy cost per unit area of interface created. That second reading explains the equation: a curved interface can lower its total energy by shrinking, and the only thing that stops it is a pressure difference across it. Minimising interfacial energy at fixed volume gives delta-P = gamma*(1/R1 + 1/R2), where R1 and R2 are the principal radii of curvature and the bracket is twice the mean curvature. A sphere has both radii equal, so delta-P = 2*gamma/R. A soap bubble carries two liquid-air interfaces, inner and outer, so its excess pressure doubles again to 4*gamma/R. Water has a surface tension of 72.8 mN/m at 20 degrees C, ethanol about 22.1 mN/m and mercury about 486 mN/m. A 1 mm water droplet therefore holds only about 146 Pa of excess pressure, but a 1 micrometre droplet holds about 146 kPa, since the effect scales as 1/R. The capillary rise form h = 2*gamma*cos(theta)/(rho*g*r) adds the contact angle, near 0 degrees for clean water on glass and about 140 degrees for mercury on glass, which is why mercury is depressed rather than raised. The main caveat is scale. Above the capillary length sqrt(gamma/(rho*g)), roughly 2.7 mm for water, gravity flattens the interface and the spherical assumption fails; below a few nanometres disjoining pressure and line tension take over. Surfactant contamination lowers gamma sharply and any gradient in it drives Marangoni flow.

Thomas Young set out the physical argument in an 1805 essay on the cohesion of fluids read to the Royal Society, relating the pressure jump to interfacial curvature and giving the contact-angle balance that still carries his name; he wrote it almost entirely in words rather than equations. Pierre-Simon Laplace supplied the mathematical treatment the following year in a supplement to the tenth book of his Mecanique Celeste, deriving the relation from a molecular theory of attraction. Carl Friedrich Gauss reworked the whole problem in 1830 from the single principle of energy minimisation, which is the derivation textbooks generally follow now. The underlying phenomenon had been studied earlier by Francis Hauksbee, whose capillary experiments date from around 1709, and by James Jurin, whose 1718 observation that rise height varies inversely with tube radius is known as Jurin's law. Practical measurement followed later. Francis Bashforth and John Adams published numerical tables of drop profiles in 1883, the ancestor of modern pendant-drop and sessile-drop tensiometry. The equation acquired a major biomedical role in the twentieth century when the mechanics of lung alveoli were analysed: John Clements identified pulmonary surfactant in 1957, and Mary Ellen Avery and Jere Mead reported in 1959 that its absence underlies hyaline membrane disease in premature infants, since without it small alveoli collapse into larger ones.