Young-Laplace Equation Solver | Interface Pressure

What is the Young-Laplace equation?

The Young-Laplace equation is a formula in fluid mechanics that gives the pressure difference across a curved interface between two fluids, such as the surface of a water droplet or a soap bubble. It shows that this pressure difference depends on the surface tension of the fluid and how sharply the interface curves. The equation explains why pressure builds up inside droplets and bubbles, and why smaller ones have higher internal pressure than larger ones.

It is named after Thomas Young and Pierre-Simon Laplace, who each described the relationship in 1805.

Young-Laplace equation formula

The general form of the equation is:

$\Delta P = \gamma \left( \frac{1}{R_1} + \frac{1}{R_2} \right)$

• $\Delta P$ is the pressure difference across the interface, in pascals (Pa)
• $\gamma$ (gamma) is the surface tension of the fluid, in newtons per meter (N/m)
• $R_1$ and $R_2$ are the two principal radii of curvature of the surface, in meters (m)

A surface curves differently in different directions. $R_1$ and $R_2$ describe the curvature along two directions at right angles to each other. For a sphere, such as a droplet or a bubble, $R_1$ equals $R_2$, so the formula simplifies to:

$\Delta P = \frac{2\gamma}{R}$

Sign convention

Pressure is always higher on the concave side of a curved interface, the side the surface bulges away from. For a droplet or bubble, that means the inside pressure is higher than the outside pressure. When a surface curves the opposite way along one direction, as on a saddle shape, that radius is treated as negative in the general formula. This calculator accepts only positive radii, so it covers surfaces that curve the same way in both directions.

Special cases

• Flat surface: as a radius grows toward infinity, its term in the formula shrinks toward zero. A perfectly flat interface has no pressure difference at all.
• Cylinder: inside a narrow tube, such as a capillary, one radius is the tube's radius and the other is effectively infinite along the tube's length. The formula reduces to $\Delta P = \gamma / R$.
• Very small scales: at the scale of a few nanometers, extra effects such as line tension become significant, and the plain equation loses accuracy.

How to calculate pressure difference with the Young-Laplace equation

1. Find the surface tension of the fluid, in N/m, at the temperature you care about. Surface tension falls as temperature rises.
2. Measure or estimate the first principal radius of curvature, $R_1$, in meters.
3. Measure or estimate the second principal radius of curvature, $R_2$, in meters. For a sphere, use the same value as $R_1$. For a cylinder, use a very large number to stand in for infinity (the calculator does not accept infinity itself).
4. Multiply the surface tension by the sum of $1/R_1$ and $1/R_2$. The result is the pressure difference in pascals.

Keep every value in SI units. Mixing millimeters with meters, for example, throws the answer off by a factor of a thousand.

Worked example

Take a spherical water droplet with a radius of 1 millimeter (0.001 m) and a surface tension of 0.072 N/m. This is the calculator's default and matches water at 25°C, 71.98 mN/m.

Because the droplet is a sphere, $R_1 = R_2 = R$, so:

$\Delta P = \frac{2\gamma}{R} = \frac{2 \times 0.072}{0.001} = 144 \text{ Pa}$

The pressure inside the droplet is 144 Pa above the surrounding air pressure. That is small next to standard atmospheric pressure, 101,325 Pa, but the effect grows fast as a droplet shrinks. A droplet one-tenth the size, 0.1 mm in radius, has ten times the pressure difference: 1,440 Pa. A droplet 0.01 mm in radius reaches 14,400 Pa, over 14% of atmospheric pressure.

Where the equation applies

• Droplets and bubbles: higher pressure inside small droplets drives Ostwald ripening, the process by which small droplets in an emulsion shrink while larger ones grow, because their contents diffuse from high pressure to low pressure.
• Capillary action: the curved surface inside a narrow tube pulls liquid upward. This is why water climbs paper towels and narrow glass tubes.
• Breathing: the tiny air sacs in the lungs, called alveoli, behave like small bubbles. A coating called pulmonary surfactant lowers surface tension so small alveoli do not collapse under the pressure the equation predicts. Premature babies who lack enough surfactant can develop respiratory distress syndrome, because without it the lungs collapse.
• Inkjet printing: printer nozzles form very small droplets, where the pressure from curvature is large enough to shape how ink leaves the nozzle.
• Materials science: instruments that measure how liquids wet a surface, and how thin coatings behave, use the equation to relate a droplet's shape to its surface energy.

History

People observed liquids climbing narrow tubes long before anyone explained why. Experiments by Francis Hauksbee and James Jurin at the Royal Society of London measured how liquid rises in narrow tubes and between plates. Jurin's law, which links the rise to the tube's width, is named after him.

In 1805, Thomas Young published an essay on the cohesion of fluids, and Laplace published a supplement to his treatise on celestial mechanics. Young's treatment was mostly descriptive, while Laplace worked out the mathematical form. Their combined work became the Young-Laplace equation. Carl Friedrich Gauss later refined the theory, and Joseph Plateau extended it through experiments on soap films.

Frequently asked questions

What does the Young-Laplace equation calculate? It calculates the pressure difference across a curved interface between two fluids, based on the fluid's surface tension and the interface's curvature.

Why is pressure higher inside a smaller droplet? Surface tension pulls the surface inward with the same intrinsic strength regardless of size, but that pull acts over a smaller area on a smaller droplet, concentrating its effect into a higher pressure. Pressure is inversely proportional to radius: halve the radius and the pressure difference doubles.

How does temperature affect the equation? Temperature changes the surface tension value, and surface tension falls as temperature rises. For water, surface tension drops from 0.0727 N/m at 20°C to 0.0627 N/m at 80°C, a decrease of about 0.17 mN/m for every degree Celsius. Using a room-temperature surface tension value for a hot system will understate the true pressure difference.

Does the equation work for shapes other than spheres? Yes. The general form, with two separate radii $R_1$ and $R_2$, covers any curved surface, including ellipsoids and saddle-shaped interfaces. Only the simplified $2\gamma/R$ version is limited to spheres. This calculator takes only positive radii, so it does not handle saddle shapes.

How is the Young-Laplace equation different from Young's equation? The Young-Laplace equation gives the pressure difference caused by a curved interface. Young's equation, a separate relationship also from Thomas Young, gives the contact angle a droplet makes where it meets a solid surface, based on the surface energies of the three materials involved. They describe different things and are sometimes used together to analyze a droplet resting on a surface.

How small can a droplet be before the equation stops working? The equation works well for everyday droplets and bubbles. At the scale of a few nanometers, additional forces that the equation ignores, including line tension and molecular-scale forces between surfaces, become large enough to matter, and more detailed models are needed.

References

1. Physics Today (American Institute of Physics). Two hundred years of capillarity research. Two hundred years of capillarity research — Young's essay and Laplace's supplement in 1805; Hauksbee and Jurin experiments; Young qualitative, Laplace mathematical.
2. International Association for the Properties of Water and Steam. Revised Release on Surface Tension of Ordinary Water Substance (2014). IAPWS surface tension release — water surface tension 72.74 mN/m at 20°C, 71.98 at 25°C, 62.68 at 80°C.
3. National Institute of Standards and Technology. NIST Guide to the SI, Appendix B.8. NIST SP 811 Appendix B.8 — standard atmosphere = 1.013 25 × 10⁵ Pa.
4. National Heart, Lung, and Blood Institute. Respiratory Distress Syndrome. NHLBI respiratory distress syndrome — RDS is caused by a lack of surfactant; without it the lungs collapse.