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 the early 1800s.

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, such as the inside surface of a bubble wall, the radius is treated as negative in the general formula.

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: below about 10 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.
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, a typical value for water near room temperature.

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 normal atmospheric pressure, about 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 breathing problems for this reason.
• Inkjet printing: printer nozzles form droplets tens of micrometers across, 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. In the early 1700s, Francis Hauksbee showed experimentally that narrower tubes cause greater capillary rise, and in 1718 James Jurin described the rise as inversely proportional to tube diameter.

Thomas Young explained the underlying physics in an 1805 paper, introducing surface tension as a property of a fluid and linking it to pressure across curved surfaces, though his treatment was mostly descriptive. In 1806, Pierre-Simon Laplace worked out the mathematical form using calculus, arriving independently at the same relationship. Their combined work became the Young-Laplace equation. Later scientists, including Carl Friedrich Gauss and Joseph Plateau, tested and extended the theory through the 1800s.

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 about 0.073 N/m at 20°C to roughly 0.063 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.

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 stays accurate down to roughly 1 micrometer. Below about 10 to 20 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. Young, T. (1805). "An Essay on the Cohesion of Fluids." Philosophical Transactions of the Royal Society of London, 95, 65–87.
2. Laplace, P.S. (1806). Traité de Mécanique Céleste, Supplement to Book 10.
3. de Gennes, P.G., Brochard-Wyart, F., & Quéré, D. (2004). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.
4. Adamson, A.W., & Gast, A.P. (1997). Physical Chemistry of Surfaces (6th ed.). Wiley-Interscience.