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Young-Laplace equation

Pronunciation: /jʌŋ ləˈplɑːs ɪˈkweɪʒən/

The Young-Laplace equation states that the pressure jump across a curved liquid surface equals the surface tension times its curvature: Δp = γ (1/R1 + 1/R2). It sets the shape of every drop and bubble, and it is the equation that drop shape analysis fits to measure surface tension.

Also called

Laplace equation (capillarity) Laplace pressure equation Young-Laplace law

Related terms

Drop shape analysis (DSA) Pendant drop method Young equation

Topics

Surface tension Drop shape Capillarity
Book a 15 minute demo Read the surface tension measurement guide

At a glance

What the Young-Laplace equation says, and why drop shape analysis depends on it.

What it is

A curved liquid surface pushes on whatever is inside it. The more tightly it curves, and the higher its surface tension, the larger the pressure inside. The Young-Laplace equation puts a number on that pressure jump.

  • Derived by Thomas Young (1805) and Pierre Simon Laplace (1806)
  • Holds at every point on a fluid interface at rest
  • Works for liquid air and liquid liquid interfaces alike

Why it matters

Gravity makes the pressure in a hanging or resting drop change with height, so the curvature changes too. That gives each drop a shape set only by γ, the density difference and its size. Fit the shape and you get γ, which is how a camera can measure surface tension, as our article on how surface tension can be measured with a smartphone camera explains.

Worked example

A water drop at 20 °C hangs with apex radius R0 = 1.5 mm, and the fit returns a Bond number Bo = 0.302. With Δρ = 997 kg/m³ and g = 9.81 m/s²:

  • Δρ g R0² = 997 × 9.81 × (1.5 × 10−3)² ≈ 0.0220 N/m
  • γ = Δρ g R0² / Bo ≈ 0.0220 / 0.302 ≈ 0.0729 N/m
  • So γ ≈ 72.9 mN/m, close to the 72.8 mN/m expected for water

Expert view

“Surface tension versus concentration is one of the few measurements that lets you watch a surfactant reorganise an interface in real time: read the curve, not just the endpoint.”

Dr. Alidad Amirfazli, Co-founder and Scientific Advisor, Droplet Lab

What the terms mean

How the Bond number tells you whether a drop shape carries enough information to fit.

Bo close to zero (nearly spherical)

Surface tension dominates and gravity barely bends the drop. Many γ values give almost the same outline, so the fit is poorly defined.

Bo of a few tenths (well deformed)

Gravity visibly stretches the drop into a pear shape. The outline now depends strongly on γ, and the fit is well conditioned.

Drop close to detachment

The drop holds nearly the largest volume the needle can carry. This is usually where pendant drop precision is best, as long as the drop stays stable.

R1 and R2 are measured in two perpendicular planes. For an axisymmetric drop, one lies in the profile plane and the other comes from rotating the profile about the axis.

Δp Pressure inside minus outside the interface, Pa
γ Surface or interfacial tension, N/m (1 N/m = 1000 mN/m)
R1, R2 Principal radii of curvature at a point on the surface, m
R0 Radius of curvature at the drop apex, m
Δρ Density difference between drop and surrounding phase, kg/m³
Bo Bond number, Δρ g R0² / γ, dimensionless
S Classical shape factor ds / de, dimensionless

Key equations

The general equation, the form used for drops under gravity, and the classical shape factor.

Equation: Δp = γ (1/R1 + 1/R2)

Variable legend:

  • Δp: pressure jump across the interface (Pa)
  • γ: surface or interfacial tension (N/m)
  • R1, R2: principal radii of curvature (m)

For a sphere R1 = R2 = R, so Δp = 2γ/R. A 1 mm radius water drop at 20 °C: Δp = 2 × 0.0728 / 0.001 ≈ 146 Pa.

Interpretation: Small drops and high surface tension mean high internal pressure. A flat surface has infinite radii and no pressure jump.

When to use: Use it for bubbles, drops, menisci and any curved fluid surface at rest.

Assumptions:

  • Interface at mechanical equilibrium
  • Uniform γ along the surface, no Marangoni flow
  • Interface thin compared with the radii

Common mistakes:

  • Using mN/m with metres and getting pressures 1000 times too large
  • Forgetting the factor of 2 for a sphere, or using it for a cylinder

Equation: γ (1/R1 + 1/R2) = 2γ/R0 + Δρ g z

Variable legend:

  • R0: radius of curvature at the apex (m)
  • Δρ: density difference between drop and surrounding fluid (kg/m³)
  • g: 9.81 m/s²
  • z: vertical distance from the apex into the drop (m); for a pendant drop with the apex at the bottom, the sign of the gravity term flips

Dividing by γ/R0 leaves one number, the Bond number Bo = Δρ g R0² / γ, which fixes the whole shape.

Interpretation: The pressure jump changes linearly with height, so curvature changes along the profile. Every drop shape belongs to one Bond number.

When to use: Use it as the model that drop shape analysis fits to a pendant or sessile drop image.

Assumptions:

  • Axisymmetric drop, not touching anything but the needle or the solid
  • Known density difference and gravity
  • Drop at rest, no vibration or evaporation during capture

Common mistakes:

  • Entering the density of the drop instead of the density difference
  • Fitting a drop so small that Bo is close to zero

Equation: S = ds / de ; γ = Δρ g de² / H

Variable legend:

  • de: largest (equatorial) diameter of a pendant drop (m)
  • ds: diameter measured at a height de above the apex (m)
  • H: correction factor read from published tables as a function of S

Interpretation: The ratio S captures how strongly gravity has stretched the drop. Before computers, two diameters and a table were enough to estimate γ.

When to use: Use it to understand older literature or as a quick check. Modern software fits the full profile instead.

Assumptions:

  • Same as the gravity form of the Young-Laplace equation

Common mistakes:

  • Measuring ds at the wrong height
  • Treating a two point estimate as precise as a full profile fit

How to read it

What a Young-Laplace fit tells you, and when to trust it.

A good fit traces the whole edge with small, random residuals. Systematic misfit means one of the assumptions is broken. Reference data such as our Droplet Lab reference measurement dataset help you check that a setup returns known values before you test unknowns.

From fit result to meaning

If you see It usually means Check next
Fitted γ changes a lot with drop volume The drop is too small and nearly spherical Use a larger drop so the Bond number rises
Residuals larger on one side of the profile The drop is not axisymmetric or the camera is tilted Needle alignment, level and focus
Water reads well above 73 mN/m at room temperature Wrong scale, density difference or needle diameter The calibration entries in the software
γ drifts down while the drop hangs Surface active material is adsorbing at the interface Whether a time resolved reading is what you need

Common misconceptions

Where the Young-Laplace equation is most often misunderstood.

Smaller drops give more precise surface tension

The opposite is true for pendant drops. A small drop is nearly spherical, and many γ values fit its outline almost equally well. Larger, well deformed drops close to detachment give the most reliable fit.

The Bond number alone tells you if a measurement is good

Bo is a useful guide, but published analysis shows that drop volume relative to the largest drop the needle can hold is a better predictor of precision. Use both, and keep the drop as large as stability allows.

The Young and Young-Laplace equations are the same thing

The Young equation balances tensions at a contact line and gives the contact angle. The Young-Laplace equation links pressure to curvature and gives the drop shape. Sessile drop software often uses both: Young-Laplace to fit the profile and the tangent at the baseline for the angle, as our free contact angle tool shows.

Any camera and ruler will do

The fit is only as good as the image scale and edge quality. You need a sharp, evenly lit profile and an accurate length reference, usually the needle diameter. Teaching labs choosing equipment can compare options in our guide to the best tensiometer for teaching labs.

Related terms

Where the Young-Laplace equation sits among other surface tension concepts.

Broader term

  • Surface tension
  • Capillarity

Built on the Young-Laplace equation

  • Drop shape analysis (DSA)
  • Pendant drop method
  • Sessile drop method

Often confused with

  • Young equation

FAQ

Short answers to the questions people ask most about the Young-Laplace equation.

What does the Young-Laplace equation tell you?

It tells you how much higher the pressure is on the inside of a curved liquid surface than on the outside. The pressure jump equals the surface tension times the curvature. That is why small bubbles have higher internal pressure than large ones, and why a drop under gravity takes a shape set by its surface tension.

How is the Young-Laplace equation used to measure surface tension?

Software traces the edge of a pendant or sessile drop in an image, then solves the Young-Laplace equation for many trial values of surface tension. The value whose computed profile best matches the traced edge is reported. You need the density difference between the two phases and an accurate image scale.

What is the Bond number in drop shape analysis?

The Bond number, Bo = Δρ g R0² / γ, compares gravity with surface tension for a drop of apex radius R0. A value near zero means the drop is almost spherical and carries little information about γ. Larger values mean gravity has deformed the drop enough for a reliable fit.

References

Primary sources and further reading.

1. Rotenberg, Y., Boruvka, L., and Neumann, A. W. (1983). Determination of surface tension and contact angle from the shapes of axisymmetric fluid interfaces. Journal of Colloid and Interface Science, 93(1), 169 to 183. https://doi.org/10.1016/0021-9797(83)90396-X
2. Berry, J. D., Neeson, M. J., Dagastine, R. R., Chan, D. Y. C., and Tabor, R. F. (2015). Measurement of surface and interfacial tension using pendant drop tensiometry. Journal of Colloid and Interface Science, 454, 226 to 237. https://doi.org/10.1016/j.jcis.2015.05.012
3. Andreas, J. M., Hauser, E. A., and Tucker, W. B. (1938). Boundary tension by pendant drops. Journal of Physical Chemistry, 42, 1001 to 1019.
4. de Gennes, P. G., Brochard-Wyart, F., and Quéré, D. (2004). Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, New York. https://doi.org/10.1007/978-0-387-21656-0
5. Surface tension measurement guide
6. Drop shape analysis (DSA)
7. Pendant drop method