Contents

Young equation

Pronunciation: /jʌŋ ɪˈkweɪʒən/

The Young equation describes the force balance at the edge of a drop resting on an ideal solid: γSV = γSL + γLV cos θY. It links the equilibrium contact angle θY to the three interfacial tensions that meet at the contact line.

Also called

Young's equation Young's law Young relation

Related terms

OWRK method Fowkes method van Oss-Chaudhury-Good method

Topics

Wetting Contact angle Surface energy
Book a 15 minute demo Read the contact angle measurement guide

At a glance

What the Young equation says, and why every surface energy method starts from it.

What it is

Three tensions pull on the contact line where solid, liquid and vapor meet. At equilibrium their horizontal components balance, and the angle that results is the Young contact angle θY.

  • Proposed by Thomas Young in 1805
  • Holds for a smooth, chemically uniform, rigid solid
  • Contact angle is read through the liquid

Worked example

Water on PTFE at 20 °C: γLV = 72.8 mN/m and θ ≈ 110°.

  • cos 110° = −0.342
  • γSV − γSL = γLV cos θ = 72.8 × (−0.342) ≈ −24.9 mN/m
  • The negative value means the solid liquid interface costs more energy than the bare solid, so the water beads up.

Reference liquid values are in our surface tension of common liquids dataset.

How it is measured

You never measure the equation itself. You measure the contact angle of a sessile drop and the surface tension of the liquid, then apply the equation. The contact angle measurement guide covers the method step by step.

Expert view

“What a liquid responds to is only the outermost few ångström of a surface: the skin, not the bulk. Change that skin and you change the wetting, even if the material underneath is identical.”

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

What the terms mean

The three tensions at the contact line and what their balance tells you.

γSV − γSL > 0 (θY < 90°)

Wetting the solid lowers the energy of the system, so the drop spreads and the angle stays below 90°. Typical of clean glass and metals with water.

γSV − γSL < 0 (θY > 90°)

The solid liquid interface costs more than the bare solid, so the drop beads up. Typical of PTFE, waxes and silicone coatings with water.

γSV − γSL ≥ γLV (no finite angle)

The balance cannot be satisfied with any angle. The liquid spreads into a film and the Young equation no longer gives a contact angle.

The contact angle is drawn through the liquid, tangent to the drop profile at the point where it meets the solid.

θY Young (equilibrium) contact angle, degrees
γSV Solid vapor interfacial tension, mN/m (equal to mJ/m²)
γSL Solid liquid interfacial tension, mN/m
γLV Liquid vapor (surface) tension, mN/m

Key equations

The Young equation and the two forms you will use most often.

Equation: γSV = γSL + γLV cos θY

Variable legend:

  • θY: equilibrium contact angle on an ideal surface (°)
  • γSV, γSL, γLV: solid vapor, solid liquid and liquid vapor tensions (mN/m)

Interpretation: The horizontal pull of the liquid surface, γLV cos θY, makes up the difference between the two solid interface tensions.

When to use: Use it to relate a measured equilibrium contact angle to the energetics of the solid, liquid and vapor.

Assumptions:

  • Smooth, rigid, chemically uniform solid
  • Thermodynamic equilibrium, no evaporation or reaction
  • No gravity or line tension effects at the contact line

Common mistakes:

  • Using an apparent angle on a rough or patterned surface as θY
  • Mixing mN/m and dyn/cm without noting they are numerically equal
  • Forgetting that cos θ turns negative above 90°

Equation: Wa = γLV (1 + cos θY)

Variable legend:

  • Wa: work of adhesion between liquid and solid (mJ/m²)
  • γLV: surface tension of the liquid (mN/m)
  • θY: Young contact angle (°)

Interpretation: The energy needed to pull a unit area of liquid off the solid. Lower angles mean stronger adhesion.

When to use: Use it when the question is how strongly a liquid sticks to a surface rather than how it spreads.

Assumptions:

  • Same as the Young equation
  • The solid does not swell or dissolve in the liquid

Common mistakes:

  • Reporting Wa without the liquid and temperature used
  • Applying it to dynamic angles measured while the line moves

Equation: γSV − γSL = γLV cos θY

Variable legend:

  • γSV − γSL: wetting tension, sometimes called adhesion tension (mN/m)

Interpretation: The one solid side quantity you can get directly from a single liquid. Splitting it into γSV and γSL needs a surface energy model.

When to use: Use it to compare surfaces with one test liquid, or as the input to the OWRK, Fowkes or van Oss-Chaudhury-Good models.

Assumptions:

  • Same as the Young equation

Common mistakes:

  • Treating the wetting tension as the surface energy of the solid

How to read it

What a measured angle tells you once you put it into the Young equation.

The Young equation is exact only on an ideal surface, so real measurements need a sanity check before the numbers go into a surface energy model. Our note on reproducible surface energy calculation from contact angles explains which angle to report and why.

From measured angle to meaning

If you see It usually means Check next
θ below 90° with water Wetting tension is positive; the surface is hydrophilic Whether the angle drifts as the drop sits
θ above 90° with water Wetting tension is negative; the surface is hydrophobic Roughness, which can push the angle higher than θY
Different angles across the same sample The surface is not uniform, so one θY does not describe it Contamination, coating coverage, handling marks
Advancing and receding angles far apart The measured angle is not θY but somewhere between the two Contact angle hysteresis on the same surface

Common misconceptions

Where the Young equation is most often misapplied.

Any measured contact angle is the Young angle

Only an equilibrium angle on a smooth, uniform solid is θY. Real surfaces show a range between the advancing and receding angle, and roughness shifts the apparent value. For coatings that are rough by design, such as those in thin film and nanocoating surface characterization, report which angle you measured.

The equation explains why oil spreads and water beads

It describes the balance, but the cause is the difference in surface tension between the liquids. Oil has a lower γLV than water, so cos θ rises and the angle falls on the same solid. Our article on why oil spreads and water beads covers the everyday version.

Small drops are always better

Very small drops evaporate quickly and are harder to image, while very large drops flatten under gravity. The Young equation assumes neither effect, so drop size is a method choice. Teaching labs weighing instruments can compare options in our guide to the best contact angle goniometer for teaching and academic labs.

Related terms

Where the Young equation sits among other wetting concepts.

Broader term

  • Wettability
  • Contact angle

Builds on the Young equation

  • Work of adhesion
  • OWRK method
  • Fowkes method
  • van Oss-Chaudhury-Good method
  • Equation of State (SFE model)

Corrections for real surfaces

  • Wenzel model
  • Cassie-Baxter model
  • Contact angle hysteresis

FAQ

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

What is the Young equation used for?

It connects a measurable contact angle to the interfacial tensions of a solid, liquid and vapor. In practice it is the first step in every contact angle based surface energy method. Combined with a model such as OWRK or Fowkes, angles from two or three liquids give the surface free energy of the solid.

Does the Young equation work on rough surfaces?

Not directly. On a rough surface the measured angle is an apparent angle. The Wenzel model corrects for liquid filling the roughness, and the Cassie-Baxter model for air trapped beneath the drop. Both use θY from the Young equation as their starting point, so it still matters on rough samples.

What is the difference between the Young and Young-Laplace equations?

The Young equation balances tensions at the contact line and gives the contact angle. The Young-Laplace equation relates the pressure across a curved liquid surface to its curvature and gives the drop shape. Drop shape analysis uses Young-Laplace to fit the profile, then reads the contact angle that the Young equation interprets.

References

Primary sources and further reading.

1. Young, T. (1805). An essay on the cohesion of fluids. Philosophical Transactions of the Royal Society of London, 95, 65 to 87. https://doi.org/10.1098/rstl.1805.0005
2. 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
3. Makkonen, L. (2016). Young's equation revisited. Journal of Physics: Condensed Matter, 28(13), 135001. https://doi.org/10.1088/0953-8984/28/13/135001
4. Contact angle measurement guide
5. OWRK method
6. Fowkes method
7. van Oss-Chaudhury-Good method