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Work of cohesion

Pronunciation: /wɜːrk əv koʊˈhiːʒən/

The work of cohesion Wc is the reversible energy per unit area needed to split a single phase into two, creating two new surfaces of the same material. For a liquid it equals twice its surface tension, Wc = 2γLV.

Also called

Cohesive work Reversible work of cohesion

Related terms

Work of adhesion Wettability

Topics

Surface tension Adhesion Wetting
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At a glance

What the work of cohesion measures and why it is the yardstick for wetting.

What it is

Picture pulling a column of liquid apart along a plane, slowly and reversibly. No interface is destroyed, and two new liquid surfaces of unit area appear. The energy needed is the work of cohesion.

  • For a liquid: Wc = 2γLV
  • Units: mJ/m², numerically equal to mN/m
  • It measures how strongly the liquid holds itself together

Worked example

At 20 °C:

  • Water, γLV = 72.8 mN/m: Wc = 2 × 72.8 = 145.6 mJ/m²
  • Diiodomethane, γLV = 50.8 mN/m: Wc = 101.6 mJ/m²
  • Hexadecane, γLV ≈ 27.5 mN/m: Wc ≈ 55 mJ/m²

Water holds itself together almost three times more strongly than hexadecane, which is why it beads on surfaces that oils wet. Compare values with our surface free energy of common polymers dataset.

Expert view

“The critical micelle concentration is not a number you look up; it is where the surface-tension curve flattens, and the only honest way to know it is to measure it.”

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

What the terms mean

How cohesion compares with adhesion, and what that comparison predicts.

Wa ≥ Wc

The liquid sticks to the solid at least as strongly as to itself. It spreads into a film and the contact angle goes to zero.

Wc/2 < Wa < Wc

The liquid prefers itself, but not by much. It forms a drop with a contact angle below 90°.

Wa < Wc/2

Cohesion clearly wins. The drop beads up with a contact angle above 90°.

Since Wa = γLV (1 + cos θ) and Wc = 2γLV, the ratio Wa/Wc = (1 + cos θ)/2. At 90° adhesion is exactly half of cohesion.

Wc Work of cohesion, mJ/m²
Wa Work of adhesion between liquid and solid, mJ/m²
γLV Surface tension of the liquid, mN/m
S Spreading coefficient, Wa − Wc, mN/m
θ Contact angle, degrees

Key equations

The definition and the two relations that make it useful.

Equation: Wc = 2γLV

Variable legend:

  • Wc: work of cohesion (mJ/m²)
  • γLV: surface tension of the liquid (mN/m)

Interpretation: Two new surfaces appear, each costing γLV per unit area, so the total is twice the surface tension.

When to use: Use it as the reference energy when comparing a liquid with its work of adhesion on a solid.

Assumptions:

  • Reversible separation at constant temperature
  • Pure liquid or a solution at equilibrium surface tension

Common mistakes:

  • Using a dynamic surface tension from a fresh surface as if it were the equilibrium value

Equation: Wa / Wc = (1 + cos θ) / 2

Variable legend:

  • Wa: work of adhesion, γLV (1 + cos θ) (mJ/m²)
  • θ: equilibrium contact angle (°)

Interpretation: A ratio of 1 means complete wetting, 0.5 means θ = 90°, and values near 0 mean the liquid barely touches the solid.

When to use: Use it to express a contact angle as a share of the liquid's own cohesion.

Assumptions:

  • Same as the Young equation

Common mistakes:

  • Reading the ratio from an advancing or receding angle without saying which

Equation: S = Wa − Wc = γLV (cos θ − 1)

Variable legend:

  • S: spreading coefficient (mN/m)

Interpretation: S is zero or positive only when adhesion matches or beats cohesion. For any finite contact angle, S is negative.

When to use: Use it to predict whether a liquid will spread into a film or stay as a drop.

Assumptions:

  • Same as the Young equation when written with θ

Common mistakes:

  • Expecting a positive S from a measured finite angle

How to read it

What a work of cohesion value tells you about how a liquid will behave.

Wc is a property of the liquid alone, so its meaning comes from comparison. Pair it with Wa on your surface to see which wins. Students can see the effect in experiment 4 on contact angle of treated and untreated surfaces, where treatment raises Wa while Wc of water stays fixed.

From Wc to meaning

If you see It usually means Check next
High Wc (water, about 146 mJ/m²) The liquid is hard to spread on low energy surfaces Surface treatment or surfactant to close the gap
Low Wc (oils, about 55 mJ/m²) The liquid spreads on most surfaces Whether spreading is wanted or a contamination risk
Wc falls after adding surfactant Surface tension has dropped, so the liquid wets more easily Whether the concentration is above the critical micelle concentration
Wa close to Wc The liquid is near the edge of complete wetting Small changes in cleanliness can tip it either way

Common misconceptions

Where the work of cohesion is most often misread.

Work of cohesion equals surface tension

It is twice the surface tension, because splitting a liquid creates two surfaces, not one. Mixing the two up halves every ratio you calculate with Wa.

Cohesive failure means the liquid had high cohesion

In adhesive testing, cohesive failure means the joint broke within the adhesive rather than at the interface. That is a mechanical result, not a measure of Wc. Standards such as DIN EN 828 on wettability of adhesives deal with the wetting side only.

A high dyne level beats any liquid cohesion

A surface wets well when its energy is high relative to the liquid's own pull. Water needs a far more energetic surface than an oil does. Our article Beyond 38 dynes covers why the target depends on the liquid.

Wc is the same for a solid

For a solid the same definition gives Wc = 2γSV, but you cannot measure γSV directly. It comes from contact angles and a surface energy model, which our surface free energy calculator applies.

Related terms

Where the work of cohesion sits among other wetting terms.

Broader term

  • Surface tension
  • Wettability

Related terms

  • Work of adhesion
  • Spreading coefficient
  • Young equation
  • Wettability

Often confused with

  • Cohesive failure (adhesive testing)
  • Surface tension

FAQ

Short answers to the questions people ask most about the work of cohesion.

Why is the work of cohesion twice the surface tension?

Splitting a liquid along a plane creates two new surfaces, each of unit area. Each surface costs γLV in energy, so the total is 2γLV. For water at 20 °C that is 2 × 72.8 = 145.6 mJ/m². No interface is destroyed, unlike the work of adhesion.

What is the difference between work of cohesion and work of adhesion?

Cohesion is the energy to split one material from itself. Adhesion is the energy to separate two different materials at their interface. When adhesion matches or exceeds cohesion, a liquid spreads on the solid. When cohesion wins, the liquid forms a drop with a finite contact angle.

How do surfactants change the work of cohesion?

Surfactants gather at the liquid surface and lower its surface tension, so Wc falls. The liquid then wets more easily. Once the surface is saturated, near the critical micelle concentration, adding more surfactant barely changes surface tension, so Wc stops falling as well.

References

Textbooks and further reading.

1. Adamson, A. W., and Gast, A. P. (1997). Physical Chemistry of Surfaces, 6th edition. Wiley, New York.
2. Israelachvili, J. N. (2011). Intermolecular and Surface Forces, 3rd edition. Academic Press.
3. 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
4. Surface tension measurement guide
5. Work of adhesion
6. Wettability