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Spreading coefficient

Pronunciation: /ˈsprɛdɪŋ ˌkoʊɪˈfɪʃənt/

The spreading coefficient S is the energy per unit area released when a liquid film covers a substrate: S = γSV − (γSL + γLV). If S is zero or positive the liquid spreads into a film; if S is negative it stays as a drop or lens with a finite contact angle.

Also called

Harkins spreading coefficient Initial spreading coefficient

Related terms

Capillary number Interfacial tension (IFT)

Topics

Wetting Interfacial tension Emulsions
Book a 15 minute demo Read the surface tension measurement guide

At a glance

What the sign of S tells you, and where the numbers come from.

What it is

S compares the energy of a bare substrate with the energy of the same substrate covered by a liquid film. It is also the difference between adhesion and cohesion: S = Wa − Wc.

  • S ≥ 0: complete wetting, the liquid spreads
  • S < 0: partial wetting, a drop or lens forms
  • Units: mN/m, equal to mJ/m²

Worked example

Hexadecane on water at about 20 °C:

  • γ water = 72.8 mN/m, γ hexadecane ≈ 27.5 mN/m
  • Hexadecane and water interfacial tension ≈ 53 mN/m
  • S = 72.8 − (27.5 + 53) ≈ −7.7 mN/m

S is negative, so hexadecane sits on water as a lens rather than a film. Liquid values are in our surface tension of common liquids dataset.

How it is measured

On a liquid substrate, all three tensions can be measured: two surface tensions and one interfacial tension, by pendant drop or a tensiometer. The surface tension measurement guide covers both. On a solid, γSV and γSL are not measurable, so S comes from the contact angle instead.

Expert view

“Plot surface tension against surfactant concentration and the interface tells you when it is saturated: the curve bends at the point where the surface can hold no more.”

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

What the terms mean

The three outcomes the sign of S predicts.

S > 0

Covering the substrate lowers the energy, so the liquid spreads spontaneously into a film. Many oils on clean metal or glass behave this way.

S = 0

The boundary case. The contact angle is zero and the liquid just wets completely.

S < 0

The liquid stays as a drop on a solid, or a floating lens on another liquid. The more negative S, the larger the contact angle.

S is the energy gained or lost per unit area when a film replaces the bare substrate surface. Its sign decides spreading.

S Spreading coefficient, mN/m (equal to mJ/m²)
γSV Surface tension of the substrate (solid or lower liquid), mN/m
γSL Interfacial tension between substrate and spreading liquid, mN/m
γLV Surface tension of the spreading liquid, mN/m
θ Equilibrium contact angle, degrees

Key equations

The definition and the two forms you will use in practice.

Equation: S = γSV − (γSL + γLV)

Variable legend:

  • S: spreading coefficient (mN/m)
  • γSV: substrate surface tension (mN/m)
  • γSL: substrate and liquid interfacial tension (mN/m)
  • γLV: liquid surface tension (mN/m)

Interpretation: The energy of the bare substrate minus the energy of the substrate covered by a film of liquid.

When to use: Use it directly for one liquid on another, where all three tensions can be measured.

Assumptions:

  • Pure liquids, not yet mutually saturated (initial S)
  • Clean interfaces

Common mistakes:

  • Using pure liquid values after the liquids have dissolved into each other, which changes S

Equation: S = γLV (cos θ − 1)

Variable legend:

  • θ: equilibrium contact angle (°)
  • γLV: liquid surface tension (mN/m)

Worked example: water at 20 °C on PTFE, θ ≈ 110°. cos 110° = −0.342, so S = 72.8 × (−1.342) ≈ −97.7 mN/m.

Interpretation: Combining with the Young equation gives S from a measured angle. Any finite angle gives a negative S.

When to use: Use it on solids, where γSV and γSL cannot be measured directly.

Assumptions:

  • Same as the Young equation: smooth, uniform, rigid solid at equilibrium

Common mistakes:

  • Expecting a positive value from this form; it can only reach zero

Equation: S = Wa − Wc

Variable legend:

  • Wa: work of adhesion, γSV + γLV − γSL (mJ/m²)
  • Wc: work of cohesion of the liquid, 2γLV (mJ/m²)

Interpretation: A liquid spreads when it sticks to the substrate at least as strongly as it sticks to itself.

When to use: Use it to explain spreading in terms of adhesion and cohesion rather than tensions.

Assumptions:

  • Same as the definition

Common mistakes:

  • Taking Wc as γLV instead of 2γLV

How to read it

What a spreading coefficient value usually means in practice.

The sign matters more than the size. Near zero, small changes in cleanliness or surfactant can flip the outcome. The everyday version of this is in our article why water forms droplets in oil but oil spreads in water, and students can measure the inputs in experiment 10 on oil and water interfacial tension.

From S to meaning

If you see It usually means Check next
S clearly positive The liquid will spread into a film Whether S stays positive after the liquids saturate each other
S slightly negative Lens or drop with a small contact angle Surfactant or cleaning that could tip it to spreading
S strongly negative Drops bead up with a high contact angle Surface treatment if wetting is wanted
Spreading starts then retracts Initial S was positive but final S is negative Mutual solubility and surfactant transfer between phases

Common misconceptions

Where the spreading coefficient is most often misapplied.

A measured contact angle can give a positive S

The contact angle form only reaches zero at θ = 0°. A positive S means no finite angle exists, so you cannot measure it from an angle. It needs the three tensions directly, or a spreading test.

The initial S decides the final state

Harkins separated the initial coefficient, using pure liquids, from the final one after the liquids saturate each other. A liquid can spread at first and then pull back into a lens as S turns negative.

Any oil spreads on water

Many alkanes, hexadecane among them, have a negative S on water and form lenses. Whether an oil spreads depends on its own surface tension and its interfacial tension with water. Emulsion work that depends on this, such as studies of glass and jamming transitions in emulsions, measures both.

S needs specialist equipment

For liquid pairs, S needs only two surface tensions and one interfacial tension. Teaching labs can compare instruments in the best tensiometer for teaching labs. On solids, a contact angle read with our free contact angle tool gives S through the angle form.

Related terms

Where the spreading coefficient sits among other wetting and interfacial terms.

Broader term

  • Spreading
  • Wettability

Related terms

  • Interfacial tension (IFT)
  • Capillary number
  • Work of adhesion
  • Work of cohesion
  • Young equation

Often confused with

  • Wetting tension
  • Dyne level

FAQ

Short answers to the questions people ask most about the spreading coefficient.

What does a negative spreading coefficient mean?

It means covering the substrate with a liquid film would raise the total surface energy. The liquid therefore stays as a drop on a solid, or as a floating lens on another liquid, with a finite contact angle. The more negative S is, the larger that contact angle becomes.

How do you calculate the spreading coefficient?

For one liquid on another, measure both surface tensions and their interfacial tension, then use S = γSV − (γSL + γLV). On a solid, measure the contact angle and use S = γLV (cos θ − 1). For water on PTFE at 110°, S ≈ −97.7 mN/m.

How do surfactants change the spreading coefficient?

Surfactants lower the liquid surface tension and often the interfacial tension too. Both terms are subtracted in the formula, so S rises and may turn positive. That is why surfactants help sprays, coatings and cleaners spread, up to the point where the interface is saturated.

References

Primary sources and further reading.

1. Harkins, W. D., and Feldman, A. (1922). Films. The spreading of liquids and the spreading coefficient. Journal of the American Chemical Society, 44, 2665 to 2685.
2. de Gennes, P. G. (1985). Wetting: statics and dynamics. Reviews of Modern Physics, 57(3), 827 to 863. https://doi.org/10.1103/RevModPhys.57.827
3. Bonn, D., Eggers, J., Indekeu, J., Meunier, J., and Rolley, E. (2009). Wetting and spreading. Reviews of Modern Physics, 81, 739 to 805. https://doi.org/10.1103/RevModPhys.81.739
4. Surface tension measurement guide
5. Interfacial tension (IFT)
6. Capillary number