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Surface free energy models

Pronunciation: /ˈsɜːrfɪs friː ˈɛnərdʒi ˈmɒdəlz/

Surface free energy models are the equations that turn measured contact angles of test liquids into the surface free energy of a solid. Each model makes a different assumption about how the liquid and solid interact, so the same angles can give different answers.

Also called

SFE models Surface energy theories Contact angle methods for surface energy

Related terms

van Oss-Chaudhury-Good method OWRK method Surface energy components

Topics

Surface energy Contact angle Adhesion
Book a 15 minute demo Read the surface energy measurement guide

At a glance

What surface free energy models do, and why the model belongs in every reported value.

What it is

You cannot measure the surface energy of a solid directly. You measure contact angles of known liquids, then a model links those angles to the solid through the Young equation.

  • Zisman: critical surface tension from a series of liquids
  • Fowkes and OWRK: dispersive and polar components
  • van Oss-Chaudhury-Good: Lifshitz-van der Waals, acid and base components
  • Equation of State: one total value, no components

Why it matters

Coating, bonding and printing all depend on whether a liquid wets the solid. A surface energy value lets you compare treatments and set limits, but only if everyone uses the same model and liquids. Our guide to reproducible surface energy calculation from contact angles explains what to fix in a method so results stay comparable.

Worked example

One hypothetical polymer, three liquids at 20 °C: water 75°, diiodomethane 50°, formamide 60°.

  • OWRK (water and diiodomethane): 34.3 + 6.7 = 41.0 mJ/m²
  • van Oss-Chaudhury-Good (all three): 34.3 + 2.0 = 36.3 mJ/m²
  • Equation of State: 38.6, 36.2 and 36.6 mJ/m² for the three liquids

Same angles, different numbers. The step by step arithmetic is on each model page, and the surface free energy calculator repeats it with your own angles.

How it is measured

Every model starts from sessile drop contact angles of two or more liquids with known surface tension. Liquid choice, drop age and which angle you read all change the answer. The surface energy measurement guide covers the procedure step by step.

Expert view

“Adhesion fails at the interface long before it fails in the joint; surface energy tells you whether that interface was ever ready to bond in the first place.”

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

What the terms mean

The three families of models and the symbols they share.

Threshold models (Zisman)

Plot cos θ against the surface tension of a series of liquids and extrapolate to cos θ = 1. The result is a wetting threshold, the critical surface tension γc, not a true surface energy.

Component models (Fowkes, OWRK, van Oss-Chaudhury-Good)

Split the surface energy into parts by the type of molecular interaction. You need at least as many liquids as unknown components, and the liquids must differ in polarity.

Equation of State (Neumann)

Treats the solid liquid interaction as a function of the two total surface tensions only. One liquid gives one total value, with no split into components.

All models share the Young equation as their starting point. They differ only in how they write the solid liquid interfacial tension γSL.

γS Surface free energy of the solid, mJ/m² (numerically equal to mN/m)
γL Surface tension of the test liquid, mN/m
θ Contact angle of the test liquid on the solid, degrees
ᵈ, ᵖ Dispersive and polar components (Fowkes, OWRK)
ᴸᵂ, ⁺, ⁻ Lifshitz-van der Waals, acid and base components (van Oss-Chaudhury-Good)

Key equations

The common starting point and the two most used model forms.

Equation: γL (1 + cos θ) = Wa

Variable legend:

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

Interpretation: The left side is what you measure. Each model supplies its own expression for Wa in terms of the solid and liquid properties.

When to use: Use it as the shared first line of every component model calculation.

Assumptions:

  • Smooth, rigid, chemically uniform solid
  • The liquid does not dissolve, swell or react with the solid

Common mistakes:

  • Using an apparent angle from a rough surface as if it were the Young angle

Equation: γL (1 + cos θ) = 2 [√(γSᵈ γLᵈ) + √(γSᵖ γLᵖ)]

Variable legend:

  • γSᵈ, γSᵖ: dispersive and polar parts of the solid (mJ/m²)
  • γLᵈ, γLᵖ: dispersive and polar parts of the liquid (mN/m)
  • Water: 72.8 = 21.8 + 51.0; diiodomethane: 50.8 = 50.8 + 0 (20 °C)

Interpretation: Two unknowns, so two liquids of different polarity are the minimum. More liquids give a straight line fit.

When to use: Use it for polymers, coatings and treated surfaces where a dispersive and polar split is enough.

Assumptions:

  • Interactions combine as a geometric mean of like components

Common mistakes:

  • Using two liquids of similar polarity, which makes the solution unstable

Equation: cos θ = −1 + 2 √(γS/γL) e^(−β (γL − γS)²)

Variable legend:

  • β = 0.0001247 (m²/mJ)², an empirical constant
  • γS, γL: total surface energies (mJ/m² and mN/m)

Interpretation: One liquid and one angle give one total γS. Results from several liquids should agree if the model holds.

When to use: Use it for low energy solids when you want a single total value rather than components.

Assumptions:

  • The solid liquid interaction depends only on the two total tensions

Common mistakes:

  • Comparing an Equation of State total with an OWRK total as if they were the same quantity

How to read it

Choosing a model and reading results from different models.

No model is universally right. Pick one that fits the question, keep the liquids and model fixed across a study, and always report both. For research groups comparing several models, our guide to the best surface energy analyzer for research with multi liquid models lists what to look for.

Which model for which question

If you need A common choice Watch for
A quick wettability threshold for films or treatments Zisman critical surface tension Curved plots with polar liquids
Dispersive and polar parts for coatings or bonding OWRK with water and diiodomethane Liquid component values from different tables
Acid and base character of a polar surface van Oss-Chaudhury-Good with three liquids Very high base values, a known artefact
One total value for a low energy solid Equation of State Spread between liquids, which signals the model does not fit

Common misconceptions

Where surface free energy models are most often misread.

Different models should give the same number

They rest on different assumptions, so they do not. In the worked example above OWRK gives 41.0 mJ/m² and van Oss-Chaudhury-Good gives 36.3 mJ/m² from the same angles. Compare values only within one model, as tables such as our surface free energy of common polymers and substrates do.

More liquids always means a better answer

Extra liquids help only if they add new information and do not swell or dissolve the solid. A liquid that reacts with the surface adds error, not precision. Experiment 6 on surface free energy with multiple liquids lets students see this effect directly.

Related terms

The individual models and the ideas they build on.

Broader term

  • Surface free energy (SFE)
  • Contact angle

Models covered here

  • Critical surface tension (Zisman)
  • Fowkes method
  • OWRK method
  • van Oss-Chaudhury-Good method
  • Equation of State (SFE model)

Related terms

  • Surface energy components
  • Work of adhesion
  • Young equation

FAQ

Short answers to the questions people ask most about surface free energy models.

Which surface free energy model should I use?

For most polymers, coatings and treated surfaces, OWRK with water and diiodomethane is the common default. Use van Oss-Chaudhury-Good when acid and base character matters, for example with biomaterials. Use the Equation of State when you only need one total value. Whatever you choose, keep it fixed and report it with the result.

Why do OWRK and van Oss give different surface energies?

They split the interaction differently. OWRK uses one polar term with a geometric mean, while van Oss-Chaudhury-Good splits polar interactions into acid and base parts that combine crosswise. With the same angles the polar or acid base part comes out different, so the totals differ. Neither is wrong; they answer slightly different questions.

How many test liquids do I need?

At least as many as the model has unknowns. Zisman needs a series of liquids, OWRK and the two step Fowkes method need two, van Oss-Chaudhury-Good needs three, and the Equation of State gives a value from each single liquid. Choose liquids that differ clearly in polarity so the equations are well conditioned.

References

Primary sources and further reading.

1. Owens, D. K., and Wendt, R. C. (1969). Estimation of the surface free energy of polymers. Journal of Applied Polymer Science, 13(8), 1741 to 1747. https://doi.org/10.1002/app.1969.070130815
2. van Oss, C. J., Chaudhury, M. K., and Good, R. J. (1988). Interfacial Lifshitz-van der Waals and polar interactions in macroscopic systems. Chemical Reviews, 88(6), 927 to 941. https://doi.org/10.1021/cr00088a006
3. Kwok, D. Y., and Neumann, A. W. (1999). Contact angle measurement and contact angle interpretation. Advances in Colloid and Interface Science, 81(3), 167 to 249. https://doi.org/10.1016/S0001-8686(98)00087-6
4. Zisman, W. A. (1964). Relation of the equilibrium contact angle to liquid and solid constitution. In Contact Angle, Wettability, and Adhesion, Advances in Chemistry 43, American Chemical Society, 1 to 51. https://doi.org/10.1021/ba-1964-0043.ch001
5. Surface energy measurement guide
6. OWRK method
7. van Oss-Chaudhury-Good method
8. Surface energy components