Contents

Equation of State (SFE model)

Pronunciation: /ɪˈkweɪʒən əv steɪt/

The Equation of State is a surface free energy model that treats the solid liquid interfacial tension as a function of only the total surface tensions of the solid and the liquid. Combined with the Young equation, it gives the total surface energy of a solid from a single contact angle and a known liquid surface tension.

Also called

Neumann method Equation of state approach EOS method

Related terms

van Oss-Chaudhury-Good method Critical surface tension (Zisman) OWRK method

Topics

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

At a glance

What the Equation of State assumes, and why it needs only one liquid.

What it is

Developed by A. W. Neumann and coworkers from 1974, and reformulated by Li and Neumann in 1990. It does not split surface energy into components.

  • One liquid, one angle, one total γS
  • Uses an empirical constant β fitted to contact angle data on low energy solids
  • If the model holds, different liquids should give the same γS

Why it matters

Component models need two or three liquids and tables of liquid components that vary between sources. The Equation of State needs only the total surface tension of each liquid, which is easy to measure. That makes it attractive for quick comparisons, and some compact instruments in our guide to the best surface energy analyzer under $5,000 report surface energy this way.

Worked example

Water (72.8 mN/m, 20 °C) at 75° on the hypothetical polymer used across our model pages. Solving numerically gives γS = 38.6 mJ/m². Check it:

  • √(38.6 / 72.8) = 0.7282
  • (72.8 − 38.6)² = 1169.64; × 0.0001247 = 0.1459; e^(−0.1459) = 0.8643
  • cos θ = −1 + 2 × 0.7282 × 0.8643 = 0.2588, so θ = 75°

Diiodomethane at 50° gives 36.2 and formamide at 60° gives 36.6 mJ/m². The surface free energy calculator does the iteration for you.

How it is measured

You measure the sessile drop contact angle of a liquid with known surface tension, ideally an advancing angle, then solve the equation for γS. Repeating with a second liquid checks the result. The surface energy measurement guide covers the full procedure.

Expert view

“Droplet adhesion is three-dimensional. Two side-view angles give an estimate; the whole story is how the angle and the contact line vary all the way around the drop.”

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

What the terms mean

How the Equation of State reads results from more than one liquid.

Liquids agree on γS

The model fits the surface. A single total value describes how the solid wets, and any of those liquids can be used for routine checks.

Liquids disagree on γS

Specific interactions, often polar or acid base, are present between the solid and some liquids. A component model may describe the surface better.

γS close to γL

The angle is small and hard to read reliably, and the result barely changes with it, so it says little about the solid. Use a liquid with a clearly higher surface tension than the solid.

Liquid values used across our model pages, 20 °C: water 72.8 mN/m, diiodomethane 50.8 mN/m, formamide 58.0 mN/m. Only the totals enter this model.

γS Total surface free energy of the solid, mJ/m²
γL Total surface tension of the liquid, mN/m
θ Contact angle, degrees (advancing angle recommended by Neumann)
β Empirical constant, 0.0001247 (m²/mJ)² in the Li and Neumann form

Key equations

The Li and Neumann equation and how it combines with the Young equation.

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

Variable legend:

  • θ: contact angle of the liquid on the solid (°)
  • γS: solid surface energy (mJ/m²); γL: liquid surface tension (mN/m)
  • β = 0.0001247 (m²/mJ)²

Interpretation: Given θ and γL, there is one γS that satisfies the equation. It has no closed form solution, so it is solved by iteration.

When to use: Use it when you want one total surface energy for a low energy solid such as a polymer or coating.

Assumptions:

  • The solid liquid interfacial tension depends only on γS and γL
  • Smooth, uniform, nonreactive solid; equilibrium or advancing angle

Common mistakes:

  • Using a receding or drifting angle, which shifts γS
  • Using a liquid that dissolves or swells the solid

Equation: γSL = γL + γS − 2 √(γL γS) e^(−β (γL − γS)²)

Variable legend:

  • γSL: solid liquid interfacial tension (mN/m)

Interpretation: The interface costs least when the two tensions are similar. The exponential term grows the penalty as they move apart.

When to use: Use it to estimate γSL once γS is known, for example to compare wetting by different liquids.

Assumptions:

  • Same as above; β is treated as universal

Common mistakes:

  • Treating the result as a measured interfacial tension rather than a model estimate

How to read it

What an Equation of State value tells you, and what it hides.

A single total is easy to track over time, but it cannot tell you why a surface changed. In the worked example water gives 38.6 mJ/m² while diiodomethane gives 36.2 mJ/m²; that spread hints at a small polar contribution. Our note on reproducible surface energy calculation from contact angles covers how to keep such comparisons fair.

Reading an Equation of State result

If you see It usually means Check next
Two liquids within about 1 mJ/m² The model describes the surface well Use the simpler liquid for routine work
Water gives a higher γS than diiodomethane Water interacts through polar forces as well A component model such as OWRK
γS falls over minutes on the same drop The angle is drifting through evaporation or absorption Read the angle at a fixed drop age
Water at about 110° on a fluoropolymer γS of about 17 mJ/m², a very low energy surface Compare with typical PTFE values of 18 to 20 mJ/m²

Common misconceptions

Where the Equation of State is most often misunderstood.

The Equation of State and OWRK measure the same number

Both report a total in mJ/m², but they rest on different physics, so the values differ. In our shared example OWRK gives 41.0 mJ/m² and the Equation of State 36.2 to 38.6 mJ/m². Standards such as ISO 19403-2 for paints and varnishes ask you to state the evaluation method for this reason.

The approach is universally accepted

It is debated. Supporters show that advancing angles on many low energy solids fall on smooth curves, while critics argue that specific interactions cannot be ignored. Treat it as one model among several and test it with a second liquid.

Related terms

Where the Equation of State sits among surface energy models.

Broader term

  • Surface free energy models
  • Surface free energy (SFE)

Related terms

  • Critical surface tension (Zisman)
  • Young equation
  • Advancing contact angle

Often confused with

  • OWRK method
  • van Oss-Chaudhury-Good method

FAQ

Short answers to the questions people ask most about the Equation of State model.

Can I get surface energy from one liquid?

With the Equation of State, yes. One contact angle and the liquid surface tension give one total surface energy. With component models such as OWRK, no; you need at least two liquids. A second liquid is still worth measuring with the Equation of State, because agreement between liquids is the check that the model fits.

Why does the Equation of State not give dispersive and polar parts?

Neumann and coworkers argued that, for most low energy solids, the interfacial tension depends only on the total tensions. If that holds, splitting into components adds parameters without adding information. Critics disagree, which is why component models remain popular. Choose based on whether you need the split.

What is the β constant in the Equation of State?

β is an empirical constant fitted to contact angle data from many liquids on several low energy solids. In the 1990 Li and Neumann form it is 0.0001247 (m²/mJ)². It controls how fast the interfacial tension grows as the solid and liquid tensions move apart, and it is treated as the same for every system.

References

Primary sources and further reading.

1. Neumann, A. W., Good, R. J., Hope, C. J., and Sejpal, M. (1974). An equation of state approach to determine surface tensions of low energy solids from contact angles. Journal of Colloid and Interface Science, 49(2), 291 to 304. https://doi.org/10.1016/0021-9797(74)90365-8
2. Li, D., and Neumann, A. W. (1990). A reformulation of the equation of state for interfacial tensions. Journal of Colloid and Interface Science, 137(1), 304 to 307. https://doi.org/10.1016/0021-9797(90)90067-X
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. Surface energy measurement guide
5. OWRK method
6. van Oss-Chaudhury-Good method
7. Critical surface tension (Zisman)