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Cassie-Baxter model

Pronunciation: /ˈkæsi ˈbækstər ˈmɒdəl/

The Cassie-Baxter model gives the apparent contact angle of a drop resting on a composite surface, such as solid tops with air trapped between them: cos θ* = fs (cos θY + 1) − 1. It explains why textured hydrophobic surfaces can reach very high angles with drops that roll off easily.

Also called

Cassie equation Cassie-Baxter equation Cassie state Fakir state

Related terms

Wenzel model Lotus effect Hydrophobic vs hydrophilic surfaces

Topics

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

At a glance

What the Cassie-Baxter model says, and why it underpins superhydrophobic surfaces.

What it is

Cassie and Baxter treated a surface as a patchwork of materials, each with its own contact angle. The apparent angle is an area weighted average of their cosines. When one of the materials is air trapped in the texture, its angle is 180° and the drop sits on a mix of solid and air.

  • Cassie and Baxter (1944): porous surfaces with trapped air
  • Cassie (1948): extended to chemically mixed surfaces

Worked example

A hydrophobic coating with θY = 110°, textured so the drop touches solid on 20 % of its footprint (fs = 0.2).

  • cos θY + 1 = −0.342 + 1 = 0.658
  • cos θ* = 0.2 × 0.658 − 1 = −0.868
  • θ* ≈ 150°

The smooth coating gives 110°; trapping air lifts the apparent angle to about 150°.

How it is measured

You measure the apparent angle on the textured sample and θY on a smooth sample of the same chemistry, then infer fs or compare with the value from the texture geometry. Texture parameters are defined in ISO 25178 for areal surface texture. The contact angle measurement guide covers the angle measurement.

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 composite state and how it compares with the Wenzel state.

Cassie-Baxter state (air trapped)

The drop rests on the tops of the texture with air pockets beneath. High apparent angle and usually low hysteresis, so drops roll.

Wenzel state (texture filled)

Liquid fills the grooves. The apparent angle is set by the roughness ratio r, and the contact line pins on many edges, so hysteresis is high.

Chemically mixed flat surface

The original Cassie form also applies to flat surfaces made of two chemistries. The apparent angle sits between the two pure angles, weighted by area fraction.

In the Cassie-Baxter state the drop bridges the texture. Under it, the liquid touches solid on the tops and air in the gaps.

θ* Apparent contact angle on the composite surface, degrees
θY Young contact angle on the smooth solid, degrees
fs Fraction of the projected footprint where liquid touches solid, 0 to 1
f1, f2 Area fractions of two surface materials, dimensionless
θ1, θ2 Contact angles on each pure material, degrees

Key equations

The general Cassie equation and its most used form for trapped air.

Equation: cos θ* = f1 cos θ1 + f2 cos θ2

Variable legend:

  • θ*: apparent angle (°)
  • f1, f2: area fractions of each material under the drop
  • θ1, θ2: contact angles on each pure material (°)

Interpretation: The cosine of the apparent angle is the area weighted average of the cosines on each material.

When to use: Use it for flat surfaces with two chemistries, or as the starting point for trapped air.

Assumptions:

  • Patches much smaller than the drop
  • Patches evenly spread, including at the contact line
  • Flat surface, so f1 + f2 = 1

Common mistakes:

  • Averaging angles instead of cosines
  • Using whole footprint fractions when the composition at the contact line differs

Equation: cos θ* = fs (cos θY + 1) − 1

Variable legend:

  • fs: wetted solid fraction of the projected area
  • θY: Young angle on the smooth solid (°)
  • Air takes the rest, with an angle of 180° (cos = −1)

Interpretation: As fs falls, θ* climbs toward 180°. A small solid fraction gives a very high angle.

When to use: Use it for textured hydrophobic surfaces where air stays trapped under the drop.

Assumptions:

  • Flat topped texture, with the liquid resting on flat tops
  • Air pockets stable; no liquid penetration
  • Texture much smaller than the drop

Common mistakes:

  • Assuming the state is stable: pressure, impact or evaporation can push it into the Wenzel state
  • Forgetting that rough tops wetted by liquid need fs multiplied by their own roughness ratio

Equation: cos θc = (fs − 1) / (r − fs)

Variable legend:

  • θc: Young angle at which both states have equal energy (°)
  • r: roughness ratio of the texture
  • fs: solid fraction

Interpretation: Trapped air is the lower energy state only when θY exceeds θc. Below it, a Cassie-Baxter drop can still exist, but only as a metastable state.

When to use: Use it to judge whether a texture is likely to keep air trapped for a given chemistry.

Assumptions:

  • Same assumptions as both underlying models

Common mistakes:

  • Treating metastable Cassie drops as permanent

How to read it

How to recognize a Cassie-Baxter drop and test whether it is stable.

A high static angle is not proof of trapped air. Hysteresis and drop mobility tell the two states apart. Students compare them in experiment 5 on Wenzel versus Cassie-Baxter wetting, and our free contact angle tool helps read high angles from images.

Signs of each wetting state

If you see It usually means Check next
Angle above 150° and drops roll at a few degrees of tilt Stable Cassie-Baxter state Whether it survives drop impact, pressing or long contact
High angle but drops stick when tilted Wenzel state, or a mixed state pinned at the edge Receding angle; it is usually much lower than the advancing angle
Angle falls suddenly as the drop evaporates Transition from Cassie-Baxter to Wenzel Texture spacing; wide gaps let liquid sag into the texture
Reflective, silvery layer under a submerged sample Air film held by the texture (a plastron) How long the film lasts under water

Common misconceptions

Where the Cassie-Baxter model is most often misapplied.

Cassie-Baxter and Wenzel are competing theories

They describe two different states of the same kind of surface. Which one you see depends on the chemistry, the texture and how the drop was placed. A drop can switch from Cassie-Baxter to Wenzel, but rarely the other way without outside help.

The area fraction under the whole drop sets the angle

The contact line decides the angle, not the footprint average. A drop centered on an unusual patch can show the angle of the surrounding surface. The model works when the texture is uniform, which is why natural surfaces in an analysis of superhydrophobic aspen leaves need care.

A high contact angle proves the surface is superhydrophobic

Superhydrophobic means a high angle and low hysteresis, so drops roll. A pinned drop at 150° does not shed water. Field data in our client collected wetting dataset illustrate why both numbers are reported.

Any goniometer image of a 160° drop is reliable

Near 180°, the baseline is hard to place and small errors change the angle a lot. Report the method and look at hysteresis or sliding angle as well. Teaching labs can compare instruments in our guide to the best contact angle goniometer for teaching and academic labs.

Related terms

Where the Cassie-Baxter model sits among other wetting concepts.

Builds on

  • Young equation
  • Surface roughness and heterogeneity

Companion model

  • Wenzel model

Related terms

  • Lotus effect
  • Hydrophobic vs hydrophilic surfaces
  • Contact angle hysteresis
  • Roll-off / sliding angle

FAQ

Short answers to the questions people ask most about the Cassie-Baxter model.

What is the difference between the Cassie-Baxter and Wenzel models?

In the Cassie-Baxter state air stays trapped in the texture, so the drop rests on a mix of solid and air. In the Wenzel state the liquid fills the texture. Cassie-Baxter drops usually have low hysteresis and roll; Wenzel drops usually pin. The same surface can show either state.

What is the solid fraction fs?

It is the fraction of the projected footprint where the liquid touches solid rather than air. For square pillars of width a and spacing b, fs = a² / (a + b)². Lower fs gives higher angles, but very low fs makes the air layer easier to collapse under pressure.

Why does a Cassie-Baxter drop collapse into the Wenzel state?

Pressure inside the drop pushes the liquid into the texture. Small drops, impacting drops and evaporating drops have higher pressure, and wide gaps between features offer less resistance. When the liquid touches the bottom of the texture, or the edge slips down the sidewalls, the state collapses.

References

Primary sources and further reading.

1. Cassie, A. B. D., and Baxter, S. (1944). Wettability of porous surfaces. Transactions of the Faraday Society, 40, 546 to 551. https://doi.org/10.1039/tf9444000546
2. Cassie, A. B. D. (1948). Contact angles. Discussions of the Faraday Society, 3, 11. https://doi.org/10.1039/DF9480300011
3. Bico, J., Thiele, U., and Quéré, D. (2002). Wetting of textured surfaces. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 206(1 to 3), 41 to 46. https://doi.org/10.1016/S0927-7757(02)00061-4
4. Lafuma, A., and Quéré, D. (2003). Superhydrophobic states. Nature Materials, 2(7), 457 to 460. https://doi.org/10.1038/nmat924
5. Gao, L., and McCarthy, T. J. (2007). How Wenzel and Cassie were wrong. Langmuir, 23(7), 3762 to 3765. https://doi.org/10.1021/la062634a
6. Contact angle measurement guide
7. Wenzel model
8. Lotus effect
9. Surface roughness and heterogeneity