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Wenzel model

Pronunciation: /ˈvɛntsəl ˈmɒdəl/

The Wenzel model gives the apparent contact angle θ* of a liquid that fully fills the roughness of a solid: cos θ* = r cos θY, where r is the roughness ratio. Roughness amplifies the intrinsic wettability, so hydrophilic surfaces wet more and hydrophobic surfaces less.

Also called

Wenzel equation Wenzel state Homogeneous wetting regime

Related terms

Cassie-Baxter model Surface roughness and heterogeneity Hydrophobic vs hydrophilic surfaces

Topics

Wetting Contact angle Roughness
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At a glance

What the Wenzel model says about rough surfaces, and when it applies.

What it is

Roughness adds real solid area under each unit of projected area. When the liquid fills every groove, that extra area multiplies the energy gain or cost of wetting. The measured angle shifts away from 90° compared with the smooth surface.

  • Proposed by Robert N. Wenzel in 1936
  • Assumes liquid fully penetrates the texture
  • r = 1 for a perfectly flat surface, r > 1 for any rough one

Worked example

Same chemistry, roughness ratio r = 1.5, fully wetted texture.

  • θY = 110°: cos θ* = 1.5 × (−0.342) = −0.513, so θ* ≈ 121°
  • θY = 70°: cos θ* = 1.5 × 0.342 = 0.513, so θ* ≈ 59°

Roughness pushes both angles away from 90°. If r cos θY falls below −1 or rises above 1, the Wenzel state cannot exist and the drop either traps air or spreads into the texture.

Expert view

“Wettability is set by the top five to ten ångström of a surface: what the liquid actually sees. It doesn't matter what lies beneath; that outermost layer is what you are really measuring.”

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

What the terms mean

How the same roughness pushes the angle in opposite directions.

θY < 90° and r > 1: θ* < θY

Wetting lowers the energy, and roughness gives the liquid more area to wet. The rough surface looks more hydrophilic than the smooth one.

θY > 90° and r > 1: θ* > θY

Wetting costs energy, and roughness adds more area that costs energy. The rough surface looks more hydrophobic, as long as the liquid still fills the texture.

r cos θY outside −1 to 1

The Wenzel equation has no solution. On hydrophobic textures air is usually trapped (Cassie-Baxter state); on hydrophilic textures liquid wicks into the grooves ahead of the drop.

In the Wenzel state the liquid follows every hill and valley under the drop. No air is trapped beneath it.

θ* Apparent contact angle on the rough surface, degrees
θY Young contact angle on a smooth surface of the same chemistry, degrees
r Roughness ratio: true area divided by projected area, dimensionless, r ≥ 1

Key equations

The Wenzel equation and the roughness ratio it depends on.

Equation: cos θ* = r cos θY

Variable legend:

  • θ*: apparent contact angle (°)
  • θY: Young angle on the smooth material (°)
  • r: roughness ratio, dimensionless

Interpretation: Roughness scales cos θ by r, so it amplifies whatever wettability the chemistry already gives.

When to use: Use it for rough surfaces where the liquid is known to fill the texture, with roughness much smaller than the drop.

Assumptions:

  • Liquid penetrates all the roughness under the drop
  • Roughness scale much smaller than the drop
  • Chemistry uniform; roughness uniform near the contact line
  • Equilibrium angle, not an advancing or receding one

Common mistakes:

  • Applying it to a surface that traps air (use Cassie-Baxter)
  • Using a single static angle when hysteresis on rough surfaces is large
  • Taking r from a 2D profile when the model needs true area

Equation: r = A(true) / A(projected) = 1 + Sdr / 100

Variable legend:

  • A(true): actual surface area including all texture (m²)
  • A(projected): flat footprint area (m²)
  • Sdr: developed interfacial area ratio from ISO 25178, in percent

Interpretation: A surface with Sdr = 50 % has r = 1.5, meaning 50 % more real area than its footprint.

When to use: Use it to put a number on r from areal metrology before applying the Wenzel equation.

Assumptions:

  • The metrology resolves the features that the liquid actually wets

Common mistakes:

  • Using Ra, which is a height average, as if it were r
  • Ignoring that the measured r depends on the instrument resolution and sampling area

How to read it

How to tell whether the Wenzel model describes your sample.

The Wenzel model is a limit, not a guarantee. Check which wetting state you have before fitting numbers to it. Students compare both states in experiment 5 on Wenzel versus Cassie-Baxter wetting, and our free contact angle tool helps read angles from images.

Wenzel or something else?

If you see It usually means Check next
High angle with large hysteresis and drops that stick Wenzel state on a hydrophobic texture Sliding angle; drops in this state pin rather than roll
High angle, low hysteresis, drops that roll off Cassie-Baxter state with trapped air Whether pressing or evaporating the drop flips it to Wenzel
Lower angle on the rough sample than on a smooth one Wenzel amplification of a hydrophilic surface Wicking of liquid into the texture beyond the drop edge
Measured angle does not match r cos θY Roughness near the contact line differs from the average, or the state is mixed Topography at the drop edge rather than across the whole sample

Common misconceptions

Where the Wenzel model is most often misapplied.

Roughness always makes a surface more hydrophobic

Only if the smooth material is already hydrophobic. On a hydrophilic material the same roughness lowers the angle. Roughness amplifies, it does not set the direction.

The whole wetted area decides the angle

Gao and McCarthy showed that the angle depends on the surface at the contact line, not on texture far inside the drop. The Wenzel equation works when the roughness is uniform, so that the average equals the edge. Data such as an analysis of superhydrophobic aspen leaves show how texture changes the wetting state.

Wenzel and Cassie-Baxter are rival theories

They describe two different wetting states. A surface can switch from Cassie-Baxter to Wenzel under pressure, impact or evaporation. Field data in our client collected wetting dataset include both kinds of rough surface.

A goniometer angle on a rough sample is the material angle

It is an apparent angle. To compare chemistry, measure a smooth reference or report the roughness alongside the angle. Teaching labs choosing equipment for these comparisons can see our guide to the best contact angle goniometer for teaching and academic labs.

Related terms

Where the Wenzel model sits among other wetting concepts.

Builds on

  • Young equation
  • Surface roughness and heterogeneity

Companion model

  • Cassie-Baxter model
  • Lotus effect

Related terms

  • Contact angle hysteresis
  • Hydrophobic vs hydrophilic surfaces
  • Wettability

FAQ

Short answers to the questions people ask most about the Wenzel model.

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

In the Wenzel state the liquid fills the roughness, so the solid under the drop is fully wetted. In the Cassie-Baxter state air stays trapped in the texture, so the drop rests on a composite of solid and air. Wenzel drops usually pin with high hysteresis; Cassie-Baxter drops often roll easily.

What is the roughness ratio r?

It is the true surface area divided by the projected, flat area. A perfectly smooth surface has r = 1. Real values depend on texture and on how finely it is measured. Areal metrology gives r from the developed interfacial area ratio Sdr, defined in ISO 25178, as r = 1 + Sdr/100.

Can the Wenzel model predict superhydrophobicity?

Only partly. It predicts higher angles for rough hydrophobic surfaces, but very high angles need large r. Most superhydrophobic surfaces instead trap air and follow Cassie-Baxter behavior, which also gives the low hysteresis needed for drops to roll. A Wenzel drop at a high angle usually still sticks.

References

Primary sources and further reading.

1. Wenzel, R. N. (1936). Resistance of solid surfaces to wetting by water. Industrial and Engineering Chemistry, 28(8), 988 to 994. https://doi.org/10.1021/ie50320a024
2. Quéré, D. (2008). Wetting and roughness. Annual Review of Materials Research, 38, 71 to 99. https://doi.org/10.1146/annurev.matsci.38.060407.132434
3. 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
4. Contact angle measurement guide
5. Cassie-Baxter model
6. Surface roughness and heterogeneity
7. Contact angle hysteresis