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Surface roughness and heterogeneity

Pronunciation: /ˈsɜːrfɪs ˈrʌfnəs ənd ˌhɛtərədʒəˈniːɪti/

Surface roughness is the physical texture of a solid, and chemical heterogeneity is variation in its surface chemistry from place to place. Both make the measured contact angle differ from the ideal Young angle and are the main causes of contact angle hysteresis.

Also called

Surface texture Topography Chemical heterogeneity Nonideal surfaces

Related terms

Cassie-Baxter model Wenzel model

Topics

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

At a glance

What roughness and heterogeneity are, and how they change a contact angle.

What it is

The Young equation assumes a flat, chemically uniform solid. Real surfaces break both assumptions.

  • Roughness: hills, valleys, pores and fibers that add real area and edges
  • Heterogeneity: patches of different chemistry, such as oxide islands, contamination or partial coatings
  • Both create many stable contact line positions, so one surface can show a range of angles

Why it matters

A contact angle on a rough or patchy sample is an apparent angle, not a pure chemistry result. Two batches with identical chemistry can differ only in texture. Roughness can also be used on purpose: flame soot on glass gives a superhydrophobic surface in experiment 1 on superhydrophobic surfaces on glass.

Worked example

A flat surface that is 70 % hydrophobic coating (θ = 110°) and 30 % bare hydrophilic patches (θ = 30°).

  • 0.7 × cos 110° = 0.7 × (−0.342) = −0.239
  • 0.3 × cos 30° = 0.3 × 0.866 = 0.260
  • cos θ* = 0.020, so θ* ≈ 89°

Partial coverage drops the apparent angle by about 21° from the full coating, and hysteresis grows as the edge pins on the patches.

Expert view

“The gap between the advancing and receding angle, the hysteresis, is where a surface reveals its roughness, its heterogeneity, and whether something is reacting.”

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

What the terms mean

Three ways a real surface departs from the ideal one in the Young equation.

Rough, liquid fills the texture (Wenzel)

Extra area amplifies the chemistry: hydrophilic surfaces wet more, hydrophobic ones less. Edges and grooves pin the contact line, so hysteresis is usually high.

Rough, air trapped in the texture (Cassie-Baxter)

The drop rests on a mix of solid tops and air. Angles rise sharply on hydrophobic textures, and hysteresis can be very low.

Smooth but chemically patchy

Different chemistries side by side. The apparent angle sits between the pure values, and patch edges pin the contact line.

Roughness and heterogeneity both act at the contact line. What lies at the edge of the drop decides the angle you measure.

r Roughness ratio: true area over projected area, dimensionless, r ≥ 1
fs Solid fraction touched by the liquid in a Cassie-Baxter state, 0 to 1
f1, f2 Area fractions of two chemistries on a heterogeneous surface
Sa, Sq Arithmetic and root mean square areal height, µm or nm (ISO 25178)
Sdr Developed interfacial area ratio, percent (ISO 25178)

Key equations

How roughness and chemical patches enter the contact angle.

Equation: cos θ* = r cos θY

Variable legend:

  • θ*: apparent angle on the rough surface (°)
  • θY: Young angle on the smooth material (°)
  • r: roughness ratio

Interpretation: Roughness scales the cosine, pushing the angle away from 90°.

When to use: Use it when the liquid fills the texture under the drop.

Assumptions:

  • Texture much smaller than the drop and uniform at the contact line

Common mistakes:

  • Applying it when air is trapped in the texture

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

Variable legend:

  • f1, f2: area fractions of each chemistry (sum to 1 on a flat surface)
  • θ1, θ2: angles on each pure material (°)

Interpretation: The apparent cosine is an area weighted average. With air as one phase (θ = 180°) it becomes the Cassie-Baxter equation.

When to use: Use it for flat surfaces with fine, evenly mixed patches of two chemistries.

Assumptions:

  • Patches much smaller than the drop and evenly spread at the contact line

Common mistakes:

  • Averaging angles instead of cosines

Equation: r = 1 + Sdr / 100

Variable legend:

  • Sdr: developed interfacial area ratio in percent (ISO 25178)

Interpretation: Sdr = 20 % means 20 % more real area than the flat footprint, so r = 1.2.

When to use: Use it to estimate r for the Wenzel equation from a measured surface.

Assumptions:

  • The instrument resolves the features the liquid wets

Common mistakes:

  • Using Ra or Sa, which describe height, as if they described area

How to read it

Telling roughness effects apart from chemistry effects.

Roughness and heterogeneity leave fingerprints in hysteresis and drop to drop scatter. Compare your numbers with smooth reference surfaces in our contact angle reference values for water, and use our free contact angle tool to read angles from images.

From symptom to cause

If you see It usually means Check next
Large scatter between drops on one sample Patchy chemistry or uneven texture at the scale of the drop Map several spots and compare with a smooth reference
Large hysteresis on a smooth looking surface Chemical heterogeneity or contamination pinning the edge Cleaning, handling and coating coverage
Rough sample angle differs from smooth reference of same material Roughness amplifying the chemistry (Wenzel) or trapping air (Cassie-Baxter) Hysteresis and whether drops roll
Drop shape is not round from above Anisotropic texture, such as grooves or machining marks Measure parallel and perpendicular to the lay

Common misconceptions

Where roughness and heterogeneity are most often misread.

Roughness always increases the contact angle

Only on surfaces that are already hydrophobic. On hydrophilic materials, roughness in the Wenzel state lowers the angle. Roughness amplifies the chemistry; it does not set the direction.

Ra tells you how roughness affects wetting

Ra and Sa describe average height, not area or shape. Two surfaces with the same Ra can have very different r and fs. Spacing, slope and feature shape matter as much as height.

A visibly smooth surface behaves ideally

Nanometer scale roughness and molecular contamination are invisible but still pin the contact line. Hysteresis of a few degrees is normal even on carefully prepared surfaces.

Average roughness under the drop decides the angle

The contact line decides it. A defect inside the footprint has little effect, while one at the edge can change the angle. Teaching labs studying this can compare equipment in our guide to the best contact angle goniometer for teaching and academic labs.

Related terms

Where roughness and heterogeneity sit among other wetting concepts.

Models

  • Wenzel model
  • Cassie-Baxter model
  • Young equation

What they cause

  • Contact angle hysteresis
  • Lotus effect
  • Roll-off / sliding angle

Broader term

  • Wettability

FAQ

Short answers to the questions people ask most about roughness, heterogeneity and wetting.

How does surface roughness affect contact angle?

It depends on the wetting state. If liquid fills the texture, the Wenzel model applies and roughness pushes the angle away from 90°. If air stays trapped, the Cassie-Baxter model applies and the angle rises sharply on hydrophobic surfaces. Either way, the measured angle is apparent, not the Young angle.

What is the difference between roughness and heterogeneity?

Roughness is physical: the shape and height of the surface. Heterogeneity is chemical: patches with different surface groups or contaminants. Both pin the contact line and cause hysteresis. Roughness is treated with the Wenzel model, and heterogeneity with the Cassie model.

How much roughness starts to matter?

There is no single threshold, because shape matters as much as height. Features that add a few percent of area usually change the angle by only a few degrees, except near complete wetting. Textures with steep slopes, deep pores or fibers can change it by tens of degrees, especially if air becomes trapped.

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. 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
3. Joanny, J. F., and de Gennes, P. G. (1984). A model for contact angle hysteresis. The Journal of Chemical Physics, 81(1), 552 to 562. https://doi.org/10.1063/1.447337
4. 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
5. Contact angle measurement guide
6. Wenzel model
7. Cassie-Baxter model
8. Contact angle hysteresis