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Dynamic contact angles

Pronunciation: /daɪˈnæmɪk ˈkɒntækt ˈæŋɡəlz/

Dynamic contact angles are the contact angles of a liquid whose contact line is moving over a solid. The advancing angle θA is measured as the liquid spreads over dry surface and the receding angle θR as it retreats from wetted surface.

Also called

Advancing and receding contact angles Moving contact angles Dynamic wetting angles

Related terms

Contact angle hysteresis

Topics

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

At a glance

What dynamic contact angles are, and the difference between slow and fast line motion.

What it is

A static drop can sit at many angles on a real surface. Moving the contact line removes that ambiguity: the edge advances at θA and recedes at θR.

  • At very low speed these are quasi static, thermodynamic limits
  • At higher speed the angle depends on velocity: advancing angles rise, receding angles fall
  • The gap between θA and θR is the contact angle hysteresis

Why it matters

Advancing and receding angles describe a surface better than one static angle, because both are reproducible limits. The receding angle is often the most sensitive to contamination and surface changes. Coating, printing and spraying processes run at speed, so the velocity dependent angle also predicts whether a film will wet or entrain air.

Worked example

Water at 20 °C (viscosity 1.0 mPa·s, γLV = 72.8 mN/m) with the line moving at 1 mm/min.

  • U = 1 mm/min = 1.67 × 10⁻⁵ m/s
  • Capillary number Ca = μU/γLV = (1.0 × 10⁻³ × 1.67 × 10⁻⁵) / 0.0728 ≈ 2.3 × 10⁻⁷

Ca is tiny, so viscous forces barely bend the angle. A liquid a thousand times more viscous at the same speed would have a thousand times larger Ca.

Expert view

“Advance or recede too fast and it is no longer a thermodynamic angle: fluid mechanics takes over. Keep below about a millimetre a minute and the thermodynamics still holds.”

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

What the terms mean

The three regimes of a contact line, from pinned to moving fast.

Pinned (θR < θ < θA)

Liquid is added or removed but the edge stays put. Only the angle changes. A static sessile drop sits somewhere in this window.

Quasi static motion (low Ca)

The edge creeps slowly. The angle stays close to θA when advancing and θR when receding, and these are the values reported as material properties.

Fast motion (higher Ca)

Viscous dissipation near the contact line changes the apparent angle with speed. Advancing angles rise and receding angles fall until air entrainment or film deposition sets in.

Dynamic angles are read where the drop profile meets the solid at the moving edge. The direction of motion decides whether you see θA or θR.

θA Advancing contact angle, degrees
θR Receding contact angle, degrees
θD Dynamic (apparent) angle at a given line speed, degrees
U Contact line speed, m/s
μ Liquid viscosity, Pa·s
Ca Capillary number μU/γLV, dimensionless

Key equations

The numbers that tell you whether an angle is quasi static or speed dependent.

Equation: Ca = μ U / γLV

Variable legend:

  • μ: viscosity (Pa·s)
  • U: contact line speed (m/s)
  • γLV: surface tension (N/m)

Interpretation: Ca compares viscous forces with surface tension. At very small Ca the dynamic angle stays close to the quasi static θA or θR.

When to use: Use it to check whether a measured angle can be treated as a material property.

Assumptions:

  • Newtonian liquid
  • Steady line motion

Common mistakes:

  • Applying a speed limit set for water to a viscous oil or polymer melt

Equation: H = θA − θR

Variable legend:

  • H: hysteresis (°)

Interpretation: The width of the window of stable static angles. It reflects roughness, heterogeneity and surface changes.

When to use: Use it to compare how strongly surfaces pin drops.

Assumptions:

  • θA and θR measured at low speed with the same liquid

Common mistakes:

  • Computing H from angles measured at high speed

Equation: θD³ ≈ θ0³ + 9 Ca ln(L / λ)

Variable legend:

  • θD: apparent dynamic angle (radians)
  • θ0: angle at the contact line, often taken as θA or θR (radians)
  • L / λ: ratio of a large length (drop size) to a molecular cutoff; ln(L/λ) is typically about 10

Interpretation: The angle grows with Ca for an advancing line. For a receding line Ca is taken as negative, and the angle falls.

When to use: Use it to estimate how fast you can move the line before the angle shifts noticeably.

Assumptions:

  • Viscous dominated regime, small Ca
  • Angles below about 135°

Common mistakes:

  • Using degrees instead of radians
  • Treating ln(L/λ) as known precisely

How to read it

What advancing and receding angles usually tell you.

Report θA and θR with the method and line speed used. Compare with typical values in our contact angle reference values for water, and use our free contact angle tool to read angles from video frames.

From dynamic angles to meaning

If you see It usually means Check next
θA and θR within a few degrees Smooth, clean, uniform surface That the line actually moved during capture
θR much lower than θA Roughness, heterogeneity or adsorption pinning the receding edge Repeat on fresh spots and after cleaning
θA rises when you speed up the needle You are in the speed dependent regime Slow down until the angle stops changing
θR drifts down over repeated cycles The surface changes on contact: swelling, reorientation or dissolution Contact time and the liquid used

Common misconceptions

Where dynamic contact angles are most often misread.

Dynamic means the drop must move fast

For surface characterization, dynamic angles are measured with the line moving as slowly as practical. Fast motion adds a viscous contribution that belongs to the flow, not the surface.

An evaporating drop gives a reliable receding angle

Evaporation can pull the edge back, but it also changes temperature, concentration and contamination near the line. Check that the edge actually moves and that the angle settles before reporting θR.

Related terms

Where dynamic contact angles sit among other wetting concepts.

Derived from them

  • Contact angle hysteresis
  • Roll-off / sliding angle

Ways to measure them

  • Sessile drop
  • Tilting plate
  • Wilhelmy plate

Related terms

  • Three-phase contact line
  • Wettability
  • Dewetting

FAQ

Short answers to the questions people ask most about dynamic contact angles.

What is the difference between static and dynamic contact angles?

A static angle is measured on a drop at rest, and on a real surface it can take any value between θR and θA. Dynamic angles are measured while the contact line moves: θA as it advances and θR as it recedes. They are better defined limits and describe the surface more completely.

Why is the receding angle always lower than the advancing angle?

Roughness and chemical patches create energy barriers that pin the contact line. Advancing liquid is held back at hydrophobic spots and receding liquid is held at hydrophilic spots. The edge must overcome these barriers to move, so it advances at a higher angle and recedes at a lower one.

How slow is slow enough for dynamic contact angle measurement?

Slow enough that the angle no longer changes when you slow down further. For water, line speeds of about a millimetre per minute keep the capillary number tiny. More viscous liquids need slower speeds. A quick check is to repeat at half the speed and compare.

References

Primary sources and further reading.

1. de Gennes, P. G. (1985). Wetting: statics and dynamics. Reviews of Modern Physics, 57(3), 827 to 863. https://doi.org/10.1103/RevModPhys.57.827
2. Cox, R. G. (1986). The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. Journal of Fluid Mechanics, 168, 169 to 194. https://doi.org/10.1017/S0022112086000332
3. Blake, T. D. (2006). The physics of moving wetting lines. Journal of Colloid and Interface Science, 299(1), 1 to 13. https://doi.org/10.1016/j.jcis.2006.03.051
4. Snoeijer, J. H., and Andreotti, B. (2013). Moving contact lines: scales, regimes, and dynamical transitions. Annual Review of Fluid Mechanics, 45, 269 to 292. https://doi.org/10.1146/annurev-fluid-011212-140734
5. Contact angle measurement guide
6. Contact angle hysteresis
7. Three-phase contact line