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Dynamic surface tension (time resolved pendant drop)

Pronunciation: /daɪˈnæmɪk ˈsɜːfɪs ˈtɛnʃən/

Dynamic surface tension is the surface tension of a liquid measured as a function of surface age, the time since that surface was created. In surfactant solutions it starts near the value of the solvent and falls towards the static value as molecules adsorb.

Also called

DST Time dependent surface tension Surface age dependent surface tension

Related terms

Surfactant Spreading Static surface tension

Topics

Surface tension Surfactants Kinetics
Book a 15 minute demo Read the surface tension measurement guide

At a glance

What dynamic surface tension means, why fast processes depend on it, and which time scales each method reaches.

What it is

When a new surface forms, surfactant has not yet reached it. Molecules must diffuse from the bulk and adsorb, so the surface tension falls over time. Dynamic surface tension is that changing value, always reported with the surface age at which it was read.

  • Units: mN/m at a stated surface age
  • Pure liquids show almost no change with age
  • The curve ends at the static surface tension

Time scales by method

  • Maximum bubble pressure: milliseconds to about a second, standardised for example in ASTM D3825 for the fast bubble technique
  • Time resolved pendant drop: from about a second up to minutes or hours
  • Plate or ring at rest: long ages, close to static

Pick the method whose range matches the speed of your process.

How it is measured

With the pendant drop approach, a fresh drop is formed and imaged repeatedly while it ages, and each frame is fitted to give γ at that moment. The workflow below gives the outline. The surface tension measurement guide covers the full procedure and compares methods.

Expert view

“The critical micelle concentration is not a number you look up; it is where the surface-tension curve flattens, and the only honest way to know it is to measure it.”

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

What the terms mean

The three stages of a typical dynamic surface tension curve.

Induction (very young surface)

Little surfactant has arrived. The tension stays close to that of the solvent, around 72 mN/m for water.

Rapid fall

Adsorption speeds up and the tension drops steeply. How early this happens depends on concentration and how fast the surfactant diffuses.

Plateau (approaching equilibrium)

The surface fills up and the tension levels off. The final level is the static surface tension.

Plotting γ against log t spreads the curve out, so the induction stage, fall and plateau are easy to see on one chart.

γ(t) Dynamic surface tension at surface age t, mN/m
γ0 Surface tension of the pure solvent, mN/m
γeq Static (equilibrium) surface tension, mN/m
t Surface age, s
c Bulk surfactant concentration, mol/m³
D Diffusion coefficient of the surfactant, m²/s

Key equations

How diffusion sets the early part of the curve.

Equation: γ(t) ≈ γ0 − 2 R T c √(D t / π)

Variable legend:

  • γ0: solvent surface tension (N/m)
  • R: gas constant, 8.314 J/(mol K); T: temperature (K)
  • c: bulk concentration (mol/m³)
  • D: diffusion coefficient (m²/s); t: surface age (s)

Worked reading: the drop below γ0 grows with √t, so ageing the surface four times longer doubles the drop, as long as the surface is still sparsely covered.

Interpretation: At early times surfactant simply diffuses onto an empty surface. Higher concentration or faster diffusion makes the tension fall sooner.

When to use: Use it to judge whether early data are diffusion controlled, by checking that γ0 − γ(t) rises linearly with √t.

Assumptions:

  • Nonionic surfactant, dilute solution
  • Adsorption limited by diffusion, not by a barrier at the surface
  • Surface still far from full

Common mistakes:

  • Applying it near the plateau, where the surface is nearly full
  • Using it for ionic surfactants without accounting for charge effects

Equation: Γ(t) = 2 √(D/π) [ c √t − ∫ cs(τ) d√(t − τ) ]

Variable legend:

  • Γ(t): surface excess, the amount adsorbed per area (mol/m²)
  • cs(τ): concentration just below the surface at earlier time τ (mol/m³)
  • The integral runs from 0 to t and accounts for molecules diffusing back into the bulk

Interpretation: It balances what diffuses to the surface against what leaves it. Combined with an adsorption isotherm, it predicts the full dynamic curve.

When to use: Use it when fitting dynamic data to extract a diffusion coefficient or test an adsorption model.

Assumptions:

  • Flat or slowly curved surface, no convection near it

Common mistakes:

  • Fitting it without an isotherm that links Γ to γ

How to read it

What a dynamic surface tension curve tells you about a formulation.

Read the whole curve, not one point. The surface age at which tension falls tells you how fast a surfactant works, and the plateau tells you how far it can go. Our article on why you need dynamic surface tension walks through real examples, and the surface tension of common liquids dataset gives solvent values for γ0.

From curve to meaning

If you see It usually means Check next
Flat curve at the solvent value No surfactant, or too little to matter Concentration and whether the additive is surface active
Tension falls early and reaches the plateau fast Fast surfactant, well suited to quick processes Whether the plateau is low enough for wetting
Tension still falling after minutes Slow surfactant, or concentration below its CMC Run a higher concentration or a faster surfactant
Two samples with equal static values, different curves They will behave differently in fast processes Match the surface age to your process speed

How it is measured

The time resolved pendant drop approach in four short steps. The surface tension measurement guide has the full procedure.

1

Form a fresh drop

Create a new drop quickly so that its surface age starts close to zero.

2

Record while it ages

Capture images at set intervals, densely at first and more sparsely later.

3

Fit each frame

Fit every image to the Young-Laplace equation to get γ at that surface age.

4

Plot against age

Plot γ against log time and read the fall and the plateau.

Common misconceptions

Where dynamic surface tension is most often misunderstood.

One method covers every surface age

No single technique spans milliseconds to hours. Bubble pressure methods reach the youngest surfaces, while pendant drop suits ages from about a second upward. Our guide to the best tensiometer for dynamic surface tension compares the options.

The static value predicts process behaviour

Only if the process is slow. In sprays, inkjet and roll coating, the surface is often milliseconds old when it matters, and its tension can sit close to the solvent value.

Dynamic tension needs specialist equipment only

For ages from seconds upward, a pendant drop imaged over time is enough. The Dropometer smartphone drop shape analysis kit measures surface tension by the pendant drop method. Millisecond ages still need a bubble pressure instrument.

A falling reading means the instrument is drifting

With pure liquids, a steady fall can point to evaporation or contamination. With surfactant solutions, a fall is the signal itself. Students can see both cases in the labs in our teaching curriculum hub, and our free contact angle tool helps when the same liquids are tested for wetting.

Related terms

Where dynamic surface tension sits among other surface tension concepts.

Broader term

  • Surface tension

Related terms

  • Surfactant
  • Critical micelle concentration (CMC)
  • Spreading
  • Pendant drop method

Often confused with

  • Static surface tension
  • Dynamic contact angles

FAQ

Short answers to the questions people ask most about dynamic surface tension.

What is the difference between dynamic and static surface tension?

Dynamic surface tension is measured while the surface is still ageing, so it depends on how long the surface has existed. Static surface tension is the final, unchanging value at equilibrium. For pure liquids the two are nearly identical. For surfactant solutions the dynamic value starts higher and falls to the static value over time.

Why is dynamic surface tension important for inks and coatings?

Printing and coating create new surface in a fraction of a second. If surfactant cannot reach that surface in time, the liquid behaves as if its tension were much higher, which can cause beading, poor levelling or missed spots. Matching the surfactant speed to the process speed avoids these defects.

How is dynamic surface tension measured with a pendant drop?

A fresh drop is formed and imaged repeatedly as it ages. Each image is fitted to the Young-Laplace equation to give the surface tension at that moment. Plotting the results against time gives the dynamic curve, typically from about a second to many minutes. Younger surfaces need a bubble pressure method.

References

Primary sources and further reading.

1. Ward, A. F. H., and Tordai, L. (1946). Time-dependence of boundary tensions of solutions I. The role of diffusion in time-effects. Journal of Chemical Physics, 14, 453 to 461. https://doi.org/10.1063/1.1724167
2. Eastoe, J., and Dalton, J. S. (2000). Dynamic surface tension and adsorption mechanisms of surfactants at the air water interface. Advances in Colloid and Interface Science, 85(2 to 3), 103 to 144. https://doi.org/10.1016/S0001-8686(99)00017-2
3. Berry, J. D., Neeson, M. J., Dagastine, R. R., Chan, D. Y. C., and Tabor, R. F. (2015). Measurement of surface and interfacial tension using pendant drop tensiometry. Journal of Colloid and Interface Science, 454, 226 to 237. https://doi.org/10.1016/j.jcis.2015.05.012
4. Surface tension measurement guide
5. Surfactant
6. Spreading