FR18 · A liquid film chooses a spacing

Thin-Film Instability

Why does a nearly flat liquid film grow a pattern instead of smoothing out? Surface tension smooths short ripples, but attraction to the substrate can amplify longer ones. Their competition selects the fastest-growing spacing. Liquid moves from thinning regions into thicker ridges, with the total volume conserved.

Release a slightly uneven film and watch liquid collect into ridges. Compare the growth of its Fourier modes with the small-amplitude prediction, or drag the film to change its starting profile.

Elapsed model time $24$

Thinnest part $0.8\,h_0$

Strongest recorded mode $12\,\ell$

Relative volume change $0$

Model assumptions

This periodic, one-dimensional lubrication model describes a nonvolatile Newtonian liquid with no slip and negligible inertia. A chosen wetting potential leaves a precursor layer instead of a dry contact line. Heights, time and horizontal distance are dimensionless; the vertical scale is enlarged. Growth rates are fitted from saved simulated height profiles during the first six time units. Finite amplitudes and the numerical grid can shift them from the continuum reference. Evolution stops if the thickness or small-slope checks fail. Three-dimensional holes, molecular rupture and thermal fluctuations are outside this model.

Which ripple is growing fastest?

Cyan bars: cosine peak amplitudes from the latest stored height profile. Amber points: growth fitted to stored profiles in the first six time units. White line: the continuum small-amplitude prediction.

Amplitude by mode number

a_n/h_0

Early growth, first seven modes

\sigma_n

Reading the comparison

Positive rates amplify a mode; negative rates damp it. The finite grid approximates spatial derivatives, and finite initial ripples couple modes. Choose Small-amplitude test to reduce the second effect. Later bars show the evolving pattern; rate points keep the early fitting window.

All displayed profiles are simulated. No experimental height measurements or molecular rupture positions are shown.

Start with the essentials

Focus question: Why does a nearly flat liquid film grow a pattern instead of smoothing out?
One-sentence intuition: Short ripples cost curvature energy; longer ones can lower the wetting energy by separating thick and thin regions.

Core mathematical model

Conserved liquid flow

\partial_t h=\partial_x\!\left(h^3\partial_x\mu\right),\qquad \mu=-\gamma\partial_{xx}h+f'(h)

Height is measured in the mean initial thickness. Horizontal distance and time use the stated model scales. A gradient of the chemical potential drives liquid along the substrate; the periodic flux conserves volume.

A wetting potential with a thin remaining layer

f(h)=-\frac{A}{2h^2}+\frac{A b^3}{5h^5}

Attraction favors thinning; the short-range repulsive term grows rapidly at small thickness and has its minimum at the precursor height. The chosen potential is a teaching model, with no fitted material parameters.

Growth of a small sinusoidal ripple

\sigma(k)=-k^2\!\left[\gamma k^2+f''(1)\right],\qquad k_{\max}=\sqrt{\frac{-f''(1)}{2\gamma}}

Wavenumber is the number of radians per model length. Positive growth amplifies a ripple. The fastest mode exists when the wetting-energy curvature is negative. The white curve below is this continuum prediction; colored points come from saved simulation profiles.

Common difficulties

Surface tension and instability

Typical misconceptionSurface tension makes the film unstable because it pulls the liquid into drops.

Better mental modelIn this flat-film model capillarity damps every Fourier mode. The wetting potential supplies the instability. Turn the substrate attraction to zero and watch all ripples shrink.

Run the experiment

01
Predict a spacing
Choose Growing ridges. Compare the tallest spectrum bars with the early growth-rate points.
What to observe: Several modes grow at different speeds. Later, nonlinear flow changes their amplitudes and couples the modes.
02
Remove the attraction
Choose Ripples flatten and watch the same seeded starting noise relax.
What to observe: The remaining capillary force smooths the film. The spectrum bars fall, and the free energy decreases.
03
Make a new starting profile
Drag across the liquid surface, then release it to resume the flow. Use Small-amplitude test to compare growth rates more closely.
What to observe: Dragging adds a localized volume-neutral perturbation and starts a new record. Smaller starting amplitudes approach the linear prediction more closely.

Thin-Film Instability

Which ripple is growing fastest?

Two tendencies choose a pattern

Core mathematical model

Conserved liquid flow

A wetting potential with a thin remaining layer

Growth of a small sinusoidal ripple

Common difficulties

Surface tension and instability

Run the experiment

Predict a spacing

Remove the attraction

Make a new starting profile

Related experiments and articles

Curvature, films, thermal flow and liquid threads