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Sandbox Physics
Dynamic interfaces

L2 · Chapter 3

Which wavelength grows out of noise first?

Sandbox Physics ·

A nearly flat liquid film can amplify selected disturbances. Curvature still suppresses short wrinkles, but interaction with the substrate may favour thinner thin regions and thicker thick regions. Their competition selects a fastest growing wavelength.

The substrate adds an energy that depends on thickness

Write the wetting energy per substrate area as f(h)f(h). Capillary energy penalizes slopes; wetting energy favours particular thicknesses. Varying height in the total energy gives a generalized pressure μ\mu. Viscosity remains η\eta throughout this chapter.

Where thinning lowers local energy, liquid can be expelled towards neighbouring thicker regions. Curvature resists sharp edges. Both effects enter one driving pressure and one transport law.

F[h]=∫[γ2hx2+f(h)]dx,μ=−γhxx+f′(h)F[h]=\int\left[\frac\gamma2 h_x^2+f(h)\right]\mathrm dx,\qquad \mu=-\gamma h_{xx}+f'(h)
The surface energy uses small slopes. Disjoining pressure is defined as the negative derivative of wetting energy.
ht=∂x ⁣[M(h)∂xμ],M(h)=h33η,Π(h)=−f′(h)h_t=\partial_x\!\left[M(h)\partial_x\mu\right],\qquad M(h)=\frac{h^3}{3\eta},\qquad \Pi(h)=-f'(h)
The positive mobility sets the rate of energy-lowering transport.

Sources: U. Thiele · Oron

Perturb the flat film with one small wave

Add a small cosine to a film of thickness h0h_0. If its amplitude changes exponentially, the exponent coefficient σ\sigma is the growth rate: positive for growth, negative for decay. Substitute this form and retain terms linear in amplitude.

The base state has no pressure gradient. Changes in mobility therefore contribute only at higher order, and we use M0=M(h0)M_0=M(h_0). Curvature supplies a fourth power of wavenumber; wetting supplies a second power. Their sum gives the dispersion relation.

h=h0+a0eσtcos⁡(kx)σ(k)=−M0k2[γk2+f′′(h0)]\begin{gathered}h=h_0+a_0e^{\sigma t}\cos(kx)\\\sigma(k)=-M_0k^2\left[\gamma k^2+f''(h_0)\right]\end{gathered}
This prediction holds while amplitude remains small compared with mean thickness.
f′′(h0)<0:kc=−f′′(h0)γk∗=kc2σ∗=M0[f′′(h0)]24γ\begin{gathered}f''(h_0)<0:\quad k_c=\sqrt{\frac{-f''(h_0)}\gamma}\\k_*=\frac{k_c}{\sqrt2}\\\sigma_*=\frac{M_0[f''(h_0)]^2}{4\gamma}\end{gathered}
Nonzero modes below the cutoff grow. The fastest one lies between zero and the cutoff.
Analytic growth rateσ^=K2(1−K2)\widehat\sigma=K^2(1-K^2)
K∗=1/2K_*=1/\sqrt2Kc=1K_c=1
  • Unstable film
  • Stable film (separate rate scale)
Newly calculated schematic of the analytic curves in Thiele (2010), Figure 1. The horizontal coordinate is wavenumber divided by the unstable cutoff. No experimental points are shown.

Sources: U. Thiele · Oron

Worked example: a finite film admits discrete modes

Choose dimensionless M0=1M_0=1, γ=1\gamma=1 and fprimeprime(h0)=−1fprimeprime(h_0)=-1. Continuous wavenumbers give k∗=1/2k_*=1/\sqrt2 and σ∗=1/4\sigma_*=1/4. A periodic domain of length L=20L=20 admits only kn=2πn/Lk_n=2\pi n/L.

For n=1,2,3n=1,2,3, the rates are approximately 0.0890,0.239,0.09920.0890,0.239,0.0992. Mode two grows fastest, with spacing L/2=10L/2=10. The infinite-domain optimum is 2π2≃8.892\pi\sqrt2\simeq8.89; the finite container selects among the wavelengths it can accommodate.

Noise contains many wavelengths with different initial amplitudes. The fastest rate need not give the first visible peak: a neighbouring mode can start larger. Test the exponential slope to separate growth from initial amplitude.

Measure growth from stored height profiles

Project each saved profile onto the sine and cosine at the same wavenumber. Combine the two coefficients to obtain a phase-independent amplitude. The slope of log⁡an(t)\log a_n(t) against time estimates growth during the small-amplitude stage.

Use records above rounding noise but below strongly nonlinear deformation. Later, modes exchange energy and drops can merge or coarsen. Save the original heights, times, seed, parameters and model version so another reader can repeat the fit with a different time interval.

an=2∣1N∑j=0N−1(hj−hˉ)e−2πinj/N∣,σ^n=log⁡an(t2)−log⁡an(t1)t2−t1a_n=2\left|\frac1N\sum_{j=0}^{N-1}(h_j-\bar h)e^{-2\pi i n j/N}\right|,\qquad \widehat\sigma_n=\frac{\log a_n(t_2)-\log a_n(t_1)}{t_2-t_1}
This normalization returns the peak amplitude of a single cosine. The mean mode separately checks mass conservation.

Predict, change and explain

Let noise select a spacing

  1. Predict the unstable wavenumbers, run the noise initial state and compare rates extracted from saved profiles with the analytic reference.
  2. Change tension, predict the direction of the wavelength shift, and test with a new record.
  3. Export the profiles and compare early and late fits. Identify which interval satisfies the small-amplitude assumption.

Exercises and explanations

  1. Why must the zero-wavenumber growth rate vanish?

    Read the explanation

    That mode changes mean thickness. A closed or periodic film conserves its volume and cannot change its mean height spontaneously.

  2. Multiply tension by four at fixed wetting energy and mobility. What happens to the fastest wavelength?

    Read the explanation

    The fastest wavenumber halves, the wavelength doubles and the maximum growth rate falls to a quarter.

Sources and calculation scope

Examples and diagrams are newly authored. The paper comparison recalculates analytic relations; the research reading distinguishes those curves from experimental photographs and molecular simulations.

  1. Oron, Davis & Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69, 931 (1997)

    Published review and framework for long-wave reductions. Specific implemented equations are cross-checked against the accessible author papers below.

  2. U. Thiele, Thin film evolution equations from (evaporating) dewetting liquid layers to epitaxial growth, J. Phys.: Condens. Matter 22, 084019 (2010)

    Author-hosted published article, equations (1)–(3), section 2 potential and equation (5), Figure 1. Recalculation covers the analytic linear dispersion only; nonlinear continuation branches in Figures 2–3 are a reading topic.

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