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Sandbox Physics
Dynamic interfaces

L2 · Chapter 4

How a drop edge moves across a solid

Sandbox Physics ·

A spreading drop must move its edge forwards. The usual no-slip condition holds liquid at rest against a stationary solid. A perfectly sharp edge brings those requirements together at zero thickness. A precursor model leaves a thin layer ahead of the drop, so its edge becomes a transition between thick and thin liquid.

The velocity changes over an ever smaller distance

A contact line is where solid, liquid and gas meet. If it advances across a stationary solid at finite speed, the flow must connect liquid at rest on the solid with the moving edge. In an ideal sharp wedge, the available distance shrinks as we approach that edge.

Let rr measure distance from the line and UU its speed. A velocity gradient scales as U/rU/r. Viscous dissipation scales with its square, whereas a wedge area element scales as r drr\,\mathrm dr. Their product gives a logarithmically divergent integral. Huh and Scriven identified this difficulty in their 1971 creeping-flow analysis.

A dynamical model must specify what replaces the zero-distance limit. Finite slip at the solid and a precursor layer ahead of the drop are two possible descriptions. This course uses a precursor layer while retaining no slip underneath the liquid.

D∼ηU2∫rmin⁡Rdrr=ηU2log⁡Rrmin⁡\mathcal D\sim\eta U^2\int_{r_{\min}}^R\frac{\mathrm dr}{r}=\eta U^2\log\frac R{r_{\min}}
This scaling argument omits a wedge-angle coefficient. Its role is to show why the smallest length matters.

Sources: Huh & Scriven

Choose a wetting energy with a thin-layer minimum

The local energy per substrate area is f(h)f(h). Long-range attraction lowers the energy of a thinner film. Short-range repulsion grows rapidly near zero thickness. The chosen powers give one minimum at h=bh=b, the precursor thickness.

Differentiating this energy gives the local cost of changing thickness. Our sign convention for disjoining pressure is Π=−fprime(h)\Pi=-fprime(h). This pressure and capillarity together determine the flux. The precursor evolves through the same height equation as the thicker liquid.

f(h)=−A2h2+Ab35h5,Π(h)=−Ah3+Ab3h6f(h)=-\frac A{2h^2}+\frac{A b^3}{5h^5},\qquad \Pi(h)=-\frac A{h^3}+\frac{A b^3}{h^6}
First interpret height and horizontal distance in the same length units, with mutually consistent energy and tension units. The browser uses separate scales, handled in the worked example.
f′(b)=0,f′′(b)=3Ab4>0,f(b)=−3A10b2f'(b)=0,\qquad f''(b)=\frac{3A}{b^4}>0,\qquad f(b)=-\frac{3A}{10b^2}
The first two statements establish a locally stable layer. The energy depth determines the large-drop equilibrium slope.

Sources: U. Thiele

Integrate across the static edge

Consider the edge of a sufficiently large stationary drop, with negligible curvature pressure. On the thin-film side, height tends to bb and slope to zero. Equilibrium requires −γhxx+fprime(h)=0-\gamma h_{xx}+fprime(h)=0. Multiply by the slope: both terms become complete spatial derivatives and can be integrated.

The precursor boundary fixes the integration constant. At h≫bh\gg b the wetting potential tends to zero, leaving the outer wedge slope. A small physical angle equals that slope in radians. A finite drop with appreciable curvature pressure requires an additional pressure term and a shifted thin-layer height; the following relation is the local large-drop limit.

−γhxxhx+f′(h)hx=0⟹γ2hx2=f(h)−f(b)-\gamma h_{xx}h_x+f'(h)h_x=0\quad\Longrightarrow\quad \frac\gamma2h_x^2=f(h)-f(b)
The precursor has zero slope, which supplies the integration constant.
θe2≃−2f(b)γ=3A5γb2\theta_{\mathrm e}^2\simeq-\frac{2f(b)}\gamma=\frac{3A}{5\gamma b^2}
This connects the energy minimum to surface tension. Check that the resulting physical angle is small.

A model slope of 1.39 can describe a shallow edge

The browser measures height in an initial film thickness and horizontal distance in a separate length. Their ratio is ε=0.05\varepsilon=0.05. For A=0.2A=0.2, b=0.25b=0.25 and γ=1\gamma=1, the dimensionless equilibrium slope is about 1.391.39.

Physical slope includes the aspect ratio. It is about 0.06930.0693, corresponding to 3.97∘3.97^\circ in the small-angle approximation. Reading the model slope directly as radians would turn a shallow edge into an apparently steep drop. An enlarged vertical drawing makes this mistake particularly easy.

Now double the precursor thickness while holding the outer angle fixed. The ratio A/γA/\gamma must increase fourfold. This controls the approximate macroscopic wetting angle while changing the contact-region scale. The flat-film growth spectrum still changes, because it depends on the energy curvature at the mean thickness.

θe≃ε3A5γb2=0.051.92≃0.0693 rad\theta_{\mathrm e}\simeq\varepsilon\sqrt{\frac{3A}{5\gamma b^2}}=0.05\sqrt{1.92}\simeq0.0693\ \mathrm{rad}
This conversion applies to the browser’s anisotropic coordinate scaling. Drawing magnification does not change the physical angle.

The edge can move while free energy falls

The energy contains capillary cost and local wetting energy. Its variational derivative is the chemical potential μ=−γhxx+fprime(h)\mu=-\gamma h_{xx}+fprime(h). Insert the conserved height equation into the time derivative of energy, then integrate by parts. Periodicity cancels the boundary terms and leaves a negative integral of a square.

Growing ridges can therefore lower total free energy. Their wetting-energy gain can exceed their increased capillary cost. Volume is conserved through a separate cancellation of end fluxes. Both checks are useful: a decreasing energy alone cannot establish that liquid is conserved.

The visible edge in this description connects thick liquid with a precursor layer. Following that transition describes spreading and retraction. Predicting a molecular rupture location requires additional small-scale physics.

dFdt=∫μht dx=∫μ ∂x(h3μx) dx=−∫h3μx2 dx≤0\frac{\mathrm dF}{\mathrm dt}=\int\mu h_t\,\mathrm dx=\int\mu\,\partial_x(h^3\mu_x)\,\mathrm dx=-\int h^3\mu_x^2\,\mathrm dx\le0
Dimensionless mobility is the height cubed. There is no evaporation, imposed surface stress or volume source.
ddt∫h dx=[h3μx]0L=0\frac{\mathrm d}{\mathrm dt}\int h\,\mathrm dx=[h^3\mu_x]_0^L=0
Periodic end fluxes agree. Closed, no-flux boundaries give the same result.

Sources: U. Thiele

Predict, change and explain

Separate contact-region scale from flat-film growth

  1. Choose Growing ridges, expand Initial noise and wetting layer, and compare two precursor thicknesses at fixed attraction. Predict the change in growth rates first.
  2. Watch the thinnest liquid approach the precursor layer and check volume. Export heights and inspect free energy in successive records.
  3. Holding an apparent angle fixed requires changing the energy strength too. Calculate the necessary parameters before choosing a pair within the control ranges.

Exercises and explanations

  1. Double precursor thickness at fixed attraction and tension. What happens to the small equilibrium angle?

    Read the explanation

    It halves: angle is inversely proportional to precursor thickness. This comparison changes macroscopic wettability as well as the contact-region scale.

  2. Why does an ever thinner precursor not automatically give a more realistic simulation?

    Read the explanation

    The grid resolves it less well, short-range pressures become steeper, and the equilibrium angle and dynamics change. State which macroscopic quantities are held fixed and check grid convergence.

  3. Can the ridges grow while free energy falls?

    Read the explanation

    Yes. Free energy includes wetting and capillary contributions. A decrease in wetting energy can exceed the increased capillary cost.

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. Huh & Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci. 35, 85–101 (1971)

    The moving sharp contact line with no slip produces divergent stress and dissipation. The course explains the scaling argument; the full wedge solution is not recalculated.

  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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