Capillary Leveling of Thin-Film Surface Roughness

Capillary leveling calculator

Estimate how a small sinusoidal roughness or coating waviness decays in a thin Newtonian liquid film.

Film and disturbance

Amplitude decay

How to use

Enter mean film thickness h₀, initial peak amplitude A₀, and sinusoidal wavelength λ in the displayed units.
Enter dynamic viscosity μ and surface tension γ at the operating temperature; do not enter kinematic viscosity.
Enter liquid density ρ. Enable Include stabilizing gravity only for a horizontal film below the gas.
Enter an elapsed time to calculate A(t), and enter a smaller target amplitude to calculate the required leveling time.
Select Calculate to update the canonical amplitude, metrics, warning, and graph. Select Reset to restore the worked example.
Interpret A(t), time to target, half-life, β, and the two validity ratios together. A gravity/capillary ratio near zero means capillarity dominates.
Use the result only when A₀/h₀ and h₀/λ are small; caution and outside-range messages identify weak or violated assumptions.
Do not use this model for finite-amplitude waves, inverted films, curing or evaporating coatings, non-Newtonian liquids, surfactant gradients, dewetting, or final process qualification.

Equations used

For h(x,t) = h₀ + A(t)cos(kx), linear long-wave theory gives:

k = 2π/λ

β_γ = γh₀³k⁴/(3μ)

β_g = ρgh₀³k²/(3μ)

β = β_γ + β_g

A(t) = A₀ exp(−βt)

t_target = ln(A₀/A_target)/β

t_1/2 = ln(2)/β

Gravity/capillary ratio = β_g/β_γ

Set β_g = 0 when gravity is disabled. Total β, β_γ, and β_g are decay rates in s⁻¹; t_1/2 is the time in seconds for the amplitude to halve. The ratio β_g/β_γ is dimensionless; it is zero when gravity is disabled, below 1 when capillary leveling is faster, and above 1 when stabilizing gravity is faster. Symbols and units: h₀, A, and λ are converted to metres; k is m⁻¹; μ is Pa·s; γ is N/m; ρ is kg/m³; g = 9.80665 m/s²; β is s⁻¹; t is seconds. Sign convention: film height and amplitude are positive upward from the substrate; gravity is stabilizing only for liquid below gas.

Assumptions and limits: incompressible Newtonian liquid, constant μ and γ, no slip, negligible gas stress and inertia, a small sinusoidal disturbance, and long-wave geometry. Prefer A₀/h₀ ≤ 0.1 and h₀/λ ≤ 0.1. The model omits evaporation, curing, disjoining pressure, surfactants, and substrate motion.

References: A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Reviews of Modern Physics 69, 931–980 (1997), Section II.D, equations 2.37–2.38, doi:10.1103/RevModPhys.69.931; S. E. Orchard, “On surface levelling in viscous liquids and gels,” Applied Scientific Research 11, 451–464 (1963), doi:10.1007/BF03184629.