LandauLevichDerjaguin

Landau–Levich–Derjaguin refers to a foundational theory in coating science that describes the thickness of a viscous liquid film entrained on a solid substrate as it is withdrawn from a liquid bath, a process known as dip coating. The theory was developed by Lev Landau and Evgeny Levich, with important extensions by B. V. Derjaguin, and provides a quantitative framework for predicting how a coating forms under steady withdrawal.

In its standard form, the film thickness h on a plate pulled vertically from a Newtonian liquid bath is governed by a balance between viscous forces in the thin coating layer and capillary forces in the dynamic meniscus. For low capillary numbers, the thickness scales as h ≈ 0.94 l_c Ca^{2/3}, where Ca = ηV/γ is the capillary number, η is the liquid viscosity, V is the withdrawal speed, γ is the surface tension, and l_c = sqrt(γ/(ρg)) is the capillary length (ρ is density, g is gravitational acceleration). This yields a practical relation h ∝ V^{2/3}, illustrating how faster withdrawal yields thicker coatings.

The theory relies on assumptions of a clean, flat substrate, no slip at the solid, negligible inertia, steady state, and gravity considered via the capillary length. It is most accurate for Ca ≪ 1 and film thickness smaller than l_c. The LLD framework has become a standard baseline for dip coating processes and has spurred numerous extensions to non-Newtonian fluids, surfactants, slip at the substrate, curved surfaces, evaporation, and disjoining pressure effects that modify the coating behavior.