Landau-Levich equation

The use of the dip-coating technique allows to obtain different overlay thicknesses by acting on the solution viscosity and extraction speed as stated by the Landau-Levich equation (see (3.14)). In particular, thicker overlays can be obtained by increasing the extraction speed and/or by increasing the solution viscosity. 

The thickness (li) of a coating obtained by dipping is given by the Landau-Levich equation ... 

Fig. 11. Coated film thickness data of the H-dip)-coated EL layer (a) and the PEDOT PSS layer (b) as a function of carrying sp>eed for two gap heights (0.9 and 0.8 mm). The solid curves show the theoretical predictions of the Landau Levich equation. (Park Han, 2009)...

The coating thickness is mainly defined by the withdrawal speed v, by the solid content and the viscosity r of the liquid. If the withdrawal speed is chosen such that the shear rates keep the system in the Newtonian regime, the coating thickness h can be calculated by the Landau-Levich equation (Landau and Levich, 1942) as... 

In the dip coating process, the substrate is immersed into a sol containing the precursor, catalyst, and the solvent and removed in a very controlled manner, resulting in a film that gels as the substrate is taken out. Subsequent heat treatment yields the desired product. If the withdrawal speed is such that the shear rate-shear stress dependence obeys Newtonian flow conditions, the coating thickness follows the Landau-Levich equation ... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

---