Rigorous derivation of a mean field model for the Ostwald ripening of thin films

In the late stage of thin liquid films, liquid droplets are connected by an ultra thin residual film. Experimental studies and numerical simulations show that the size distributions of liquid droplets approach a self-similar form. However, theoretical study of the size distributions is lacking because it has been a challenge to retrieve statistical information from the mathematical PDE model of thin films. To facilitate the study of the statistical information, we rigorously derive a mean field model for the Ostwald ripening of thin liquid films through homogenization. This model corresponds to the dilute limit when the droplets are far away from each other and occupy a very small part of the thin film. Our analysis captures the screening effect of the droplets and shows that the mean field spatially varies in a length scale proportional to the screening length.

Keywords

thin film, coarsening, mean field model, homogenization

2010 Mathematics Subject Classification

35B27, 35K25, 35K55, 76A20

Full Text (PDF format)

The author acknowledges support by the US NSF through grants DMS-1802863 and DMS-1815746.

Received 13 December 2018

Accepted 9 September 2019

Published 20 June 2022