A mathematical model for drying paint layers

Many industrial processes involve the coating of substrates with thin layers of paint. This paper is concerned with modelling the variations in layer thickness which may occur as a paint layer dries. Firstly, a systematic derivation is provided of a model based on classical lubrication theory for a drying paint layer consisting of a non-volatile resin and a volatile solvent. The effects of variable surface tension, viscosity, solvent diffusivity and solvent evaporation rate are all included in the model. This analysis makes explicit the validity of the physically intuitive approximations made by earlier authors and hence clarifies when the model is appropriate. Secondly, the model is used to analyse the evolution of small perturbations to the thickness of, and the concentration of solvent in, a drying paint layer. This analysis provides an analytical description of the 'reversal' of an initial perturbation to the thickness of the layer and the appearance of a perturbation to an initially flat layer caused by an initial perturbation to the concentration of solvent. Thirdly, it is shown how a simplified version of the model applicable to the case of surface-tension-gradient-dominated flow can be derived and solved as an initial-value problem. Fourthly, the applicability of the present theory developed for solvent-based high-gloss alkyd paints to waterborne coatings is discussed. Finally, the results obtained are summarised and the practical implications of the work are discussed.