Chemo-mechanical model for degradation of oil paintings by amorphous and crystalline metal soaps

Metal soap formation is recognised as a critical degradation mechanism in historical oil paintings, which threatens the preservation of museum collections worldwide. Metal soaps form via a complex sequence of chemical reactions between metal ions released by the pigments and saturated fatty acids originating from the drying oil. The latest advances in chemistry research suggest that metal ions and saturated fatty acids may initially react by means of a reversible reaction, which leads to the formation of metal soaps in an amorphous state. Metal soaps may subsequently crystallise via an irreversible reaction into large aggregates that deform the paint layers, potentially triggering delamination, cracking, and ultimately flaking of the paint. This paper proposes a chemo-mechanical model to predict metal soap formation and the consequent mechanical damage in historical oil paintings. The chemical process is described in terms of a set of diffusion-reaction equations, which account for both the reversible reaction between free saturated fatty acids and metal ions forming amorphous metal soap, and the subsequent irreversible reaction to crystalline metal soap. The chemical model is two-way coupled with a mechanical model that effectively describes the cracking processes caused by metal soap formation and growth. The coupling is generated from the mechanical model by accounting for the development of a chemically-induced growth strain in the crystalline metal soap. In addition, the presence of cracks locally hampers the diffusion of chemical species, which is taken into account in the chemical model through a dependency of the diffusion parameter at the crack faces on the amount of mechanical damage generated. The spatial development of the crystalline metal soap phase is simulated by using a tailor-made scanning algorithm that identifies the reaction zone in which metal soap formation takes place. The proposed model is calibrated on experimental data presented in the literature. The model is subsequently applied to analyse two numerical examples that are representative of typical metal soap-related degradation processes observed in historical oil paintings, revealing that the growth process of crystalline metal soap, the deformation of the paint surface, and the consequent cracking and delamination patterns are predicted in a realistic fashion.