Pressure–deformation relations of elasto-capillary drops (droploons) on capillaries†
An increasing number of multi-phase systems exploit complex interfaces in which capillary stresses are coupled with solid-like elastic stresses. Despite growing efforts, simple and reliable experimental characterisation of these interfaces remains a challenge, especially of their dilational properties. Pendant drop techniques are convenient, but suffer from complex shape changes and associated fitting procedures with multiple parameters. Here we show that simple analytical relationships can be derived to describe reliably the pressure–deformation relations of nearly spherical elasto-capillary droplets (“droploons”) attached to a capillary. We consider a model interface in which stresses arising from a constant interfacial tension are superimposed with mechanical extra-stresses arising from the deformation of a solid-like, incompressible interfacial layer of finite thickness described by a neo-Hookean material law. We compare some standard models of liquid-like (Gibbs) and solid-like (Hookean and neo-Hookean elasticity) elastic interfaces which may be used to describe the pressure–deformation relations when the presence of the capillary can be considered negligible. Combining Surface Evolver simulations and direct numerical integration of the drop shape equations, we analyse in depth the influence of the anisotropic deformation imposed by the capillary on the pressure–deformation relation and show that in many experimentally relevant circumstances either the analytical relations of the perfect sphere may be used or a slightly modified relation which takes into account the geometrical change imposed by the capillary. Using the analogy with the stress concentration around a rigid inclusion in an elastic membrane, we provide simple non-dimensional criteria to predict under which conditions the simple analytical expressions can be used to fit pressure–deformation relations to analyse the elastic properties of the interfaces via “Capillary Pressure Elastometry”. We show that these criteria depend essentially on the drop geometry and deformation, but not on the interfacial elasticity. Moreover, this benchmark case shows for the first time that Surface Evolver is a reliable tool for predictive simulations of elastocapillary interfaces. This opens doors to the treatment of more complex geometries/conditions, where theory is not available for comparison. Our Surface Evolver code is available for download in the ESI.