Analytical description of elastocapillary membranes held by needles

Fluid objects bounded by elastocapillary membranes display intriguing physical properties due to the interplay of capillary and elastic stresses arising upon deformation. Increasingly exploited in foam or emulsion science, the mechanical properties of elastocapillary membranes are commonly characterised by the shape analysis of inflating/deflating bubbles or drops held by circular needles. These impose complex constraints on the membrane deformation, requiring the shape analysis to be done using elaborate numerical fitting procedures of the shape equations. While this approach has proven quite reliable, it obscures insight into the underlying physics of the problem. We therefore propose here the first fully theoretical approach to this problem using the elastic theory for a membrane with additive contributions of capillary and Hookean-type elastic stresses. We exploit this theory to discuss some of the key features of the predicted pressure-deformation relations. Interestingly, we highlight a breakdown of the quadratic approximation at a well-defined value of the elastocapillary parameter depending on the shape of the reference state, which is regularized by the non-quadratic terms. Additionally, we provide an analytical relationship which allows experimentalists to obtain the elastocapillary properties of a membrane by simple measurement of the height and the width of a deformed bubble (or a drop).