Volume–controlled buckling of thin elastic shells: Application to crusts formed on evaporating partially–wetted droplets

Motivated by the buckling of glassy crusts formed on evaporating droplets of polymer and colloid solutions, we numerically model the deformation and buckling of spherical elastic caps controlled by varying the volume between the shell and the substrate. This volume constraint mimics the incompressibility of the unevaporated solvent. Discontinuous buckling is found to occur for sufficiently thin and/or large contact angle shells, and robustly takes the form of a single circular region near the boundary that ‘snaps’ to an inverted shape, in contrast to externally pressurised shells. Scaling theory for shallow shells is shown to well approximate the critical buckling volume, the subsequent enlargement of the inverted region and the contact line force. PACS numbers: 46.32.+x, 46.70.De, 81.15.-z The properties of fluid interfaces are strongly modified by the presence of a permeable solid layer [1]. This is deliberately exploited in solid–stabilised or ‘Pickering’ emulsions, where added colliodal particles become pinned at the liquid– liquid interfaces, inhibiting dewetting and vastly reducing droplet coalescence [2, 3]. Solid interfacial layers may also form spontaneously, as in the observation of a glassy ‘crust’ on the surface of evaporating polymer or colloid solutions in thin film [4, 5], suspended droplet [6] and partially–wetted droplet [7, 8, 9, 10, 11, 12] geometries. In all of the above examples, it has been observed that, as the fluid volume(s) change with time, either by evaporation or draining (and might also be expected to occur during Ostwald ripening of emulsions), the interfacial area contracts, compressing the solid layer which then wrinkles or buckles as an elastic sheet. Predicting the onset and nature of this buckling is thus crucial to controlling system evolution and preventing the occurrence of undesired features in any given application. The deformation and buckling of thin elastic shells is a classic problem; see e.g. [13, 14, 15, 16]. However, the systems mentioned above introduce a complication that has not, to the best of our knowledge, been properly treated, namely that the presence of the incompressible fluid imposes a volume constraint on the space of allowed deformations (on time scales shorter than e.g. the evaporation time) [7]. It might be thought that controlling the volume V would be identical to imposing some corresponding pressure P , but this is not necesarily true with regards the stability of the shell: an S–shaped P–V curve would reveal different limit points, and hence distinct buckling events [17, 18], depending on which quantity is being controlled (similar to S–shaped flow curves in non–linear rheology, which may be stable under stress control but unstable under an imposed flow rate or vice versa; see e.g. [19]). Volume–controlled buckling of spherical elastic caps 2 Below we describe numerical simulations of a thin elastic shell with the same geometry as the crust on a partially–wetted spherical droplet, with the shell deformation driven by incrementally reducing the volume between it and the substrate. This geometry was chosen for its industrial relevance: the drying of inks and paints [20, 21] and ‘bottom–up’ manufacturing of microscale electronic devices via microfluidic technology [22] all involve the evaporation of partially–wetted solutions. Our primary finding is of a single class of buckling event for thin and/or steep shells, in which a region of the shell near the fixed boundary ‘snaps’ to an inverted configuration [see Fig. 1], in contrast to the range of axisymmetry preserving and breaking buckling modes already documented for the pressure–controlled case [15]. For shallow shells, the critical buckling volume Vc extracted from the simulations, as well as the subsequent enlargement of the inverted region, is shown to be well estimated by predictions of a simple scaling theory. We first describe the employed simulation method. Methodology: The crust is modelled as a thin elastic shell described by a 2–dimensional mid–surface S parameterised by a thickness h, such that the crust surfaces lie at a distance ±h/2 normal to S. As the droplet surface is spherical, we take S to be a section of sphere of radius R that intersects the substrate at an angle θ0; see Fig. 1(c). Deviations from this unstressed state are described by a pair of rank–2 tensors; uij , the in–surface or membrane strains, and the change in curvature ∆χij (with corrections for the difference between centroidal and mid–surfaces [13]). A standard closure due to Kirchoff and Love then gives the following expression for the strain energy, partitioned into a membrane (or ‘stretch’) contribution Ustretch, and a flexural (or ‘bend’) term Ubend [14],