From drop-shape analysis to stress-fitting elastometry

Drop-shape analysis using pendant or sessile drops is a well-established experimental technique for measuring the interfacial or surface tension, and changes thereof. The method relies on deforming a drop by either gravity or buoyancy and fitting the Young-Laplace equation to the drop shape. Alternatively one can prescribe the shape and measure the pressure inside the drop or bubble using pressure tensiometry. However, when an interface with a complex microstructure is present, extra and anisotropic interfacial stresses may develop due to lateral interactions between the surface-active moieties, leading to deviations of the drop shape or even a wrinkling of the interface. To extract surface-material properties of these complex interfaces using drop-shape analysis or pressure tensiometry, the Young-Laplace law needs to be generalized in order to account for the extra and anisotropic stresses at the interface. In the present work, we review the different approaches that have been proposed to date to extract the surface tension as the thermodynamic state variable, as well as other rheological material properties such as the compression and the shear modulus. To evaluate the intrinsic performance of the methods, computer generated drops are subjected to step-area changes and then subjected to analysis using the different methods. Shape-fitting methods, now combined with an adequate constitutive method, do however perform rather poorly in determining the elastic stresses, especially at small area strains. An additional measurement of the pressure or capillary-pressure tensiometry is required to improve the sensitivity. However, pressure-based methods still require the knowledge of the undeformed reference state, which may be difficult to achieve in practice. Moreover, it is not straightforward to judge from what point onwards one needs to go beyond the Young-Laplace equation. To overcome these limitations, a method based on stress fitting, which uses a local force balance method, is introduced here. One aspect of this new method is the use of the Chebyshev transform to numerically describe the contour shape of the drop interface. For all methods we present a detailed error analysis to evaluate if, and with what precision, surface material parameters can be extracted. Depending on the desired information, different ideal experimental conditions and most suitable methods are discussed, in addition to having a criterion to investigate if extra and anisotropic stresses matter.