Elastic Langmuir layers and membranes subjected to unidirectional compression: wrinkling and collapse

We apply the two-dimensional elastic continuum model to describe the wrinkling of elastic Langmuir layers (membranes) subjected to unidirectional compression. The effects of the dilatational, shear, and bending elasticities are taken into account. Among the numerous solutions of the generalized Laplace equation, corresponding to different membrane tensions, we determine the membrane shape as the profile that minimizes the energy of the system. In the case of small deformations, the problem can be linearized. Its solution predicts a wavelike shape of the compressed membrane. At negligibly small bending elasticity, the energy of the system is minimal for a sinusoidal profile, whose amplitude and wavelength tend to zero. In the opposite limiting case, where the effect of bending elasticity prevails over the effect of gravity, the membrane has a half-wave profile. When the two effects are comparable, the membrane shape exhibits multiple periodic wrinkles (ripples). An expression is derived for calculating the bending elasticity (rigidity) from the wavelength, and reasonable values are obtained from available experimental data. To determine the membrane shape at larger out-of-plane deformations, we solved numerically the respective nonlinear problem. Depending on the values of the physical parameters, the theory predicts various shapes: nonharmonic oscillations, toothed profiles, and profiles with two characteristic wavelengths. The results can be used for determining the bending elastic modulus of Langmuir films (membranes) as well as for the interpretation of buckling and collapse of monolayers.