Abstract

We present complete analytical solutions describing the deformations of both rectangular and circular lipid membranes subjected to local inflammations and coordinate-dependent (nonuniform) property distributions. The membrane energy potential of the Helfrich type is refined to accommodate the coordinate-dependent responses of the membranes. Within the description of the superposed incremental deformations and Monge parametrization, a linearized version of the shape equation describing coordinate-dependent membrane morphology is obtained. The local inflammation of a lipid membrane is accommodated by the prescribed uniform internal pressure and/or lateral pressure. This furnishes a partial differential equation of Poisson type from which a complete analytical solution is obtained by employing the method of variation of parameters. The solution obtained predicts the smooth and coordinate-dependent morphological transitions over the domain of interest and is reduced to those from the classical uniform membrane shape equation when the equivalent energy potential is applied. In particular, the obtained model closely assimilated the pressure-induced inflammations of lipid membranes where only quantitatively equivalent analyses were reported via the impositions of equivalent edge moments. Lastly, we note that the principle of superposition remains valid even in the presence of coordinate-dependent membrane properties.

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