Conditions for overall planarity in membranes: Applications to multicomponent membranes with lamellar morphology

The coupling of geometric shapes and local composition in multicomponent membranes results in the formation of complex structures. These membranes can form structures that are planar at large length scales while retaining complex morphologies at smaller scales. We explicitly construct the overall planarity condition for membranes and identify it with the presence of an average tension at the membrane boundary with nonzero components only along the plane defined by the average height of the membrane. The explicit construction of the condition simplifies the analysis of morphologies of the planar membrane. We apply this method to the case of a bicomponent membrane with lamellar morphology. We determine the possible shapes of membranes, their stabilities and the thermodynamic equations of state satisfied by their intensive variables.