Abstract.
We minimize a discrete version of the fourth-order curvature-based Landau free energy by extending Brakke's Surface Evolver. This model predicts spherical as well as non-spherical shapes with dimples, bumps and ridges to be the energy minimizers. Our results suggest that the buckling and faceting transitions, usually associated with crystalline matter, can also be an intrinsic property of non-crystalline membranes.
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We avoid using the word ``sphere'' on purpose, because theory predicts that a perfect sphere is not an equilibrium shape of $\mathcal{F}_4$ but we did not pursue the analysis of the shapes for $A<0$ in detail. However, within our numerical accuracy, deviations from a perfect sphere are larger for $\mathcal{F}_4$ than for the Willmore functional
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Manyuhina, O.V., Hetzel, J.J., Katsnelson, M.I. et al. Non-spherical shapes of capsules within a fourth-order curvature model. Eur. Phys. J. E 32, 223–228 (2010). https://doi.org/10.1140/epje/i2010-10631-2
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DOI: https://doi.org/10.1140/epje/i2010-10631-2