{'fraction_non_alphanumeric': 0.031007751937984496, 'fraction_numerical': 0.013289036544850499, 'mean_word_length': 4.986754966887418, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Shape and stability of two-dimensional lipid domains with dipole-dipole interactions.

We study the general energy and shape of the two-dimensional solid monolayer domains with the dipole-dipole interactions. Compared with the domain energy without tilted dipole moments [M. Iwamoto and Z. C. Ou-Yang, Phys. Rev. Lett. 93, 206101 (2004)], the general dipolar energy is not only shape and size but also boundary orientation dependent. The general shape equation derived by this energy using variational approach predicts a circular solution and an equilibrium shape grown from this circle. In particular, the latter is composed of two branches: a translation-induced growth of all odd harmonic modes and a pressure-induced cooperative deformation by all even harmonic modes. The good qualitative agreement between our prediction and the experimental observations shows the validity of the present theory.