What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component “invades” the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.

中文翻译：

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两组分二十面体壳的屈曲过渡和软相侵入

二十面体壳（例如许多病毒衣壳）表面上两种晶相的最佳分布是什么？我们在这里研究了软材料薄层在凸面二十面体凸壳上的分布。我们展示了如何从薄的两组分壳的弹性理论的角度理解球形病毒的形状。我们在表面上添加了较软但仍为结晶的材料后，开发了二十面体壳的形状转换理论。我们显示了软组分如何“侵入”具有最高弹性能量和表面上12种拓扑缺陷所施加的应力的区域。我们探索相图作为软质材料表面分数，壳尺寸的函数，以及刚性和软性相的弹性模量的不可度量性。我们发现，正如预期的那样，软相逐渐填充刚性壳是从二十面体变形最大的区域开始的。随着软相覆盖率的逐渐提高，首先填充圆顶的球形部分（壳的12个顶点），然后入侵连接圆顶的圆柱部分（30个边缘），最终，二十面体的20个平面外壳往往被柔软的材料占据。我们目前对该入侵过程的前两个阶段进行详细的理论研究，并开发了允许非圆形横截面的锥体结构形态变化的模型。总之，我们讨论了根据我们的计算预测的某些结构的生物学相关性，