Geometrical exact solutions for confocal lamellar textures yielding confinement and faceting phenomena
Geometrical exact solutions for confocal lamellar textures yielding confinement and faceting phenomena
复制标题
产生限制和刻面现象的共焦层状纹理的几何精确解决方案
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
E. Virga
中科院分区：
文献类型：
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作者：
J. Fournier;E. Virga
我们研究了由两个平行板限制的细胞中层状体的平衡纹理，这些平行板仅与层的方向相互作用，而不与它们的位置相互作用。对于这种“松散”的锚定，需要共焦纹理，即等距和平行弯曲层的纹理。当所有缺陷都位于近晶体外部时，借助变量的幸运变化，我们找到了描述纹理的非线性平衡方程的精确解。平衡纹理不是由微分方程控制，而是由有限方程控制，该方程给出各层的共同渐开线 f。然后可以通过几何构造从 f 推导出平衡纹理。这种方法让人想起 Wulff (1901) 为确定固体晶体的平衡形状而引入的方法。我们的主要结果是，在某些边界条件下，根据表面能量函数的形状，存在有限的扭曲和多面纹理。多面纹理表征了表面能量中角点的存在，这是目前争论的一个特征。
We study the equilibrium textures of lamellar bodies in cells limited by two parallel plates interacting only with the orientations of the layers but not with their positions. For such ‘loose’ anchorings, confocal textures are expected, i.e. textures of equidistant and parallel curved layers. When all defects lie outside the smectic body, with the aid of a fortunate change of variables, we find an exact solution of the nonlinear equilibrium equation describing the textures. Instead of being governed by a differential equation, the equilibrium textures are governed by a finite equation that gives the common evolute f of the layers. The equilibrium texture can then be deduced from f by a geometrical construction. This approach is reminiscent of that introduced by Wulff (1901) in order to determine the equilibrium shape of solid crystals. Our main result is the existence, for certain boundary conditions and according to the shape of the surface energy function, of confined distortions and faceted textures. Faceted textures characterize the presence of an angular point in the surface energy, a feature currently debated.