Constant spacing in filament bundles
Constant spacing in filament bundles
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DOI：
10.1088/1367-2630/ab1c2d
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发表时间：
2019-06-06
影响因子：
3.3
通讯作者：
Grason, Gregory M.
中科院分区：
文献类型：
--
作者：
Atkinson, Dania W.;Santangelo, Christian D.;Grason, Gregory M.
一维细丝的组合出现在广泛的物理系统中：从生物聚合物束、柱状液晶和超导涡旋阵列；到熟悉的宏观材料，如绳索、电缆和纺织品。在这样的体系中，组成细丝之间的相互作用对中心曲线之间的最接近距离最敏感，这些中心曲线近似它们的配置，使这些不同的组装受到共同的几何约束。在这篇文章中，我们考虑了R-3中多丝堆积中两个不同的等距概念：等距，其中最近的接近距离沿纤维对的长度是恒定的；以及等距，其中所有相邻的纤维之间的最近接近的距离是恒定和相等的。我们证明了，尽管R-3中的任何光滑曲线都允许一维共线等距曲线族属于直纹曲面，但R-3中只有两族切场具有相互等距的积分曲线。这些族中曲线的相对形状和配置受到高度约束：它们必须是(等距)可展域，可以弯曲，但不能扭曲；或者(非等距)等距螺旋束，可以扭曲，但不能弯曲。因此，同时弯曲和扭曲的细丝纹理，如浓缩DNA质粒的扭曲环状或钢丝绳，受到双重阻碍：扭曲阻碍横截面上恒定的相邻间距，而不等距需要沿细丝的额外纵向间距变化。为了说明等距失效的后果，我们比较了三个“几乎等距”的扭曲环丛的间距，并利用我们的等距公式构造了间距的纵向变化随丛厚度增长的上界。
Assemblies of one-dimensional filaments appear in a wide range of physical systems: from biopolymer bundles, columnar liquid crystals, and superconductor vortex arrays; to familiar macroscopic materials, like ropes, cables, and textiles. Interactions between the constituent filaments in such systems are most sensitive to the distance of closest approach between the central curves which approximate their configuration, subjecting these distinct assemblies to common geometric constraints. In this paper, we consider two distinct notions of constant spacing in multi-filament packings in R-3: equidistance, where the distance of closest approach is constant along the length of filament pairs; and isometry, where the distances of closest approach between all neighboring filaments are constant and equal. We show that, although any smooth curve in R-3 permits one dimensional families of collinear equidistant curves belonging to a ruled surface, there are only two families of tangent fields with mutually equidistant integral curves in R-3. The relative shapes and configurations of curves in these families are highly constrained: they must be either (isometric) developable domains, which can bend, but not twist; or (non-isometric) constant-pitch helical bundles, which can twist, but not bend. Thus, filament textures that are simultaneously bent and twisted, such as twisted toroids of condensed DNA plasmids or wire ropes, are doubly frustrated: twist frustrates constant neighbor spacing in the cross-section, while non-equidistance requires additional longitudinal variations of spacing along the filaments. To illustrate the consequences of the failure of equidistance, we compare spacing in three 'almost equidistant' ansatzes for twisted toroidal bundles and use our formulation of equidistance to construct upper bounds on the growth of longitudinal variations of spacing with bundle thickness.