Abstract
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. However, little is known about how fractal stimuli such as fractal surfaces interact with their local environment if it exhibits order. Here we show geometry-induced formation of fractal defect states in Koch nematic colloids, exhibiting fractal self-similarity better than 90% over three orders of magnitude in the length scales, from micrometers to nanometres. We produce polymer Koch-shaped hollow colloidal prisms of three successive fractal iterations by direct laser writing, and characterize their coupling with the nematic by polarization microscopy and numerical modelling. Explicit generation of topological defect pairs is found, with the number of defects following exponential-law dependence and reaching few 100 already at fractal iteration four. This work demonstrates a route for generation of fractal topological defect states in responsive soft matter.
Highlights
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales
Geometrical fractals are characterized by the self-similarity of a system to a part of itself across different length scales[6,7], which in case of an ideally selfsimilar fractal means that the system is invariant over all length scales and has no characteristic length scale
Geometrical fractal patterns of different scaling behaviours are generally determined by the fractal dimension D, which quantifies the change in the geometrical details of the fractal relative to the change in the scales[7]
Summary
Fractals are remarkable examples of self-similarity where a structure or dynamic pattern is repeated over multiple spatial or time scales. The second iteration Koch star particles show again two stable orientations in the planar nematic cell, as shown in Fig. 2e,f and Supplementary Figs 6 and 7.
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