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The author of this paper and coauthors in 2022 studied a family of self-affine tiles in R d with noncollinear digit sets, and gave a sufficient and necessary condition for such tiles to be tame balls. We in this paper mainly present a simpler proof of such equivalent condition. We replace quadric surfaces by some zigzag planes, and redefine the quasi-invariant plane which plays a key role in the construction of the desired homeomorphism. This adjustment greatly simplifies the proof. [ABSTRACT FROM AUTHOR]

Polylactic acid (PLA) aerogels with spider network structure, bead-like connected microsphere structure, and cluster petal structure were fabricated by precisely controlling the phase separation behavior (nucleation growth and spinodal separation) of the ternary solution system. The PLA aerogel with ideal spider network structure achieved an extremely high porosity of 96% and a high specific surface area of 114 m2/g, which rendered it with excellent triboelectric energy generation performance. [Display omitted] Aerogels with 3D porous structures have been attracting increasing attention among functional materials due to their advantages of being lightweight and high specific surface area. Precise control of the porous structure of aerogel is essential to improve its performance. In this work, polylactic acid (PLA) aerogels with distinctly different microstructures were fabricated by precisely controlling the phase separation behavior of the ternary solution system. Rheological and theoretical analyses have revealed that the interactions between polymer molecules, solvents and non-solvents play a crucial role in determining the nucleation and growth of poor olymer and rich polymer phases. By adjusting the non-solvent type and the solution composition, aerogels with spider network structure, bead-like connected microsphere structure, and cluster petal structure were obtained. Ideal spinodal phase separation conditions were obtained to produce aerogels with a homogeneous fiber network structure. The optimum PLA aerogel achieved an extremely porosity of 96 % and a high specific surface area of 114 m2/g, which rendered it with excellent triboelectric generation performance. Thus, this work provides fundamental insights into the precise regulation of the phase separation behavior and the structure of the aerogel, which can help boost the performance and expand the applications of PLA aerogels. [ABSTRACT FROM AUTHOR]