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There are two opposite mechanisms for the growth of anodic TiO2 nanotubes. One is the field-assisted dissolution and ejection theory, the other is the oxygen bubble mold. In order to prove the latter mechanism is right, we constructed a three-layer nanotube structure (upper layer nanotubes + dense oxide film + lower layer nanotubes), which was obtained by three steps anodization. The upper and lower layers nanotubes are separated by a dense oxide film. When the dense oxide film is thin, the lower layer nanotubes is longer than the upper layer nanotubes. On the contrary, when the dense oxide film is thick, the lower layer nanotubes is shorter than the upper layer nanotubes. In either case, the anodization conditions of the upper and lower layer nanotubes are the same. This could not be explained by field-assisted dissolution theory. By analyzing the curve, we introduce a new method to decide the time of entering the third stage of the anodization, which is the main stage of nanotubes growth. With the second anodization voltage increases, the time edge entering the third stage become longer, which means the time to growth the nanotubes becomes shorter, and finally leading the different length of the nanotubes.

This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the sixth root of unity ω = 1+i √ 3 2 (and its symmetric entry is ω = 1−i √ 32 ); the entry corresponding to an undirected edge is equal to 1, and 0 otherwise. The main results of this paper include the following: Some interesting properties are discovered about the characteristic polynomial of this novel matrix. Cospectral problems among mixed graphs, including mixed graphs and their underlying graphs, are studied. We give equivalent conditions for a mixed graph that shares the same spectrum of its Hermitian adjacency matrix of the second kind (H S -spectrum for short) with its underlying graph. A sharp upper bound on the H Sspectral radius is established and the corresponding extremal mixed graphs are identified. Operations which are called three-way switchings are discussed-they give rise to a large number of H S -cospectral mixed graphs. We extract all the mixed graphs whose rank of its Hermitian adjacency matrix of the second kind (H S -rank for short) is 2 (resp. 3). Furthermore, we show that all connected mixed graphs with H S -rank 2 can be determined by their H S -spectrum. However, this does not hold for all connected mixed graphs with H S -rank 3. We identify all mixed graphs whose eigenvalues of its Hermitian adjacency matrix of the second kind (H S -eigenvalues for short) lie in the range (−α, α) for α ∈ √ 2, √ 3, 2 .