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One of the major challenges towards understanding and further utilizing the properties and functional behaviors of grain boundaries (GB) is the complexity of general GBs with mixed tilt and twist characters. Here, we report the correlations between mixed GBs and their tilt and twist components in terms of structure, energy and stress field by computationally examining 7040 silicon GBs. Such correlations indicate that low angle mixed GBs are formed through the reconstruction mechanisms between their superposed tilt and twist components, which are revealed as the energetically favorable dissociation, motion and reaction of dislocations and stacking faults. In addition, various complex disconnection network structures are discovered near the conventional twin and structural unit GBs, implying the role of disconnection superposition in forming high angle mixed GBs. By unveiling the energetic correlation, an extended Read-Shockley model that predicts the general trends of GB energy is proposed and confirmed in various GB structures across different lattices. Finally, this work is validated in comparison with experimental observations and first-principles calculations., Comment: 25 pages 10 figures

Understanding how dislocation structures vary with grain boundary (GB) characters enables accurate controls of interfacial nano-patterns. In this atomistic study, we report the structure-property correlations of Si (001) small angle mixed grain boundaries (SAMGBs) under three macroscopic GB characters (tilt character, twist character, and an implicit rotation character between them). Firstly, the SAMGB energies are computed as a function of tilt angle, twist angle and rotation angle, based on which a revised Read-Shockley relationship capable of precisely describing the energy variations span the three-dimensional GB character space is fitted. Secondly, GB structural transitions from dislocation to amorphous structures are given as a function of tilt angle, twist angle and dislocation core radii. The proportion, topology and structural signatures of different SAMGB types defined from the ratio between the tilt and twist angles are also presented. Thirdly, by extracting the transformation of metastable SAMGB phases, the formation mechanisms of SAMGB structures are characterized as energetically favorable dislocation glide and reaction, from which the dislocation density function is derived. The relevant results about SAMGB energies and structures are validated and supported by theoretical calculations and experimental observations, respectively., Comment: 26 pages, 10 figures and 2 tables

Velocity gradient is the basis of many vortex recognition methods, such as Q criterion, $\Delta$ criterion, $\lambda_{2}$ criterion, $\lambda_{ci}$ criterion and $\Omega$ criterion, etc.. Except the $\lambda_{ci}$ criterion, all these criterions recognize vortices by designing various invariants, based on the Helmholtz decomposition that decomposes velocity gradient into strain rate and spin. In recent years, the intuition of 'no vortex in straight flows' has promoted people to analyze the vortex state directly from the velocity gradient, in which vortex can be distinguished from the situation that the velocity gradient has couple complex eigenvalues. A specious viewpoint to adopt the simple shear as an independent flow mode was emphasized by many authors, among them, Kolar proposed the triple decomposition of motion by extracting a so-called effective pure shearing motion; Li et al. introduced the so-called quaternion decomposition of velocity gradient and proposed the concept of eigen rotation; Liu et al. further mined the characteristic information of velocity gradient and put forward an effective algorithm of Liutex, and then developed the vortex recognition method. However, there is another explanation for the increasingly clear representation of velocity gradient, that is the local streamline pattern based on critical-point theory. In this paper, the tensorial expressions of the right/left real Schur forms of velocity gradient are clarified from the characteristic problem of velocity gradient. The relations between the involved parameters are derived and numerically verified. Comparing with the geometrical features of local streamline pattern, we confirm that the parameters in the right eigen-representation based on the right real Schur form of velocity gradient have good meanings to reveal the local streamline pattern. Some illustrative examples from the DNS data are presented., Comment: 16 pages, 9 figures, 1 appendix