A Cayley graph description of the symmetry breaking associated with deformation and structural phase transitions in metallic materials

Deformation and structural phase transitions are typical physical processes, which can produce variety types of defects (e.g., twinning and dislocation) in metals and alloys. Those crystalline defects are critical in determining the mechanical and functional properties of materials. Theoretically, crystalline defects belong to the so-called topological defects, which are generated by symmetry breaking processes. In spite of the well-established group theory description of crystal symmetry (for an individual crystal), a theoretical framework to characterize multiple symmetry breaking processes (e.g., multiple transformation pathways between two crystals, multiple deformation twinning modes) is still lacking. In the literature, a graph method, phase transition graph, has been introduced to describe the pathway connectivity in specific martensitic transformations. However, a general framework capturing the coupling of deformation and phase transitions is unavailable, partially due to the lack of rigorous connection between group theory (description of crystal symmetry) and graph theory (description of deformation pathway connectivity). In this paper, a theoretical connection between crystal symmetry and pathway connectivity is rigorously established for the first time. By examining the broken and preserved symmetries during crystal deformation and structural phase transitions through group theory, we employ a Cayley graph method to analyze symmetry groups, which leads to a new theoretical description of symmetry breaking based on group theory and graph theory. Our graph approach establishes a mathematical foundation to investigate the symmetry breaking associated with deformation and structural phase transitions in crystals, which also provides a new insight into the physical origin of crystalline defects.