Extended defects such as dislocations and grain boundaries in real materials are significantly more complex than the idealized nano-systems studied in the past. 

Different topology classes exist for extended defects in crystalline phases, and each topological defect class has a unique thermodynamic description that is specific to the defect geometry. The free energies of two extended defects will therefore have qualitatively different functional forms due to distinct differences in their geometric degrees of freedom.  We will develop a systematic enumeration and cataloguing of different topology classes of extended defects and subsequently formulate the topology allowed functional forms of the defect free energy in terms of mathematical descriptions of geometry. 

Each topology class likely allows for geometric variations that can serve as a gateway to the heterogeneous nucleation of a new crystalline phase. It is well-known that crystallographic pathways that connect different crystal structures, e.g., the Bain (BCC to FCC) and Burgers (BCC to HCP) pathways, play a crucial role in determining phase transformation kinetics. For each topology class we will develop new mathematical/crystallographic approaches to systematically enumerate the pathways to new crystal structures emerging from the crystallographic disturbance of a parent crystals.

Specializing to dislocations, they belong to different topology classes that determine their geometrical degrees of freedom as well as the functional form of their free energy. Each topology class has a particular type of "dislocation phase diagram" that maps stability domains of dislocation phases whose structures depend on solute chemical potentials, temperature and local stress state. Initial theoretical efforts will focus on developing crystallographic methods to enumerate topology classes in different crystal structures and to identify their degrees of freedom and free energy forms.

The initial experimental focus will be on dislocations in solvents that form FCC to systematically characterize dislocation structure through variations in alloy composition. We will consider HCP-forming solutes to determine whether they preferentially segregate to stacking faults, solutes that form FCC based ordered phases to determine the role of long and short-range order on dislocation structure, and solutes that form complex intermetallic compounds, (e.g., Laves phases or B2).