Domain pattern formation is one of the most common phenomena in nature and is a topic of immense interest in many fields including materials science, physics, chemistry, and biology. The main objective of this research program is to explore the basic sciences concerning the thermodynamic stability of mesoscale polarization domain patterns and their temporal evolution mechanisms during formation and subsequent switching in ferroelectric nanostructures and heterostructures. This research employs the phase-field method in combination of microelasticity and electrostatic theories. An important aspect of the work involves the validation of computational predictions through closely working together with experimental collaborators performing microscopic analysis and observations of high-quality ferroelectric films and nanostructures synthesized with atomic scale control. The proposed research is motivated by the recent discoveries that new thermodynamically stable mesoscale polar states might emerge from ferroelectric heterostructures at the nanoscale. Specifically, the project will be focused on the basic understanding of (1) the spatial length scales, temperature ranges, and electromechanical conditions leading to the emergence of both transient and stable novel polarization states containing vortex lattices in ferroelectric superlattices; and (2) the switching and relaxation mechanisms of these polarization vortex lattices under electrical and mechanical excitations and the impact of polarization vortex formation and switching on other forms of domain states such as magnetic spin configurations. The proposed research ideas are conceived through recent discussions and close collaborations between the PI’s group and a number of world-class experimental groups who use High Resolution Transmission Electron Microscopy (HRTEM), in situ TEM with Scanning Probe Microscopy (SPM), or Piezoresponse Force Microscopy (PFM) to characterize the atomic/mesoscale domain states and dynamics in high-quality ferroelectric thin films. Fundamental understandings of these exotic polarization domain states will not only significantly advance the basic science on the stability of mesoscale polar states and pattern evolution but also improve our ability to control and engineer ferroelectric thin films and heterostructures for potential applications in nanoscale electronic devices.