Geometric‐Bifurcation Methods for Dual Order‐Disorder Transitions: Applications to the Isotropic‐to‐Smectic A Phase Transformation

Herein, the focus is on equilibrium self‐assembly of smectic A liquid crystal (LC) phases, characterized by partial orientational and 1D positional order. This LC material organization is observed in the protein solution precursor phases of mussel byssus and its emergence is of importance to biological material science, biomimetics, and green manufacturing. A purely thermodynamic model is intrinsically complex due to proliferation of unknown parameters and computational predictions with low information content. This work extends previous work on smectic self‐assembly using an integrated theory and computational platform based on polynomial conservation laws, differential geometry of thermodynamic surfaces, and soft matter shape algebra. The predictions include two zones (nucleation and growth and spinodal decomposition), the stability predicted for single‐order states (nematic, plastic), and the birth and death of the possible phases under varying quenches. A systematic theory‐computation loop yields conservation laws for smectic ordering, the geometry description (shape and curvedness) of bifurcations, and the response functions to thermal quenches. The outputs of this study form the foundation for characterization of drop‐like spatial‐temporal self‐assembly and colloidal nonequilibrium self‐organization of rod‐like protein precursors into functional biological fibrous materials.