Solution landscapes of the diblock copolymer-homopolymer model under two-dimensional confinement

Abstract

We investigate the solution landscapes of the confined diblock copolymer and homopolymer in twodimensional domain by using the extended Ohta-Kawasaki model. The projection saddle dynamics method is developed to compute the saddle points with mass conservation and construct the solution landscape by coupling with downward and upward search algorithms. A variety of stationary solutions are identified and classified in the solution landscape, including Flower class, Mosaic class, Core-shell class, and Tai-chi class. The relationships between different stable states are shown by either transition pathways connected by index-1 saddle points or dynamical pathways connected by a high-index saddle point. The solution landscapes also demonstrate the symmetry-breaking phenomena, in which more solutions with high symmetry are found when the domain size increases.

Revealing excited states of rotational Bose-Einstein condensates

“…This methodology has been successfully applied to liquid crystals, 32 , 33 , 34 , 35 quasicrystals, 36 and diblock copolymers. 37 Using the solution landscape approach, we reveal four excitation mechanisms of BECs: vortex addition, rearrangement, merging, and splitting. We further demonstrate how the ground state changes with increasing rotational frequencies and the evolution of the stability of the ground states.…”

Probabilistic Error Estimate for Numerical Discretization of High-Index Saddle Dynamics with Inaccurate Models

“…Here x ∈ R d represents the state variable, {v i } k i=1 are directional variables, β, γ > 0 are relaxation parameters, F (x) : R d → R d represents the force generated from the energy E(x) by F (x) = −∇E(x) and J(x) is the negative Hessian of E(x), i.e., J(x) = −∇ 2 E(x). This high-index saddle dynamics could be further combined with the downward and upward algorithms [42] to construct solution landscapes of complex systems, the pathway map consisting of all stationary points and their connections [36], that arises several successful applications [13,14,24,30,38,40,41,44,45,48,49,51].…”

Discretization and index-robust error analysis for constrained high-index saddle dynamics on high-dimensional sphere

“…High-index saddle dynamics provides a powerful instrument in systematically finding the high-index saddle points and construction of the solution landscape [3,8,29,30,31,36]. It has been applied to study various applications in physical and engineering problems [12,13,14,25,26,29,32,34,35,37].…”