[Paper Review] Solution Landscapes of the Simplified Ericksen--Leslie Model and its Comparison with the Reduced Landau--de Gennes Model

Yucen Han, Jianyuan Yin+3 more|Research Explorer (The University of Manchester)|Jun 9, 2021

Liquid Crystal Research Advancements45 references7 citations

TL;DR

This paper develops a numerical solution landscape framework to compare the simplified Ericksen–Leslie (sEL) vector model and the reduced Landau–de Gennes (rLdG) tensor model for nematic liquid crystals in a 2D square domain with tangent boundary conditions. It reveals that while both models share stable diagonal (D) and rotated (R) states and similar bifurcation behaviors, the sEL model exhibits unique features such as stable C± states with interior defects, high-index 'fake defects,' tessellating solutions, and more flexible dynamical pathways, highlighting fundamental differences in saddle point structure and defect stabilization mechanisms.

ABSTRACT

We investigate the solution landscapes of a simplified Ericksen--Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau--de Gennes (rLdG) models can be viewed as Ginzburg--Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare with the solution landscapes of the corresponding rLdG models. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviors, and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution $C\pm$ with interior defects, high-index "fake defects" solutions, novel tessellating solutions, and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

Motivation & Objective

• To systematically compare the solution landscapes of the simplified Ericksen–Leslie (sEL) vector model and the reduced Landau–de Gennes (rLdG) tensor model for nematic liquid crystals.
• To investigate how model differences—particularly in order parameter representation and symmetry—impact the stability and connectivity of equilibrium and saddle states.
• To explore the role of boundary conditions and model choice in stabilizing interior defects, such as ±1 and ±1/2 vortices.
• To identify and characterize novel solution types unique to the sEL model, including C± states and tessellating patterns.
• To assess the flexibility of dynamical pathways in the sEL model and compare them with those in the rLdG model.

Proposed method

• Developed an efficient numerical algorithm leveraging symmetry properties of the 2D square domain and the sEL model to construct full solution landscapes.
• Treated both the sEL and rLdG models as Ginzburg–Landau-type functionals and computed their energy landscapes across varying domain sizes.
• Identified stable states, index-1 saddle points (e.g., J states), and higher-index saddle solutions (e.g., C± with index 2 in rLdG) via numerical continuation and bifurcation analysis.
• Utilized weak boundary conditions in the sEL model to enhance connectivity between states and assess pathway flexibility.
• Performed direct comparison of solution structures, including defect types, indices, and energy barriers, between sEL and rLdG models.
• Explored the impact of elastic anisotropy and boundary conditions on defect stabilization, particularly for interior ±1 defects.

Experimental results

Research questions

• RQ1 How do the solution landscapes of the sEL and rLdG models compare in terms of stable states, bifurcations, and saddle point structures?
• RQ2 What are the key differences in defect types and their stability between the sEL and rLdG models, particularly regarding interior ±1 and ±1/2 defects?
• RQ3 Why does the C± saddle point have index 2 in the rLdG model but not in the sEL model, and what does this imply about defect splitting mechanisms?
• RQ4 How does the choice of order parameter (vector vs. tensor) and boundary conditions affect the connectivity and flexibility of dynamical pathways between stable states?
• RQ5 To what extent can the solution landscape framework be generalized to compare other continuum models in soft matter and phase field theory?

Key findings

• The sEL and rLdG models exhibit strong qualitative similarity in their stable D and R states and index-1 J saddle points, especially for large domain sizes.
• The sEL model supports stable C± states with interior ±1 defects, which are absent in the rLdG model under the same boundary conditions.
• The C± saddle point has index 2 in the rLdG model due to two distinct splitting channels into ±1/2 defects along diagonals, while it is not a saddle point in the sEL model, indicating a fundamental difference in defect topology and stability.
• The sEL model features high-index 'fake defects' and novel tessellating solutions not observed in the rLdG model, suggesting richer solution complexity.
• Dynamical pathways in the sEL model are more flexible and less sensitive to initial state than in the rLdG model, likely due to weaker boundary constraints enabling more connections between saddle and stable states.
• The solution landscape approach reveals that saddle point indices for interior defects can differ by 2 between sEL and rLdG models, and this difference may persist for very large domains, suggesting a topological origin in the order parameter representation.