[Paper Review] Flexoelectricity versus Electrostatics in Polar Nematic Liquid Crystals

Lincoln Paik, Jonathan V. Selinger|arXiv (Cornell University)|Aug 19, 2024

Liquid Crystal Research Advancements6 citations

TL;DR

The paper builds a unified theory combining flexoelectricity and electrostatics in polar nematic liquid crystals, yielding a phase diagram with ferroelectric, antiferroelectric, and conventional nematic phases and clarifying opposing effects on splay.

ABSTRACT

Polar nematic liquid crystals have two special features, compared with conventional nematic liquid crystals. First, because of flexoelectricity, the combination of polar order and splay reduces the free energy. Second, because of electrostatics, any splay generates a bound charge density, which increases the free energy. To assess the competition between these two effects, we develop a theory that combines flexoelectricity and electrostatics. The theory predicts a phase diagram that includes ferroelectric, antiferroelectric, and conventional nematic phases.

Motivation & Objective

• Motivate and understand how flexoelectricity and electrostatics compete in polar nematic liquids.
• Develop a free-energy framework that couples Frank elasticity, Landau polarization terms, flexoelectric coupling, and electrostatic interactions.
• Analyze spontaneous cholesteric twist and the emergence of splay nematic structures within this combined theory.
• Derive a phase diagram showing ferroelectric, antiferroelectric, and conventional nematic phases and identify regimes of weak versus strong electrostatics.
• Discuss mean-field critical behavior and how electrostatics and flexoelectricity shift transition boundaries.

Proposed method

• Construct a free energy F = F_Frank + F_Landau + F_flexo + F_elec with explicit expressions for each term.
• Treat polarization P with parallel and perpendicular components to the director to capture anisotropy in the Landau energy.
• Incorporate flexoelectric coupling as F_flexo = -λ(P·n)(∇·n).
• Model electrostatics with a screened Coulomb interaction F_elec = (1/2) ∫∫ ρ(r1)ρ(r2) e^{-|r12|/Λ} / (4πε|r12|) and define bound and surface charges.
• Explore two applications: spontaneous cholesteric twist and a 1D splay nematic via variational ansatz for n(r) and P(r), leading to effective free energies (10–18) and mean-field scalings.

Experimental results

Research questions

• RQ1 How do flexoelectricity and electrostatics compete to determine polar order in polar nematic liquids?
• RQ2 What are the predicted phase boundaries and characteristic modulations (ferroelectric, antiferroelectric, conventional nematic) when both effects are included?
• RQ3 How does the Debye screening length influence the onset and nature (uniform vs modulated) of polar order?
• RQ4 What is the nature of the phase transitions and the associated critical scaling in the weak and strong electrostatic regimes?

Key findings

• A combined free-energy framework yields a phase diagram with conventional nematic, ferroelectric nematic, and antiferroelectric (modulated) polar phases.
• Flexoelectricity lowers the effective splay elastic constant and promotes polar order, broadening the polar phase region.
• Electrostatics penalizes splay and polar order, increasing the effective splay elastic constant and reducing the polar phase region.
• Two regimes emerge: in weak electrostatics, polar order begins at zero wavevector (uniform ferroelectric nematic); in strong electrostatics, order nucleates with a finite wavevector (antiferroelectric/splay nematic).
• Mean-field analysis shows distinct scaling near transitions: P0 ~ (μ_c - μ) for weak electrostatics and P0 ~ (μ_c - μ)^{1/2} for strong electrostatics, with θ0 and q behavior described by (15)–(18).
• The theory explains how flexoelectricity and electrostatics oppose each other in setting the transition temperature and modulation wavelength, offering a framework to interpret experiments on polar nematic materials.