# A Novel Landau-de Gennes Model with Quartic Elastic Terms

## TL;DR

This paper introduces a new quartic elastic Landau-de Gennes model that ensures well-posedness across a broader range of elastic constants and rigorously connects to the Oseen-Frank theory via Gamma-convergence.

## Contribution

It proposes a quartic elastic energy in Landau-de Gennes theory, extending the model's applicability and establishing a rigorous link to Oseen-Frank theory.

## Key findings

- The quartic model is well-posed for a wider range of elastic constants.
- The model accurately describes nematic-to-isotropic phase transitions.
- Strong convergence of minimizers is established.

## Abstract

Within the framework of the generalized Landau-de Gennes theory, we identify a $Q$-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the $Q$-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimization problem is well-posed for a significantly wider choice of elastic constants. In particular, quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen-Frank counterpart via a $\Gamma$-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimizers.