A Modified Landau-de Gennes Theory for Smectic Liquid Crystals: Phase Transitions and Structural Transitions
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arXiv
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| Format: | Preprint |
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2024
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| author | Shi, Baoming Han, Yucen Ma, Chengdi Majumdar, Apala Zhang, Lei |
| author_facet | Shi, Baoming Han, Yucen Ma, Chengdi Majumdar, Apala Zhang, Lei |
| contents | We mathematically model Smectic-A (SmA) phases with a modified Landau-de Gennes (mLdG) model. The orientational order of the SmA phase is described by a tensor-order parameter $\mathbf{Q}$, and the positional order is described by a real scalar $u$, which models the deviation from the average density of liquid crystal molecules. Firstly, we prove the existence and regularity of global minimisers of the mLdG free energy in three-dimensional settings. Then, we analytically prove that the mLdG model can capture the Isotropic-Nematic-Smectic phase transition as a function of temperature, under some assumptions. Further, we explore stable smectic phases on a square domain, with edge length $λ$, and tangent boundary conditions. We use heuristic arguments to show that defects repel smectic layers and strong nematic ordering promotes layer formation. We use asymptotic arguments in the $λ\to 0$ and $λ\to\infty$ limits which reveal the correlation between the number and thickness of smectic layers, the amplitude of density fluctuations with the phenomenological parameters in the mLdG energy. For finite values of $λ$, we numerically recover BD-like and D-like stable smectic states observed in experiments. We also study the frustrated mLdG energy landscape and give numerical examples of transition pathways between distinct mLdG energy minimisers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_03343 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Modified Landau-de Gennes Theory for Smectic Liquid Crystals: Phase Transitions and Structural Transitions Shi, Baoming Han, Yucen Ma, Chengdi Majumdar, Apala Zhang, Lei Soft Condensed Matter Mathematical Physics We mathematically model Smectic-A (SmA) phases with a modified Landau-de Gennes (mLdG) model. The orientational order of the SmA phase is described by a tensor-order parameter $\mathbf{Q}$, and the positional order is described by a real scalar $u$, which models the deviation from the average density of liquid crystal molecules. Firstly, we prove the existence and regularity of global minimisers of the mLdG free energy in three-dimensional settings. Then, we analytically prove that the mLdG model can capture the Isotropic-Nematic-Smectic phase transition as a function of temperature, under some assumptions. Further, we explore stable smectic phases on a square domain, with edge length $λ$, and tangent boundary conditions. We use heuristic arguments to show that defects repel smectic layers and strong nematic ordering promotes layer formation. We use asymptotic arguments in the $λ\to 0$ and $λ\to\infty$ limits which reveal the correlation between the number and thickness of smectic layers, the amplitude of density fluctuations with the phenomenological parameters in the mLdG energy. For finite values of $λ$, we numerically recover BD-like and D-like stable smectic states observed in experiments. We also study the frustrated mLdG energy landscape and give numerical examples of transition pathways between distinct mLdG energy minimisers. |
| title | A Modified Landau-de Gennes Theory for Smectic Liquid Crystals: Phase Transitions and Structural Transitions |
| topic | Soft Condensed Matter Mathematical Physics |
| url | https://arxiv.org/abs/2408.03343 |