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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2501.14043 |
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| _version_ | 1866910797319372800 |
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| author | Bronsard, Lia Lamy, Xavier Stantejsky, Dominik Venkatraman, Raghavendra |
| author_facet | Bronsard, Lia Lamy, Xavier Stantejsky, Dominik Venkatraman, Raghavendra |
| contents | We establish, as $ρ\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $ρ$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where $μ_j$ is the minimal energy after zooming in at scale $ρ$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_14043 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Interaction energies in nematic liquid crystal suspensions Bronsard, Lia Lamy, Xavier Stantejsky, Dominik Venkatraman, Raghavendra Analysis of PDEs Materials Science Mathematical Physics We establish, as $ρ\to 0$, an asymptotic expansion for the minimal Dirichlet energy of $\mathbb S^2$-valued maps outside a finite number of three-dimensional particles of size $ρ$ with fixed centers $x_j\in\mathbb{R}^3$, under general anchoring conditions at the particle boundaries. Up to a scaling factor, this expansion is of the form \begin{align*} E_ρ= \sum_j μ_j -4πρ\sum_{i\neq j} \frac{\langle v_i,v_j\rangle}{|x_i-x_j|} +o(ρ)\,, \end{align*} where $μ_j$ is the minimal energy after zooming in at scale $ρ$ around each particle, and $v_j\in\mathbb{R}^3$ is a torque determined by the far-field behavior of the corresponding single-particle minimizer. The above expansion highlights Coulomb-like interactions between the particle centers. This agrees with the \textit{electrostatics analogy} commonly used in the physics literature for colloid interactions in nematic liquid crystal. That analogy was pioneered by Brochard and de Gennes in 1970, based on a formal linearization argument. We obtain here for the first time a precise estimate of the energy error introduced by this linearization procedure. |
| title | Interaction energies in nematic liquid crystal suspensions |
| topic | Analysis of PDEs Materials Science Mathematical Physics |
| url | https://arxiv.org/abs/2501.14043 |