We study the small particle limit of the Landau–de Gennes model around two spherical colloids with strong homeotropic anchoring. We obtain an explicit representation of the minimizing $Q$-tensor. We then investigate the structure of defect lines and its dependence on the particle distance and orientation. In particular, for certain orientations and for small distances, we observe a line singularity disconnected from the singular line surrounding both particles, similar to the entangled hyperbolic defect configuration observed in experiments, while for larger distances, the two lines merge and eventually the inner singular line disappears.
Nous étudions la limite à petite échelle du modèle de Landau–de Gennes autour de deux colloïdes sphériques avec un ancrage homéotrope fort. Nous obtenons une représentation explicite du $Q$-tenseur minimisant. Nous étudions ensuite la structure des lignes de défaut et sa dépendance par rapport à la distance et à l’orientation entre les particules. En particulier, pour certaines orientations et pour de petites distances, nous observons une ligne singulière déconnectée de la ligne singulière entourant les deux particules, similaire à celle des défauts hyperboliques enchevêtrés observée dans les expériences, alors que pour de plus grandes distances, les deux lignes fusionnent et finalement la ligne singulière intérieure disparaît.
Revised:
Accepted:
Published online:
Keywords: Liquid crystals, explicit solution, line defect, bispherical coordinates
Mots-clés : Cristaux liquides, solution explicite, défaut de ligne, coordonnées bisphériques
Lia Bronsard 1 ; Spencer Locke 2 ; Hayley Monson 1 ; Dominik Stantejsky 3
CC-BY 4.0
@article{CRMATH_2025__363_G13_1517_0,
author = {Lia Bronsard and Spencer Locke and Hayley Monson and Dominik Stantejsky},
title = {Structure of {Saturn} ring defects for two small spherical colloidal particles},
journal = {Comptes Rendus. Math\'ematique},
pages = {1517--1532},
year = {2025},
publisher = {Acad\'emie des sciences, Paris},
volume = {363},
doi = {10.5802/crmath.801},
language = {en},
}
TY - JOUR AU - Lia Bronsard AU - Spencer Locke AU - Hayley Monson AU - Dominik Stantejsky TI - Structure of Saturn ring defects for two small spherical colloidal particles JO - Comptes Rendus. Mathématique PY - 2025 SP - 1517 EP - 1532 VL - 363 PB - Académie des sciences, Paris DO - 10.5802/crmath.801 LA - en ID - CRMATH_2025__363_G13_1517_0 ER -
%0 Journal Article %A Lia Bronsard %A Spencer Locke %A Hayley Monson %A Dominik Stantejsky %T Structure of Saturn ring defects for two small spherical colloidal particles %J Comptes Rendus. Mathématique %D 2025 %P 1517-1532 %V 363 %I Académie des sciences, Paris %R 10.5802/crmath.801 %G en %F CRMATH_2025__363_G13_1517_0
Lia Bronsard; Spencer Locke; Hayley Monson; Dominik Stantejsky. Structure of Saturn ring defects for two small spherical colloidal particles. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 1517-1532. doi: 10.5802/crmath.801
[1] Saturn ring defect around a spherical particle immersed in a nematic liquid crystal, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 6, 225, 50 pages | DOI | Zbl | MR
[2] Minimizers of the Landau–de Gennes energy around a spherical colloid particle, Arch. Ration. Mech. Anal., Volume 222 (2016) no. 1, pp. 427-450 | Zbl | DOI | MR
[3] The Saturn ring effect in nematic liquid crystals with external field: effective energy and hysteresis, Arch. Ration. Mech. Anal., Volume 241 (2021) no. 3, pp. 1403-1457 | DOI | Zbl | MR
[4] Convergence to line and surface energies in nematic liquid crystal colloids with external magnetic field, Calc. Var. Partial Differ. Equ., Volume 63 (2024) no. 5, 129, 62 pages | DOI | Zbl
[5] The ParaView guide: a parallel visualization application, Kitware, Inc., 2015
[6] Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, Springer, 2006 no. 10 | Zbl
[7] On a divergence penalized Landau–de Gennes model, SeMA J. (2025) | DOI | MR
[8] An analytic solution for low-frequency scattering by two soft spheres, SIAM J. Appl. Math., Volume 58 (1998) no. 2, pp. 370-386 | DOI | MR | Zbl
[9] An inequality for Legendre polynomials, J. Math. Phys., Volume 35 (1994), pp. 1348-1360 | DOI | Zbl | MR
[10] Scalings and limits of Landau–de Gennes models for liquid crystals: a comment on some recent analytical papers, Math. Model. Anal., Volume 23 (2018) no. 3, pp. 414-432 | DOI | MR | Zbl
[11] Inequalities for the associated Legendre functions, J. Approx. Theory, Volume 95 (1998), pp. 178-193 | Zbl | MR | DOI
[12] Field theory handbook, Springer, 1988 | MR | DOI
[13] Two-dimensional nematic colloidal crystals self-assembled by topological defects, Science, Volume 313 (2006) no. 5789, pp. 954-958 | DOI
[14] Novel colloidal interactions in anisotropic fluids, Science, Volume 275 (1997) no. 5307, pp. 1770-1773 | DOI
[15] Entangled nematic colloidal dimers and wires, Phys. Rev. Lett., Volume 99 (2007), 247801, 4 pages | DOI
[16] Landau–de Gennes modelling of nematic liquid crystal colloids, Liq. Cryst., Volume 36 (2009), pp. 1201-1214 | DOI
[17] Essential mathematical methods for physicists, Academic Press Inc., 2004
Cited by Sources:
Comments - Policy