On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco

Commentationes Mathematicae Universitatis Carolinae (2017)

  • Volume: 58, Issue: 2, page 153-171
  • ISSN: 0010-2628

Abstract

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In a cholesteric liquid crystal the director field n ( x , y , z ) tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz transition (without external fields). We consider the classical situation of a thin layer between two glass sheet assuming the Oseen-Frank model for the energy, and that the director field depend only on the direction z orthogonal to the layer; then we focus on two kinds of boundary conditions: the planar case and the orthogonal case. In the first, we impose that n ( 0 ) = ( 1 , 0 , 0 ) , n ( d ) = ( cos α , sin α , 0 ) (where z = 0 and z = d > 0 are, respectively, the bottom and the top of the layer), and search for the couples ( d , α ) such that oblique helicoid appear. In the case K 1 > 0 , K 2 = K 3 = 1 for the elastic constants of the Oseen-Frank energy, we completely characterize these couples. In the second case it is a classical result that oblique helicoid bifurcates from the trivial solution n ( z ) = ( 0 , 0 , 1 ) for suitable values of d ; then we study the exact number of these nontrivial solutions and their stability.

How to cite

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Greco, Carlo. "On a bifurcation problem arising in cholesteric liquid crystal theory." Commentationes Mathematicae Universitatis Carolinae 58.2 (2017): 153-171. <http://eudml.org/doc/288216>.

@article{Greco2017,
abstract = {In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz transition (without external fields). We consider the classical situation of a thin layer between two glass sheet assuming the Oseen-Frank model for the energy, and that the director field depend only on the direction $z$ orthogonal to the layer; then we focus on two kinds of boundary conditions: the planar case and the orthogonal case. In the first, we impose that $n(0)=(1,0,0)$, $n(d)=(\cos \alpha ,\sin \alpha ,0)$ (where $z=0$ and $z=d>0$ are, respectively, the bottom and the top of the layer), and search for the couples $(d,\alpha )$ such that oblique helicoid appear. In the case $K_1>0$, $K_2=K_3=1$ for the elastic constants of the Oseen-Frank energy, we completely characterize these couples. In the second case it is a classical result that oblique helicoid bifurcates from the trivial solution $n(z)=(0,0,1)$ for suitable values of $d$; then we study the exact number of these nontrivial solutions and their stability.},
author = {Greco, Carlo},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {bifurcation; time map; cholesteric liquid crystals; Freedericksz transition},
language = {eng},
number = {2},
pages = {153-171},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a bifurcation problem arising in cholesteric liquid crystal theory},
url = {http://eudml.org/doc/288216},
volume = {58},
year = {2017},
}

TY - JOUR
AU - Greco, Carlo
TI - On a bifurcation problem arising in cholesteric liquid crystal theory
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2017
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 58
IS - 2
SP - 153
EP - 171
AB - In a cholesteric liquid crystal the director field $n(x,y,z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known as Freedericksz transition (without external fields). We consider the classical situation of a thin layer between two glass sheet assuming the Oseen-Frank model for the energy, and that the director field depend only on the direction $z$ orthogonal to the layer; then we focus on two kinds of boundary conditions: the planar case and the orthogonal case. In the first, we impose that $n(0)=(1,0,0)$, $n(d)=(\cos \alpha ,\sin \alpha ,0)$ (where $z=0$ and $z=d>0$ are, respectively, the bottom and the top of the layer), and search for the couples $(d,\alpha )$ such that oblique helicoid appear. In the case $K_1>0$, $K_2=K_3=1$ for the elastic constants of the Oseen-Frank energy, we completely characterize these couples. In the second case it is a classical result that oblique helicoid bifurcates from the trivial solution $n(z)=(0,0,1)$ for suitable values of $d$; then we study the exact number of these nontrivial solutions and their stability.
LA - eng
KW - bifurcation; time map; cholesteric liquid crystals; Freedericksz transition
UR - http://eudml.org/doc/288216
ER -

References

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  1. Bedford S., Global minimisers of cholesteric liquid crystal systems, arXiv:1411.3599v2, 2014. 
  2. Bedford S., Analysis of local minima for constrained minimization problems, arXiv:1411.3595, 2014. 
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  5. Leslie F.M., Distorted twisted orientation patterns in nematic liquid crystals, Pramana Suppl. 1 (1975), 41–55. 
  6. Leslie F.M., Some topics in equilibrium theory of liquid crystals, in Theory and Applications of Liquid Crystals, J. Ericksen and D. Kinderlehrer, Eds., Springer, New York, 1987, pp. 211–234. MR0900827
  7. Schaaf R., 10.1007/BFb0098346, Lecture Notes in Mathematics, 1458, Springer, New York, 1990. Zbl0780.34010MR1090827DOI10.1007/BFb0098346
  8. Stewart I.W., The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction, Taylor & Francis, London and New York, 2004. 
  9. Zel'dovic B.Ya., Tabiryan N.V., Freedericksz transition in cholesteric liquid crystals without external fields, JETP Lett. 34 (1981), no. 8, 406–408. 
  10. Zel'dovic B.Ya., Tabiryan N.V., Equilibrium structure of a cholesteric with homeotropic orientation on the walls, Sov. Phys. JETP 56 (1982), no. 3, 563–566. 

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