### Abstract

We evaluate the azimuthal anchoring energy of a grooved surface by calculating numerically the Frank elastic energy of a nematic cell composed of the grooved surface and a flat one with rigid azimuthal anchoring, where the director is fixed along the φ direction. We pay attention to the surface anchoring induced by elastic distortions of the director due to its contact with a nonflat surface, which impose local planar degenerate anchoring. Surface anchoring of this kind was analyzed analytically for shallow grooves by Berreman [Phys. Rev. Lett. 28, 1683 (1972)] and critically reexamined by the present authors. We consider two types of surface. one is a surface with one-dimensional sinusoidal parallel grooves, and the other is a surface with two-dimensional square patterns whose surface height is given by a sum of two sinusoidal functions with orthogonal wave vectors. The total energy is the sum of the anchoring energy and the twist energy in the bulk. For the calculation of the twist energy to be eliminated and the evaluation of the azimuthal-angle dependence of the anchoring energy, the "average" azimuthal angle at the bottom, φ (0), must be determined. We adopt two methods to determine φ (0). One is a simple extrapolation of the twist deformation in the bulk. The other relates φ (0) to the variation of the total Frank elastic energy with respect to φ. Our calculations indicate that both methods give essentially the same results, which indicates the consistency of those two methods. We also show that, for a surface with square patterns, the agreement between theory and numerical calculations is quite good even when the maximum of the surface slope is around 0.4, which theory assumes is much smaller than unity. When the surface slope is of order unity, the deviation of numerical results from theory crucially depends on the the surface elastic constant K24.

Original language | English |
---|---|

Article number | 011705 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 79 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 5 2009 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*79*(1), [011705]. https://doi.org/10.1103/PhysRevE.79.011705

**Consistent numerical evaluation of the anchoring energy of a grooved surface.** / Fukuda, Jun Ichi; Yoneya, Makoto; Yokoyama, Hiroshi.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 79, no. 1, 011705. https://doi.org/10.1103/PhysRevE.79.011705

}

TY - JOUR

T1 - Consistent numerical evaluation of the anchoring energy of a grooved surface

AU - Fukuda, Jun Ichi

AU - Yoneya, Makoto

AU - Yokoyama, Hiroshi

PY - 2009/1/5

Y1 - 2009/1/5

N2 - We evaluate the azimuthal anchoring energy of a grooved surface by calculating numerically the Frank elastic energy of a nematic cell composed of the grooved surface and a flat one with rigid azimuthal anchoring, where the director is fixed along the φ direction. We pay attention to the surface anchoring induced by elastic distortions of the director due to its contact with a nonflat surface, which impose local planar degenerate anchoring. Surface anchoring of this kind was analyzed analytically for shallow grooves by Berreman [Phys. Rev. Lett. 28, 1683 (1972)] and critically reexamined by the present authors. We consider two types of surface. one is a surface with one-dimensional sinusoidal parallel grooves, and the other is a surface with two-dimensional square patterns whose surface height is given by a sum of two sinusoidal functions with orthogonal wave vectors. The total energy is the sum of the anchoring energy and the twist energy in the bulk. For the calculation of the twist energy to be eliminated and the evaluation of the azimuthal-angle dependence of the anchoring energy, the "average" azimuthal angle at the bottom, φ (0), must be determined. We adopt two methods to determine φ (0). One is a simple extrapolation of the twist deformation in the bulk. The other relates φ (0) to the variation of the total Frank elastic energy with respect to φ. Our calculations indicate that both methods give essentially the same results, which indicates the consistency of those two methods. We also show that, for a surface with square patterns, the agreement between theory and numerical calculations is quite good even when the maximum of the surface slope is around 0.4, which theory assumes is much smaller than unity. When the surface slope is of order unity, the deviation of numerical results from theory crucially depends on the the surface elastic constant K24.

AB - We evaluate the azimuthal anchoring energy of a grooved surface by calculating numerically the Frank elastic energy of a nematic cell composed of the grooved surface and a flat one with rigid azimuthal anchoring, where the director is fixed along the φ direction. We pay attention to the surface anchoring induced by elastic distortions of the director due to its contact with a nonflat surface, which impose local planar degenerate anchoring. Surface anchoring of this kind was analyzed analytically for shallow grooves by Berreman [Phys. Rev. Lett. 28, 1683 (1972)] and critically reexamined by the present authors. We consider two types of surface. one is a surface with one-dimensional sinusoidal parallel grooves, and the other is a surface with two-dimensional square patterns whose surface height is given by a sum of two sinusoidal functions with orthogonal wave vectors. The total energy is the sum of the anchoring energy and the twist energy in the bulk. For the calculation of the twist energy to be eliminated and the evaluation of the azimuthal-angle dependence of the anchoring energy, the "average" azimuthal angle at the bottom, φ (0), must be determined. We adopt two methods to determine φ (0). One is a simple extrapolation of the twist deformation in the bulk. The other relates φ (0) to the variation of the total Frank elastic energy with respect to φ. Our calculations indicate that both methods give essentially the same results, which indicates the consistency of those two methods. We also show that, for a surface with square patterns, the agreement between theory and numerical calculations is quite good even when the maximum of the surface slope is around 0.4, which theory assumes is much smaller than unity. When the surface slope is of order unity, the deviation of numerical results from theory crucially depends on the the surface elastic constant K24.

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U2 - 10.1103/PhysRevE.79.011705

DO - 10.1103/PhysRevE.79.011705

M3 - Article

AN - SCOPUS:59949085043

VL - 79

JO - Physical Review E

JF - Physical Review E

SN - 2470-0045

IS - 1

M1 - 011705

ER -