中国物理B ›› 2008, Vol. 17 ›› Issue (3): 1060-1069.doi: 10.1088/1674-1056/17/3/052

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Theory of nine elastic constants of biaxial nematics

刘红   

  1. School of Physical Science and Technology, Nanjing Normal University, Nanjing 210097, China
  • 收稿日期:2006-07-03 修回日期:2007-08-09 出版日期:2008-03-04 发布日期:2008-03-04
  • 基金资助:
    Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No 07JKD140095).

Theory of nine elastic constants of biaxial nematics

Liu Hong(刘红)   

  1. School of Physical Science and Technology, Nanjing Normal University, Nanjing 210097, China
  • Received:2006-07-03 Revised:2007-08-09 Online:2008-03-04 Published:2008-03-04
  • Supported by:
    Project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No 07JKD140095).

摘要: In this paper, a rotational invariant of interaction energy between two biaxial-shaped molecules is assumed and in the mean field approximation, nine elastic constants for simple distortion patterns in biaxial nematics are derived in terms of the thermal average $\langle {D_{mn}^{(l)} } \rangle \langle {D_{{m}'{n}'}^{({l}')} } \rangle $, where $D_{mn}^{(l)} $ is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients $\Ga$, ${\Ga'}$, $\lambda $, order parameters $\bar {Q}_0 = Q_0 \langle {D_{00}^{(2)} } \rangle + Q_2 \langle {D_{02}^{(2)} + D_{0 - 2}^{(2)} } \rangle $ and $\bar {Q}_2 = Q_0 \langle {D_{20}^{(2)} } \rangle + Q_2 \langle {D_{22}^{(2)} + D_{2 - 2}^{(2)} } \rangle $. Here $\Ga $ and ${\Ga'}$ depend on the function form of molecular interaction energy $v_{j'j''j} ( {r_{12} } )$ and probability function $f_{k'k''k} ( {r_{12} } )$, where $r_{12} $ is the distance between two molecules, and $\lambda $ is proportional to temperature. $Q_0 $ and $Q_2 $ are parameters related to multiple moments of molecules. Comparing these results with those obtained from Landau--de Gennes theory, we have obtained relationships between coefficients, order parameters used in both theories. In the special case of uniaxial nematics, both results are reduced to a degenerate case where $K_{11} = K_{33}$.

关键词: biaxial nematic liquid crystal, elastic theory

Abstract: In this paper, a rotational invariant of interaction energy between two biaxial-shaped molecules is assumed and in the mean field approximation, nine elastic constants for simple distortion patterns in biaxial nematics are derived in terms of the thermal average $\langle {D_{mn}^{(l)} } \rangle \langle {D_{{m}'{n}'}^{({l}')} } \rangle $, where $D_{mn}^{(l)} $ is the Wigner rotation matrix. In the lowest order terms, the elastic constants depend on coefficients $\varGamma$, ${\Gamma'}$, $\lambda $, order parameters $\bar {Q}_0 = Q_0 \langle {D_{00}^{(2)} } \rangle + Q_2 \langle {D_{02}^{(2)} + D_{0 - 2}^{(2)} } \rangle $ and $\bar {Q}_2 = Q_0 \langle {D_{20}^{(2)} } \rangle + Q_2 \langle {D_{22}^{(2)} + D_{2 - 2}^{(2)} } \rangle $. Here $\varGamma$ and $\varGamma'$ depend on the function form of molecular interaction energy $v_{j'j''j} ( {r_{12} } )$ and probability function $f_{k'k''k} ( {r_{12} } )$, where $r_{12} $ is the distance between two molecules, and $\lambda $ is proportional to temperature. $Q_0 $ and $Q_2 $ are parameters related to multiple moments of molecules. Comparing these results with those obtained from Landau--de Gennes theory, we have obtained relationships between coefficients, order parameters used in both theories. In the special case of uniaxial nematics, both results are reduced to a degenerate case where $K_{11} = K_{33}$.

Key words: biaxial nematic liquid crystal, elastic theory

中图分类号:  (Mechanical properties of liquids)

  • 62.10.+s
61.30.Cz (Molecular and microscopic models and theories of liquid crystal structure)