Biscari Paolo, Napoli Gaetano, Turzi Stefano
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Sep;74(3 Pt 1):031708. doi: 10.1103/PhysRevE.74.031708. Epub 2006 Sep 21.
Nematic liquid crystals possess three different phases: isotropic, uniaxial, and biaxial. The ground state of most nematics is either isotropic or uniaxial, depending on the external temperature. Nevertheless, biaxial domains have been frequently identified, especially close to defects or external surfaces. In this paper we show that any spatially varying director pattern may be a source of biaxiality. We prove that biaxiality arises naturally whenever the symmetric tensor S=(Vn)(Vn)T possesses two distinct nonzero eigenvalues. The eigenvalue difference may be used as a measure of the expected biaxiality. Furthermore, the corresponding eigenvectors indicate the directions in which the order tensor Q is induced to break the uniaxial symmetry about the director n. We apply our general considerations to some examples. In particular we show that, when we enforce homeotropic anchoring on a curved surface, the order tensor becomes biaxial along the principal directions of the surface. The effect is triggered by the difference in surface principal curvatures.
各向同性相、单轴相和双轴相。大多数向列相的基态取决于外部温度，要么是各向同性相，要么是单轴相。然而，双轴畴经常被识别出来，特别是在缺陷或外表面附近。在本文中，我们表明任何空间变化的指向矢图案都可能是双轴性的来源。我们证明，只要对称张量S = (▽n)(▽n)T具有两个不同的非零本征值，双轴性就会自然产生。本征值差可以用作预期双轴性的度量。此外，相应的本征向量指示了序张量Q被诱导打破关于指向矢n的单轴对称性的方向。我们将我们的一般考虑应用于一些例子。特别是，我们表明，当我们在曲面上施加垂面锚定时，序张量沿曲面的主方向变为双轴性。这种效应是由表面主曲率的差异触发的。
