Interfacial crystal symmetry

The question of how to terminate the box is fundamental to all the calculations of interfacial energy in compounds, including the calculation of surface energies. It has been addressed previously for particular cases by Chetty and Martin [11,12]. These authors pointed out that a suitable termination is one which is on a symmetry plane of the crystal, or which follows symmetry planes if it is not parallel to the boundary. However, it may not always be possible to find a symmetry plane. I offer a solution here which is more general. It reconciles the atomistic picture with the thermodynamic limit. 

Surface and Double-layer Properties Valette [19] has analyzed earlier experimental data on the inner-layer capacity at PZC for Ag(lll), Ag(lOO), and Ag(llO) surfaces in order to estimate the surface area and capacitance contributions of superficial defects for real electrodes, as compared to ideal faces. Considering the application of surface spectroscopy techniques to single-crystal Ag electrodes, one should note that anisotropy of the SHG response for metal electrode allows one to analyze and correlate its pattern with interfacial symmetries and its variations by changing nonlinear susceptibility and the surface structure. Early studies on Ag(lll) single-crystal electrodes have... 

A repeat of Fig. 2.4.) The primitive direct-lattice unit cell in a triclinic (lowest-symmetry) crystal is an oblique parallelopiped with sidesa, b, c, interfacial angles ot, ft, and y and unit vectors ea, eb, and ec. 

The configuration of natural amino acids has led to studies on the possibility that homochirality emerged at a prebiotic stage, which may be supported by the presence of amino acids as non-racemic mixtures in meteorites (32). This enantiomeric excess may have resulted from the exposition of extraterrestrial matter to circularly polarized light (42). Whatever the origin of this enantiomeric excess, it may have initiated stereoselective processes through different catalytic pathways (7). Symmetry breaking may also have resulted from reactivity in connection with other processes such as crystallization or interfacial chemistry (43, 44) and polymerization of amino acids (45). 

S.D. Lee and J.S. Patel, Symmetry-breaking effect of interfacial interactions on electro-optical properties of hquid crystals, Phys. Rev. Lett. 65(1), 56-59, (1990). doi 10.1103/PhysRevLett.65.56... 

Now we are interested in phenomena at an interface between a liquid crystal and another phase (gas, liquid or sohd) [1,2]. Why is it important First, the structure of a liquid crystal in a thin interfacial layer is different from that in the bulk and manifests many novel features. Second, the interface plays a decisive role in applications, because liquid crystals are always used in a cmifined geometry. There are two approaches to the surface problems, microscopic and macroscopic. In the first approach, we are interested in a structure and properties of interfacial liquid crystal layers at the molecular level in the second one, we ignore the microscopic details and use only symmetry properties and the concept of the director. 

Note that the polar vector reflects only polar symmetry of the interfacial layer and may be associated with the conical (not rod-like) form of the molecules. However, when the electric charges are involved in the game, the same polar order may results in appearance of the macroscopic surface electric polarization Psurf that is the dipole moment of a unit volume [units CGS(charge) cm/cm = CGSQ/cm = StatV/cm, or C/m in SI system]. When an electric field is applied to a liquid crystal the surface polarization contributes to the free energy of a surface layer... 

In the present volume we discuss a relatively new and rapidly developing branch of the field, namely nonlinear optical effects in liquid crystals. Optical studies have always played a significant role in liquid crystal science. Research of optical nonlinearities in liquid crystals began at the end of the sixties. Since then it became a powerful tool in the investigation of symmetry properties, interfacial phenomena or dynamic behaviour. Furthermore, several new aspects of nonlinear processes were demonstrated and studied extensively in liquid crystals. The subject covered in this book is therefore of importance both for liquid crystal research and for nonlinear optics itself. 

When a mesophase is in contact with another phase an interfacial layer is formed whose symmetry is, generally speaking, different from the original one. First of all, the interface itself must be characterized by the normal h which is a polar vector. Fig. 3.1. Thus, an interfacial liquid crystal layer is, in principle, polar. 

In addition, an anisotropic substrate can change the in-plane symmetry of the interfacial layer. For example, it may induce a weak biaxiality of the nematic phase or a positional order in the plane of the interface (a type of epitaxy). Hence, in general, the symmetry group of the interfacial layer (Synun 1-2) is a subgroup of the synunetry groups of a liquid crystal (Symm 2) and a substrate (Synun 1), Fig. 3.1. In other cases, a substrate can induce a multistable orientation of a nematic when the director can choose between different equivalent directions in the plane of the substrate (2, 3],... 

The science of geometric crystallography was concerned with the outward spatial arrangement of crystal planes and the geometric shape of crystals. Workers of that day arrived at three fundamental laws (a) the law of constancy of interfacial angles, (b) the law of rationality of indexes, and (c) the law of symmetry (14). 

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