Long-range order of molecules

Liquid crystal polymers (LCP) are polymers that exhibit liquid crystal characteristics either in solution (lyotropic liquid crystal) or in the melt (thermotropic liquid crystal) [Ballauf, 1989 Finkelmann, 1987 Morgan et al., 1987]. We need to define the liquid crystal state before proceeding. Crystalline solids have three-dimensional, long-range ordering of molecules. The molecules are said to be ordered or oriented with respect to their centers of mass and their molecular axes. The physical properties (e.g., refractive index, electrical conductivity, coefficient of thermal expansion) of a wide variety of crystalline substances vary in different directions. Such substances are referred to as anisotropic substances. Substances that have the same properties in all directions are referred to as isotropic substances. For example, liquids that possess no long-range molecular order in any dimension are described as isotropic. 

The order parameter S = 0.5(3(cos ) — I) characterizes the long-range order of molecules in a mesophase, where 0 is the momentary angle between the long axis of the molecule and the director. In an ideal crystal the order parameter S equals 1, and it equals 0 in an isotropic liquid. In a nematic phase the order parameter hes in the range 0.5-0.7. 

Mesophase composed of board-like molecules in which there are long-range orderings of both the longer and the shorter molecular axes. 

As the cholesteric phase is a twisted nematic phase, the orientational long range order of the mesogenic molecules, characterized by Eq. (3) ... 

It will be useful now to review some elementary facts regarding the structure of liquids at equilibrium. When a crystalline solid melts to form a liquid, the long range order of the crystal is destroyed. However, a residue of local order persists in the liquid state with a range of several molecular diameters. The local order characteristic of the liquid state is described in terms of a pair correlation function, g-i(R)> defined as the ratio of the average molecular density, p(R), at a distance R from an arbitrary molecule to the mean bulk density, p, of the liquid... 

The liquid crystal state (LCS) shows order in one or two dimensions it lacks the three-dimensional long-range order of the crystalline state. LCS has characteristics intermediate between those of the crystalline and the disordered amorphous states. These phases are called liquid crystals because many of them can flow like ordinary liquids but they display-birefringence and other properties characteristic of crystalline soHds. In liquid crystal phases the molecules can move but the orientational order is conserved in at least ne direction. The LCS can be displayed by small molecules and by polymersj but in both cases a characteristic chemical structure is needed. The existence of the liquid crystal state is related to the molecular asymmetry and the presence of strong anisotropic intermolecular interactions (19-21). Thus, molecules with a rigid rod structure can form highly ordered... 

The theoretical calculation of the thermodynamic properties of liquids presents an even more difficult problem than that of solids, for in a liquid the molecules are in a state of disorder intermediate between the random motion of a gas, and the long range order of a solid. We must therefore be content with an even cruder approximation than for solids. ) A model which leads to a useful semi-quantitative correlation is based upon the idea of the free volume. 

We have already said that the solid differs firom the other states of matter in that long range ordering of atoms or molecules has appeared. To achieve long range order in any solid, one must stack atoms in a S3rmmetrical way to completely fill space. If the atoms are all of one kind,... 

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