Long-range positional order

Since the development of grazing incidence x-ray diffraction, much of the convincing evidence for long-range positional order in layers has come from this technique. Structural relaxations from distorted hexagonal structure toward a relaxed array have been seen in heneicosanol [215]. Rice and co-workers combine grazing incidence x-ray diffraction with molecular dynamics simulations to understand several ordering transitions [178,215-219]. 

It is, however, important to note tliat individual columns are one-dimensional stacks of molecules and long-range positional order is not possible in a one-dimensional system, due to tlieniial fluctuations and, therefore, a sliarji distinction between colj. and colj. g is not possible [20]. Phases where tlie columns have a rectangular (col. ) or oblique packing (col j of columns witli a disordered stacking of mesogens have also been observed [9, 20, 25,... 

In the second section a classification of the different kinds of polymorphism in polymers is made on the basis of idealized structural models and upon consideration of limiting models of the order-disorder phenomena which may occur at the molecular level. The determination of structural models and degree of order can be made appropriately through diffraction experiments. Polymorphism in polymers is, here, discussed only with reference to cases and models, for which long-range positional order is preserved at least in one dimension. 

At one extreme, one has the structural models of perfect crystals, which have long-range positional order for all the atoms (apart thermal motion). A diffraction experiment on a set of such crystals oriented in one direction (corresponding, in most real cases of polymeric materials, to an oriented fiber) would result in a pattern of sharp reflections organized in layer lines. 

Within the class of polymer crystals having, ideally, long-range positional order for all the atoms, different crystalline forms (polymorphs) may arise as a result of having different almost isoenergetic macromolecular conformations (of the main chain, in most known cases) or as a result of different, almost isoenergetic modes of packing of macromolecules with identical conformations [1-3]. 

Long-range positional order in three dimensional is maintained only for structural features which are not point-centered (e.g., for the chain axes, for which two periodicities only are sufficient to define a three-dimensional repetition) iii) Long-range positional order of some feature is maintained only in two or in one dimension (e.g., only along each chain axis). 

An important sub-case of this kind corresponds to the occurrence of long-range positional order of all the atoms in two dimensions within layers of macromolecules (which may be single layers or bilayers, etc.) and disorder in the stacking of such layers, whereas some characterizing points of the layers maintain long-range periodicity and a well defined 3-D lattice. 

The examples of polymer crystal structures shown in the previous sections are ideal structures, which can be described with the traditional concepts of the principles of equivalence and close packing or the new concepts of symmetry breaking146 and frustration.154 The models of perfect crystals are characterized by a long-range positional order for all the atoms (disregarding thermal motion). The X-ray diffraction patterns of such crystals, oriented with the chain axes along one direction (as in oriented fibers), present sharp reflections organized in layer lines. 

It is worth noting that in polymer structures the various kinds of long-range positional order of the equilibrium positions of the structural elements may be lost after not too big numbers of repetitions owing to the presence of lattice distortions (different from the thermal one), which have been called distortions of the second kind. According to Hosemann and Bagchi,171 these forms are called paracrystalline modifications. 

With further cooling, the SmA LC, which is more ordered than the nematic, becomes the thermodynamic minimum. In the SmA, there is a spontaneous formation of layers, with long-range positional order normal to the layer planes. Thus, the SmA can be considered a stack of two-dimensional fluid layers with crystalline (long-range positional) order in the third dimension, but no... 

Mesophase formed by a non-chiral compound or by the racemate of a chiral compound in which the spatial distribution of the molecular centers of mass is devoid of long-range positional order and the molecules are, on average, orientationally ordered about a common axis defined as the director and represented by the unit vector n. 

Smectic mesophase involving a parallel arrangement of the molecules within layers in which the long axes of the molecules tend to be perpendicular to the layer planes and the molecular centers of mass have no long-range positional order parallel to the layer planes. Note 1 See Fig. 5 for the molecular organization in a smectic A mesophase Note 2 Each layer approximates to a true two-dimensional liquid. The system is optically uniaxial and the optic axis, Z, is normal to the layer planes. 

In liquid crystals or LC-glasses one looks for orientational order and an absence of three-dimensional, long-range, positional order. In liquid crystals, large scale molecular motion is possible. In LC-glasses the molecules are fixed in position. The orientational order can be molecular or supermolecular. If the order rests with a supermolecular structure, as in soap micelles and certain microphase separated block copolymers, the molecular motion and geometry have only an indirect influence on the overall structure of the material. 

Plastic crystals The long-range orientational order is lost, the long-range positional order is preserved, i.e. a plastic quasi-crystalline lattice is still present. 

Molecules Orthogonal to the OiKuse Layers No Long Range Positional Order... 

Many computers, such as laptops, have flat-panel screens or displays that contain liquid crystals (Bahadur 1990). Liquid crystals are composed of small semirigid molecules that are elliptical or oblong in shape and spontaneously orient, forming anisotropic fluids (Collings 1990). Liquid crystals commonly used in flat-panel displays have orientational order but do not have long-range positional order. The direction of the preferred orientation in a liquid... 

Within the family of untilted smectics, there is a hierarchy of phases with order ranging from that of smectic A, in which there is no positional order within each layer, to the crystalline smectics B and E, which have long-range positional order within each layer, namely hexagonal and orthorhombic for the B and E phases, respectively. This inplane order, represented by the dots within circles or ellipses of Fig. 10-27, propagates from layer to layer, producing fully three-dimensional crystalhne order. The family of tilted smectics includes similar ordered phases J, G, K, and H. 

Within a smectic layer, a state of order called hexatic can exist that is intermediate between the liquid order of the smectic A, and the crystalline order of crystalhne smectic B. This state of order characterizes the hexatic smectic-B phase. Flexatic order consists of long-range bond orientational order, but no long-range positional order. This is illustrated... 

While all of the previously described phases are genuine liquid-crystalline phases, there are other phases in which there is long-range positional order. They differ from true crystal phases in one important aspect the molecules in them have the freedom of (at least) rotation around their long axis their thermal motion is not completely frozen out. These phases are labelled crystal smectic phases (denoted by the letters B, E, G, H, J, K). 

There is a long-range orientational order in the plane but no long-range positional order. In the smectic B phase, only the first harmonic q )... 

Liquid crystalline phases can show not only long-range orientational order as nematic phases do but also long-range positional order. When this positional order is one-dimensional, the mesophase is called lamellar or smectic when it is two-dimensional, it is called columnar. The latter case is often found with thermotropic liquid-crystal disk-like molecules. Such molecules stack in columns that assemble on a 2-D lattice of hexagonal, rectangular, or oblique symmetry. The molecules in a given column only show 1-D liquid-Hke order and the uncorrelated columns are free to slide past each other, which ensures the mesophase fluidity [73]. 

A parameter that cannot be over looked is the dispersity in size of the particles. Indeed, polydispersity usually prevents long range positional ordering. For example, it is crucial in order to obtain (i) smectic phases to have nanoparticles of fairly homogeneous length and (ii) hexagonal phases to have nanoparticles with diameters as monodisperse as possible. 

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