Paracrystalline distortions

Figure 8.44. Effect of paracrystalline distortions on a series of reflections in a scattering diagram after compensation of the decay according to POROD s law (lattice factor (1 /N) Z 2). The quadratic increase of integral breadths of the reflections is indicated by boxes of equal area and increasing integral breadth. L is the average long period...

A consideration of the paracrystalline distortions, primarily the limited lateral correlations of axial chain coordinates ( chain shift disorder ), is found in the paper by Kawaguchi and Petermann [174]. According to the analysis of their electron diffraction patterns of flow-oriented PPP films, the diffraction on the third and higher layer lines is essentially single-chain diffraction. Although indications for an off-meridional position of the (002) reflection are found in some patterns, the authors favour the orthorhombic unit cell. 

Kulshreshtha et al. [83], as well as Mitra and Mukherjee [118], postulate the existence of paracrystallinity in ramie, jute, and hemp. The latter claim to have developed refinements to the techniques of measurement of parameters of the three-phase model for cellulose I [117]. Subsequently, by measuring the structural parameters relative to [002], [101], and [101] reflections, they show that the degrees of crystallinity and paracrystallinity determined for each of the three reflections are different (anisotropy) for each of the three fibers (ramie, jute, and hemp) [122]. The results also confirm that the greater the paracrystalline distortion, the smaller the paracrystallite size. 

Because of the presence of structural disorder, the X-ray diffraction patterns of mesophases show a large amount of diffuse scattering and need a special care for a quantitative evaluation. Paracrystalline distortions of the lattice [59,60] usually affect the shape and the width of the diffraction peaks to a large extent. The analysis of disorder necessarily implies a multidisciplinary approach, in order to unravel the complicated nature of disorder in disordered crystalline materials [61]. 

The presence of lattice distortions of the first kind induces a decrease of the intensities of X-ray diffraction reflections with increasing diffraction order, whereas the width of reflection spots does not vary. Lattice distortions of the second kind, instead, result in both a diminution of reflection intensity and an increase in reflection breadth, with increasing reflection order [59,60]. They are also called paracrystalline distortions [60]. 

As shown in Fig. 4, the X-ray diffraction patterns of nylon 6 fibers in the P form (Fig. 4A) present only two equatorial reflections (Fig. 4B) and several meridional reflections (up to the 7-th order. Fig. 4C) [163]. This indicates a pseudo-hexagonal unit cell with a(= b) = 4.80 A and a mean periodicity along the chain axis equal to c = 8.35 A. The substantial absence of diffraction off the equator and off the meridian indicates that a high degree of conformational disorder is present. From the meridional profile reported in Fig. 4C it is apparent that the half-height widths of the meridional reflections increase with increasing diffraction order, indicating paracrystalline distortions of the lattice parameter c. 

The results of these analyses have indicated that the mesomorphic form of iPP is characterized by small bundles of parallel chains in ordered 3/1 helical conformation with disorder in the lateral packing [262]. The relative heights of neighboring chains within each bundle are mainly correlated. The local correlations are near to those which characterize the crystal structure of monoclinic a form [262,263]. Any correlation about the relative position of atoms is lost at distances higher then 30-40 A [262]. Since the lateral order is maintained at very short range, it was proposed that the term paracrystalline could be used for description of this form, if the paracrystalline distortion is intended to mean that a correlation exists within small nuclei of chains and fades away at longer interchain distances [262]... 

If one examines Figure 6a closely, one will notice that the rings are not really homogeneous rather they are composed of numerous spots, indicating that the liquid crystalline domains are of considerable size. The dependence of the reflection width on the order of the reflection was analyzed according to the theory of paracrystalline distortions of the second kind. Information about the crystallite or domain size and distortion parameter can be obtained from such analysis. The breadth due to instrumental... 

Width of the peaks depends not only on the size of the crystallite but also on other factors, such as instrumental effects and paracrystalline distortions of crystalline lattice caused by dislocations, twinning, internal stresses, surfaces and boundaries of crystallites, etc. (Singh, 2005],... 

To determine the actual sizes of crystallites, an improved WAXS method should be used taking into consideration the contribution both of instrumental factor (h] and also of paracrystalline distortions (A] in the experimental width [5] of the peak ... 

The instrumental factor, b, was measured using crystalline standard, e.g., D-cellobiose. Theoretical contribution of the paracrystalline distortions in the width of the peak is A = 2dA/... 

The knowledge of the morphology of the block copolymers makes a great progress. Numerous studies are theoretical, but they are often coupled with one or several experimental techniques. Hashimoto et al. [114] used SAXS to investigate the hexagonally packed cylindrical particles with paracrystalline distortion by comparing experimental and theoretical values. Stocker et al. [115] associated transmission electron microscopy to atomic force microscopy or SAXS and theoretical calculations or electron microscopy to treat surface problems [116] (see also Chapters 6 and 7). 

Hashimoto T, Kawamura T, Harada M and Tanaka H (1994) Small-angle scattering from hexagonally packed cylindrical particles with paracrystalline distortion. Macromolecules 27 3063-3072. 

In wood, the cross-sectional diameters of the smallest basic units, often referred to as elementary fibrils, are in the range 2-4 nm. These fibrils are associated in higher systems with diameters of 10-30 nm (microfibrils) [3]. The microfibrils are also known as nanofibrils, protofibrils, and so on. Here the traditional and well-established term microfibril is used. Microfibrils possess a high degree of crystallinity, but along their transverse direction there are localized (paracrystalline) distortions, and they also contain amorphous domains [1, 5, 6]. 

The word particle has become so widely used ia the technical mbber and carbon black Hterature that it is convenient to retain the term when ia fact nodule is meant. The layer planes are curved, distorted, and of varyiag size. They also iatersect and interconnect one particle or nodule with its neighbors. This type of stmcture has been termed paracrystalline. It is obvious that iadividual particles do not exist ia carbon blacks, with the exception of thermal... 

It is worth to be noted that these definitions of first- and second-order distortions according to Warren-Averbach are model-free. From a linear or a quadratic increase of peak breadths it can neither be concluded in reverse that strain broadening, nor that paracrystalline disorder were detected. 

Distortions of the Second Kind and Linear Paracrystallinity. The idea of lattice distortions of the second kind goes back to the famous work of Zernike... 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

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. 

It is an unusual problem to construct a homogeneously distorted two-dimensional domain structure which allows us to cover the range of the hexagonal closely packed crystal up to a two-dimensional gas. For example, the concept of the paracrystal is based on the philosophy that every state of condensed matter has at least a micro-paracrystalline arrangement of segments within a distorted lattice of arbitrary symmetry with a defined coordination number. 

Fibrous polymers are, at best, paracrystalline, exhibiting lattice distortions of the second kind which destroy long range order (1, 2). 

Methods for estimating lattice distortion generally require two or more orders of a particular reflection to be present, and most polymers have only one order available. A method for estimating both crystallinity and lattice disorder, which does not need higher orders of a reflection, and indeed takes into account the whole of the diffraction trace, is that due to Buland (27). This method has been applied to many different fibres by Sotton and his colleagues, who have discussed their results both here (28) and elsewhere (12). The major problem with Ruland s method is that an arbitrary separation of the crystalline scatter from the non-crystalline scatter must be made other restrictions are that the method cannot be used to measure crystallite size and cannot give any indication of the presence of paracrystalline or intermediate-phase material. 

Distortions of the Second Kind and Linear Paracrystallinity. The idea... 

PARACRYSTALLINE STRUCTURE OF POLYMER-CRYSTAL LAHICE DISTORTION INDUCED BY ELECTRON IRRADIATION. 

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