Lamellar crystal geometry

In most spherulitic polymers, touching spherulites occupy whole of the space. Their microstructure is too complex to be completely modelled, especially if there is twisting of lamellar stacks about spherulite radii. Consequently, models simplify the structure, and use composite micromechanics concepts. A stack of parallel lamellar crystals with interleaved amorphous layers (Fig. 3.20) has a similar geometry to a laminated rubber/metal spring (Fig. 4.1). The crystals have different Young s moduli E, Eb and E (Section 3.4.3), and different shear moduli when the... 

As can be seen from Table 2.3, in the case of athermal nucleation the Avrami exponent equals the dimension of the geometry of the growing crystal entities 1 for fibrillar crystals, 2 for lamellar crystals, and 3 for spherulites. In the case of thermal nucleation, the Avrami exponent equals the geometry of the growing entities plus 1 (however, this is not true for diffusion-controlled processes). 

The model has been successfully used to describe wetting behavior of the microemulsion at the oil-water interface [12,18-20], to investigate a few ordered phases such as lamellar, double diamond, simple cubic, hexagonal, or crystals of spherical micelles [21,22], and to study the mixtures containing surfactant in confined geometry [23]. 

FIG. 3. Geometry of hydrated molecules cylinders associate to a lamellar liquid crystal, cones to a hexagonal and an inverse hexagonal. Adapted from The Physical Chemistry ofMembranes (Silver, B., ed.), Allen Unwin, Inc. Solomon Press, Winchester, MA, 1985. 

Several investigators have used radioactive tracer methods to determine diffusion rates. Bangham et al. (32) and Papahadjopoulos and Watkins (33) studied transport rates of radioactive Na+, K+, and Cl" from small particles or vesicles of lamellar liquid crystal to an aqueous solution in which the particles were dispersed. Liquid crystalline phases of several different phospholipids and phospholipid mixtures were used. Because of uncertainties regarding particle geometry and size distribution, diffusion coefficients could not be calculated. Information was obtained, however, showing that the transport rates of K+ and Cl" in a given liquid crystal could differ by as much as a factor of 100. Moreover, relative transport rates of K+ and Cl" were quite different for different phospholipids. The authors considered that ions had to diffuse across platelike micelles to reach the aqueous phase. 

Thermotropic liquid crystals can then be furflier subdivided into high molecular mass, main and side-chain polymers [10] and low molecular mass, the latter class of compounds being one of the areas of this review. The phases exhibited by the low molecular mass molecules are then properly described with reference to the symmetry and/or supramolecular geometry of the phases, which are briefly introduced here and are discussed in more detail further below. Thus, the most disordered mesophase is the nematic (N), which is found for calamitic molecules (N), discoidal molecules (Nq) and columnar aggregates (Nc), among others. The more ordered lamellar or smectic phases (S) [11, 12] are commonly shown by calamitic molecules, and there exists a variety of such phases distinguished by a subscripted letter (e. g. Sa, Sb)- Columnar phases (often, if incorrectly, referred to as discotic phases) may be formed from stacks of disc-like molecules, or from... 

In presence of a solvent. e.g water, it is known that mesopha.se can be lamellar (zero curvature), hexagonal, or cubic (20). In the presence of both oil and water, both pha.ses may penetrate between (he surfactant bilayers, and liquid crystals of the curved type could occur with the water or the oil toward concavity. In between. i.e for the K = 1 case, a zero curvature, that may be attained either with a lamellar structure of the IX type, or a biconlinuous microemulsion with monkey saddle zero curvature geometry everywhere, is expected. 

FIGURE 17 Schematic representation of (a) the lamellar geometry of PTT single crystal and (b) the twisting mechanism of the intralamellar model in PTT, as proposed by Ho et... 

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