Fractional-order rods

The intensity variation along the rod (i.e. as a function of or /) is solely contained in the structure factor it is thus related to the z-co-ordinates of the atoms within the unit-cell of this quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements. This is the case of a reconstructed surface, for which rods are found for fractional order values of h and k, i.e. outside scattering from the bulk. 

From the free energy (see Equation 2.23), the equation of state at equilibrium and the chemical potential can be determined accordingly. For instance, when the free energy is differentiated with respect to volume V, P = —dF/dV, the pressure-volume relationship shown in Figure 2.3 is obtained. The ordinate is the reduced pressure PVo/k T (Vo is the volume of rod) and the abscissa is the reduced volume V/NVq (or the reciprocal of volume fraction of rods in solution). In a system of axial ratio 10, the change of density at the transition is 0.15 while the order parameter jumps by 0.7361. 

In both theories the aspect ratio of the rod (the shape anisotropy of the solute) determines the critical volume fraction of rods which, if exceeded, results in a spontaneously ordered phase of quasiparallel rods (a nematic lyotropic phase). 

Onsager treated the LC state by a virial expansion method. He deduced the relationship between the volume fraction of rods, the rod length, and the rod diameter for both ordered and isotropic phases [16]. 

The N-N transition is predicted to occur at quite high volume fractions of rods. At these high volume fractions the N-N transition may be superseded by more highly ordered (liquid) crystal phases such as the colloidal smectic phase. Experimentally, this colloidal smectic phase has been observed [28, 44, 45] in suspensions of monodisperse rods. Simulations confirmed that hard rods can form a thermodynamically stable smectic phase [2-A]. 

Figure 8 Bragg peaks measured by GIXD on a self-assembled uncompressed monolayer of arachidic acid on CdCl2 (10 M, pH = 8.8 (ammonia), T= 9°C), from ref [122, 123]. The corresponding Bragg rods (not shown) show that the integer order peaks result mainly from the close-packed molecules while only the metal ions, laterally ordered in a thin layer, contribute to the fractional order peaks.

In order to facilitate contact with the original literature we use the rod s aspect ratio to transform from the number concentration c to the volume fraction of rods (j> = (j>/A)cd L. 

All terms in Eq. (49) are purely entropi-cal in nature because the system is athermal. The second term is the orientational entropy and the third one is the packing entropy that is related to the second virial coefficient for two rigid rods. The expansion in Eq. (49) is actually performed in powers of the packing fraction i]=pVQ pipD L< if LID > 1, where Vq is the volume of a sphe-rocylinder. At a very low volume fraction of rods the higher order terms in the expansion can be neglected [29]. 

Figure 3.4.2.9 Real and reciprocal space of a (2x1) reconstructed surface. The two times larger periodicity in real space gives a two times smaller periodicity in reciprocal space and leads to fractional-order (half-order in this example) or superstructure rods that have no bulk Bragg peaks.

The polyamides are soluble in high strength sulfuric acid or in mixtures of hexamethylphosphoramide, /V, /V- dim ethyl acetam i de and LiCl. In the latter, compHcated relationships exist between solvent composition and the temperature at which the Hquid crystal phase forms. The polyamide solutions show an abmpt decrease in viscosity which is characteristic of mesophase formation when a critical volume fraction of polymer ( ) is exceeded. The viscosity may decrease, however, in the Hquid crystal phase if the molecular ordering allows the rod-shaped entities to gHde past one another more easily despite the higher concentration. The Hquid crystal phase is optically anisotropic and the texture is nematic. The nematic texture can be transformed to a chiral nematic texture by adding chiral species as a dopant or incorporating a chiral unit in the main chain as a copolymer (30). 

Figure 15 Morphological map of linear polyethylene fractions. Plot of molecular weight against crystallization temperature. The types of supermolecular structures are represented by symbols. Patterns a, b and c represent spherulitic structures with deteriorating order from a to c. Patterns g and d represent rods or sheet-like structures whose breadth is comparable to their length g or display a different aspect ratio d. Pattern h represents randomly oriented lamellae. Neither h nor g patterns have azimuthal dependence of the scattering. Reproduced with permission from Ref. [223]. Copyright 1981 American Chemical Society. (See Ref. [223] for full details.) Note the pattern a is actually located as o in the figure this was an error on the original.

Acetyl chloride (54 g. =075 mole) is allowed to run drop by drop from a tap funnel on to 80 g. of finely powdered anhydrous sodium acetate prepared in the manner described below. When about half of the chloride has been added the experiment is interrupted for a short time in order to stir the pasty mass of material with a bent glass rod, the lower end of which has been flattened. The rest of the acetyl chloride is then run in at such a rate that none passes over unchanged. The anhydride is now distilled from the residual salt by mean of a luminous flame kept constantly in motion. Complete conversion of the last traces of unchanged acetyl chloride to acetic anhydride is attained by adding 3 g. of finely powdered anhydrous sodium acetate to the distillate, which is finally fractionally distilled. Boiling point of acetic anhydride 138°. Yield 55-60 g. Use for acetylation, in Perkin s synthesis (Chap. V. 8, p. 232), preparation of acetophenone (Chap. IX. 3 6, p. 346). 

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