Cubic phase representation

Fig. 3 Schematic representation of the self-assembling of conical-like monodendrons into spherical supramolecular dendrimers and their self-organization into cubic phases (here Pm3n symmetry)...

According to electron micrographic evidence (principally comparison with bicontinuous cubic phases), phases are an accurate representation of so-called "non-bilayer" conformations of membranes. It must be stressed, however, that this conformation is a true bilayer. 

Figure 1.92 Structural representations of the main cubic mesophases formed by lipid assemblies. The mesophases shown are (a) Ia3d (Q °) and (b) Pn3m both of which are Qn fluid cubic phases ...

Figure 7. Polyhedral representations showing the micellar arrangements for (a) Pm3n, (b) Im3m and (c) Fm3m of I cubic phases (reproduced from [46]). Micelles are located at the centres of the polyhedra...

Figure 9. Schematic representations of cubic phases (reproduced from [42]).

Other more complex morphologies also arise for A-B mixtures. In particular, domains A and B may enclose each other, forming entangled networks, separated by a hyperbolic interface. Those cases include mesh , bicontinuous microemulsions, bicontinuous cubic phases and their disordered counterparts, sponge phases, which are discussed below. In these cases too, the sign (convex/concave) of the interfacial mean curvature sets the Type . A representation of the disordered mesostructure in a Type 2 bicontinuous microemulsion is shown in Figure 16.3. A hyperbolic interface may be equally concave and convex (a minimal surface, e.g. see Figure 16.2(c)) so that the mesophase is neither Type 1 nor Type 2. Lamellar mesophases ( smectics or neat phases) are the simplest examples. Bicontinuous balanced microemulsions, with equal polar and apolar volume fractions are further examples. 

This phase is built up of the regular packing of small micelles, which have similar properties to small micelles in the solution phase. However, the micelles are short prolates (axial ratio 1-2) rather than spheres since this allows a better packing. The micellar cubic phase is highly viscous. A schematic representation of the micellar cubic phase [25] is shown in Fig. 1.19. 

Figure 7.9 Schematic representations of the three inverse bicontinuous cubic phases. Reproduced from Reference 6 with permission of R. Oldenburg Verlag, GmbH.

Fig. 5.8. (a) Schematic representation of the repeating unit of a bicontinuous cubic phase (Vj). (b) Stacking of repeat units. (From Ref. 28 with permission of the American Chemical Society.)... 

Schwartzentruber J., F. Galivel-Solastiuk and H. Renon, "Representation of the Vapor-Liquid Equilibrium of the Ternary System Carbon Dioxide-Propane-Methanol and its Binaries with a Cubic Equation of State. A new Mixing Rule", Fluid Phase Equilibria, 38,217-226 (1987). 

WoWj/2 the body-centred cubic structure of W (1 atom in 0, 0, 0 and 1 atom in A, A, /) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and A, respectively. Note, however, that for a fall description of the structure, either in the hexagonal or the tetragonal case, the inter-layer distance must be taken into account not only in terms of the fractional coordinates (that is, the c/a axial ratio must be considered). For more complex polygonal nets, their symbolic representation and use in the description, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). 

Figure 11.10. Structure of dispersed self-assembly particles. The original cryo-TEM image is shown at the top with schematic representation of the structure below the images. Particles shown are (a) micelles, (b) vesicle, (c) inverted bicontinuous cubic and (d) reverted hexagonal phase particle. Adapted from Sagalowicz et al. 2006a.

Fig. 12 Schematic representation of the self-assembly of dendromesogens into various type of mesophases (smectic, hexagonal and rectangular columnar, micellar cubic, and tetragonal phases) by the control of the molecular shape conformation (from flat tapered to cylindrical to conical and to spherical shape). From [126]...

Apb is the scattering length density difference, Q is Porod s invariant, and Y the mean chord length. For the calculation of Yo(r) we approximated I(q) hy a cubic spline. The equations used for the calculation of " pore and " soUd are to be found in [8,30,39-41,47]. Analytical expressions for the descriptors of RES were published in [10,11,13,42,43]. In its most simple variant, the stochastic optimization procedure evolves the two-point probability S2 (r) of a binary representation of the sample towards S2(r) by randomly excWiging binary ceUs of different phases, starting from a random configuration which meets the preset volume fractions. After each exchange the objective function... 

Electron-density map A contour representation of electron density in a crystal structure. Peaks appear at atomic positions. The map is computed by a Fourier synthesis, that is, the summation of waves of known amplitude, periodicity, and relative phase. The electron density is expressed in electrons per cubic A. 

A two-parameter mixing rule is used with several cubic equations of state and is shown to be relatively successful in correlating the phase equilibrium behavior of biomolecules that cannot be correctly represented by conventional one-parameter mixing rules. The modification is related to the idea of local composition, which has been shown to improve the representation of the phase equilibrium in asymmetric mixtures. However, further improvement is still needed. 

Figure 13. Schematic representation of supramolecular structures of rod-coil molecules 10. (a) Smectic A, (b) bicontinuous cubic, and (c) hexagonal columnar phases. (Reprinted with permission from ref 64. Copyright 1998 American Chemical Society).

Ashour I, Almehaideb R, Fateen SE, Aly G. Representation of solid fluid phase equilibria using cubic equations of state. Fluid Phase Equilibria 2000 167 41-61. 

The electron density (p(xyz) at a point x,y,z in the unit cell is expressed in electrons per cubic A and is highest near atomic centers. Electron density values can be plotted at constant intervals to give a representation of features of such a map. The equation for the electron density at any point x,y,z in the unit cell then involves F hkl, the structure factor amplitude, and a /, the relative phase angle. 

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