Cubic sphere phase

The morphology of the ABA-type linear block copolymers is strongly influenced by the volume fraction of the two components. For example, in PS-EB-PS-type block copolymer as the volume fraction of PS is increased, the shape of the dispersed PS phase changes from spherical (comprising body-centered cubic spheres of PS dispersed in continuous soft phase) to cylindrical form (hexagonal packed cylinders of PS) [10,133,134]. When the volume fraction of the two phases... 

Here, we have shown an "A" ion (shown as a black sphere) which approaches the surface of "AB" and displaces an "A" ion in this solid phase. A series of "hops", i.e.- from "1" to "7", then occurs in the AB-phase with the final "A" ion ending within the B-phase where a displacement in the normally cubic "B" phase occurs. At the same time, the displaced "B" ion is diffusing in the opposite direction by a series of "hops", i.e.- from "a" to "e" to the interface of "AB" with "A". Note that the "A" phase is shown as a hexagonal phase while "B" is a cubic phase (as is the "AB" phase). It should be clear that the rate of diffusion of "A" will differ firom that of "B". 

PE-PEP diblock were similar to each other at high PE content (50-90%). This was because the mechanical properties were determined predominantly by the behaviour of the more continuous PE phase. For lower PE contents (7-29%) there were major differences in the mechanical properties of polymers with different architectures, all of which formed a cubic-packed sphere phase. PE-PEP-PE triblocks were found to be thermoplastic elastomers, whereas PEP-PE-PEP triblocks behaved like particulate filled rubber.The difference was proposed to result from bridging of PE domains across spheres in PE-PEP-PE triblocks, which acted as physical cross-links due to anchorage of the PE blocks in the semicrystalline domains. No such arrangement is possible for the PEP-PE-PEP or PE-PEP copolymers (Mohajer et al. 1982). 

Although a face-centered cubic-ordered phase appears at equilibrium for (f> > 0.50, disordered dispersions of hard spheres can persist for... 

We should now consider the factor 6 in the numerator of the pre-exponential in Eq. (14) which has the physical meaning of either a form factor, packing factor or coordination number of the closest spherical (cubic) sphere packing. The points of Fig. 3 are somewhat scattered probably because of the equal proportions of open cells (9J in samples of different volume weights. Eq. (14) has a maximum when 9(,/9p = 1 (Fig. 3), i.e. when the volume ratio of polymer in a sample is equal to that in a plastic foam with closest spherical packing the gas phase volume is then 74% which, for polyurethane foam, corresponds to a volumetric weight of 315 kg/m. ... 

Figure 7 A series of theoretieal rod-coil phase diagrams, where the degree of polymerization is kept constant and the balance of crystallization and order-disorder driving forces ((w/x) is increased from (a) to (e). Bcc = body-centered cubic (spheres). Hex = hexagonally packed cylinders, Lam = lamellae and the angles associated with the smectic C phase describe the tilted angle of the rod blocks, 6, and the dashed lines are contour lines for 6, separating intervals of 5°. (Adapted from Ref. 16. American Chemical Society, 2002.)...

For surfactant aggregates of the interfacial complex type under discussion, Eq. (205) will be fulfilled for the same sequence of elementary shapes as the one we have already discussed for microemulsions, that is, spheres and cylinders with radii in correspondence with the spontaneous curvature, infinite periodical CMC structures for which tOsO H—Hq are equal to zero but where K is less than zero, yielding cubic surfactant phases and lamellar structures. However, it is probably worth stressing once more that noninteracting surfactant-laden interfaces are in focus here. With this in mind, we can discuss just interactionless aggregate geometries but not the whole issue about the possible formation of three-dimensional structures of these aggregates. 

Fig. 58. The RAlj compounds with the cubic Laves phase crystal structure. Spheres with and without pattern show the A1 atoms and the R atoms, respectively.

Figure 5 Phase diagram of diblock copolymers with equal segmental lengths and segmental volumes of both block components. % Flory-Huggins-Staverman interaction parameter, N degree of polymerization, ( ) volume fraction, D disordered phase, CPS close packed spheres, BCC body-centered cubic spheres, H hexagonally packed cylinders, G gyroid, L lamellae. (From Ref. 110, Copyright 1996 American Chemical Society.)...

Charged particles in polar solvents have soft-repulsive interactions (see section C2.6.4). Just as hard spheres, such particles also undergo an ordering transition. Important differences, however, are that tire transition takes place at (much) lower particle volume fractions, and at low ionic strengtli (low k) tire solid phase may be body centred cubic (bee), ratlier tlian tire more compact fee stmcture (see [69, 73, 84]). For tire interactions, a Yukawa potential (equation (C2.6.11)1 is often used. The phase diagram for the Yukawa potential was calculated using computer simulations by Robbins et al [851. 

Fig. 1 Morphologies of diblock copolymers cubic packed spheres (S), hexagonal packed cylinders (C or Hex), double gyroid (G or Gyr), and lamellae (L or Lam). Inverse phases not shown. From [8], Copyright 2000 Wiley...

Usually the discussion of the ODT of highly asymmetric block copolymers in the strong segregation limit starts from a body-centred cubic (bcc) array of the minority phase. Phase transitions were calculated using SOFT accounting for both the translational entropy of the micelles in a disordered micelle regime and the intermicelle free energy [129]. Results indicate that the ODT occurs between ordered bcc spheres and disordered micelles. 

Fig-1 Mean-field predication of the morphologies for conformationally symmetric diblock melts. Phases are labeled as S (bcc spheres), C (hexagonal cylinders), G (bicontin-uous la 3 d. cubic), L (lamellar)./a is the volume fraction... 

The supramolecular transformation from sphere to cylinder is supported by X-ray data indicating that the spherical polymers adopt a cubic phase, whereas the cylindrical polymers adopt a hexagonal phase [23b]. Further studies involving a library of dendritic macromonomers led to the conclusion that the effect of DP on polymer shape is a general phenomenon [24], More recently, scanning... 

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