Body-centered 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... 

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.)...

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.)...

Figure 1. Phase diagram for a structurally symmetric coil—coil block copolymer (Lam = lamellae, Hex = hexago-najly packed cylinders, Q/a3d = bicontinuous cubic with laid symmetry, Q/m3m = body-centered cubic, CPS = close packed sphere).

The development of patterns is not necessarily a manifestation of a nonequilibrium process. A spatially non-uniform state can correspond to the minimum of the free-energy functional of a system in thermodynamic equihbrium, as Abrikosov vortex lattices, stripe ferromagnetic phases and periodic diblock-copolymer phases mentioned above. In the latter, a hnear chain molecule of a diblock-copolymer consists of two blocks, say, A and B. Above the critical temperature Tc, there is a mixture of both types of blocks. Below Tc, the copolymer melt undergoes phase separation that leads to the formation of A-rich and B-rich microdomains. In the bulk, these microdomains typically have the shape of lamellae, hexagonally ordered cylinders or body-centered cubic (bee) ordered spheres. On the surface, one again observes stripes or hexagonally ordered spots. 

Under the influence of such a potential, the spheres easily settle into a crystalline form when the temperature is lowered or the pressure is raised. The solid phase can be either a face-centered cubic (fee) lattice or a body-centered cubic (bee) lattice, depending on the detailed nature of the potential. In the case of argon, the crystal freezes into an fee lattice, while liquid sodium freezes into a bee lattice. The transformation of simple spherical molecules (such as argon and sodium) into the crystalline solid phase is now rather well understood, largely because of the extensive use of computer simulation studies, accompanied by theoretical analysis using methods of statistical mechanics. 

Figure 2.5 Equilibrium phase morphologies of AB diblock copolymers as a function of component A volume fraction (S) Spheres (body-centered cubic lattice), (C) Cylinders (hexagonal lattice), (G) Gyroid, (L) Lamellar. Reproduced with permission from Ref. [7].

Fig. 8. Phase diagram of a diblock copolymer following from SOFT assuming similar segment lengths of both blocks. Scpi spheres arranged on a face-centered cubic lattice, S spheres arranged on a body-centered cubic lattice, C hexagonally packed cylinders, G double gyroid, L lamellae. From Ref 92. Copyright (1997) American Institute of Physics.

Early morphological studies to determine the nature of multiphase polymers and blends were reviewed by Folkes and Keller [363]. Many studies were of extruded block copolymers of materials such as SBS where the dispersed phase, an unsaturated rubber stained with OSO4 (see Section 4.4.2), was observed in the form of spheres, cylinders, or lamellae [364]. An excellent example is shown in a TEM micrograph of a thin section of a poly(styrene-butadiene) diblock copolymer, stained with OSO4 [365], which depicts the (100) projection of a body centered cubic lattice (Fig. 5.79). 

Discontinuous 3D cubic structures commonly observed in BCP SA are spherical morphologies as shown in Fig. 2. In this case, the minority block of the BCP forms spheres and the spatial arrangement of the spheres varies from body-centered cubic (Im3 m, Q229 space group) to closed packed spheres as the volume fraction of the minority block and/or the product iflV decrease [18]. Furthermore, the so-called A15 phase with Pm3 n symmetry was observed in amphiphilic di-BCPs containing a linear block and a dendron block [17]. 

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