Packing frustration

However, a PS-fo-PI/PI blend shows direct L G transitions without appearance of the PL phase. The L microdomain is more favourable than the PL phase since the volume fraction of the PI block component and the symmetry of microdomains is increased by the addition of PI homopolymer. Hence, the PL phase may not be formed as an intermediate structure if relatively high molecular weight PI homopolymer is added. The latter is not able to effectively fill the corners of the Wigner-Seitz cells in consequence packing frustration cannot be released and the PL phase is not favoured [152]. In contrast, the addition of low molecular weight PI homopolymer to the minor component of the PL phase reduces the packing frustration imposed on the block copolymers and stabilizes it [153]. Hence, transition from the PL to the G phase indicates an epitaxial relationship between the two structures, while the direct transition between L and G yields a polydomain structure indicative of epitaxial mismatches in domain orientations [152]. 

Fig. 6.26 Schematic illustrating packing frustration induced by addition of a short diblock (s) to a microphase of a long diblock (I) (Hasegawa and Hashimoto 1996). If the long diblock forms a spherical or cylindrical domain, for example, (a) segregation of the short diblock to the interface leads to regions of low packing density (region X in (b)). This frustration can be relieved if the interface adopts a planar geometry, driving a transition to a lamellar phase (with a bimodal copolymer brush at the interface) (c).

Compound 189 with a shorter terphenyl rod-like core also has two Colhex phases, formed by cylinders, but with a hexagonal cross sectional area (Fig. 63). In this case a complete segregation of the Rp-chains from the RSi-chains is impossible as it would lead to a packing frustration and therefore formation of cylinders incorporating mixed chains cannot be avoided. Based on simulation results [346], XRD data, and AFM results, the two color tiling with a p6mm superlattice... 

Oligofluorenes studies have been performed by Wegner and coworkers [18,57,86,87] and others [88,89]. The packing frustration evident in the PF2/6 polymer is already present in the chains as short as a trimer [89]. Even at this short length thin films exhibit mesotropic-type phases [89] as opposed to the crystalline monomer samples. 

The phase diagrams of copolymer-homopolymer blends can be complex. Matsen has examined binary blends of AB copolymers with A homopolymers, that is, AB/A, and presented phase diagrams for various values of /a for the copolymer, examining the overall phase behavior, including the stability of different microphases and how they are affected by the homopolymer [44,45]. A notable conclusion of this work is that the addition of homopolymer can stabilize the OBDD and PL phases. This is attributed to the relief of packing frustration, discussed above in Section VI.C, which results from the nonuniform distribution of the homopolymer, with preferential localization at the comers of the unit cells. 

We note that there are hints in Table I, and structural fluctuation quantities such as Ag(r) discussed elsewhere, that as N or stiffness mismatch increase the excess free energy of mixing also increases and the blend is less stable. Thus, the possibility of entropy-driven phase separation due to packing frustration of dissimilar flexibility chains as N increases beyond 200 remains open based on the simulation studies of Ref. 97. 

Although a few 3D isotropic cubic network phases have already been observed, other cubic network phases have stayed elusive in BCP SA, such as the I-WP [25], Neovius [26], K surface [27], and Lidinoid [28] structures (see Fig. 2). Packing frustration of polymer chains in the network nodes of these structures is a primary hurdle for formation of these phases. However, as shown in the cases of the plumber s nightmare and double diamond structures, BCP co-assembly with additive molecules or ordered local packing of monomer units may open new routes to such cubic structures. 

There have been few reports to date on quasicrystaUine structure foimatimi from BCPs. Hayashida et al. obtained a 2D quasicrystaUine structure, a tUing pattern with 12-fold symmetry, from an ABC star BCP and homopolymer blend (see Fig. 3) [34]. The Bates group at the University of Minnesota generated a 3D dodecagonal quasicrystal from diblock and tetrablock copolymers [33]. These unique structures are mediated by macromolecular packing frustration. However, a substantial number of unanswered questions about the stmcture formation await further in-depth studies of these interesting morphologies. 

Within the mean-field approximation, the reciprocal-space method of Matsen and Schick [14] provides an efficient and accurate numerical technique to solve the SCMFT equations for given ordered structures. This has led to a comprehensive understanding of the equilibrium phase behavior of simple block copolymer systems. Valuable insights into the physics of the self-assembly in block copolymer systems have been obtained from the numerical solutions. In particular, the formation of different structures can be explained using the concepts of spontaneous interfadal curvature and packing frustration [3]. 

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