Geometric Theory of Diblock Copolymer Phases

We analyze the energetics of spherelike micellar phases in diblock copolymers in terms of well-studied, geometric quantities for their lattices. We argue that the $A15$ lattice with $Pm\overline{3}n$ symmetry should be favored as the blocks become more symmetric and corroborate this through a self-consistent field theory. Because phases with columnar or bicontinuous topologies intervene, the $A15$ phase, though metastable, is not an equilibrium phase of symmetric diblocks. We investigate the phase diagram of branched diblocks and find that the $A15$ phase is stable.

Figure 1

In (a) we show the cubic unit cell of the $A15$ lattice. The black spheres form a bcc lattice and the $A15$ lattice has, in addition, dimers on each face, in grey. The diblock topologies we consider are (b) linear and (c) multiply branched with three generations.

Figure 2

A plot of the $AB$ interfaces in the bcc ($Im\overline{3}m$) phase obtained by extracting the surface where the concentration of each block is equal. Both figures were extracted for $\chi N=40$. In (a) $\varphi =0.222$ and the interface is almost spherical. In (b) $\varphi =0.45$ and the interface takes on the shape of the Voronoi cell. The solids lines represent the entire bcc cell, while the dashed lines represent the reduced Voronoi cell that would arise in the $\varphi \to 1$ limit. The insets depict (a) the spherical interface limit (small $\varphi$) and (b) the flat interface limit (large $\varphi$).

Figure 3

The SCFT phase diagram for melts composed of (a) linear chains. The dotted line is where the free energy of the bcc phase is equal to that of the $A15$ phase, while the dashed line is the corresponding boundary for the fcc phase. In the phase diagram (b) for the three-generation diblocks shown in Fig. 1c the $A15$ phase is a ground state for a significant range of $\varphi$ and $\chi N$. All the $B$ blocks are of equal length. Note the critical point on the lamellar/disordered boundary.