Formation of Nonclassical Ordered Phases of $AB$-Type Multiarm Block Copolymers

The formation of ordered phases from block copolymers is driven by a delicate balance between the monomer-monomer interaction and chain configurational entropy. The configurational entropy can be regulated by designed chain architecture, resulting in a new entropy-driven mechanism to control the self-assembly of ordered phases from block copolymers. An effective routine to regulate the configurational entropy is to utilize multiarm architecture, in which the entropic contribution to the free energy could be qualitatively controlled by the fraction of bridging configurations. As an illustration of this mechanism, the phase behavior of two $AB$-type multiarm block copolymers, ${B_0-(B_i-A_i)}_m$ and ${(B_1-A_i-B_2)}_m$ where the minority $A$ blocks form cylindrical or spherical domains, are examined using the self-consistent field theory (SCFT). The SCFT results demonstrate that the packing symmetry of the cylinders or spheres can be controlled by the length of the bridging $B$ blocks. Several nonclassical ordered phases, including a novel square array cylinder with $p4mm$ symmetry, are predicted to form from the $AB$-type multiarm block copolymers.

Figure 1

Left: schematic plot of the architectures of the two designed $AB$-type multiblock copolymers, $M_1$ and $M_2$. $A$ and $B$ blocks are drawn in red and blue colors, respectively. Center: illustrative plot of chain configurations in three typical cylindrical morphologies with different coordination numbers demonstrating the formation of bridges among neighboring domains and their impact on the domain arrangement (i.e., packing lattice). Right: density profiles of the stable morphologies self-assembled from the two block copolymers, including the classical phase of hexagonal cylinders (denoted as $C_{p6mm}^6$, where the subscript and superscript indicate the plane symmetry and coordination number, respectively) and three nonclassical phases of square array of cylinders (similarly denoted as $C_{p4mm}^4$), graphenelike cylinders ($C_{p3mm}^3$), and honeycomblike network phase ($L_{p3m}$). More candidate phases are listed in Fig. S1 [32].

Figure 2

Phase diagram of molecule $M_1$ in the $f_{A_i}-f_{B_0}$ plane with $m=5$ and $\chi N=100$. The red dashed line indicates the phase path of $f_{A_i}=0.05$.

Figure 3

Phase diagram of molecule $M_2$ in the $f_{A_i}-f_{B_1}$ plane with $m=5$ and $\chi N=140$. In the upper region of the diagram, the $C_{p6mm}^6$ phase transforms continuously into a core-shell cylinder phase denoted as $C_{p6mm}^6(\mathrm{cs})$, and thereby no boundary is plotted.

Figure 4

Density profiles of the junction point of $M_1$ in the $C_{p4mm}^4$ phase for various $f_{B_0}$ with $f_{A_i}=0.05$. The density distribution along the middle line of the cell, i.e., $y=L_y/2$, is plotted in Fig. S2 [32].