Microstructures of A4(BC)4 star copolymers and the influence of architecture

The self-consistent field theory is used to investigate the microstructures of A4(BC)4 star copolymers. The phase diagram is mapped out to show different phase regions. Compared with A(BC)4 star copolymers, several microstructures are lost, and more lamellar structures are found due to the stronger topological constraint of the junction. The results may be helpful to understand the influence of molecular architectures on microstructures and provide an effective way to design controllable microstructures.


Introduction
The self-assembly of block copolymers has attracted much attention in recent years for its promising applications in material science, nanotechnology and biology. For the simplest linear AB diblock copolymers, the fascinating periodic morphologies have been deeply understood both experimentally and theoretically [1][2][3][4]. As the distinct components increase, say ABC triblock copolymers, the variety and complexity of microstructures may increase dramatically [5,6]. Especially, once the topological architecture changes from line shape to star, comb-like or other complicated shapes, more novel structures can be found [7][8][9][10]. The star architecture imposes a strong topological constraint on the polymer chain and may lead to new phenomena. In particular, the development of synthesis technology allows us to obtain more complex copolymers, which provides an opportunity to find new periodic microstructures due to the additional entropic effect. By far, systematic investigations on how the microstructures relate to the architectures are still desired. In this work, we use the self-consistent field theory (SCFT) to investigate the microstructures of A 4 (BC) 4 star copolymers, where four A chains and four BC chains are attached to a junction point. Furthermore, we make a comparison between the microstructures of A(BC) 4 star copolymers, where one A chain and four BC chains are attached to a junction point. It is very helpful to explore the microstructure differences between A 4 (BC) 4 and A(BC) 4 star copolymers to understand the influence of molecular architecture on microstructures. Figure 1 gives a schematic of A 4 (BC) 4 and A(BC) 4   As an important mean-field theory for polymers, the SCFT has been widely used in the prediction of phase behavior of block copolymers and other related systems [11,12]. For the A 4 (BC) 4 or A(BC) 4 copolymer melts in a volume V, the free energy F can be written as

Model and method
In the equation (1), T is the temperature and k B is the Boltzmann's constant. The variables φ i , W i (i = A, B, C) and ζ are the functions of the position r . φ i is the local densities of different components. χ ij (= χ ij , i, j = A, B, C) is the Flory-Huggins interaction parameter between two different components. W i is the effective chemical potential field acting on the different component. ζ is a Lagrange-multiplier potential field ensuring the system incompressibility. Ω is the partition function of a single star copolymer and is defined by ( In the equation (2), q i and q i + (i = A or BC) are the propagators and denote the probability that finds the segment s at the position r . The propagators satisfy the modified diffusion equations: Due to the different architectures of the A 4 (BC) 4 and A(BC) 4 copolymers, the initial conditions of above equations are different.
For the A 4 (BC) 4 copolymers, the initial conditions are given by   . (8) Minimizing the free energy F, we obtain a set of self-consistent equations. The field W i and the local density φ i at the equilibrium can be determined by using the combinatorial screening method [12] to solve the self-consistent equations with periodic boundary conditions.

Results and discussion
To compare the microstructures of two kinds of copolymers, we ensure all factors are same except for the architectures. The Flory-Huggins interaction parameters and the polymerization in both copolymers are set as χ AB N = χ BC N = χ AC N = 140 and N = 600. All the sizes are in units of a.   To further examine the influence of architecture, we compare the microstructures with those of the A(BC) 4 star copolymers. For clarity, the phase diagram of the A(BC) 4 star copolymers is shown in figure 2(b). Obviously, the microstructure number of the A 4 (BC) 4 star copolymers is smaller than that of the A(BC) 4 star copolymers. Other structures such as the two-phase lamellar structure with alternating blocks inside a layer (L2AL), the two-phase lamellar structure with beads at the interface (L2BD) and the interpenetrating hexagonal microstructures (INHE) observed in the A(BC) 4 star copolymers(see Ref. [8]) cannot be found in the A 4 (BC) 4   microstructures show that the mixed blocks in the lamellar structures may further separate inside the mixed layer which can be attributed to the flexibility of one arm A. In addition, we find that most of the grid points in the phase diagram are occupied by the lamellar structures in the A 4 (BC) 4 star copolymers. In comparison with the phase diagram of the A(BC) 4 star copolymers, most of the hexagonal microstructures and the core-shell hexagonal microstructures in the A(BC) 4 star copolymers are replaced by all kinds of the lamellar structures in the A 4 (BC) 4 star copolymers. Due to the junction constraint of four chains A, the formation of the lamellar structures may help to lower the free energy by relieving the interfacial crowding. The symmetry of four blocks A and four blocks BC also favours the formation of symmetrical structures. Therefore more lamellar structures are observed in the A 4 (BC) 4 star copolymers.

Conclusion
We studied the microstructures of the A 4 (BC) 4 star copolymers by using self-consistent field theory. Six kinds of different microstructures are observed by varying the average densities of distinct components when χ AB N = χ BC N = χ AC N = 140 and N = 600. Compared with the microstructures of the A(BC) 4 star copolymers with the same parameters, the number of the microstructures decreases and more lamellar structures are found due to a stronger junction constraint of blocks A. It is helpful to understand the influence of architectures on microstructure and provide an effective way to design controllable microstructures with these results.