^{1,a)}, Wenjuan Qiu

^{1}, Linxi Zhang

^{1}and Haojun Liang

^{2}

### Abstract

The nanostructures and phase diagrams of ABC star triblock copolymers in pore geometries are investigated using the real-space self-consistent field theory in two-dimensional space. Two types of pores with neutral surfaces, namely, pores with small and large diameters, are considered. A rich variety of nanostructures are exhibited by the ABC star triblock copolymers in these two types of pores, which differ from those observed in bulk and in other confinements. These structures include perpendicular undulating lamellae, concentric core-shell cylinders, polygonal tiling with cylindrical arrangements, and other complex structures. Triangular phase diagrams for the ABC star triblock copolymers are constructed. The small pores clearly affect the corner and central space of the phase diagrams by distorting the bulk structures into concentric arrangements. Meanwhile, the large pores induce the transformation of bulk structures into concentric structures in most of the phase space, but slightly affect the structures at the center of the phase diagrams. Furthermore, the order–order and order–disorder phase transitions, as well as the stable and metastable phases, in the triangular phase diagrams are examined by analyzing their free energies. These observations on the ABC star triblock copolymers in the pore geometries provide a deeper insight into the behavior of macromolecules in a confined system.

This research was supported by the National Natural Science Foundation of China (Nos. 91127046, 20934004, 20974081, 21174131, and 21074096), the Outstanding Youth Fund of China (No. 20525416), and the National Basic Research Program of China (No. 2005CB623800). The authors thank Professor Jeff. Z. Y. Chen for helpful discussion of related topics, and also thank the referees for their critical reading for the manuscript and their good ideas.

I. INTRODUCTION

II. THEORETICAL METHOD

III. RESULTS AND DISCUSSION

A. Structures and phase diagrams in bulk and small pores

1. Bulk structures and phase diagram

2. Structures and phase diagram in small pores

B. Structures and phase diagrams in large pores

1. Structures in large pores

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 74.0
- Phase diagrams
- 54.0
- Free energy
- 32.0
- Phase transitions
- 21.0
- Lamellae
- 17.0

## Figures

Nanostructures and phase diagram of ABC star triblock copolymers in bulk. The interaction parameters are selected to be symmetric, i.e., χ_{ AB } *N* = χ_{ BC } *N* = χ_{ AC } *N* = 32. The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. (a) Several typical nanostructure with the designations and periods in *x* and *y* directions list below. (b) The triangular phase diagram. The red dash line represents the isopleths of AC boundary.

Nanostructures and phase diagram of ABC star triblock copolymers in bulk. The interaction parameters are selected to be symmetric, i.e., χ_{ AB } *N* = χ_{ BC } *N* = χ_{ AC } *N* = 32. The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. (a) Several typical nanostructure with the designations and periods in *x* and *y* directions list below. (b) The triangular phase diagram. The red dash line represents the isopleths of AC boundary.

Structures of ABC star triblock copolymers confined in a pore geometry with small diameter, *d* = 9.2 *R* _{ g }. The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. The phase symbols and designations are also shown below the corresponding structures, respectively.

Structures of ABC star triblock copolymers confined in a pore geometry with small diameter, *d* = 9.2 *R* _{ g }. The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. The phase symbols and designations are also shown below the corresponding structures, respectively.

Triangular phase diagrams of ABC star triblock copolymers confined in a pore geometry with small diameter, *d* = 9.2 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 2. The phase with symbol and designation, whose free energy is very close, is list near the phase diagram where the phase point is also labeled. The red dashed line is the isopleths of AC boundary.

Triangular phase diagrams of ABC star triblock copolymers confined in a pore geometry with small diameter, *d* = 9.2 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 2. The phase with symbol and designation, whose free energy is very close, is list near the phase diagram where the phase point is also labeled. The red dashed line is the isopleths of AC boundary.

Typical examples of order-order phase transitions with volume fraction of A block in the pores with small diameters, *d* = 9.2 *R* _{ g }, when *f* _{ C } = 0.4. The order-order transition points are denoted as OOT, and the nanostructures and the symbols are also inserted in the corresponding positions.(a). The free energies as functions of volume fraction. We subtract a linear function from the scaled free energies in order to highlight where the curve cross, and plot Δ*F*/*nk* _{ B } *T* = *F* _{min}/*nk* _{ B } *T* − [7.05(*f* _{ C } − 0.17) + 8.23]. (b). The phase stability regions as function of volume fraction.

Typical examples of order-order phase transitions with volume fraction of A block in the pores with small diameters, *d* = 9.2 *R* _{ g }, when *f* _{ C } = 0.4. The order-order transition points are denoted as OOT, and the nanostructures and the symbols are also inserted in the corresponding positions.(a). The free energies as functions of volume fraction. We subtract a linear function from the scaled free energies in order to highlight where the curve cross, and plot Δ*F*/*nk* _{ B } *T* = *F* _{min}/*nk* _{ B } *T* − [7.05(*f* _{ C } − 0.17) + 8.23]. (b). The phase stability regions as function of volume fraction.

Structures of ABC star triblock copolymers confined in a pore geometry with large diameter, *d* = 20 *R* _{ g } The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. The phase symbols and designations are also shown below the corresponding structures, respectively.

Structures of ABC star triblock copolymers confined in a pore geometry with large diameter, *d* = 20 *R* _{ g } The red, green, and blue colors represent the A-block, B-block, and C-block rich domains, respectively. The phase symbols and designations are also shown below the corresponding structures, respectively.

Triangular phase diagrams and phase transitions of ABC star triblock copolymers in a pore geometry with large diameter, *d* = 20 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 5. (a). Triangular phase diagram with the phase symbols and designations, whose free energy is very close to one phase point, is also shown. (b) The free energy as function of volume fraction of A-block blocks, when *f* _{ C } = 0.4. The nanostructures are also inserted in the corresponding positions.

Triangular phase diagrams and phase transitions of ABC star triblock copolymers in a pore geometry with large diameter, *d* = 20 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 5. (a). Triangular phase diagram with the phase symbols and designations, whose free energy is very close to one phase point, is also shown. (b) The free energy as function of volume fraction of A-block blocks, when *f* _{ C } = 0.4. The nanostructures are also inserted in the corresponding positions.

Refined triangular phase diagrams and phase transitions of ABC star triblock copolymers in a pore with large diameter, *d* = 20 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 5. The order-order transition points are denoted as OOT, while the order-disorder transition is denoted as ODT. The nanostructures are also inserted in the corresponding positions. (a) The refined triangular phase diagram for the zone labeled as blue line in Figure 6. (b) The order-order and order-disorder phase transitions with volume fractions of C blocks, when *f* _{ A } = *f* _{ B }. We subtract a linear function from the scaled free energies in order to highlight where the curves cross, and plot Δ*F*/*nk* _{ B } *T* = *F* _{min}/*nk* _{ B } *T* − [18.31 − 15.63 *f* _{ C }].

Refined triangular phase diagrams and phase transitions of ABC star triblock copolymers in a pore with large diameter, *d* = 20 *R* _{ g }, arranged as the volume fractions. All the phase denotations are the same as those in Figure 5. The order-order transition points are denoted as OOT, while the order-disorder transition is denoted as ODT. The nanostructures are also inserted in the corresponding positions. (a) The refined triangular phase diagram for the zone labeled as blue line in Figure 6. (b) The order-order and order-disorder phase transitions with volume fractions of C blocks, when *f* _{ A } = *f* _{ B }. We subtract a linear function from the scaled free energies in order to highlight where the curves cross, and plot Δ*F*/*nk* _{ B } *T* = *F* _{min}/*nk* _{ B } *T* − [18.31 − 15.63 *f* _{ C }].

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