^{1,a)}, Yuci Xu

^{1}, Guojie Zhang

^{1}, Feng Qiu

^{1}, Yuliang Yang

^{1}and An-Chang Shi

^{2}

### Abstract

The phase behavior of ABC star triblock copolymers is examined using real-space self-consistent mean-field theory. The central part of the triangular phase diagram for ABC triblock copolymers with equal A/B, B/C, and C/A interactions is determined by comparing the free energy of a number of candidate ordered phases. In this region of the phase diagram, the dominant microstructures are cylinders with polygonal cross sections or two-dimensional polygon-tiling patterns. Most of the known polygon-tiling patterns observed in experiments and simulations, plus some neighboring morphologies, are considered in the construction of the phase diagram. The resulting phase behavior is consistent with experiments and computer simulations.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 20974026 and 20625413). W.L. gratefully acknowledges support from the Shanghai Pujiang Program (Program No. 08PJ1402000), the Shanghai Educational Development Foundation (Program No. 2008CG02), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

I. INTRODUCTION

II. THEORY

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 55.0
- Free energy
- 41.0
- Phase diagrams
- 40.0
- Mean field theory
- 28.0
- Spectral methods
- 23.0

## Figures

Monomer density plots of typical ordered microstructures formed in ABC star triblock copolymers with . The colors of red, green, and blue, indicate the regions where the most components are A, B, and C, respectively. There are seven 2D cylindrical structures of polygon-tiling patterns with translational symmetry along the third direction: (a) [6.6.6], (b) [8.8.4], (c) [8.6.4; 8.6.6], (d) [10.6.4; 10.8.4], (e) [10.6.4; 10.6.6], (f) [12.6.4], and (j) [8.6.4; 8.6.6; 12.6.4] (these integers indicate the sequence of polygons meeting at a vortex in each pattern); and three other 2D structures, including (g) L3, (h) HL and hexagonally arranged HC, and one HHC. The basic vectors of these periodic structures are labeled as .

Monomer density plots of typical ordered microstructures formed in ABC star triblock copolymers with . The colors of red, green, and blue, indicate the regions where the most components are A, B, and C, respectively. There are seven 2D cylindrical structures of polygon-tiling patterns with translational symmetry along the third direction: (a) [6.6.6], (b) [8.8.4], (c) [8.6.4; 8.6.6], (d) [10.6.4; 10.8.4], (e) [10.6.4; 10.6.6], (f) [12.6.4], and (j) [8.6.4; 8.6.6; 12.6.4] (these integers indicate the sequence of polygons meeting at a vortex in each pattern); and three other 2D structures, including (g) L3, (h) HL and hexagonally arranged HC, and one HHC. The basic vectors of these periodic structures are labeled as .

Typical Fourier spectra of the density profiles of (a)–(f) and (j) in Fig. 1. The size of the filled circles denotes the peak intensities. For (a), the picture is for A (red), B (green), or C (blue) component; for (b), the left picture is for A or B, and the right one is for C; for others, the pictures from left to right are for A, B, and C, respectively. For each pattern, some typical diffraction peaks are labeled.

Typical Fourier spectra of the density profiles of (a)–(f) and (j) in Fig. 1. The size of the filled circles denotes the peak intensities. For (a), the picture is for A (red), B (green), or C (blue) component; for (b), the left picture is for A or B, and the right one is for C; for others, the pictures from left to right are for A, B, and C, respectively. For each pattern, some typical diffraction peaks are labeled.

(a) Free energy differences from the value of the homogeneous phase as a function of the volume fraction of A composition for ABC star triblock copolymers with symmetric B and C arms. With increasing, the phase structure sequence is from L3, through [8.8.4], [6.6.6], [8.6.4; 8.6.6], [10.6.4; 10.6.6], [12.6.4], HL, to HHC. The inset shows the transition between HL and HHC. (b) Phase stability regions as a function of the arm-length ratio of . Note that one break is applied with the x axis for the reason of clarity of the figure.

(a) Free energy differences from the value of the homogeneous phase as a function of the volume fraction of A composition for ABC star triblock copolymers with symmetric B and C arms. With increasing, the phase structure sequence is from L3, through [8.8.4], [6.6.6], [8.6.4; 8.6.6], [10.6.4; 10.6.6], [12.6.4], HL, to HHC. The inset shows the transition between HL and HHC. (b) Phase stability regions as a function of the arm-length ratio of . Note that one break is applied with the x axis for the reason of clarity of the figure.

The magnitudes of basic vectors in unit of the radius of gyration of these 2D structures along the phase path in Fig. 2. The unfilled (including the symbols of cross) and filled symbols denote the vector lengths of and , respectively. For the reason of clarity, the of HL is not shown.

The magnitudes of basic vectors in unit of the radius of gyration of these 2D structures along the phase path in Fig. 2. The unfilled (including the symbols of cross) and filled symbols denote the vector lengths of and , respectively. For the reason of clarity, the of HL is not shown.

(a) Internal part and (b) entropic part of the free energy difference in Fig. 2(a) as a function of .

(a) Internal part and (b) entropic part of the free energy difference in Fig. 2(a) as a function of .

The free energy difference as a function of for fixed . The phase of [8.6.4; 8.6.6; 12.6.4], denoted as filled squares, does not appear as stable one in this parameter region.

The free energy difference as a function of for fixed . The phase of [8.6.4; 8.6.6; 12.6.4], denoted as filled squares, does not appear as stable one in this parameter region.

The lengths of basic vectors as a function of along the phase path of fixed .

The lengths of basic vectors as a function of along the phase path of fixed .

(a) Internal energy of and (b) entropic energy of of various structures as a function on the phase path of fixed .

(a) Internal energy of and (b) entropic energy of of various structures as a function on the phase path of fixed .

The triangular phase diagram of ABC star triblock copolymers with equal interaction parameters of . The phase diagram is composed of a set of transition points shown as black dots.

The triangular phase diagram of ABC star triblock copolymers with equal interaction parameters of . The phase diagram is composed of a set of transition points shown as black dots.

## Tables

The phase transition boundaries between the bcc and hex phases obtained by the OpS2 with discretizations and , together with the free energy difference of the bcc phase between the two discretizations in AB diblock copolymers. The transition data read from Fig. 2 in Ref. 38 is shown as a reference.

The phase transition boundaries between the bcc and hex phases obtained by the OpS2 with discretizations and , together with the free energy difference of the bcc phase between the two discretizations in AB diblock copolymers. The transition data read from Fig. 2 in Ref. 38 is shown as a reference.

The free energies of [10.6.4; 10.6.6] and [12.6.4], as well as their free energy difference calculated with and , respectively.

The free energies of [10.6.4; 10.6.6] and [12.6.4], as well as their free energy difference calculated with and , respectively.

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