^{1,a)}, Toshihiro Kawakatsu

^{2}, Peilong Chen

^{3}and Chun-Yi David Lu

^{4,b)}

### Abstract

A density functional theory is developed for the diblock copolymer melt, where one block contains the segment orientation dependent chiral interaction. In addition to the standard (scalar) pair interaction between the two types of monomers, the chiral block has the additional pairwise interaction, which is linear in the tangent vectors of the segments. We construct a density functional, which contains both the scalar density field and the vector chain alignment field. The quadratic part of the density functional comes from the mean field theory of the microscopic model, whereas the fourth order terms are introduced phenomenologically in the spatially local form. From the stability analysis of this model, we find that the additional chiral interaction shifts the order-disorder transition, which is consistent with the behavior of experimental system. Further numerical calculation reveals a new metastable chiral helical cylinder structure, which is similar to the one found experimentally. Another similar metastable structure but with zigzag modulation is also observed. As the helical and zigzag structures disappear when the chiral interaction is switched off, we understand that the chiral effect is the driving force for the formation of these exotic metastable structures.

We thank Professor R.-M. Ho for suggesting this problem and providing helpful discussions. The present study is supported by National Science Council (Grant Nos. NSC97-2112-M-002-008-MY3 and NSC98-2917-I-002-122) and partially supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology of Japan. We gratefully acknowledge financial support by the Summer Program of Interchange Association, Japan.

I. INTRODUCTION

II. THE MODEL

III. THE DENSITY FUNCTIONAL

A. Mean field theory

B. The free energydensity functional

IV. LINEAR STABILITY ANALYSIS

V. NUMERICAL SOLUTION

VI. SUMMARY AND DISCUSSION

### Key Topics

- Chiral symmetries
- 67.0
- Density functional theory
- 25.0
- Free energy
- 25.0
- Mean field theory
- 16.0
- Block copolymers
- 12.0

## Figures

An A*B copolymer chain and its labels.

An A*B copolymer chain and its labels.

A shift in the spinodal point between the SOP and VSOP model (α = 0.0001), where ΔχN is the difference in the spinodal point between the VSOP and the SOP models. Here, we use N = 600, a = 0.1 and the behavior is rather insensitive to the change in α under the α ≪ Γ JJ limit.

A shift in the spinodal point between the SOP and VSOP model (α = 0.0001), where ΔχN is the difference in the spinodal point between the VSOP and the SOP models. Here, we use N = 600, a = 0.1 and the behavior is rather insensitive to the change in α under the α ≪ Γ JJ limit.

The metastable H* phase. The distance between the centers of the two neighboring cylinders is 4.7R g . In this figure, the cylinders are plotted with the isosurface ψ = 0.65. The helical amplitude and the pitch of helical cylinders are 0.5R g and 8.5R g , respectively.

The metastable H* phase. The distance between the centers of the two neighboring cylinders is 4.7R g . In this figure, the cylinders are plotted with the isosurface ψ = 0.65. The helical amplitude and the pitch of helical cylinders are 0.5R g and 8.5R g , respectively.

The chiral interaction free energy density. The white circular curves are the A*-B interface (ψ = 0 isosurface). The superimposed gray plot indicates the chiral interaction free energy density in the H* structure. The dark region shows the strong chiral interaction near the A*-B interfaces.

The chiral interaction free energy density. The white circular curves are the A*-B interface (ψ = 0 isosurface). The superimposed gray plot indicates the chiral interaction free energy density in the H* structure. The dark region shows the strong chiral interaction near the A*-B interfaces.

The chain alignment strength. The interface ψ = 0 is at r = 1.47R g .

The chain alignment strength. The interface ψ = 0 is at r = 1.47R g .

The angle between the chain alignment and the z axis. The interface ψ = 0 is at r = 1.47R g .

The angle between the chain alignment and the z axis. The interface ψ = 0 is at r = 1.47R g .

The zigzag H* phase. The cylinders have right-handed helical shape and sharp turns in a zigzag manner. The distance between the centers of the two neighboring cylinders is 4.5R g . The zigzag structure has a longer pitch P zigzag = 12.7R g , while the cylinder also has a shorter chiral helical pitch P* = 8R g . Note that the period of the zigzag undulation is just as twice as the helical pitch.

The zigzag H* phase. The cylinders have right-handed helical shape and sharp turns in a zigzag manner. The distance between the centers of the two neighboring cylinders is 4.5R g . The zigzag structure has a longer pitch P zigzag = 12.7R g , while the cylinder also has a shorter chiral helical pitch P* = 8R g . Note that the period of the zigzag undulation is just as twice as the helical pitch.

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