^{1}, Xuehao He

^{2,a)}and Long Wang

^{1}

### Abstract

In this study, we apply a self-consistent field theory of polymers to study the structures of a symmetric diblock copolymer in parallel substrates filled with square-pillar arrays in which the substrates and pillars exhibit a weak preference for one block of the copolymer. Three classes of structures, i.e., lamellae, perpendicular cylinders, and bicontinuous structures, are achieved by varying the polymer film thickness, the pillar pitch (the distance between two centers of the nearest neighboring pillars), the gap and rotation of the pillars. Because of the confinement along horizontal directions imposed by the pillar array, eight novel types of perpendicular lamellar structures and eight novel types of cylindrical structures with various shapes and distributions occur. In the hybridization states of the parallel and perpendicular lamellar structures, several novel bicontinuous structures such as the double-cylinder network, pseudo-lamellae, and perforated lamellar structure are also found. By comparing the free energies of the various possible structures, the antisymmetric parallel lamellae are observed to be stable with the larger pillar gap at a certain film thickness. The structural transformations between the alternating cylindrical structures (alternating cross-shaped, square-shaped, and octagonal perpendicular cylinders) and parallel lamellae with increasing film thickness or pillar gap are well explained by the modified strong separation theory. Our results indicate that array confinement can be an effective method to prepare novel polymeric nanopattern structures.

The project is supported by NSF of China (NSFC) (Nos. 20974078 and 91127046), KPCME (No. 109043). X.H. thanks the Alexander von Humboldt Foundation for an equipment grant.

I. INTRODUCTION

II. THEORETICAL MODEL

III. RESULTS AND DISCUSSION

A. Normal square-pillar array

B. Rotated square-pillar array

IV. CONCLUSIONS

### Key Topics

- Lamellae
- 76.0
- Block copolymers
- 22.0
- Free energy
- 18.0
- Polymer structure
- 14.0
- Thin film structure
- 14.0

## Figures

Schematic illustration of 3D square lattices of square pillars. (a) Vertical view of this confinement uncovered by the substrate above composed of a 3D normal square-pillar array. (b) Lateral view of 3D normal square-pillar array. (c) Vertical view of 3D rotated square-pillar array. The blue components represent the square pillars (*P*), and the parallel yellow substances represent the substrates (*S*). The variables *D*, *W*(*G*), *Z*, and *L* represent the pillar pitch, the gap between two nearest neighboring square pillars, the copolymer film thickness, and the side length of the pillar, respectively.

Schematic illustration of 3D square lattices of square pillars. (a) Vertical view of this confinement uncovered by the substrate above composed of a 3D normal square-pillar array. (b) Lateral view of 3D normal square-pillar array. (c) Vertical view of 3D rotated square-pillar array. The blue components represent the square pillars (*P*), and the parallel yellow substances represent the substrates (*S*). The variables *D*, *W*(*G*), *Z*, and *L* represent the pillar pitch, the gap between two nearest neighboring square pillars, the copolymer film thickness, and the side length of the pillar, respectively.

Lamellar structures found in normal square-pillar array. The color bars in Fig. 2(a) represent the component density profiles above 0.40 (the density has been normalized to 0.00–1.00). The red region represents the *A* block, and the green region represents another block. For simplicity and analysis, the listed values of the parameters *Z*, *D*, and *W* are rescaled in units of *R* _{0} in a later figure. Parallel lamellae: (a) symmetric lamellae () (*Z* = 0.9, *W* = 0.36, and *D* = 0.52); (b) antisymmetric lamellae () (*Z* = 1.5, *W* = 0.36, and *D* = 0.52); (c) symmetric lamellae () (*Z* = 1.81, *W* = 0.36, and *D* = 0.52). Perpendicular lamellae: (d) simple lamellae (*L* _{⊥}) (*Z* = 0.47, *W* = 0.55, and *D* = 0.78); (e) diagonal wavy lamellae () (*Z* = 0.45, *W* = 0.9, and *D* = 1.29); (f) knitting-pattern lamellae () (*Z* = 0.47, *W* = 0.62, and *D* = 1.55); (g) square lamellae () (*Z* = 0.47, *W* = 0.78, and *D* = 1.55); (h) semi-circle-section lamellae () (*Z* = 0.47, *W* = 0.93, and *D* = 1.55); (i) wavy-circle-section lamellae () (*Z* = 0.47, *W* = 1.09, and *D* = 1.55).

Lamellar structures found in normal square-pillar array. The color bars in Fig. 2(a) represent the component density profiles above 0.40 (the density has been normalized to 0.00–1.00). The red region represents the *A* block, and the green region represents another block. For simplicity and analysis, the listed values of the parameters *Z*, *D*, and *W* are rescaled in units of *R* _{0} in a later figure. Parallel lamellae: (a) symmetric lamellae () (*Z* = 0.9, *W* = 0.36, and *D* = 0.52); (b) antisymmetric lamellae () (*Z* = 1.5, *W* = 0.36, and *D* = 0.52); (c) symmetric lamellae () (*Z* = 1.81, *W* = 0.36, and *D* = 0.52). Perpendicular lamellae: (d) simple lamellae (*L* _{⊥}) (*Z* = 0.47, *W* = 0.55, and *D* = 0.78); (e) diagonal wavy lamellae () (*Z* = 0.45, *W* = 0.9, and *D* = 1.29); (f) knitting-pattern lamellae () (*Z* = 0.47, *W* = 0.62, and *D* = 1.55); (g) square lamellae () (*Z* = 0.47, *W* = 0.78, and *D* = 1.55); (h) semi-circle-section lamellae () (*Z* = 0.47, *W* = 0.93, and *D* = 1.55); (i) wavy-circle-section lamellae () (*Z* = 0.47, *W* = 1.09, and *D* = 1.55).

Square lattices of perpendicular cylinders found in normal square-pillar array. (a) Alternating cross-shaped cylinders () (Z = 0.23, *W* = 0.21, and *D* = 0.52); (b) cross-shaped cylinders () (*Z* = 0.21, *W* = 0.41, and *D* = 1.03); (c) simple cylinders (*C* _{⊥}) (*Z* = 0.21, *W* = 0.62, and *D* = 1.03); (d) elliptical cylinders () (*Z* = 0.26, *W* = 0.77, and *D* = 1.29).

Square lattices of perpendicular cylinders found in normal square-pillar array. (a) Alternating cross-shaped cylinders () (Z = 0.23, *W* = 0.21, and *D* = 0.52); (b) cross-shaped cylinders () (*Z* = 0.21, *W* = 0.41, and *D* = 1.03); (c) simple cylinders (*C* _{⊥}) (*Z* = 0.21, *W* = 0.62, and *D* = 1.03); (d) elliptical cylinders () (*Z* = 0.26, *W* = 0.77, and *D* = 1.29).

Comparison between the modified strong segregation theory (curve lines) and the simulation results (symbols). The inset is a vertical view of the unit cell of the alternating cross-shaped cylindrical structure (). The SST_{1}, SST_{2}, and SST_{3} curves illustrate the structural transformations between the three types of the parallel lamellar structures, i.e., the , , and , and alternating cross-shaped cylindrical structure (), respectively.

Comparison between the modified strong segregation theory (curve lines) and the simulation results (symbols). The inset is a vertical view of the unit cell of the alternating cross-shaped cylindrical structure (). The SST_{1}, SST_{2}, and SST_{3} curves illustrate the structural transformations between the three types of the parallel lamellar structures, i.e., the , , and , and alternating cross-shaped cylindrical structure (), respectively.

Possible structures at *Z* = 1.50 and *D* = 0.52. (a) Asymmetric parallel lamellae and its lateral view (); (b) antisymmetric parallel lamellae (); (c) alternating circle-shaped perpendicular lamellae (); (d) alternating cross-shaped cylinders ().

Possible structures at *Z* = 1.50 and *D* = 0.52. (a) Asymmetric parallel lamellae and its lateral view (); (b) antisymmetric parallel lamellae (); (c) alternating circle-shaped perpendicular lamellae (); (d) alternating cross-shaped cylinders ().

Bicontinuous structures found in normal square-pillar array. Double-cylinder network structure (): (a) theat *Z* = 0.93, *W* = 0.72, and *D* = 1.03; (a-1) the *A* structure of the ; (b) the at *Z* = 1.81, *W* = 0.72, and *D* = 1.03; (b-1) the *A* structure of the . Pseudo-lamellae (): (c) the at *Z* = 0.93, *W* = 1.09, and *D* = 1.55; (c-1) the *A* structure of the ; (d) the at *Z* = 1.78, *W* = 1.09, and *D* = 1.55; (d-1) the *A* structure of the .

Bicontinuous structures found in normal square-pillar array. Double-cylinder network structure (): (a) theat *Z* = 0.93, *W* = 0.72, and *D* = 1.03; (a-1) the *A* structure of the ; (b) the at *Z* = 1.81, *W* = 0.72, and *D* = 1.03; (b-1) the *A* structure of the . Pseudo-lamellae (): (c) the at *Z* = 0.93, *W* = 1.09, and *D* = 1.55; (c-1) the *A* structure of the ; (d) the at *Z* = 1.78, *W* = 1.09, and *D* = 1.55; (d-1) the *A* structure of the .

Bicontinuous structures found in normal square-pillar array. (a) Similar pseudo-lamellae () (*Z* = 0.71, *W* = 0.90, and *D* = 1.29); (a-1) the *A* structure of the ; (b) perforated lamellae (*BpL* _{×}) (*Z* = 0.93, *W* = 0.62, and *D* = 1.55); (b-1) the *A* structure of the *BpL* _{×}and its cut view; (c) dislocation of the elliptical cylinders () (*Z* = 1.4, *W* = 0.62, and *D* = 1.55); (c-1) the *A* structure of the ; (d) curved-lamellae network (*BclL* _{×}) (*Z* = 0.7, *W* = 1.09, and *D* = 1.55); (d-1) vertical view of the *A* structure of the *BclL* _{×}; (e) diagonal-wavy-lamellae network () (*Z* = 0.89, *W* = 0.55, and *D* = 0.78); (e-1) vertical view of the *A* structure of the .

Bicontinuous structures found in normal square-pillar array. (a) Similar pseudo-lamellae () (*Z* = 0.71, *W* = 0.90, and *D* = 1.29); (a-1) the *A* structure of the ; (b) perforated lamellae (*BpL* _{×}) (*Z* = 0.93, *W* = 0.62, and *D* = 1.55); (b-1) the *A* structure of the *BpL* _{×}and its cut view; (c) dislocation of the elliptical cylinders () (*Z* = 1.4, *W* = 0.62, and *D* = 1.55); (c-1) the *A* structure of the ; (d) curved-lamellae network (*BclL* _{×}) (*Z* = 0.7, *W* = 1.09, and *D* = 1.55); (d-1) vertical view of the *A* structure of the *BclL* _{×}; (e) diagonal-wavy-lamellae network () (*Z* = 0.89, *W* = 0.55, and *D* = 0.78); (e-1) vertical view of the *A* structure of the .

Antisymmetric and cross-network bicontinuous structures found in normal square-pillar array. (a) Hybridization of the *BpL* _{×} and the () (*Z* = 1.81, *W* = 0.77, and *D* = 1.29); (a-1) the *A* structure of the ; (a-2) the top part of the *A* structure; (a-3) the bottom part of the *A* structure; (b) cross-network (*BcnL* _{×}) (*Z* = 1.82, *W* = 0.39, and *D* = 0.78); (b-1) the *A* structure of the *BcnL* _{×}; (b-2) cut view of the *A* structure; (b-3) cut view of the *A* structure.

Antisymmetric and cross-network bicontinuous structures found in normal square-pillar array. (a) Hybridization of the *BpL* _{×} and the () (*Z* = 1.81, *W* = 0.77, and *D* = 1.29); (a-1) the *A* structure of the ; (a-2) the top part of the *A* structure; (a-3) the bottom part of the *A* structure; (b) cross-network (*BcnL* _{×}) (*Z* = 1.82, *W* = 0.39, and *D* = 0.78); (b-1) the *A* structure of the *BcnL* _{×}; (b-2) cut view of the *A* structure; (b-3) cut view of the *A* structure.

Lamellar structures found in rotated square-pillar array. Parallel lamellae: (a) symmetric lamellae () (*Z* = 0.9, *G* = 0.29, and *D* = 0.52). Perpendicular lamellae: (b) simple lamellae (*L* _{⊥}) (*Z* = 0.23, *G* = 0.43, and *D* = 0.78); (c) novel knitting-pattern lamellae () (*Z* = 0.26, *G* = 0.71, and *D* = 1.29); (d) wavy lamellae () (*Z* = 0.23, *G* = 0.85, and *D* = 1.55).

Lamellar structures found in rotated square-pillar array. Parallel lamellae: (a) symmetric lamellae () (*Z* = 0.9, *G* = 0.29, and *D* = 0.52). Perpendicular lamellae: (b) simple lamellae (*L* _{⊥}) (*Z* = 0.23, *G* = 0.43, and *D* = 0.78); (c) novel knitting-pattern lamellae () (*Z* = 0.26, *G* = 0.71, and *D* = 1.29); (d) wavy lamellae () (*Z* = 0.23, *G* = 0.85, and *D* = 1.55).

Perpendicular cylinders found in rotated square-pillar array. (a) Alternating square-shaped cylinders () (*Z* = 0.23, *G* = 0.08, and *D* = 0.52); (b) alternating octagonal cylinders () (*Z* = 0.23, *G* = 0.23, and *D* = 0.52); (c) distorted cylinders (*dC* _{⊥}) (*Z* = 0.23, *G* = 0.12, and *D* = 0.78); (d) simple cylinders (*C* _{⊥}) (*Z* = 0.26, *G* = 0.32, and *D* = 1.29); (e) elliptical cylinders () (*Z* = 0.26, *G* = 0.58, and *D* = 1.29); (f) square-shaped cylinders () (*Z* = 0.23, *G* = 0.23, and *D* = 1.55).

Perpendicular cylinders found in rotated square-pillar array. (a) Alternating square-shaped cylinders () (*Z* = 0.23, *G* = 0.08, and *D* = 0.52); (b) alternating octagonal cylinders () (*Z* = 0.23, *G* = 0.23, and *D* = 0.52); (c) distorted cylinders (*dC* _{⊥}) (*Z* = 0.23, *G* = 0.12, and *D* = 0.78); (d) simple cylinders (*C* _{⊥}) (*Z* = 0.26, *G* = 0.32, and *D* = 1.29); (e) elliptical cylinders () (*Z* = 0.26, *G* = 0.58, and *D* = 1.29); (f) square-shaped cylinders () (*Z* = 0.23, *G* = 0.23, and *D* = 1.55).

Comparison between the modified strong segregation theory (curve lines) and the simulation results (symbols). The inset is a vertical view of the unit cell of the alternating square-shaped or octagonal cylindrical structures. The SST_{1} and SST_{2} curves illustrate the structural transformations between the alternating cylindrical structure ( or ) and the parallel lamellar structures, i.e., and , respectively.

Comparison between the modified strong segregation theory (curve lines) and the simulation results (symbols). The inset is a vertical view of the unit cell of the alternating square-shaped or octagonal cylindrical structures. The SST_{1} and SST_{2} curves illustrate the structural transformations between the alternating cylindrical structure ( or ) and the parallel lamellar structures, i.e., and , respectively.

Bicontinuous structures found in rotated square-pillar array. Double-cylinder network structure (): (a) the at *Z* = 0.93, *G* = 0.57, and *D* = 1.03; (a-1) the *A* structure of the ; (b) the at *Z* = 1.81, *G* = 0.57, and *D* = 1.03; (b-1) the *A* structure of the . Pseudo-lamellae (): (c) the at *Z* = 0.93, *G* = 0.85, and *D* = 1.55; (c-1) the *A* structure of the ; (d) the at *Z* = 1.78, *G* = 0.85, and *D* = 1.55; (d-1) the *A* structure of the . (e) Perforated lamellae (*BpL* _{×}) (*Z* = 0.93, *G* = 0.39, and *D* = 1.55); (e-1) the *A* structure of the *BpL* _{×}.

Bicontinuous structures found in rotated square-pillar array. Double-cylinder network structure (): (a) the at *Z* = 0.93, *G* = 0.57, and *D* = 1.03; (a-1) the *A* structure of the ; (b) the at *Z* = 1.81, *G* = 0.57, and *D* = 1.03; (b-1) the *A* structure of the . Pseudo-lamellae (): (c) the at *Z* = 0.93, *G* = 0.85, and *D* = 1.55; (c-1) the *A* structure of the ; (d) the at *Z* = 1.78, *G* = 0.85, and *D* = 1.55; (d-1) the *A* structure of the . (e) Perforated lamellae (*BpL* _{×}) (*Z* = 0.93, *G* = 0.39, and *D* = 1.55); (e-1) the *A* structure of the *BpL* _{×}.

## Tables

Structures in normal square-pillar array.

Structures in normal square-pillar array.

Free energies for several possible structures at *Z* = 1.5 and *D* = 0.52.

Free energies for several possible structures at *Z* = 1.5 and *D* = 0.52.

Structures in rotated square-pillar array.

Structures in rotated square-pillar array.

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