^{1}, Pingchuan Sun

^{2}, Tiehong Chen

^{2}, Qinghua Jin

^{3}, Datong Ding

^{3}, Baohui Li

^{3,a),b)}and An-Chang Shi

^{4,a),c)}

### Abstract

Self-assembly of diblock copolymers confined in cylindrical nanopores is studied using a simulated annealing technique. The pore diameter and surface preference are systematically varied to examine their effects on the self-assembled morphologies and the chain conformations. For bulk lamella-forming and cylinder-forming diblock copolymers, novel structures such as helices and concentric (perforated) lamellae spontaneously form when the copolymers are confined in cylindrical pores. The observed equilibrium morphologies are compared with that obtained from experiments, theory, and other simulations. A simple model is proposed for symmetric diblock copolymers, which gives a reasonable description of the layer thickness for the concentric lamellae. It is found that chains near the pore surfaces are compressed relative to the bulk chains, which can be attributed to the existence of the surfaces. The dependence of the chain conformation on the degree of confinement and strength of the surface preference are reasonably explained. The energetics is discussed qualitatively and used to account for the appearance of the complex phase behavior observed for certain intermediate conditions.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 20474034, 20374031, and 20373029), by the Chinese Ministry of Education with the Program of New Century Excellent Talents in Universities and the Program of the Joint-Research Foundation of Nankai and Tianjin Universities, and by Nankai University ISC. One of the authors (A.C.S.) gratefully acknowledges the supports from the Natural Sciences and Engineering Research Council (NSERC) of Canada.

I. INTRODUCTION

II. MODEL AND METHOD

III. RESULTS AND DISCUSSION

A. Confined self-assembly of symmetric diblock copolymers with

B. Confined self-assembly of asymmetric diblock copolymers with

IV. CONCLUSION

### Key Topics

- Lamellae
- 108.0
- Block copolymers
- 63.0
- Surface morphology
- 40.0
- Exoelectron emission
- 20.0
- Copolymers
- 16.0

## Figures

(Color) Self-assembled morphologies for symmetric diblock copolymers with confined in pores (a) the morphologies as functions of the ratio and . For the S helices or imperfect S helices, morphologies of only the A blocks (or B blocks) are also given, and for some concentric lamellae, a cross section view is also given. The boundary of the concentric lamellae (or the perpendicular lamellae) is given with two identical structures for each value when . [(b)–(d)] Morphologies formed with : (b) , ; (c) , , and (d) , .

(Color) Self-assembled morphologies for symmetric diblock copolymers with confined in pores (a) the morphologies as functions of the ratio and . For the S helices or imperfect S helices, morphologies of only the A blocks (or B blocks) are also given, and for some concentric lamellae, a cross section view is also given. The boundary of the concentric lamellae (or the perpendicular lamellae) is given with two identical structures for each value when . [(b)–(d)] Morphologies formed with : (b) , ; (c) , , and (d) , .

(Color online) The order parameter for concentric lamellae at . (a) The value of the order parameter at the pore center as a function of and (b) at different values.

(Color online) The order parameter for concentric lamellae at . (a) The value of the order parameter at the pore center as a function of and (b) at different values.

Schematics of concentric lamellae with . The outermost solid circle represents the pore surfaces, the inside solid circles represent interfaces, and the dashed circles represent the assumed or interfaces in the strong segregations limit.

Schematics of concentric lamellae with . The outermost solid circle represents the pore surfaces, the inside solid circles represent interfaces, and the dashed circles represent the assumed or interfaces in the strong segregations limit.

(Color online) Comparison of the dimensionless thicknesses of the alternating -rich and -rich layers in concentric lamellae for strongly preferential surfaces obtained from simulations (symbols) with those predicted from the equations (lines). In the figure, three parts according to from small to larger correspond to , 2, and 3. In each part, the squares and the lower lines represent the thicknesses of the outermost -rich layer . The up lines represent the thicknesses of the inner - or -rich layers. The up triangles, down triangles, and right triangles represent the thickness of , , , respectively.

(Color online) Comparison of the dimensionless thicknesses of the alternating -rich and -rich layers in concentric lamellae for strongly preferential surfaces obtained from simulations (symbols) with those predicted from the equations (lines). In the figure, three parts according to from small to larger correspond to , 2, and 3. In each part, the squares and the lower lines represent the thicknesses of the outermost -rich layer . The up lines represent the thicknesses of the inner - or -rich layers. The up triangles, down triangles, and right triangles represent the thickness of , , , respectively.

(Color online) The mean-square end-to-end distance, , as a function of for symmetric diblock copolymers.

(Color online) The mean-square end-to-end distance, , as a function of for symmetric diblock copolymers.

(Color online) The mean-square end-to-end distances of and chains and respectively, as a function of for symmetric diblock copolymers with (a) strongly preferential surfaces, (b) weakly preferential surfaces, and (c) neutral surfaces.

(Color online) The mean-square end-to-end distances of and chains and respectively, as a function of for symmetric diblock copolymers with (a) strongly preferential surfaces, (b) weakly preferential surfaces, and (c) neutral surfaces.

(Color online) The components of the mean-square end-to-end distances of and chains, respectively, as a function of for symmetric diblock copolymers with strongly preferential surfaces. (a) and (b)

(Color online) The components of the mean-square end-to-end distances of and chains, respectively, as a function of for symmetric diblock copolymers with strongly preferential surfaces. (a) and (b)

(Color online) The components, and , of the mean-square end-to-end distance as a function of for symmetric diblock copolymers with neutral surfaces.

(Color online) The components, and , of the mean-square end-to-end distance as a function of for symmetric diblock copolymers with neutral surfaces.

(Color online) The mean-square end-to-end distances as a function of . (a) and [(b) and (c)] and , where (b) and (c) .

(Color online) The mean-square end-to-end distances as a function of . (a) and [(b) and (c)] and , where (b) and (c) .

(Color online) (a) The local concentration profile of monomers near the surface, (b) the average contact numbers for an monomer with monomers, and (c) the free energy per chain as a function of .

(Color online) (a) The local concentration profile of monomers near the surface, (b) the average contact numbers for an monomer with monomers, and (c) the free energy per chain as a function of .

(Color) Self-assembled morphologies for asymmetric diblock copolymers with as a function of . Only the blocks are shown. The outermost red circle in each top view indicates the surface of the cylindrical pore. (a) and (b) .

(Color) Self-assembled morphologies for asymmetric diblock copolymers with as a function of . Only the blocks are shown. The outermost red circle in each top view indicates the surface of the cylindrical pore. (a) and (b) .

(Color online) Comparison of the radial order parameter profiles for different structures: (a) The degenerated structures of concentric perforated lamellae and S helices at and (b) the two-ring structure at .

(Color online) Comparison of the radial order parameter profiles for different structures: (a) The degenerated structures of concentric perforated lamellae and S helices at and (b) the two-ring structure at .

(Color online) The mean-square end-to-end distances as a function of for asymmetric diblock copolymers, where the values for the degenerated structures are all shown. (a) , (b) , and (c) .

(Color online) The mean-square end-to-end distances as a function of for asymmetric diblock copolymers, where the values for the degenerated structures are all shown. (a) , (b) , and (c) .

(Color online) The average contact numbers for an monomer with monomers as a function of , where the values for the degenerated structures are all shown.

(Color online) The average contact numbers for an monomer with monomers as a function of , where the values for the degenerated structures are all shown.

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