^{1,a)}, Nan Xie

^{1}, Feng Qiu

^{1}, Yuliang Yang

^{1}and An-Chang Shi

^{2}

### Abstract

The ordering kinetics of directed assembly of cylinder-forming diblock copolymers is investigated by cell dynamics simulation of the time-dependent Ginzburg–Landau theory. The directing field, mimicking chemically or topologically patternedsurfaces, is composed of a rectangular array of potential wells which are attractive to the minority blocks. The period of the templating fields is commensurate with the hexagonal lattice of the block copolymer domains. The ordering kinetics is described by the time evolution of the defect concentration, which reveals that the rectangular field of [1 m] for a given density multiplication has the best directing effect, and the reversed case of [m 1] has the worst. Compared with a hexagonal directing field, the rectangular field provides a better directing efficiency for a fixed high density multiplication. The difference of the directing effect can be understood by analyzing the ordering mechanisms in the two types of directing fields. The study reveals that the rectangular pattern is an alternative candidate to direct block copolymer assembly toward large-scale ordered domains.

This work is supported by the National Natural Science Foundation of China (Grants 20974026, 20704010, 20990231). W.L. gratefully acknowledges supports from the Shanghai Pujiang Program (Programs No. 08PJ1402000) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

I. INTRODUCTION

II. MODEL AND THEORY

III. RESULTS AND DISCUSSIONS

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 12.0
- Surface patterning
- 8.0
- Dislocations
- 7.0
- Field emitters
- 7.0
- Free energy
- 6.0

## Figures

Schematic plot of periodic rectangular array of potential wells on a substrate. The blue color indicates the regions where the minority block is preferred, and the other regions do not have preference to any block. Each rectangular cell is denoted by two periods of and on x and y directions, respectively.

Schematic plot of periodic rectangular array of potential wells on a substrate. The blue color indicates the regions where the minority block is preferred, and the other regions do not have preference to any block. Each rectangular cell is denoted by two periods of and on x and y directions, respectively.

Defect concentrations of cylinders assembled on patterned templates as function of time. Figures (a) and (b) are results of four rectangular fields for each of density multiplications of 12 and 16, respectively. In (b), the filled symbols denote the results of periodic hexagonal field with the same density multiplication of 16 from Ref. 21.

Defect concentrations of cylinders assembled on patterned templates as function of time. Figures (a) and (b) are results of four rectangular fields for each of density multiplications of 12 and 16, respectively. In (b), the filled symbols denote the results of periodic hexagonal field with the same density multiplication of 16 from Ref. 21.

Monomer density plots of cylinder patterns directed by periodic fields of [1 6] (left column) and [6 1] (right column) at *t* = 10^{4} (upper row) and *t* = 10^{5} (bottom row). Each figure exhibits a 512^{2} portion of the entire sample. Insets give the Fourier spectrums of the density.

Monomer density plots of cylinder patterns directed by periodic fields of [1 6] (left column) and [6 1] (right column) at *t* = 10^{4} (upper row) and *t* = 10^{5} (bottom row). Each figure exhibits a 512^{2} portion of the entire sample. Insets give the Fourier spectrums of the density.

Distributions of local lattice orientation of the entire block copolymer domains in Fig. 3. The colors of the spectrum indicate the range of lattice orientation from 0 to 60 degrees.

Distributions of local lattice orientation of the entire block copolymer domains in Fig. 3. The colors of the spectrum indicate the range of lattice orientation from 0 to 60 degrees.

Orientation distribution plots for the density multiplication of 16. From top to bottom, the periodic field is [1 8], [8 1], and 〈40〉, and from left to right, the corresponding time is *t* = 10^{4}, 10^{5}, and 2 × 10^{5}, respectively.

Orientation distribution plots for the density multiplication of 16. From top to bottom, the periodic field is [1 8], [8 1], and 〈40〉, and from left to right, the corresponding time is *t* = 10^{4}, 10^{5}, and 2 × 10^{5}, respectively.

Small portion of density plots around the locations of a pair of dislocations for the rectangular [1 6]-field. From (a) to (h), the time is *t* = 2 × 10^{5}, 2.2 × 10^{5}, 2.4 × 10^{5}, 2.6 × 10^{5}, 3.2 × 10^{5}, 3.4 × 10^{5}, 3.6 × 10^{5}, and 3.8 × 10^{5}, respectively. In (a), the two dislocations are indicated by short color lines, and the Delaunay triangles around the bottom one are plotted. Black and white circles indicate where a new domain is going to appear and where a new domain just comes out, respectively.

Small portion of density plots around the locations of a pair of dislocations for the rectangular [1 6]-field. From (a) to (h), the time is *t* = 2 × 10^{5}, 2.2 × 10^{5}, 2.4 × 10^{5}, 2.6 × 10^{5}, 3.2 × 10^{5}, 3.4 × 10^{5}, 3.6 × 10^{5}, and 3.8 × 10^{5}, respectively. In (a), the two dislocations are indicated by short color lines, and the Delaunay triangles around the bottom one are plotted. Black and white circles indicate where a new domain is going to appear and where a new domain just comes out, respectively.

Time evolution of defect concentrations for the type [1 m] of rectangular field, where m = 10, 12, 14, 16, and 18. The red and blue solid lines are the results of hexagonal field, 〈50〉 and 〈60〉, respectively. The green solid line indicates the relation of 1/3 power law.

Time evolution of defect concentrations for the type [1 m] of rectangular field, where m = 10, 12, 14, 16, and 18. The red and blue solid lines are the results of hexagonal field, 〈50〉 and 〈60〉, respectively. The green solid line indicates the relation of 1/3 power law.

Orientation distribution plots for the rectangular [1 18]-field (upper row) and the hexagonal 〈60〉-field (bottom row). Left and right columns correspond to evolving time, *t* = 10^{5} and *t* = 10^{6}, respectively.

Orientation distribution plots for the rectangular [1 18]-field (upper row) and the hexagonal 〈60〉-field (bottom row). Left and right columns correspond to evolving time, *t* = 10^{5} and *t* = 10^{6}, respectively.

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