^{1,a)}, Nan Xie

^{1,a)}, Weihua Li

^{1,b)}, Feng Qiu

^{1,2}and An-Chang Shi

^{3}

### Abstract

The thermodynamics and kinetics of the self-assembly of cylinder-forming diblock copolymers directed by the lateral confinement of hexagons have been studied by the combination of self-consistent field theory (SCFT) calculation and time-dependent Ginzburg-Landau (TDGL) theory simulation. The SCFT calculations are used to determine the stability of candidate 2D and 3D equilibrium phases formed in small-size hexagons. Our phase diagram predicts the existence of stable phase regions with respect to the hexagonal size, which is centered around the optimal size with an extent of about a period, for the phases of perfect hexagonal cylinders. Our TDGL simulations reveal that the ordering event, in which the structure evolves toward the perfect state, occurs stochastically according to the Poisson distribution, and the ordering time grows roughly with a power-law relation of the hexagonal size. This prediction is helpful to estimate the annealing time for larger systems with the knowledge of the annealing time of a small system in experiments.

This work was supported by the National Natural Science Foundation of China (NNSFC) (Grants Nos. 20974026, 21174031, and 20990231); the National High Technology Research and Development Program of China (863 Grant No. 2008AA032101); and the Natural Science and Engineering Research Council (NSERC) of Canada.

I. INTRODUCTION

II. THEORY AND MODEL

A. Self-consistent field theory

B. Time-dependent Ginzburg-Landau theory

III. RESULTS AND DISCUSSION

A. SCFT calculations of the phase behaviors

B. TDGL simulations of the ordering kinetics

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 18.0
- Annealing
- 17.0
- Self assembly
- 17.0
- Phase diagrams
- 11.0
- Probability theory
- 11.0

##### C21D1/26

## Figures

(a) Density plots of stable morphologies observed in the cylinder-forming diblock copolymer melts, with fixed *f* = 0.26 and χ*N* = 20, under the confinement of two-dimensional hexagons with varying diagonal size of D, in units of the bulk cylinder-to-cylinder distance of L_{0}; (b) stable phase sequence of morphologies as a function of D/L_{0}; (c) density plots of observed metastable 2D morphologies.

(a) Density plots of stable morphologies observed in the cylinder-forming diblock copolymer melts, with fixed *f* = 0.26 and χ*N* = 20, under the confinement of two-dimensional hexagons with varying diagonal size of D, in units of the bulk cylinder-to-cylinder distance of L_{0}; (b) stable phase sequence of morphologies as a function of D/L_{0}; (c) density plots of observed metastable 2D morphologies.

(a) Isosurface density plots and the density profiles of cross section of three-dimensional morphologies. The major component inside the green isosurface is A block indicated by red-color region in the cross section profiles. (b) Stable phase sequence as a function of D/L_{0} identified by considering both 2D and 3D phases.

(a) Isosurface density plots and the density profiles of cross section of three-dimensional morphologies. The major component inside the green isosurface is A block indicated by red-color region in the cross section profiles. (b) Stable phase sequence as a function of D/L_{0} identified by considering both 2D and 3D phases.

Free-energy comparisons between all candidate 2D and 3D morphologies for hexagons with the diagonal size between L_{0} and 6L_{0}. For the reason of clarity, the free energy relative to that of a selected reference phase is plotted, and the plot is divided into four pieces along the size.

Free-energy comparisons between all candidate 2D and 3D morphologies for hexagons with the diagonal size between L_{0} and 6L_{0}. For the reason of clarity, the free energy relative to that of a selected reference phase is plotted, and the plot is divided into four pieces along the size.

Typical snapshots of density isosurfaces at *t* = 5 × 10^{4}Δ*t* (a), *t* = 10^{5}Δ*t* (b), and *t* = 2 × 10^{5}Δ*t*, respectively, for the diagonal size of the hexagon, D/L_{0} = 15.

Typical snapshots of density isosurfaces at *t* = 5 × 10^{4}Δ*t* (a), *t* = 10^{5}Δ*t* (b), and *t* = 2 × 10^{5}Δ*t*, respectively, for the diagonal size of the hexagon, D/L_{0} = 15.

Histogram plots of the distributions of the ordering time for the diagonal size of D/L_{0} = 13 (a) and D/L_{0} = 15 (b), respectively. The Poisson distribution is used to fit the results. (a) The time interval is chosen as δ*t* = 10^{4} time steps, and the Poisson distribution curves of three parameters of λ = 3, 4, and 5, are plotted; (b) The time interval is chosen as δ*t* = 3 × 10^{4}, and λ = 2, 3, and 4, respectively.

Histogram plots of the distributions of the ordering time for the diagonal size of D/L_{0} = 13 (a) and D/L_{0} = 15 (b), respectively. The Poisson distribution is used to fit the results. (a) The time interval is chosen as δ*t* = 10^{4} time steps, and the Poisson distribution curves of three parameters of λ = 3, 4, and 5, are plotted; (b) The time interval is chosen as δ*t* = 3 × 10^{4}, and λ = 2, 3, and 4, respectively.

Ordering time as a function of the diagonal size of hexagons, plotted in a double-logarithm plot. The solid line is obtained by a linear fitting. Note that, the filled symbol for D = 19L_{0} indicates that the ordering time is beyond 10^{6} time steps in one or more runs of our simulated eight samples.

Ordering time as a function of the diagonal size of hexagons, plotted in a double-logarithm plot. The solid line is obtained by a linear fitting. Note that, the filled symbol for D = 19L_{0} indicates that the ordering time is beyond 10^{6} time steps in one or more runs of our simulated eight samples.

(a) The time evolution of the space fluctuation of bonds, θ, for D/L_{0} = 21. (b) Four typical snapshots of the structures are shown for *t* = 50 × 10^{4}, *t* = 77 × 10^{4}, *t* = 78 × 10^{4}, and *t* = 85 × 10^{4}, respectively (indicated by color circles and numbers 1, 2, 3, and 4) (a link to the corresponding movie of the structure evolution is provided at the end of this figure caption). (c) The corresponding distributions of bonds with respect to their positions are presented for the above four times. The bond length is indicated by both the filled circle size and the color spectrum. However, for the reason of clarity, different length ranges are used in the four plots: L/L_{0} ≈ 0.794–1.158 (c1), 0.714–1.050 (c2), 0.976–1.077 (c3), and 0.982–1.030 (c4), respectively (enhanced online). [URL: http://dx.doi.org/10.1063/1.4765098.1]10.1063/1.4765098.1

(a) The time evolution of the space fluctuation of bonds, θ, for D/L_{0} = 21. (b) Four typical snapshots of the structures are shown for *t* = 50 × 10^{4}, *t* = 77 × 10^{4}, *t* = 78 × 10^{4}, and *t* = 85 × 10^{4}, respectively (indicated by color circles and numbers 1, 2, 3, and 4) (a link to the corresponding movie of the structure evolution is provided at the end of this figure caption). (c) The corresponding distributions of bonds with respect to their positions are presented for the above four times. The bond length is indicated by both the filled circle size and the color spectrum. However, for the reason of clarity, different length ranges are used in the four plots: L/L_{0} ≈ 0.794–1.158 (c1), 0.714–1.050 (c2), 0.976–1.077 (c3), and 0.982–1.030 (c4), respectively (enhanced online). [URL: http://dx.doi.org/10.1063/1.4765098.1]10.1063/1.4765098.1

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