^{1}and Jianzhong Wu

^{1,a)}

### Abstract

Morphology control is important for practical applications of composite materials that consist of functional polymers and nanoparticles. Toward that end, block copolymers provide useful templates to arrange nanoparticles in the scaffold of self-organized polymer microdomains. This paper reports theoretical predictions for the distribution of nanoparticles in the lamellar structures of symmetric diblock copolymers on the basis of a polymerdensity functional theory(DFT) and the potential distribution theorem (PDT). The DFT predicts periodic spacing of lamellar structures in good agreement with molecular dynamics simulations. With the polymer structure from DFT as the input, the PDT is used to examine the effects of particle size, surface energy, polymer chain length, and compressibility on the distribution of nanoparticles in the limit of low particle density. It is found that the nanoparticle distribution depends not only on the particle size and surface energy but also on the local structure of the microdomain interface,polymer chain length, and compressibility. The theoretical predictions are compared well with experiments and simulations.

The authors are thankful to Zhidong Li for insightful discussions, to Carol K. Hall and Andrew J. Schultz for providing the simulation results, and to Mark Matsen for providing results from the self-consistent theory. This research is sponsored by the U.S. Department of Energy (DE-FG02-06ER46296) and uses the computational resources from the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC03-76SF0009.

I. INTRODUCTION

II. MOLECULAR MODEL AND THEORETICAL BACKGROUND

III. RESULTS AND DISCUSSION

A. Lamellar structure

B. Particle distributions

1. Distribution of neutral particles

2. Distribution of energetically selective particles

IV. CONCLUSIONS

### Key Topics

- Polymers
- 57.0
- Nanoparticles
- 36.0
- Density functional theory
- 31.0
- Block copolymers
- 28.0
- Lamellae
- 17.0

## Figures

(a) The density profiles of the polymeric segments (“A” and “B”) within one periodic spacing of the lamellar structure. Here , , and the microdomain spacing is . (b) Effect of the polymer incompatibility on the total local polymer density. For , the microdomain spacing is . In both cases, the polymer chain length is , and the average packing fraction is .

(a) The density profiles of the polymeric segments (“A” and “B”) within one periodic spacing of the lamellar structure. Here , , and the microdomain spacing is . (b) Effect of the polymer incompatibility on the total local polymer density. For , the microdomain spacing is . In both cases, the polymer chain length is , and the average packing fraction is .

The reduced periodic spacing of the lamellae vs . (a) and (b) . Closed symbols are simulation results from Ref. 15, open symbols are from the DFT, and the solid lines are from the SCFT.

The reduced periodic spacing of the lamellae vs . (a) and (b) . Closed symbols are simulation results from Ref. 15, open symbols are from the DFT, and the solid lines are from the SCFT.

(a) The density profiles of the polymeric segments (A and B) and the total polymer density within one period of the lamellae. Here the average packing fraction of the polymer is , the energy parameters are , , and the periodic spacing is . (b) The distribution probability of neutral particles with different radii in the block-copolymer lamellae.

(a) The density profiles of the polymeric segments (A and B) and the total polymer density within one period of the lamellae. Here the average packing fraction of the polymer is , the energy parameters are , , and the periodic spacing is . (b) The distribution probability of neutral particles with different radii in the block-copolymer lamellae.

(a) Dependence of the biased one-particle potential on the ratio of the particle radius to the lamellar periodic spacing . All parameters for the copolymers are identical to those shown in Fig. 3(a). (b) Effect of the chain length on the biased one-particle potential . (c) Effect of the polymer density on .

(a) Dependence of the biased one-particle potential on the ratio of the particle radius to the lamellar periodic spacing . All parameters for the copolymers are identical to those shown in Fig. 3(a). (b) Effect of the chain length on the biased one-particle potential . (c) Effect of the polymer density on .

Effect of the chain length on the local structure of lamellar interface (the interface is ).

Effect of the chain length on the local structure of lamellar interface (the interface is ).

Effect of the chain length on the distribution of small and large neutral particles. The peaks in the particle distribution functions are aligned at . The embedded figure represents the reduced total density profile at the interface.

Effect of the chain length on the distribution of small and large neutral particles. The peaks in the particle distribution functions are aligned at . The embedded figure represents the reduced total density profile at the interface.

Effect of the overall packing density on the local structure of lamellar interface.

Effect of the overall packing density on the local structure of lamellar interface.

(a) The probability distribution function for a small neutral particle at the lamellar interface . (b) The probability distribution function for a large neutral particle at the center of the A domain .

(a) The probability distribution function for a small neutral particle at the lamellar interface . (b) The probability distribution function for a large neutral particle at the center of the A domain .

Effect of the energetic selectivity on the particle-distribution probability. Here the particle radius is ; all parameters for the polymers are identical to those shown in Fig. 3(a).

Effect of the energetic selectivity on the particle-distribution probability. Here the particle radius is ; all parameters for the polymers are identical to those shown in Fig. 3(a).

(a) The probability distribution functions for particles of different sizes at . (b) Effect of the selectivity parameter on the particle distribution at . All parameters for the polymers are the same as those shown in Fig. 3(a).

(a) The probability distribution functions for particles of different sizes at . (b) Effect of the selectivity parameter on the particle distribution at . All parameters for the polymers are the same as those shown in Fig. 3(a).

Effect of the chain length on the distributions of energetically selective nanoparticles .

Effect of the chain length on the distributions of energetically selective nanoparticles .

Effect of the chain length on the probability distribution functions for nanoparticles with different selectivity parameters. Here the particle size is fixed relative to the periodic spacing .

Effect of the chain length on the probability distribution functions for nanoparticles with different selectivity parameters. Here the particle size is fixed relative to the periodic spacing .

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