^{1}, Victor Pryamitsyn

^{1}and Venkat Ganesan

^{1,a)}

### Abstract

We use polymer self-consistent field theory to quantify the interfacial properties of random copolymer brushes (AB) in contact with a homopolymer melt chemically identical to one of the blocks (A). We calculate the interfacial widths and interfacial energies between the melt and the brush as a function of the relative chain sizes, grafting densities, compositions of the random copolymer in the brush, and degree of chemical incompatibility between the A and B species. Our results indicate that the interfacial energies between the melt and the brush increase (signifying expulsion of the free chains from the brush) with increasing grafting density, chemical incompatibility between A and B components, and size of the free chains relative to the grafted chains. We also compare the interfacial energies of random copolymers of different sequence characteristics and find that, except for the case of very blocky or proteinlike chains, blockiness of the copolymer has only little effect on interfacial properties. Our results for interfacial energies are rationalized based on the concept of an “effective volume fraction” of the brush copolymers,, which quantifies the chemical composition of the brush segments in the interfacial zone between the brush and melt copolymers. Using this concept, we modify the strong-stretching theory of brush–melt interfaces to arrive at a simple model whose results qualitatively agree with our results from self-consistent field theory. We discuss the ramifications of our results for the design of neutral surfaces.

The authors would like to acknowledge Dr. Manas Shah and Dr. Landry Khounlavong for helpful discussions. We also appreciate useful discussions from Professor Jan Genzer and Ravish Malik on the generation of proteinlike sequences and with Professor Alfred Crosby on the contact angle measurements for random copolymer layers. This work was partially supported by National Science Foundation under Award Number 1005739, by a grant from Robert A. Welch Foundation (Grant No. F1599) and the US Army Research Office under Grant No. W911NF-10-1-0346. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing computing resources that have contributed to the research results reported within this work.

I. INTRODUCTION

II. THEORY AND NUMERICAL METHODS

A. Self-consistent field theory

B. Strong-stretching theory

III. RESULTS

A. Brush–melt interfacial width

B. Brush–melt interfacial energies

IV. DISCUSSION

A. Implications for particle dispersion and neutral surfaces

B. Characterization of blockiness using interfacial properties

V. SUMMARY

### Key Topics

- Polymers
- 44.0
- Copolymers
- 34.0
- Block copolymers
- 29.0
- Interfacial properties
- 19.0
- Field theory models
- 10.0

## Figures

(a) Enrichment of segments in the brush for different *f* (, and χ*N* = 10); and (b) different χ*N* (*f* = 0.5, , and α = 1.0).

(a) Enrichment of segments in the brush for different *f* (, and χ*N* = 10); and (b) different χ*N* (*f* = 0.5, , and α = 1.0).

Effective volume fraction of A segments in the interface as a function of *w* _{ b/h } calculated from SCFT (a) and SST (b) for different values of . Overall volume fraction of A segments in the brush is *f* = 0.5. Parameters α = 1.0 and χ*N* = 10.

Effective volume fraction of A segments in the interface as a function of *w* _{ b/h } calculated from SCFT (a) and SST (b) for different values of . Overall volume fraction of A segments in the brush is *f* = 0.5. Parameters α = 1.0 and χ*N* = 10.

Interfacial widths from SCFT (solid) and SST (dashed) as a function of the key parameters. Unless otherwise stated, chains are purely random (λ = 0); (a) Varying *f*, χ*N* = 10, α = 1.0, and ; (b) Varying χ*N*, *f* = 0.5, α = 1.0, and ; (c) Varying α, *f* = 0.5, χ*N* = 10, and ; and (d) Varying , *f* = 0.5, χ*N* = 10, and α = 1.0.

Interfacial widths from SCFT (solid) and SST (dashed) as a function of the key parameters. Unless otherwise stated, chains are purely random (λ = 0); (a) Varying *f*, χ*N* = 10, α = 1.0, and ; (b) Varying χ*N*, *f* = 0.5, α = 1.0, and ; (c) Varying α, *f* = 0.5, χ*N* = 10, and ; and (d) Varying , *f* = 0.5, χ*N* = 10, and α = 1.0.

(a) Interfacial widths between the brush and the melt as a function of , with *f* = 0.5, χ*N* = 10, and α = 1.0 for different blockiness of sequences; and (b) Enrichment of segments in the brush structure for different blockiness of sequences. Parameters *f* = 0.5, , and χ*N* = 10.

(a) Interfacial widths between the brush and the melt as a function of , with *f* = 0.5, χ*N* = 10, and α = 1.0 for different blockiness of sequences; and (b) Enrichment of segments in the brush structure for different blockiness of sequences. Parameters *f* = 0.5, , and χ*N* = 10.

Interaction potentials for different values of *f*. χ*N* = 10, α = 1.0, and .

Interaction potentials for different values of *f*. χ*N* = 10, α = 1.0, and .

Interfacial energy of the brush–melt interface for purely random chains (λ = 0) (a) for different values of *f* (χ*N* = 10, α = 1.0, and ); (b) for different values of χ*N* (*f* = 0.5, α = 1.0, and ); (c) for different values of α (*f* = 0.5, χ*N* = 10, and ); and (d) for different values of (*f* = 0.5, χ*N* = 10, and α = 1.0). The dotted lines in (c) and (d) show predictions for the autophobic scenario. The insets show corresponding predictions from SST.

Interfacial energy of the brush–melt interface for purely random chains (λ = 0) (a) for different values of *f* (χ*N* = 10, α = 1.0, and ); (b) for different values of χ*N* (*f* = 0.5, α = 1.0, and ); (c) for different values of α (*f* = 0.5, χ*N* = 10, and ); and (d) for different values of (*f* = 0.5, χ*N* = 10, and α = 1.0). The dotted lines in (c) and (d) show predictions for the autophobic scenario. The insets show corresponding predictions from SST.

(a) Interfacial energy and (b) effective concentration of A segments in the interfacial region displayed as a function of for different kinds of randomness. *f* = 0.5, χ*N* = 10, and α = 1.0.

(a) Interfacial energy and (b) effective concentration of A segments in the interfacial region displayed as a function of for different kinds of randomness. *f* = 0.5, χ*N* = 10, and α = 1.0.

(a) Plot of γ(*f*) − γ(1 − *f*) vs *f* as a function of . χ*N* = 10, and α = 1.0. (b) Plot of γ(*f*) − γ(1 − *f*) vs *f* as a function of type of randomness. α = 1.0, χ*N* = 10, and . In both plots, the dotted lines correspond to the theoretically reported limits of the neutral window from Meng and Wang (Ref. 62).

(a) Plot of γ(*f*) − γ(1 − *f*) vs *f* as a function of . χ*N* = 10, and α = 1.0. (b) Plot of γ(*f*) − γ(1 − *f*) vs *f* as a function of type of randomness. α = 1.0, χ*N* = 10, and . In both plots, the dotted lines correspond to the theoretically reported limits of the neutral window from Meng and Wang (Ref. 62).

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