^{1}, Marcus Müller

^{2}and Juan J. de Pablo

^{1,a)}

### Abstract

We perform a theoretically informed coarse grain Monte-Carlo simulation in the *nPT*-ensemble and the Gibbs ensemble on symmetric ternary mixtures of *AB*-diblock copolymers with the corresponding homopolymers. We study the lamellar period by varying the length and amount of homopolymers. The homopolymer distribution within the lamellar morphology is determined as is the maximum amount of homopolymer within the lamellae. Gibbs ensemble simulations are used to locate the three-phase coexistence between two homopolymer-rich phases and a lamellar phase.

The authors are grateful for the financial support from the National Science Foundation (NSF) through the Nanoscale Science and Engineering Center (NSEC) at the University of Wisconsin (DMR-0425880). Support from the Semiconductor Research Corporation (SRC) is also gratefully acknowledged. The calculations presented in this work were performed in the Grid Laboratory of Wisconsin (GLOW). Finally, two anonymous reviewers provided comments which helped improve this article.

I. INTRODUCTION

II. METHOD

III. RESULTS

IV. CONCLUSION

### Key Topics

- Block copolymers
- 24.0
- Polymers
- 17.0
- Copolymers
- 16.0
- Chemical potential
- 14.0
- Lamellae
- 13.0

## Figures

Average density vs. homopolymer fraction in the lamellae for various homopolymer length. The reference state, pure homopolymers with α = 1, has an average density of ρ_{ o }.

Average density vs. homopolymer fraction in the lamellae for various homopolymer length. The reference state, pure homopolymers with α = 1, has an average density of ρ_{ o }.

The equilibrium period, *L* _{ B }, of the ternary mixture increases as the volume fraction of homopolymers, ϕ_{ H }, increases. This increase is more pronounced with longer homopolymers. The inset shows a grand-canonical SCFT study with similar results. The Fourier spectral method limits the amount of metastable data attainable.

The equilibrium period, *L* _{ B }, of the ternary mixture increases as the volume fraction of homopolymers, ϕ_{ H }, increases. This increase is more pronounced with longer homopolymers. The inset shows a grand-canonical SCFT study with similar results. The Fourier spectral method limits the amount of metastable data attainable.

The effective value of β (cf. Eq. (3)) is computed for different relative homopolymer lengths, α, as a function of the volume fraction, ϕ_{ H }. The horizontal lines are at the values indicated in the legend. These are the values of β used to create the curves in Fig. 2.

The effective value of β (cf. Eq. (3)) is computed for different relative homopolymer lengths, α, as a function of the volume fraction, ϕ_{ H }. The horizontal lines are at the values indicated in the legend. These are the values of β used to create the curves in Fig. 2.

The distribution of copolymers and homopolymers in a ternary system with α = 0.25 and ϕ_{ H } = 0.5 is shown through a snapshot (a) and density profiles (b). In the snapshot, copolymer beads of type *A* are red and type *B* are blue. Homopolymers of both types are green. In the density profile plot, the solid lines show the copolymer density and the dotted line shows the homopolymer density.

The distribution of copolymers and homopolymers in a ternary system with α = 0.25 and ϕ_{ H } = 0.5 is shown through a snapshot (a) and density profiles (b). In the snapshot, copolymer beads of type *A* are red and type *B* are blue. Homopolymers of both types are green. In the density profile plot, the solid lines show the copolymer density and the dotted line shows the homopolymer density.

The distribution of copolymers and homopolymers in a ternary system with α = 1.0 and ϕ_{ H } = 0.5. The conventions are the same as in Fig. 4.

The distribution of copolymers and homopolymers in a ternary system with α = 1.0 and ϕ_{ H } = 0.5. The conventions are the same as in Fig. 4.

The number of copolymers, *n* _{ AB }, per interfacial area, *A*, as a function of the homopolymer volume fraction, ϕ_{ H }, measured for various homopolymer lengths, α. The inset is a grand-canonical SCFT calculation with a polymer density equivalent to the Monte-Carlo reference state. The SCFT values are larger due to a greater polymer density and period length.

The number of copolymers, *n* _{ AB }, per interfacial area, *A*, as a function of the homopolymer volume fraction, ϕ_{ H }, measured for various homopolymer lengths, α. The inset is a grand-canonical SCFT calculation with a polymer density equivalent to the Monte-Carlo reference state. The SCFT values are larger due to a greater polymer density and period length.

The Gibbs free energy of mixing per molecule, Δ*g* ^{ mixing }, as a function of the relative length of the homopolymers, α, as extracted from the *nPT*-simulations according to Eq. (7).

The Gibbs free energy of mixing per molecule, Δ*g* ^{ mixing }, as a function of the relative length of the homopolymers, α, as extracted from the *nPT*-simulations according to Eq. (7).

In the Gibbs ensemble simulation, the equilibrium value of ϕ_{ H } = 0.39 is approached for α = 1.0.

In the Gibbs ensemble simulation, the equilibrium value of ϕ_{ H } = 0.39 is approached for α = 1.0.

The boundary ϕ_{ H } of the symmetric three-phase coexistence between two homopolymer-rich phases and a lamellar phase is obtained by comparing the chemical potential of the *A* homopolymer, μ_{ A }, in the pure *A* phase and the lamellae swollen by both homopolymers.

The boundary ϕ_{ H } of the symmetric three-phase coexistence between two homopolymer-rich phases and a lamellar phase is obtained by comparing the chemical potential of the *A* homopolymer, μ_{ A }, in the pure *A* phase and the lamellae swollen by both homopolymers.

The difference in chemical potential between the lamellar and pure phase for homopolymers of length α = 0.25 is shown as a function of lamellar period, *L* _{ B }. The data points are the same as those used in Fig. 2, with ϕ_{ H } ranging from 0.1 to 0.9 in increments of 0.1. The curves fit the data to one or more exponentials, as noted in the text.

The difference in chemical potential between the lamellar and pure phase for homopolymers of length α = 0.25 is shown as a function of lamellar period, *L* _{ B }. The data points are the same as those used in Fig. 2, with ϕ_{ H } ranging from 0.1 to 0.9 in increments of 0.1. The curves fit the data to one or more exponentials, as noted in the text.

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