^{1,a)}, Marianne Heckmann

^{1}and Barbara Drossel

^{1}

### Abstract

We investigate the microphases of asymmetric *AB*diblock copolymers confined to thin films in the strong segregation limit under the application of electric fields. We evaluate the free energy of a given set of possible phases and present phase diagrams for diblock copolymers with a cylindrical bulk phase in dependence of the film thickness and the attraction between the confining walls and the *A* or *B*monomers. This is done for different field strengths and volume fractions. We find that with increasing field strength structures show a preference for alignment with the field. The alignment is stronger when the permittivity of the minority monomer is larger than that of the majority monomer. Depending on the strength of the wall potential and the film thickness, the walls can become completely wetted by the minority monomer.

This work was supported by the German Research Foundation (DFG) Grant Nos. Dr300/5 and Dr300/11. We thank Tiago de Paula Peixoto for discussions and help, especially concerning programming and algorithms. Furthermore, we benefitted from discussions with Manuel Bach during the early stages of this project.

I. INTRODUCTION

II. MODEL

A. Contributions to the free energy

1. Interfacial energy

2. Free energy within a domain

3. Surface energy

4. Electrostatic energy

B. Morphology of considered phases

III. RESULTS AND DISCUSSION

A. Configuration of cylindrical domains

B. Phase diagrams

C. Different ratios of permittivities

D. Accuracy of the numerical evaluation

E. The possible occurrence of other phases

IV. CONCLUSIONS

### Key Topics

- Electric fields
- 53.0
- Free energy
- 50.0
- Polymers
- 43.0
- Lamellae
- 40.0
- Electrostatics
- 32.0

## Figures

Example for the discretization of the interface between the *A* and *B* domain. This figure explains also the notation used in Eq. (15).

Example for the discretization of the interface between the *A* and *B* domain. This figure explains also the notation used in Eq. (15).

Example for the phase: A thin film containing two rows of hexagons. The deformation of the inner hexagons is quantified by the parameter *h*, the side length of the hexagons is . The horizontal semi-axes of the outer and inner elliptical *A*-domains *a* can be varied. The outer hexagons contain more volume than the inner hexagons, the slope *p* can be varied.

Example for the phase: A thin film containing two rows of hexagons. The deformation of the inner hexagons is quantified by the parameter *h*, the side length of the hexagons is . The horizontal semi-axes of the outer and inner elliptical *A*-domains *a* can be varied. The outer hexagons contain more volume than the inner hexagons, the slope *p* can be varied.

Example for the phase: A thin film containing lamellae at the film boundaries and hexagons in the center. In this instance, we have diverted some volume to the inner hexagon, such that each lamellae is slightly smaller than 1.5 hexagons. The lamellae can be undulated, and the amplitude varied.

Example for the phase: A thin film containing lamellae at the film boundaries and hexagons in the center. In this instance, we have diverted some volume to the inner hexagon, such that each lamellae is slightly smaller than 1.5 hexagons. The lamellae can be undulated, and the amplitude varied.

Example for : A thin film containing two rows of hexagons. The height of the vertical hexagon-boundary is given by *l* _{ o } (can be varied independently for outer and inner hexagons), the width of the hexagons by *w*. The horizontal semi-axis *a* and vertical height *l* _{ i } of the *A*-domain can be varied independently for outer and inner domains.

Example for : A thin film containing two rows of hexagons. The height of the vertical hexagon-boundary is given by *l* _{ o } (can be varied independently for outer and inner hexagons), the width of the hexagons by *w*. The horizontal semi-axis *a* and vertical height *l* _{ i } of the *A*-domain can be varied independently for outer and inner domains.

Example for : A thin film containing three rows of hexagons. Only the lower half of the symmetric configuration is shown. All hexagons have the same vertical height 2*l* _{ o }, the lower (upper) half of lower (upper) outer hexagon is converted to a rectangular shape while keeping the same volume. The horizontal semi-axis *a* and vertical height *l* _{ i } of the *A*-domains can be varied independently for outer and inner hexagons. The outer hexagons have two independently adjustable vertical parts of the *A*-domain.

Example for : A thin film containing three rows of hexagons. Only the lower half of the symmetric configuration is shown. All hexagons have the same vertical height 2*l* _{ o }, the lower (upper) half of lower (upper) outer hexagon is converted to a rectangular shape while keeping the same volume. The horizontal semi-axis *a* and vertical height *l* _{ i } of the *A*-domains can be varied independently for outer and inner hexagons. The outer hexagons have two independently adjustable vertical parts of the *A*-domain.

Different configurations of the hexagonal cell and *A*-domain with resulting minimal free energy densities for a volume fraction of *f* = 0.13. Light gray: perfectly circular/hexagonal, dark gray: *C* _{∥,hs}, black: *C* _{∥,vs}.

Different configurations of the hexagonal cell and *A*-domain with resulting minimal free energy densities for a volume fraction of *f* = 0.13. Light gray: perfectly circular/hexagonal, dark gray: *C* _{∥,hs}, black: *C* _{∥,vs}.

Phase diagram for volume fraction *f* = 0.13 and *E* = 0.0. For the *C* _{∥,vs} phase, the gray shades indicate the percentage to which the walls are wetted by the *A* domain.

Phase diagram for volume fraction *f* = 0.13 and *E* = 0.0. For the *C* _{∥,vs} phase, the gray shades indicate the percentage to which the walls are wetted by the *A* domain.

Phase diagrams for *E* = 0.0 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 0.0 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 1.66 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 1.66 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 3.33 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 3.33 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 5.0 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Phase diagrams for *E* = 5.0 and volume fractions *f* = 0.13, *f* = 0.2, and *f* = 0.27.

Ratio of electrostatic energies of three parallel phases with the *C* _{⊥} phase, for eight different values of λ, and for *f* = 0.2. The three parallel phases are shown in the pictures on the left. The gray shades indicate undeformed *C* _{∥,vs} (, light gray), vertically stretched *C* _{∥,vs} (, dark gray), and *L* _{∥} (black) phases.

Ratio of electrostatic energies of three parallel phases with the *C* _{⊥} phase, for eight different values of λ, and for *f* = 0.2. The three parallel phases are shown in the pictures on the left. The gray shades indicate undeformed *C* _{∥,vs} (, light gray), vertically stretched *C* _{∥,vs} (, dark gray), and *L* _{∥} (black) phases.

Analytical approximation of the electrostatic energies for *C* _{∥} and *S* in a cube of unit volume for a volume fraction of *f* = 0.13 with a field strength *E* _{0}.

Analytical approximation of the electrostatic energies for *C* _{∥} and *S* in a cube of unit volume for a volume fraction of *f* = 0.13 with a field strength *E* _{0}.

## Tables

Wetting capabilities of undeformed *C* _{⊥}, *C* _{∥}, and *S*, with *C* stacked hexagonally and *S* in a bcc lattice.

Wetting capabilities of undeformed *C* _{⊥}, *C* _{∥}, and *S*, with *C* stacked hexagonally and *S* in a bcc lattice.

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