^{1}, Thomas P. Russell

^{1,a)}and Gregory M. Grason

^{1,b)}

### Abstract

We study the phase behavior of diblock copolymer melts with one block possessing orientation-dependent segmental interactions using self-consistent field theory. A generalized coarse-grained description is introduced based on the local (polar) orientational order parameter and *K*, an effective Frank elastic constant for orientational gradients. To explore the role played by orientational interactions in assembly thermodynamics, we apply the theory to two-dimensional melt morphologies for a range of *K*. As microphase segregation necessarily introduces splay deformations of the segment orientation, we find that increasing the stiffness *K* raises the critical χ*N* at the onset of microphase separation. Furthermore, we find that strong orientational interactions in one block give rise to highly asymmetric phase diagrams due to the large penalty for high-splay morphologies, such as the cylindrical phase. Finally, we analyze the costs of inter-segmental splay as well as the size dependence of domain spacing on *K* based on a strong-segregation picture of morphologies.

The authors are especially grateful to H. Johnston for invaluable discussions regarding the numerical implementation of this theory. This work was supported by the Materials Research Science and Engineering Center on Polymers at the University of Massachusetts (W.Z. and G.G.) and the Department of Energy Office of Basic Energy Sciences DOE-DE-FG02-45612 (T.P.R.).

I. INTRODUCTION

II. THEORY

III. RESULTS AND DISCUSSION

IV. CONCLUSION

### Key Topics

- Block copolymers
- 22.0
- Mean field theory
- 22.0
- Elasticity
- 16.0
- Free energy
- 15.0
- Phase diagrams
- 9.0

## Figures

Schematic of a chiral BCP molecule, in which the A block is formed by a series of segments with orientation sensitive segment interactions. Vector **t**(**x**) is defined as local volume average of segment orientation.

Schematic of a chiral BCP molecule, in which the A block is formed by a series of segments with orientation sensitive segment interactions. Vector **t**(**x**) is defined as local volume average of segment orientation.

Phase boundaries between cylinder-disorder phases and lamellae-cylinder phases for different orientational stiffness values, .

Phase boundaries between cylinder-disorder phases and lamellae-cylinder phases for different orientational stiffness values, .

Critical compositions *f* (left) and critical χ*N* values (right) as a function of reduced Frank constant .

Critical compositions *f* (left) and critical χ*N* values (right) as a function of reduced Frank constant .

Plots of (a) the interfacial width *w* as a function of χ*N*, and (b), (c) the domain spacing *D* as functions of χ*N* and Frank constant , respectively, for the lamella phase (L). The two lengths are reduced to dimensionless numbers by comparing with root-mean-square end-to-end distance *N* ^{1/2} *a* of a Gaussian chain.

Plots of (a) the interfacial width *w* as a function of χ*N*, and (b), (c) the domain spacing *D* as functions of χ*N* and Frank constant , respectively, for the lamella phase (L). The two lengths are reduced to dimensionless numbers by comparing with root-mean-square end-to-end distance *N* ^{1/2} *a* of a Gaussian chain.

Plots of the profiles (a) ϕ_{ A }(**x**) and (b) **t**(**x**) across a symmetric (*f* = 0.5) lamellar phase in the strong-segregation limit (χ*N* = 32) and for three different values of Frank elastic constant, , and 0.8. Here, is the direction normal to the lamellar layers.

Plots of the profiles (a) ϕ_{ A }(**x**) and (b) **t**(**x**) across a symmetric (*f* = 0.5) lamellar phase in the strong-segregation limit (χ*N* = 32) and for three different values of Frank elastic constant, , and 0.8. Here, is the direction normal to the lamellar layers.

Schematic representations of the unit-cell approximation for C_{ inner }, L, and C_{ outer } phases. The unit cell and the AB interface for C_{ inner } and C_{ outer } phases are assumed to be uniformly round. As for L phase, it is assumed to be uniformly flat for the unit-cell and AB interface. The polymer chains extend radially in the C_{ inner } and C_{ outer } phases, while perpendicularly to the interface in the L phase. The blocks with orientational segment interactions are marked red, which are formed by sequentially connected vector arrows as shown in the highlighted molecules of each individual phase. The other blocks are marked blue, the interfaces are marked as green dashed lines, and the junction points are marked as purple dots.

Schematic representations of the unit-cell approximation for C_{ inner }, L, and C_{ outer } phases. The unit cell and the AB interface for C_{ inner } and C_{ outer } phases are assumed to be uniformly round. As for L phase, it is assumed to be uniformly flat for the unit-cell and AB interface. The polymer chains extend radially in the C_{ inner } and C_{ outer } phases, while perpendicularly to the interface in the L phase. The blocks with orientational segment interactions are marked red, which are formed by sequentially connected vector arrows as shown in the highlighted molecules of each individual phase. The other blocks are marked blue, the interfaces are marked as green dashed lines, and the junction points are marked as purple dots.

Schematic of a chiral BCP molecule, on which two adjacent unit tangent vectors **t** _{α} and **t** _{β} with distance **r** _{αβ} interacting with each other.

Schematic of a chiral BCP molecule, on which two adjacent unit tangent vectors **t** _{α} and **t** _{β} with distance **r** _{αβ} interacting with each other.

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