^{1,a)}and J.-U. Sommer

^{1,2}

### Abstract

Pure melts of asymmetric diblock copolymers are studied by means of the off-lattice Gaussian disphere model with Monte-Carlo kinetics. In this model, a diblock copolymer chain is mapped onto two soft repulsive spheres with fluctuating radii of gyration and distance between centers of mass of the spheres. Microscopic input quantities of the model such as the combined probability distribution for the radii of gyration and the distance between the spheres as well as conditional monomernumber densities assigned to each block were derived in the previous work of F. Eurich and P. Maass [J. Chem. Phys. **114**, 7655 (2001)] within an underlying Gaussian chain model. The polymerization degree of the whole chain as well as those of the individualblocks are freely tunable parameters thus enabling a precise determination of the regions of stability of various phases. The model neglects entanglement effects which are irrelevant for the formation of ordered structures in diblock copolymers and which would otherwise unnecessarily increase the equilibration time of the system. The gyroid phase was reproduced in between the cylindrical and lamellar phases in systems with box sizes being commensurate with the size of the unit cell of the gyroid morphology. The region of stability of the gyroid phase was studied in detail and found to be consistent with the prediction of the mean-fieldtheory. Packing frustration was observed in the form of increased radii of gyration of both blocks of the chains located close to the gyroid nodes.

We acknowledge fruitful discussions with Gerd E. Schroeder-Turk.

I. INTRODUCTION

II. GAUSSIAN DISPHERE MODEL

III. RESULTS

A. Order–disorder transition point

B. Ordered phase

C. Region of stability of the gyroid phase

D. Packing Frustration

IV. CONCLUSIONS

### Key Topics

- Block copolymers
- 35.0
- Mean field theory
- 28.0
- Polymers
- 25.0
- Phase diagrams
- 14.0
- Free energy
- 13.0

## Figures

Phase diagram for diblock copolymers calculated with the SCFT showing regions of stability for disordered (DIS), lamellar (L), bicontinuous gyroid with symmetry (), cylindrical (H), *BCC* packed spheres (), and closed packed spheres (CPS) phases (Ref. 4).

Phase diagram for diblock copolymers calculated with the SCFT showing regions of stability for disordered (DIS), lamellar (L), bicontinuous gyroid with symmetry (), cylindrical (H), *BCC* packed spheres (), and closed packed spheres (CPS) phases (Ref. 4).

Schematic representation of a diblock copolymer chain modeled as two soft spheres with fluctuating radii of gyration and distance between their centers of mass (Ref. 36).

Schematic representation of a diblock copolymer chain modeled as two soft spheres with fluctuating radii of gyration and distance between their centers of mass (Ref. 36).

Isotropically averaged structure factor of the minority component *A* calculated in the disordered phase for various χ*N* in the system with *M* = 4000 chains. Continuous curves are data fits obtained with a modified Leibler structure factor, Eq. (3.15).

Isotropically averaged structure factor of the minority component *A* calculated in the disordered phase for various χ*N* in the system with *M* = 4000 chains. Continuous curves are data fits obtained with a modified Leibler structure factor, Eq. (3.15).

Inverse of the peak intensity of the structure factor for various χ*N* in the system with *M* = 4000 chains. The ODT as evaluated by a linear extrapolation of the data toward is

Inverse of the peak intensity of the structure factor for various χ*N* in the system with *M* = 4000 chains. The ODT as evaluated by a linear extrapolation of the data toward is

Mean-square radii of gyration of the *A* and *B* blocks, , , and mean-square radius of gyration of the whole chain at various χ*N* both in the disordered and ordered phases (*M* = 4000). Values are normalized with respect to χ*N* = 0.

Mean-square radii of gyration of the *A* and *B* blocks, , , and mean-square radius of gyration of the whole chain at various χ*N* both in the disordered and ordered phases (*M* = 4000). Values are normalized with respect to χ*N* = 0.

Snapshot of the cylindrical morphology shown as isosurfaces obtained in the simulation, χ*N* = 30.0, (), and *M* = 4000 chains.

Snapshot of the cylindrical morphology shown as isosurfaces obtained in the simulation, χ*N* = 30.0, (), and *M* = 4000 chains.

Structure factor of the minority component calculated for the system with *M* = 4000 chains at χ*N* = 30.0 . The structure factor has two higher order peaks characteristic of the cylindrical morphology.

Structure factor of the minority component calculated for the system with *M* = 4000 chains at χ*N* = 30.0 . The structure factor has two higher order peaks characteristic of the cylindrical morphology.

Structure factor of the minority component calculated for the system with *M* = 4000 chains at χ*N* = 70.0 . The structure factor has five higher order peaks characteristic of the gyroid morphology. In addition, there are two peaks between and which are not typical for the gyroid.

Structure factor of the minority component calculated for the system with *M* = 4000 chains at χ*N* = 70.0 . The structure factor has five higher order peaks characteristic of the gyroid morphology. In addition, there are two peaks between and which are not typical for the gyroid.

Region of stability of the gyroid phase as calculated with the SCFT (Ref. 3).

Region of stability of the gyroid phase as calculated with the SCFT (Ref. 3).

Snapshot of one unit cell of the gyroid morphology obtained in the simulation. Here χ*N* = 40.0, (), and *M* = 2430 chains.

Snapshot of one unit cell of the gyroid morphology obtained in the simulation. Here χ*N* = 40.0, (), and *M* = 2430 chains.

Structure factor calculated for the simulated gyroid morphology showing ten higher order peaks characteristic of the gyroid. The structure factor was averaged over the last 20 000 MCS. The system parameters are the same as in Fig. 10.

Structure factor calculated for the simulated gyroid morphology showing ten higher order peaks characteristic of the gyroid. The structure factor was averaged over the last 20 000 MCS. The system parameters are the same as in Fig. 10.

Simulated phase diagram for the pure diblock copolymer system (G gyroid, H cylindrical, L lamellar, Co disordered continuous network, and Dis disordered). The gyroid phase is limited by the cylindrical phase on the left and a metastable perforated morphology on the right that converged into the lamellar phase as the simulation proceeded. Below χ*N* = 26.0 a narrow region of the cylindrical morphology is encountered, whereas above χ*N* = 44.0, a disordered continuous network is found resembling a defect gyroid phase. Dashed straight lines depict approximately the region of stability of the gyroid phase. Five independent runs were performed to obtain the structures at each set of parameters and χ*N* on the phase diagram.

Simulated phase diagram for the pure diblock copolymer system (G gyroid, H cylindrical, L lamellar, Co disordered continuous network, and Dis disordered). The gyroid phase is limited by the cylindrical phase on the left and a metastable perforated morphology on the right that converged into the lamellar phase as the simulation proceeded. Below χ*N* = 26.0 a narrow region of the cylindrical morphology is encountered, whereas above χ*N* = 44.0, a disordered continuous network is found resembling a defect gyroid phase. Dashed straight lines depict approximately the region of stability of the gyroid phase. Five independent runs were performed to obtain the structures at each set of parameters and χ*N* on the phase diagram.

Structure factor calculated for the simulated gyroid morphology in the vicinity of the order–order transition point between cylindrical and gyroid phases, χ*N* = 26.0 ( *M* = 1701 chains). The structure factor has only two higher order peaks of the gyroid located at relative positions and , the second order peak is absent. The structure factor was averaged over the last 20 000 MCS.

Structure factor calculated for the simulated gyroid morphology in the vicinity of the order–order transition point between cylindrical and gyroid phases, χ*N* = 26.0 ( *M* = 1701 chains). The structure factor has only two higher order peaks of the gyroid located at relative positions and , the second order peak is absent. The structure factor was averaged over the last 20 000 MCS.

Mean-square radii of gyration of the minority *A*-blocks and the majority *B*-blocks near the center of the node. The system parameters are the same as in Fig. 10.

Mean-square radii of gyration of the minority *A*-blocks and the majority *B*-blocks near the center of the node. The system parameters are the same as in Fig. 10.

Mean-square distance between blocks *A* and *B* of the chain near the center of the node. The system parameters are the same as in Fig. 10.

Mean-square distance between blocks *A* and *B* of the chain near the center of the node. The system parameters are the same as in Fig. 10.

## Tables

Values of fitting parameters α, , δ and the radius of gyration of the whole chain .

Values of fitting parameters α, , δ and the radius of gyration of the whole chain .

Free energy per chain in units calculated for various simulated morphologies: gyroid (G), cylindrical (H), lamellar (L), and disordered continuous network (CN).

Free energy per chain in units calculated for various simulated morphologies: gyroid (G), cylindrical (H), lamellar (L), and disordered continuous network (CN).

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