^{1}, Zheng Wang

^{1}, Yuhua Yin

^{1,a)}, Baohui Li

^{1}and An-Chang Shi

^{2}

### Abstract

The phase behavior of gradient copolymers is studied theoretically using random phase approximation (RPA) and self-consistent field theory (SCFT), focusing on the effects of monomer sequence distribution, or compositional polydispersity, of the polymer chains. The stability of the disordered phase is examined using RPA analysis, whereas the ordered phases of the system are studied using SCFT calculations. It is discovered that the critical domain spacing increases and the disorder-order transition temperature moves to higher values with the increase of the compositional polydispersity. SCFT results reveal that, depending on the value of the degree of segregation, structural change due to the different chain-to-chain monomer sequence distribution is controlled by two different mechanisms.

The research was financially supported by the National Natural Science Foundation of China (Grant Nos. 20904026, 20904027, and 20990234), and by the National Science Fund for Distinguished Young Scholars of China (20925414). A.-C.S. acknowledges the supports by the Natural Science and Engineering Research Council (NSERC) of Canada.

I. INTRODUCTION

II. THEORETICAL FRAMEWORK

A. Multiblock model for gradient copolymers with compositional polydispersity

B. Perturbation method

C. Random phase approximation

D. Self-consistent mean-field theory

III. RESULTS AND DISCUSSION

A. Critical point and critical periods

B. Microphase-separated gradient copolymers

IV. CONCLUSIONS

### Key Topics

- Copolymers
- 60.0
- Block copolymers
- 52.0
- Polymers
- 42.0
- Free energy
- 14.0
- Perturbation methods
- 14.0

## Figures

Schematic illustration of the composition profile for the linear gradient copolymer. The solid line denotes the average composition profile. The dash line denotes a deviation from the average composition profile. *t* is the segment index and is normalized by the degree of polymerization of the polymer chain. A schematic representation of different monomer sequences in gradient copolymers with the same chain length is provided in the inset. The open circles denote monomer A, and the closed circles denote monomer B.

Schematic illustration of the composition profile for the linear gradient copolymer. The solid line denotes the average composition profile. The dash line denotes a deviation from the average composition profile. *t* is the segment index and is normalized by the degree of polymerization of the polymer chain. A schematic representation of different monomer sequences in gradient copolymers with the same chain length is provided in the inset. The open circles denote monomer A, and the closed circles denote monomer B.

Inverse structure factor of the linear gradient copolymer melts with the average composition profile, *G*(*t*) = *t*, as a function of *qR* _{ g } for three compositional polydispersity: *k* _{ y } = 0, 0.1, 0.2.

Inverse structure factor of the linear gradient copolymer melts with the average composition profile, *G*(*t*) = *t*, as a function of *qR* _{ g } for three compositional polydispersity: *k* _{ y } = 0, 0.1, 0.2.

Plots of the critical point segregation, (χ*N*)_{ c } (solid line), and the critical domain spacing, *L* _{ c } (dash line), as a function of the parameter *k* _{ y }.

Plots of the critical point segregation, (χ*N*)_{ c } (solid line), and the critical domain spacing, *L* _{ c } (dash line), as a function of the parameter *k* _{ y }.

Equilibrium density profiles of A monomers at (a) χ*N* = 30 and (b) χ*N* = 100 for different values of *k* _{ y }. The period of the lamellar structure is *D*.

Equilibrium density profiles of A monomers at (a) χ*N* = 30 and (b) χ*N* = 100 for different values of *k* _{ y }. The period of the lamellar structure is *D*.

Interfacial width *W* plotted as a function of *k* _{ y } for the lamellar phases at (a) χ*N* = 30 and (b) χ*N* = 100.

Interfacial width *W* plotted as a function of *k* _{ y } for the lamellar phases at (a) χ*N* = 30 and (b) χ*N* = 100.

Period of the lamellar phase as a function of χ*N* for different values of *k* _{ y }.

Period of the lamellar phase as a function of χ*N* for different values of *k* _{ y }.

Equilibrium probability distribution functions of selected points along a single polymer chain for χ*N* = 30. *J* = 0, 2, and 1 indicate the two chain ends and the mid-point of chain, respectively. The solid and dash lines correspond to the results from *k* _{ y } = 0 and *k* _{ y } = 0.15, respectively.

Equilibrium probability distribution functions of selected points along a single polymer chain for χ*N* = 30. *J* = 0, 2, and 1 indicate the two chain ends and the mid-point of chain, respectively. The solid and dash lines correspond to the results from *k* _{ y } = 0 and *k* _{ y } = 0.15, respectively.

Schematic comparison of the domain spacings formed by gradient copolymers in the region with a relatively small value of χ*N*. In this and the following figures, D and D^{′} represent the domain spacings for system with compositional monodispersity and polydispersity, respectively.

Schematic comparison of the domain spacings formed by gradient copolymers in the region with a relatively small value of χ*N*. In this and the following figures, D and D^{′} represent the domain spacings for system with compositional monodispersity and polydispersity, respectively.

Equilibrium probability distribution functions of selected points along a single polymer chain for χ*N* = 100. The index *J* (*J* = 0 ∼ 10) indicates 11 points (including two chain ends) selected uniformly along the polymer. The solid lines and dash lines correspond to the results for *k* _{ y } = 0 and *k* _{ y } = 0.15, respectively.

Equilibrium probability distribution functions of selected points along a single polymer chain for χ*N* = 100. The index *J* (*J* = 0 ∼ 10) indicates 11 points (including two chain ends) selected uniformly along the polymer. The solid lines and dash lines correspond to the results for *k* _{ y } = 0 and *k* _{ y } = 0.15, respectively.

Schematic comparison of the domain spacings formed by gradient copolymers in the region with a relatively large value of χ*N*.

Schematic comparison of the domain spacings formed by gradient copolymers in the region with a relatively large value of χ*N*.

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