^{1}and Qiang Wang

^{1,a)}

### Abstract

We have investigated several effects due to the confinement of polymer melts by impenetrable (hard) surfaces in the self-consistent field calculations. To adequately represent such confinement, the total (normalized) polymer segmental density (volume fraction) is usually constrained to an imposed profile that continuously decreases from 1 in the interior of confined melts to 0 at the surfaces over a short distance. The choice of this profile strongly influences the numerical performance of the self-consistent field calculations. In addition, for diblock copolymers the hard-surface confinement has both energetic and entropic effects: On one hand, the decrease of polymer density from 1 reduces repulsion and favors morphologies with more interfaces near the surfaces. On the other hand, the enrichment of chain ends and depletion of middle segments near the surfaces favor parallel morphologies where chains orient mainly perpendicular to the surfaces. These two effects are comparable in magnitude, and for asymmetric diblock copolymers result in an entropic preference of a neutral surface for the shorter block as proposed previously [Q. Wang *et al.*, Macromolecules34, 3458 (2001)]. The hard-surface effects are weak in practice and thus manifested only when the surfaces are nearly neutral.

Q.W. thanks Professor Glenn Fredrickson, Professor Carlos Garcia-Cervera, and Professor Eric Cochran for helpful discussions on the fourth-order implicit-explicit scheme. Financial support for this work was provided by Colorado State University, which is gratefully acknowledged.

I. INTRODUCTION

II. MODELS AND NUMERICAL METHODS

A. Confined homopolymer melts

B. Confined diblock copolymer melts

III. CONFINED HOMOPOLYMERS

A. Choice of

B. Segmental distributions

IV. CONFINED SYMMETRIC DIBLOCK COPOLYMER MELTS

A. Surface-induced compatabilization

B. Surface-induced entropy loss

C. Influence of on phase behavior

V. CONFINED ASYMMETRIC DIBLOCK COPOLYMERS

A. Hard-surface effects

B. Effectively neutral surface

VI. CONCLUSIONS

### Key Topics

- Block copolymers
- 34.0
- Polymers
- 27.0
- Lamellae
- 17.0
- Surface morphology
- 15.0
- Boundary value problems
- 10.0

## Figures

Pressure field for confined homopolymer melts, with the linear profile P1 used in the surface layer. , , and .

Pressure field for confined homopolymer melts, with the linear profile P1 used in the surface layer. , , and .

Pressure field for confined homopolymer melts, where and the values are chosen to give the same in all cases: (a) The quadratic profile P2 and the cosine profile COS are used in the surface layer, respectively. and . In the P2 case, two data points located around with values less than are not shown. (b) The fifth-order polynomial P5 is used in the surface layer. . (c) The profile EXP is used in the surface layer. For the case of and , a minimum located around with value about is not shown.

Pressure field for confined homopolymer melts, where and the values are chosen to give the same in all cases: (a) The quadratic profile P2 and the cosine profile COS are used in the surface layer, respectively. and . In the P2 case, two data points located around with values less than are not shown. (b) The fifth-order polynomial P5 is used in the surface layer. . (c) The profile EXP is used in the surface layer. For the case of and , a minimum located around with value about is not shown.

Influence of spatial-domain discretization on the accuracy of free-energy calculations for confined homopolymers. and . The values are given in the caption of Fig. 2. Here we approximate the exact value of (obtained when ) by that calculated with the largest , .

Influence of spatial-domain discretization on the accuracy of free-energy calculations for confined homopolymers. and . The values are given in the caption of Fig. 2. Here we approximate the exact value of (obtained when ) by that calculated with the largest , .

Reduced distributions of (a) the end segments and (b) the middle segments in confined homopolymer melts, where , , and . The values are given in the caption of Fig. 2. The results of using COS profile are hardly distinguishable from those of using EXP, thus not shown. In the log-log plot shown in part (a), the curves approach a slope of as . In part (b), the curves are symmetric about . Under the bulk condition a constant value of 1 is expected for these distributions.

Reduced distributions of (a) the end segments and (b) the middle segments in confined homopolymer melts, where , , and . The values are given in the caption of Fig. 2. The results of using COS profile are hardly distinguishable from those of using EXP, thus not shown. In the log-log plot shown in part (a), the curves approach a slope of as . In part (b), the curves are symmetric about . Under the bulk condition a constant value of 1 is expected for these distributions.

Free-energy differences between parallel and perpendicular lamellae of symmetric diblock copolymers ( and ) between two neutral surfaces separated at . Here we use (with ) such that adjusting from 0 to 1 changes from 1 to COS profile, and accordingly the Neumann boundary conditions in the direction. The dot at represents calculated using the Dirichlet boundary conditions, where the fast sine transforms are used accordingly.

Free-energy differences between parallel and perpendicular lamellae of symmetric diblock copolymers ( and ) between two neutral surfaces separated at . Here we use (with ) such that adjusting from 0 to 1 changes from 1 to COS profile, and accordingly the Neumann boundary conditions in the direction. The dot at represents calculated using the Dirichlet boundary conditions, where the fast sine transforms are used accordingly.

Reduced distributions of (a) ends and (b) joints in parallel lamellae of symmetric diblock copolymers ( and ) of 1.5 periods confined between two neutral surfaces separated at . Here we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition . The -end distribution can be obtained by the symmetry between and .

Reduced distributions of (a) ends and (b) joints in parallel lamellae of symmetric diblock copolymers ( and ) of 1.5 periods confined between two neutral surfaces separated at . Here we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition . The -end distribution can be obtained by the symmetry between and .

Reduced distributions of (a) ends and (b) joints in perpendicular lamellae of symmetric diblock copolymers ( and , microphase separated along the direction) confined between two neutral surfaces separated at . Here we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition . The -end distribution can be obtained by the symmetry between and . In the legend, the -rich domain corresponds to a value where reaches a maximum of 0.9403, the -rich domain corresponds to a value where reaches a minimum of 0.0597, and the interface is where .

Reduced distributions of (a) ends and (b) joints in perpendicular lamellae of symmetric diblock copolymers ( and , microphase separated along the direction) confined between two neutral surfaces separated at . Here we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition . The -end distribution can be obtained by the symmetry between and . In the legend, the -rich domain corresponds to a value where reaches a maximum of 0.9403, the -rich domain corresponds to a value where reaches a minimum of 0.0597, and the interface is where .

Free-energy differences between parallel and perpendicular cylinders of asymmetric diblock copolymers confined by EXP profile between two neutral surfaces separated at . , 4.1266, 4.2832, and 4.4068 for , 0.25, 0.28, and 0.3, respectively. The data points are actual calculations, and the curves are guide to the eyes only. and (for perpendicular cylinders) are used in our calculations.

Free-energy differences between parallel and perpendicular cylinders of asymmetric diblock copolymers confined by EXP profile between two neutral surfaces separated at . , 4.1266, 4.2832, and 4.4068 for , 0.25, 0.28, and 0.3, respectively. The data points are actual calculations, and the curves are guide to the eyes only. and (for perpendicular cylinders) are used in our calculations.

interfaces [where ] in (a) parallel cylinders (whose axis is along the direction) and (b) perpendicular cylinders of asymmetric diblock copolymers ( and ) confined between two neutral surfaces separated at . The solid curves show the results under confinement by EXP profile , and the dotted curves show corresponding results under the bulk condition . and (for perpendicular cylinders) are used in our calculations.

interfaces [where ] in (a) parallel cylinders (whose axis is along the direction) and (b) perpendicular cylinders of asymmetric diblock copolymers ( and ) confined between two neutral surfaces separated at . The solid curves show the results under confinement by EXP profile , and the dotted curves show corresponding results under the bulk condition . and (for perpendicular cylinders) are used in our calculations.

Reduced distributions of (a) ends, (b) ends, and (c) joints in parallel cylinders of asymmetric diblock copolymers ( and ) confined between two neutral surfaces separated at . In parts (a) and (b) we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition , and use two values corresponding to Fig. 9(a). In part (c) only the results of hard-surface confinement are shown for clarity. Part (d) shows these distributions in perpendicular cylinders, where represents the average over directions, and under the bulk condition a constant value of 1 is expected for these distributions. In all cases, the distributions are symmetric about , and and (for perpendicular cylinders) are used in our calculations.

Reduced distributions of (a) ends, (b) ends, and (c) joints in parallel cylinders of asymmetric diblock copolymers ( and ) confined between two neutral surfaces separated at . In parts (a) and (b) we compare the case of hard-surface confinement (EXP profile with ) with that under the bulk condition , and use two values corresponding to Fig. 9(a). In part (c) only the results of hard-surface confinement are shown for clarity. Part (d) shows these distributions in perpendicular cylinders, where represents the average over directions, and under the bulk condition a constant value of 1 is expected for these distributions. In all cases, the distributions are symmetric about , and and (for perpendicular cylinders) are used in our calculations.

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