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Symmetric diblock copolymers in nanopores: Monte Carlo simulations and strong-stretching theory
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10.1063/1.2406078
/content/aip/journal/jcp/126/2/10.1063/1.2406078
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/2/10.1063/1.2406078

Figures

Image of FIG. 1.
FIG. 1.

Schematics of concentric cylinders with . The outermost thick circle represents the pore surface, the inside solid circles represent A-B interfaces, and the dashed circle represents the location where all the B ends reside in the strong-stretching theory.

Image of FIG. 2.
FIG. 2.

Schematics of the A-B interface undulations near the pore surface preferring A segments in the perpendicular morphology. The dotted line represents the A-B interface in the undistorted state. The two dashed lines represent the invariant interfaces between adjacent chain layers. It is assumed that chain trajectories are perpendicular to the A-B interface and that all chain ends are at the interfaces between adjacent chain layers. See text for more details.

Image of FIG. 3.
FIG. 3.

(a) Representative configurations of concentric cylinders from Monte Carlo simulations at . A segments are shown in dark (blue) and B segments in light (yellow) colors. From the left to right, the dimensionless pore diameter , 2.134, and 3. (b) Radial order parameter profiles for concentric cylinders at .

Image of FIG. 4.
FIG. 4.

Ensemble-averaged total polymer segmental density profiles at . The slab morphology forms at , while the concentric cylinders form at .

Image of FIG. 5.
FIG. 5.

The first-order phase transition between concentric cylinders at . (a) Values of the order parameter at the pore center and the constant-volume heat capacity . Sharp transitions between concentric cylinders with different numbers of A-B interfaces are clearly seen. (b) The ensemble-averaged total polymer segmental density and mean square end-to-end distances of A and B blocks, and , respectively, in nanopores. The dotted horizontal line marks the bulk value of 0.8032 for , and the solid horizontal line marks the bulk value of 17.29 for .

Image of FIG. 6.
FIG. 6.

(a) Comparison of the dimensionless thicknesses of the alternating A-rich and B-rich layers in concentric cylinders obtained from Monte Carlo simulations at (symbols) with those from the strong-stretching theory (curves). The squares and the solid curve represent the thickness of the outermost A-rich layer , the circles and the dotted curve represent thickness of the outermost B-rich layer , the up-triangles and the dashed curve represent the thickness of the second A-rich layer from the surface when or 3 , and the down-triangles and the dash-dotted curve represent the radius of the innermost B cylinder when . (b) SST results of the dimensionless free energy per chain for concentric cylinders with A-B interfaces, indicating that the transition between these concentric cylinders is of the second order. In both plots, the vertical line at marks the transition between concentric cylinders with and 4 predicted by SST. Since in MC the concentric cylinders have at most three A-B interfaces up to , the SST results for are calculated using accordingly, which are somewhat larger than the values obtained using more A-B interfaces.

Image of FIG. 7.
FIG. 7.

Snapshots of the system configuration (plane cut along the pore axis) (a) at and , and (b) at , where, from the left to right, , 2.672, 2.687, 2.693, and 2.749 and , 0.824, , , and , respectively. In both plots, A segments are shown in dark (blue) and B segments in light (yellow) colors.

Image of FIG. 8.
FIG. 8.

Representative configurations of (a) the slab morphology at and (b) the mixed morphology at . A segments are shown in dark (blue) and B segments in light (yellow) colors. The dimensionless pore diameter .

Image of FIG. 9.
FIG. 9.

Qualitative comparison of the undulations of A-B interfaces in the slab morphology obtained from (a) MC simulations at with those from (b) SST at .

Image of FIG. 10.
FIG. 10.

The order parameters at the pore center and surface, and , respectively, and the constant-volume heat capacity at (a) , (b) , and (c) .

Image of FIG. 11.
FIG. 11.

The ensemble-averaged (a) total polymer segmental density , (b) mean square end-to-end distance of A blocks , and (c) mean square end-to-end distance of B blocks in nanopores. The bulk value of is 0.8032, and the bulk value of is 17.29, as marked by the solid horizontal line in (c).

Image of FIG. 12.
FIG. 12.

(a) Block-averaged energy curve for a part of the simulation run at and . Each point is averaged over one simulation block of MCS, i.e., consecutively collected configurations. The snapshot of the system configuration (plane cut along the pore axis) shown on the left is at the end of the 929th block (marked by the left dotted line), while the one shown on the right is at the end of the 993th block (marked by the right dotted line). (b) Block-averaged total segmental density curve for the simulation run at and . The representative configuration of the mixed morphology is at the end of the 540th block (marked by the left dotted line), while that of the slab morphology is at the end of the 1040th block (marked by the right dotted line). In both configurations, A segments are shown in dark (blue) and B segments in light (yellow) colors.

Tables

Generic image for table
Table I.

List of some quantities used in this work.

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/content/aip/journal/jcp/126/2/10.1063/1.2406078
2007-01-11
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Symmetric diblock copolymers in nanopores: Monte Carlo simulations and strong-stretching theory
http://aip.metastore.ingenta.com/content/aip/journal/jcp/126/2/10.1063/1.2406078
10.1063/1.2406078
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