^{1,a)}and A. Cacciuto

^{1,a)}

### Abstract

Using numerical simulations and a simple scaling theory, we study the microphase separation of a mixture of polymer brushes with different chain lengths tethered to surfaces with nonuniform curvature. We measure the free energy difference of the phase separated configurations as a function of spheroid eccentricity and ordering of the microdomains formed on them. We find that there is a preference for the longer chains to locate in high curvature regions, and identify and quantify the driving forces associated with this phenomenon. We also find that nonuniform curvature typically limits the number of striped domains that would normally form on a spherical surface under identical physical conditions. Finally, we generalize the scaling theory developed for brushes on spherical surfaces to include prolate and oblate spheroids, and show explicitly that while immiscibility between the chains is required for phase separation to occur on spheroids, it is unnecessary for certain surfaces with regions of positive and negative curvature. We present a phase diagram showing the conditions under which curvature-driven phase separation of miscible, but lengthwise asymmetric chains is expected to occur.

This work was supported by the National Science Foundation under CAREER Grant No. DMR-0846426. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation, Grant No. OCI-1053575.

INTRODUCTION

METHODS

RESULTS

CONCLUSIONS

### Key Topics

- Polymers
- 48.0
- Free energy
- 44.0
- Phase separation
- 12.0
- Nanoparticles
- 7.0
- Solubility
- 7.0

## Figures

Diagram of the model used in our simulations.

Diagram of the model used in our simulations.

(a) and (b) Snapshots of microphase-separated configurations taken from MD trajectories. Short and long brushes have heights of 4 and 8 monomers, respectively. From left to right: oblate spheroid, same configuration showing only the adsorbed monomers; prolate spheroid, same configuration showing only the adsorbed monomers. Long brushes concentrate on regions of high curvature, short brushes concentrate on flatter regions.

(a) and (b) Snapshots of microphase-separated configurations taken from MD trajectories. Short and long brushes have heights of 4 and 8 monomers, respectively. From left to right: oblate spheroid, same configuration showing only the adsorbed monomers; prolate spheroid, same configuration showing only the adsorbed monomers. Long brushes concentrate on regions of high curvature, short brushes concentrate on flatter regions.

Pairs of phase-separated, three-striped configurations from thermodynamic integration simulations. Left pair (a and b): Oblate long-on-pole, oblate short-on-pole. Right pair (c and d): Prolate long-on-pole, prolate short-on-pole.

Pairs of phase-separated, three-striped configurations from thermodynamic integration simulations. Left pair (a and b): Oblate long-on-pole, oblate short-on-pole. Right pair (c and d): Prolate long-on-pole, prolate short-on-pole.

Free energy differences from simulation between opposite pairs of striped configurations on oblate and prolate spheroids. Snapshots of these pairs are shown in Fig. 3 . Pairs of oblate spheroids have positive values and pairs of prolate spheroids have negative values because the free energy difference is defined as ΔF = F LOP − F SOP . The line at zero height difference illustrates that the free energy for two equal-length brushes on different spheroids is the same regardless of the configuration.

Free energy differences from simulation between opposite pairs of striped configurations on oblate and prolate spheroids. Snapshots of these pairs are shown in Fig. 3 . Pairs of oblate spheroids have positive values and pairs of prolate spheroids have negative values because the free energy difference is defined as ΔF = F LOP − F SOP . The line at zero height difference illustrates that the free energy for two equal-length brushes on different spheroids is the same regardless of the configuration.

Scaling picture of a polymer brush on a spheroid. ξ, the blob size, and h, the brush height, are functions of the polar angle θ, and depend on the radius of curvature at the grafting point.

Scaling picture of a polymer brush on a spheroid. ξ, the blob size, and h, the brush height, are functions of the polar angle θ, and depend on the radius of curvature at the grafting point.

(a) Free energy differences between pairs of three-striped configurations on oblate and prolate spheroids, using the analytical expression in Eq. (6) for the free energy of brushes on spheroids. Differences increase with eccentricity and with brush heights, in agreement with our results from simulations shown in Fig. 4 . (b) Free energy differences between pairs of striped configurations as a function of eccentricity for different number of stripes. With more stripes, the overall free energy gain decreases.

(a) Free energy differences between pairs of three-striped configurations on oblate and prolate spheroids, using the analytical expression in Eq. (6) for the free energy of brushes on spheroids. Differences increase with eccentricity and with brush heights, in agreement with our results from simulations shown in Fig. 4 . (b) Free energy differences between pairs of striped configurations as a function of eccentricity for different number of stripes. With more stripes, the overall free energy gain decreases.

(a)–(c) Snapshots of phase-separated configurations taken from equilibrium MD trajectories. Only the adsorbed monomers are shown. Red monomers are from the long brush, yellow monomers are from the short brush. Systems spontaneously microphase-separate into configurations with long brushes on high curvature regions and short brushes on low curvature regions.

(a)–(c) Snapshots of phase-separated configurations taken from equilibrium MD trajectories. Only the adsorbed monomers are shown. Red monomers are from the long brush, yellow monomers are from the short brush. Systems spontaneously microphase-separate into configurations with long brushes on high curvature regions and short brushes on low curvature regions.

(a) Drawing of a Pac-Man-like surface with positive and negative curvature obtained by overlapping a large sphere of radius R 1 and a smaller sphere of radius R 2. The location of the center of the smaller sphere is such that R 2 = R 1sin α. (b) Phase diagram for the Pac-Man-like object depicted in Fig. 8 , showing phase-separated and mixed coexistence curves as a function of chain length ratio and chain mixing ratio. Phase separation occurs for values of to the right of the coexistence curves. As the length of the short chain decreases, phase separation occurs at lower volume fractions.

(a) Drawing of a Pac-Man-like surface with positive and negative curvature obtained by overlapping a large sphere of radius R 1 and a smaller sphere of radius R 2. The location of the center of the smaller sphere is such that R 2 = R 1sin α. (b) Phase diagram for the Pac-Man-like object depicted in Fig. 8 , showing phase-separated and mixed coexistence curves as a function of chain length ratio and chain mixing ratio. Phase separation occurs for values of to the right of the coexistence curves. As the length of the short chain decreases, phase separation occurs at lower volume fractions.

Snapshot from simulations of two miscible chains having different lengths while they phase separate on a surface with regions of positive and negative curvature. Red monomers correspond to the long chain, yellow to the short chain. The surface is defined by Eq. (11) .

Snapshot from simulations of two miscible chains having different lengths while they phase separate on a surface with regions of positive and negative curvature. Red monomers correspond to the long chain, yellow to the short chain. The surface is defined by Eq. (11) .

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