^{1,a)}, Venkat Padmanabhan

^{2}and Michael E. Mackay

^{2}

### Abstract

In an athermal blend of nanoparticles and homopolymer near a hard wall, there is a first order phase transition in which the nanoparticles segregate to the wall and form a densely packed monolayer above a certain nanoparticle density. Previous investigations of this phase transition employed a fluids density functional theory(DFT) at constant packing fraction. Here we report further DFT calculations to probe the robustness of this phase transition. We find that the phase transition also occurs in athermal systems at constant pressure, the more natural experimental condition than constant packing fraction. Adding nanoparticle-polymer attractions increases the nanoparticletransition density, while sufficiently strong attractions suppress the first-order transition entirely. In this case the systems display a continuous transition to a bulk layered state. Adding attractions between the polymers and the wall has a similar effect of delaying and then suppressing the first-order nanoparticle segregation transition, but does not lead to any continuous phase transitions.

We thank the U.S. Department of Energy for funding this research (Contract No. DE-FG02-05ER46211). This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility, under a CINT User Project. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration (Contract No. DE-AC04-94AL85000).

I. INTRODUCTION

II. MODEL AND METHODS

A. Model

B. DFT

C. Arc-length and constant pressure calculations

III. RESULTS

A. Constant pressure versus constant packing fraction

B. Effect of attractions

1. Nanoparticle-monomer attractions

2. Polymer-wall attractions

C. Relation to experiment

IV. CONCLUSIONS

### Key Topics

- Nanoparticles
- 139.0
- Polymers
- 79.0
- Free energy
- 43.0
- Surface phase transitions
- 40.0
- Phase transitions
- 33.0

##### B82B1/00

## Figures

Pressure for calculations at constant packing fraction (dashed blue curve, left axis) and packing fraction for calculations at constant pressure (solid brown curve, right axis), for *N* = 80 and σ_{ n } = 2σ_{ p }.

Pressure for calculations at constant packing fraction (dashed blue curve, left axis) and packing fraction for calculations at constant pressure (solid brown curve, right axis), for *N* = 80 and σ_{ n } = 2σ_{ p }.

Surface free energy as a function of nanoparticle density for a system with *N* = 80 and σ_{ n } = 2σ_{ p }. Open squares correspond to constant pressure and open triangles correspond to constant packing fraction calculations.

Surface free energy as a function of nanoparticle density for a system with *N* = 80 and σ_{ n } = 2σ_{ p }. Open squares correspond to constant pressure and open triangles correspond to constant packing fraction calculations.

Density profiles for the system with N = 80 and σ_{ n } = 2σ_{ p } for constant pressure calculations. (a) Converged from a lower density profile and (b) Converged from a higher density profile. The polymers (x) scale is along the left axis and the nanoparticle (+) scale is on the right axis.

Density profiles for the system with N = 80 and σ_{ n } = 2σ_{ p } for constant pressure calculations. (a) Converged from a lower density profile and (b) Converged from a higher density profile. The polymers (x) scale is along the left axis and the nanoparticle (+) scale is on the right axis.

Nanoparticle density at the first phase transition as a function of chain length *N* for blends with σ_{ n } = 2σ_{ p }. Squares are for constant pressure and triangles are for constant packing fraction.

Nanoparticle density at the first phase transition as a function of chain length *N* for blends with σ_{ n } = 2σ_{ p }. Squares are for constant pressure and triangles are for constant packing fraction.

Transition density as a function of nanoparticle diameter for blends with *N* = 80. Squares are for constant pressure and triangles are for constant packing fraction.

Transition density as a function of nanoparticle diameter for blends with *N* = 80. Squares are for constant pressure and triangles are for constant packing fraction.

Surface free energy as a function of nanoparticle density for a system with *N* = 80 and σ_{ n } = 3σ_{ p }. Open squares correspond to constant pressure and open triangles correspond to constant packing fraction calculations. Note: The continuation curves are cutoff to show the phase transition point more clearly.

Surface free energy as a function of nanoparticle density for a system with *N* = 80 and σ_{ n } = 3σ_{ p }. Open squares correspond to constant pressure and open triangles correspond to constant packing fraction calculations. Note: The continuation curves are cutoff to show the phase transition point more clearly.

Density profiles of polymer (dotted magenta) and nanoparticles (solid brown) at constant pressure for N = 80 and σ_{ n } = 2σ_{ p } at .

Density profiles of polymer (dotted magenta) and nanoparticles (solid brown) at constant pressure for N = 80 and σ_{ n } = 2σ_{ p } at .

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and α_{ np } = 0.5σ_{ np } at ε_{ np } = 0 (blue), ε_{ np } = 0.05 (dashed brown), ε_{ np } = 0.1 (black), ε_{ np } = 0.2 (red), ε_{ np } = 0.4 (dash-dot green), and ε_{ np } = 0.5 (magenta).

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and α_{ np } = 0.5σ_{ np } at ε_{ np } = 0 (blue), ε_{ np } = 0.05 (dashed brown), ε_{ np } = 0.1 (black), ε_{ np } = 0.2 (red), ε_{ np } = 0.4 (dash-dot green), and ε_{ np } = 0.5 (magenta).

Nanoparticle density at the phase transition for monomer-particle attractions with *N* = 80 and σ_{ n } = 2σ_{ p }, as a function of ε_{ np } at α_{ np } = 0.5σ_{ p } (lower axis, blue squares) and as a function of α_{ np } at ε_{ np } = 0.05 kT (upper axis, brown circles).

Nanoparticle density at the phase transition for monomer-particle attractions with *N* = 80 and σ_{ n } = 2σ_{ p }, as a function of ε_{ np } at α_{ np } = 0.5σ_{ p } (lower axis, blue squares) and as a function of α_{ np } at ε_{ np } = 0.05 kT (upper axis, brown circles).

Density profiles for polymer (dashed curves, left axis) and nanoparticles (solid curves, right axis), for *N* = 80, σ_{ np } = 2σ_{ p }, α_{ np } = 0.5σ_{ p }, and ε_{ np } = 0.5, at (top), (middle), and (bottom).

Density profiles for polymer (dashed curves, left axis) and nanoparticles (solid curves, right axis), for *N* = 80, σ_{ np } = 2σ_{ p }, α_{ np } = 0.5σ_{ p }, and ε_{ np } = 0.5, at (top), (middle), and (bottom).

Density profiles for polymer (dashed curve) and nanoparticles (solid curve), for *N* = 80, σ_{ np } = 2σ_{ p }, α_{ np } = 0.5σ_{ p }, and ε_{ np } = 0.5, at , with periodic boundary conditions (no wall).

Density profiles for polymer (dashed curve) and nanoparticles (solid curve), for *N* = 80, σ_{ np } = 2σ_{ p }, α_{ np } = 0.5σ_{ p }, and ε_{ np } = 0.5, at , with periodic boundary conditions (no wall).

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and ε_{ np } = 0.5 kT at α_{ np } = 0.5 (solid blue), α_{ np } = 0.7 (dashed brown), α_{ np } = 0.9 (solid black), α_{ np } = 1.0 (solid red), α_{ np } = 1.1 (dashed-dot green), and α_{ np } = 1.4 (solid magenta).

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and ε_{ np } = 0.5 kT at α_{ np } = 0.5 (solid blue), α_{ np } = 0.7 (dashed brown), α_{ np } = 0.9 (solid black), α_{ np } = 1.0 (solid red), α_{ np } = 1.1 (dashed-dot green), and α_{ np } = 1.4 (solid magenta).

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and ε_{ np } = 0.5 kT at ε_{ wp } = 0.05 (solid blue), ε_{ wp } = 0.1 (dashed brown), ε_{ wp } = 0.2 (solid black), and ε_{ wp } = 0.3 (solid magenta).

Surface free energy as a function of nanoparticle density for *N* = 80, σ_{ np } = 2σ_{ p }, and ε_{ np } = 0.5 kT at ε_{ wp } = 0.05 (solid blue), ε_{ wp } = 0.1 (dashed brown), ε_{ wp } = 0.2 (solid black), and ε_{ wp } = 0.3 (solid magenta).

Nanoparticle density at the phase transition for monomer-particle attractions with *N* = 80 and σ_{ n } = 2σ_{ p }, as a function of ε_{ wp } at α_{ wp } = 0.5σ_{ p } (lower axis, blue squares) and as a function of α_{ wp } at ε_{ wp } = 0.05 kT (upper axis, brown circles).

Nanoparticle density at the phase transition for monomer-particle attractions with *N* = 80 and σ_{ n } = 2σ_{ p }, as a function of ε_{ wp } at α_{ wp } = 0.5σ_{ p } (lower axis, blue squares) and as a function of α_{ wp } at ε_{ wp } = 0.05 kT (upper axis, brown circles).

Nanoparticle transition density as a function of *N* at constant pressure, for hard sphere systems (closed blue squares), weakly attractive particle-monomer interactions (open brown squares), and weakly attractive polymer-wall attractions (purple triangles).

Nanoparticle transition density as a function of *N* at constant pressure, for hard sphere systems (closed blue squares), weakly attractive particle-monomer interactions (open brown squares), and weakly attractive polymer-wall attractions (purple triangles).

## Tables

Weight functions associated with various parts of the DFT. In all cases, |*r*| = |**r** − **r** ^{′}|. Here *R* _{α} = σ_{α}/2 is the radius of site α, δ is the Dirac delta function and Θ is the Heaviside step function.

Weight functions associated with various parts of the DFT. In all cases, |*r*| = |**r** − **r** ^{′}|. Here *R* _{α} = σ_{α}/2 is the radius of site α, δ is the Dirac delta function and Θ is the Heaviside step function.

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