^{1}and Kenneth S. Schweizer

^{1,a)}

### Abstract

The microscopic polymer reference interaction site model theory is generalized and applied to study intermolecular pair correlation functions and collective structure factors of dense solutions and melts of spherical nanoparticles carrying a single tethered chain. The complex interplay of entropy (translational, conformational, and packing) and enthalpy (particle-particle attraction) leads to different structural arrangements with distinctive small and wide angle scattering signatures. Strong concentration fluctuations, indicative of aggregate formation and/or a tendency for microphase separation, occur as the total packing fraction and/or particle-particle attraction strength increase. In analogy with block copolymers, the microphase spinodal curve is estimated by extrapolation of the inverse of the amplitude of the small angle scattering peak. For nanoparticles that are twice the diameter of monomers, the microphase separation boundary spinodal occurs at higher particle-particle attraction strength (or lower temperature) as compared to the macrophase demixing curve for nanoparticles with no tethers when the packing fraction is below 0.45, while the opposite trend is observed above 0.45. Increasing nanoparticle diameter results in a reduction in the microphase spinodal temperature and a qualitative change in its packing fraction dependence.

This work was supported by the Nanoscale Science and Engineering Initiative of the National Science foundation under NSF Award No. DMR-0642573. We thank Chris Iacovella and Sharon Glotzer for discussions of their simulation results.

I. INTRODUCTION

II. MODEL AND THEORY

A. Model and interaction potentials

B. Polymer reference interaction site model theory

C. System parameters

III. ATHERMAL SYSTEMS

IV. EFFECT OF PARTICLE-PARTICLE ATTRACTIONS, TETHER LENGTH, AND PARTICLE SIZE

A. Scattering patterns and real space correlations

B. Aggregation and mesoscale ordering

C. Apparent microphase transition

D. Comparison with the liquid-vapor transition and a microphase ordering simulation

E. Effect of particle size

F. Potential of mean force at contact

V. SUMMARY AND DISCUSSION

### Key Topics

- Nanoparticles
- 51.0
- Polymers
- 41.0
- Correlation functions
- 14.0
- Block copolymers
- 13.0
- Mean field theory
- 13.0

## Figures

Colloid Lennard–Jones potential (black solid curve) as a function of with , shifted Lennard–Jones-like potential (red dotted curve) as a function of used in the simulation of Iacovella *et al.* (Ref. 18), and standard Lennard–Jones potential (green dashed curve). All potentials are normalized to at the depth of their respective attractive minimum. A schematic of a tethered nanoparticle is also shown.

Colloid Lennard–Jones potential (black solid curve) as a function of with , shifted Lennard–Jones-like potential (red dotted curve) as a function of used in the simulation of Iacovella *et al.* (Ref. 18), and standard Lennard–Jones potential (green dashed curve). All potentials are normalized to at the depth of their respective attractive minimum. A schematic of a tethered nanoparticle is also shown.

Collective nanoparticle and polymer structure factors as a function of dimensionless wave vector (a) and (b) for tether length of 8 and nanoparticle size at a total packing fraction (solid line), 0.22 (dotted line), 0.50, (dashed line), and 0.60 (dotted-dashed line) under athermal conditions.

Collective nanoparticle and polymer structure factors as a function of dimensionless wave vector (a) and (b) for tether length of 8 and nanoparticle size at a total packing fraction (solid line), 0.22 (dotted line), 0.50, (dashed line), and 0.60 (dotted-dashed line) under athermal conditions.

Site-site radial distribution functions as a function of dimensionless separation under athermal conditions for (a) with as inset, and (b) with as inset, for tether length of 8 and nanoparticle size at total packing fractions (solid line), 0.22 (dotted line), 0.50, (dashed line), and 0.60 (dotted-dashed line).

Site-site radial distribution functions as a function of dimensionless separation under athermal conditions for (a) with as inset, and (b) with as inset, for tether length of 8 and nanoparticle size at total packing fractions (solid line), 0.22 (dotted line), 0.50, (dashed line), and 0.60 (dotted-dashed line).

(a) Particle potential of mean force (units of the thermal energy) under athermal conditions as a function of surface-to-surface separation in units of the monomer diameter, and (b) monomer-particle radial distribution function for tether length of 8 and nanoparticle size at total packing fractions (solid line), 0.22 (dotted line), 0.50 (dashed line), and 0.60 (dotted-dashed line).

(a) Particle potential of mean force (units of the thermal energy) under athermal conditions as a function of surface-to-surface separation in units of the monomer diameter, and (b) monomer-particle radial distribution function for tether length of 8 and nanoparticle size at total packing fractions (solid line), 0.22 (dotted line), 0.50 (dashed line), and 0.60 (dotted-dashed line).

Inverse of the dimensionless isothermal compressibility as a function of total packing fraction for tether length of 8 and nanoparticle size , and particle-particle attraction strengths of (circles), (squares) and (crosses). The inset shows the same results but for .

Inverse of the dimensionless isothermal compressibility as a function of total packing fraction for tether length of 8 and nanoparticle size , and particle-particle attraction strengths of (circles), (squares) and (crosses). The inset shows the same results but for .

(a) , (b) , (c) , and (inset), and (d) and (inset) for varying tether lengths (solid line), 10 (cross-solid line), 50 (dashed line), 100 (dotted line), and 200 (dotted-dashed line), under athermal conditions and a total packing fraction and nanoparticle size .

(a) , (b) , (c) , and (inset), and (d) and (inset) for varying tether lengths (solid line), 10 (cross-solid line), 50 (dashed line), 100 (dotted line), and 200 (dotted-dashed line), under athermal conditions and a total packing fraction and nanoparticle size .

Collective structure factors (a) and (b) for tether length and nanoparticle size at packing fraction and attraction strengths (solid line), (circle-solid line), (dashed line), (dashed dotted line), and (dotted line).

Collective structure factors (a) and (b) for tether length and nanoparticle size at packing fraction and attraction strengths (solid line), (circle-solid line), (dashed line), (dashed dotted line), and (dotted line).

Radial distribution functions (a) with as inset (b) with as inset, and (c) with as inset, for tether length of 8 and nanoparticle size at a packing fraction and particle-particle attraction strengths (solid line), (circle-solid line), (dashed line), and (dotted line).

Radial distribution functions (a) with as inset (b) with as inset, and (c) with as inset, for tether length of 8 and nanoparticle size at a packing fraction and particle-particle attraction strengths (solid line), (circle-solid line), (dashed line), and (dotted line).

(a) , (b) , (c) and (inset), and (d) and (inset) for tether length and nanoparticle size , packing fraction and particle-particle attraction strengths (solid line), (circle-solid line), and (dashed line).

(a) , (b) , (c) and (inset), and (d) and (inset) for tether length and nanoparticle size , packing fraction and particle-particle attraction strengths (solid line), (circle-solid line), and (dashed line).

Number of particle nearest neighbors as a function of attraction strength for with tethers of length 8 at the five indicated packing fractions. The small angle peak position of is for all packing fractions at the lowest indicated temperature.

Number of particle nearest neighbors as a function of attraction strength for with tethers of length 8 at the five indicated packing fractions. The small angle peak position of is for all packing fractions at the lowest indicated temperature.

Microphase order parameter (a) , (b) plotted as a function of particle-particle attraction strength at packing fractions (circles), 0.10 (crosses), 0.22 (upward triangle), 0.30 (squares), 0.40 (downward triangle), and 0.50 (diamonds) for a tether of length 8 and nanoparticle size . The dotted lines indicate the linear extrapolation to obtain the spinodal . The horizontal dashed line in a) indicates the Verlet–Hansen value of .

Microphase order parameter (a) , (b) plotted as a function of particle-particle attraction strength at packing fractions (circles), 0.10 (crosses), 0.22 (upward triangle), 0.30 (squares), 0.40 (downward triangle), and 0.50 (diamonds) for a tether of length 8 and nanoparticle size . The dotted lines indicate the linear extrapolation to obtain the spinodal . The horizontal dashed line in a) indicates the Verlet–Hansen value of .

(a) Microphase spinodal attraction strength based on (solid line-solid circles) and Verlet–Hansen attraction strength (dashed lines–solid squares) as a function of packing fraction for tether length of 8 and nanoparticle size . For comparison, the macrophase separation spinodal (dotted line–solid triangles) for nanoparticles with no tethers are also shown. The order-disorder phase transition obtained from the simulations of Iacovella *et al.* (Ref. 18) are shown as the crosses and solid line. (b) Corresponding temperatures of all quantities shown in (a).

(a) Microphase spinodal attraction strength based on (solid line-solid circles) and Verlet–Hansen attraction strength (dashed lines–solid squares) as a function of packing fraction for tether length of 8 and nanoparticle size . For comparison, the macrophase separation spinodal (dotted line–solid triangles) for nanoparticles with no tethers are also shown. The order-disorder phase transition obtained from the simulations of Iacovella *et al.* (Ref. 18) are shown as the crosses and solid line. (b) Corresponding temperatures of all quantities shown in (a).

(a) , (b) , (c) with as inset, and (d) with as inset, at (solid line), (circle-solid lines), (dashed lines), (dotted lines), (cross-solid line) for a packing fraction of 0.30, tether length of 8, and larger nanoparticle size .

(a) , (b) , (c) with as inset, and (d) with as inset, at (solid line), (circle-solid lines), (dashed lines), (dotted lines), (cross-solid line) for a packing fraction of 0.30, tether length of 8, and larger nanoparticle size .

(a) Microphase order parameter as a function of particle-particle attraction strength at packing fractions , 0.20, 0.25, 0.30, 0.35, and 0.45 for tether length 8, and larger nanoparticle size . Dotted lines indicate the extrapolations used to obtain the spinodal . The horizontal dashed line is the Verlet–Hansen value of . (b) Spinodal temperature as a function of packing fraction for larger nanoparticle size (solid line) and (dashed line) and tether length of 8.

(a) Microphase order parameter as a function of particle-particle attraction strength at packing fractions , 0.20, 0.25, 0.30, 0.35, and 0.45 for tether length 8, and larger nanoparticle size . Dotted lines indicate the extrapolations used to obtain the spinodal . The horizontal dashed line is the Verlet–Hansen value of . (b) Spinodal temperature as a function of packing fraction for larger nanoparticle size (solid line) and (dashed line) and tether length of 8.

## Tables

Potential of mean force at contact (units of kT), , at the particle-particle attraction strength closest to the extrapolated spinodal value (shown in parentheses) for varying total packing fraction and particle sizes and 3 with a tether length .

Potential of mean force at contact (units of kT), , at the particle-particle attraction strength closest to the extrapolated spinodal value (shown in parentheses) for varying total packing fraction and particle sizes and 3 with a tether length .

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