^{1}, Jack F. Douglas

^{2}and Francis W. Starr

^{3,a)}

### Abstract

The clustering of nanoparticles (NPs) in solutions and polymer melts depends sensitively on the strength and directionality of the NP interactions involved, as well as the molecular geometry and interactions of the dispersing fluids. Since clustering can strongly influence the properties of polymer-NP materials, we aim to better elucidate the mechanism of reversible self-assembly of highly symmetric NPs into clusters under equilibrium conditions. Our results are based on molecular dynamics simulations of icosahedral NP with a long-ranged interaction intended to mimic the polymer-mediated interactions of a polymer-melt matrix. To distinguish effects of polymer-mediated interactions from bare NP interactions, we compare the NP assembly in our coarse-grained model to the case where the NP interactions are purely short ranged. For the “control” case of NPs with short-ranged interactions and no polymer matrix, we find that the particles exhibit ordinary phase separation. By incorporating physically plausible long-ranged interactions, we suppress phase separation and qualitatively reproduce the thermally reversible cluster formation found previously in computations for NPs with short-ranged interactions in an explicit polymer-melt matrix. We further characterize the assembly process by evaluating the cluster properties and the location of the self-assemblytransition. Our findings are consistent with a theoretical model for equilibrium clustering when the particle association is subject to a constraint. In particular, the density dependence of the average cluster mass exhibits a linear concentration dependence, in contrast to the square root dependence found in freely associating systems. The coarse-grained model we use to simulate NP in a polymer matrix shares many features of potentials used to model colloidal systems. The model should be practically valuable for exploring factors that control the dispersion of NP in polymer matrices where explicit simulation of the polymer matrix is too time consuming.

We thank S. V. Buldyrev for the code that generate the spinodal and coexistence lines shown in Fig. 1. We thank the NSF for support under Grant No. DMR-0427239.

I. INTRODUCTION

II. MODEL AND SIMULATIONS

III. PURE NANOPARTICLES WITH ONLY SHORT-RANGE ATTRACTION

IV. MIMICKING POLYMERIC EFFECTS ON NANOPARTICLE BEHAVIOR

A. Effective polymeric interactions

B. Basic thermodynamic and cluster properties

V. CONCLUSION

### Key Topics

- Polymers
- 44.0
- Cluster analysis
- 34.0
- Nanoparticles
- 27.0
- Phase separation
- 22.0
- Self assembly
- 14.0

## Figures

Equation of state for coarse-grained nanoparticle dispersion model. (a) The phase diagram for pure NPs. The inset is a blowup of the region close to the critical point to emphasize the shapes of the coexistence (heavy solid black line) and spinodal (heavy dashed black line) lines. We show sample data for (symbols) that were used to determine the isotherms (in solid lines), and the coexistence (red dashed line) and spinodal lines (dashed blue line). See text for the fitting procedure used to obtain these lines. (b) phase diagram showing the coexistence line (solid line) and the spinodal line (dashed line), from which we get the critical point , , and .

Equation of state for coarse-grained nanoparticle dispersion model. (a) The phase diagram for pure NPs. The inset is a blowup of the region close to the critical point to emphasize the shapes of the coexistence (heavy solid black line) and spinodal (heavy dashed black line) lines. We show sample data for (symbols) that were used to determine the isotherms (in solid lines), and the coexistence (red dashed line) and spinodal lines (dashed blue line). See text for the fitting procedure used to obtain these lines. (b) phase diagram showing the coexistence line (solid line) and the spinodal line (dashed line), from which we get the critical point , , and .

Snapshots illustrating the particle dispersion under different thermodynamic conditions: (a) a gas phase system at , ; (b) a gas-liquid phase system at , ; and (c) a liquid phase system at , .

Snapshots illustrating the particle dispersion under different thermodynamic conditions: (a) a gas phase system at , ; (b) a gas-liquid phase system at , ; and (c) a liquid phase system at , .

Isotherms of vs for the modified NP with the Yukawa potential. The lack of a “loop” indicatives that the long-ranged repulsion suppresses phase separation down to at least . Each isotherm is separated by 0.2 in temperature, and curves are progressively shifted by a factor of 1.5 on the pressure axis for clarity. The lines are intended as a guide for the eye.

Isotherms of vs for the modified NP with the Yukawa potential. The lack of a “loop” indicatives that the long-ranged repulsion suppresses phase separation down to at least . Each isotherm is separated by 0.2 in temperature, and curves are progressively shifted by a factor of 1.5 on the pressure axis for clarity. The lines are intended as a guide for the eye.

The pair distribution function between the centers of NP. We separate into the contributions between pairs of small or large NP, and also the cross correlations between small and large NPs. We define particles to be nearest neighbors if , where is the location of the minimum of . (Inset) An expanded view of near . The arrows show the locations of the for the three possible pairs.

The pair distribution function between the centers of NP. We separate into the contributions between pairs of small or large NP, and also the cross correlations between small and large NPs. We define particles to be nearest neighbors if , where is the location of the minimum of . (Inset) An expanded view of near . The arrows show the locations of the for the three possible pairs.

Fraction of clustered NP as a function of . At low we observe that most NPs are in a clustered state. As we increase , the system becomes dispersed, except for large density (low ) where clusters are unavoidable. The lines are only intended as a guide to the eye, and are the results of spline fits to the data.

Fraction of clustered NP as a function of . At low we observe that most NPs are in a clustered state. As we increase , the system becomes dispersed, except for large density (low ) where clusters are unavoidable. The lines are only intended as a guide to the eye, and are the results of spline fits to the data.

(a) Average cluster mass as a function of along isochores. (b) The same data with a reduced temperature scale, where is normalized by , show that the data collapse to a single master curve.

(a) Average cluster mass as a function of along isochores. (b) The same data with a reduced temperature scale, where is normalized by , show that the data collapse to a single master curve.

A comparison between the particle clustering transition temperatures and estimated from and , respectively. The nonmonotonic behavior of suggests that is not a good measure of the assembly transition temperature for this system. The inset shows that conforms reasonably well to an Arrhenius relation, as expected from simple theoretical models of equilibrium particle assembly. The black line indicates the best fit of Eq. (7); the red dotted line is only a guide for the eye.

A comparison between the particle clustering transition temperatures and estimated from and , respectively. The nonmonotonic behavior of suggests that is not a good measure of the assembly transition temperature for this system. The inset shows that conforms reasonably well to an Arrhenius relation, as expected from simple theoretical models of equilibrium particle assembly. The black line indicates the best fit of Eq. (7); the red dotted line is only a guide for the eye.

The average cluster mass as a function of density for various . Note that for our system, similar to the behavior of wormlike micelles, colloidal fluids, protein suspensions, and dipolar fluids (Refs. 47, 54, and 55). The lines simply connect data points.

The average cluster mass as a function of density for various . Note that for our system, similar to the behavior of wormlike micelles, colloidal fluids, protein suspensions, and dipolar fluids (Refs. 47, 54, and 55). The lines simply connect data points.

Potential energy as a function of density, showing a double-well structure in the potential energy. If there were phase separation, the free energy would have a double-well dependence on density; thus the density dependence of the entropy must be responsible for the lack of phase separation. The lines simply linearly interpolate between points.

Potential energy as a function of density, showing a double-well structure in the potential energy. If there were phase separation, the free energy would have a double-well dependence on density; thus the density dependence of the entropy must be responsible for the lack of phase separation. The lines simply linearly interpolate between points.

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