^{1}and Jayant K. Singh

^{1,a)}

### Abstract

In this work, we study the influence of polymer chain length (*m*), based on Lennard-Jones potential, and nanoparticle (NP)-polymer interaction strength (ɛ_{ np }) on aggregation and dispersion of soft repulsive spherically structured NPs in polymer melt using coarse-grain molecular dynamics simulations. A phase diagram is proposed where transitions between different structures in the NP-polymer system are shown to depend on *m* and ɛ_{ np }. At a very weak interaction strength ɛ_{ np } = 0.1, a transition from dispersed state to collapsed state of NPs is found with increasing *m*, due to the polymer's excluded volume effect. NPs are well dispersed at intermediate interaction strengths (0.5 ⩽ ɛ_{ np } ⩽ 2.0), independent of *m*. A transition from dispersion to agglomeration of NPs, at a moderately high NP-polymer interaction strength ɛ_{ np } = 5.0, for *m* = 1–30, is identified by a significant decrease in the second virial coefficient, excess entropy, and potential energy, and a sharp increase in the Kirkwood-Buff integral. We also find that NPs undergo the following transitions with increasing *m* at ɛ_{ np } ⩾ 5.0: string-like → branch-like → sphere-like → dispersed state.

This work is supported by the Department of Science and Technology (DST), Government of India. The computational facilities are provided from the Centre for Development of Advance Computing (CDAC), Pune, India. T.K.P. thanks the Council of Scientific and Industrial Research (CSIR), Government of India for supporting this work in terms of a senior research fellowship. J.K.S. wishes to thank Evangelos Voyiatzis for fruitful discussions.

I. INTRODUCTION

II. MODEL AND METHOD

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Polymers
- 60.0
- Entropy
- 22.0
- Aggregation
- 15.0
- Polymer melts
- 11.0
- Polymer phase transitions
- 11.0

##### B82B1/00

## Figures

The NP-NP RDFs, g_{nn}(*r*), for *m* = 30, at different NP-polymer interaction strengths (ɛ_{ np }).

The NP-NP RDFs, g_{nn}(*r*), for *m* = 30, at different NP-polymer interaction strengths (ɛ_{ np }).

The MD snapshots at different NP-polymer interaction strengths (ɛ_{ np }) for *m* = 30. The connectivity between adjacent beads of chains is not shown for clarity.

The MD snapshots at different NP-polymer interaction strengths (ɛ_{ np }) for *m* = 30. The connectivity between adjacent beads of chains is not shown for clarity.

The NP-NP RDFs, g_{nn}(*r*), for *m* = 1, at different NP-polymer interaction strengths (ɛ_{ np }). Representative snapshots corresponding to RDFs are shown in the inset of all the panels. The connectivity between adjacent beads of the chains is not shown for clarity.

The NP-NP RDFs, g_{nn}(*r*), for *m* = 1, at different NP-polymer interaction strengths (ɛ_{ np }). Representative snapshots corresponding to RDFs are shown in the inset of all the panels. The connectivity between adjacent beads of the chains is not shown for clarity.

The NP-NP RDFs, g_{nn}(*r*), for *m* = 50, at different NP-polymer interaction strengths (ɛ_{ np }). Representative snapshots corresponding to RDFs are shown in the inset of all the panels. The connectivity between adjacent beads of chains is not shown for clarity.

The NP-NP RDFs, g_{nn}(*r*), for *m* = 50, at different NP-polymer interaction strengths (ɛ_{ np }). Representative snapshots corresponding to RDFs are shown in the inset of all the panels. The connectivity between adjacent beads of chains is not shown for clarity.

The potential energies and entropic contributions of different pairs for *m* = 30 as a function of NP-polymer interaction strength (ɛ_{ np }). Energy and entropy are shown on the left and right y-axes, respectively, in all the panels. *E* _{nn}, S_{nn}: NP-NP; *E* _{pp}, S_{pp}: polymer-polymer; *E* _{np}, S_{np}: NP-polymer. *E* = *E* _{nn} + *E* _{pp} + *E* _{np}. The symbols are the calculated values and lines serve as a guide to the eye. Closed symbols correspond to energy values and open symbols correspond to excess entropy values. The error bars are of the order of symbol sizes.

The potential energies and entropic contributions of different pairs for *m* = 30 as a function of NP-polymer interaction strength (ɛ_{ np }). Energy and entropy are shown on the left and right y-axes, respectively, in all the panels. *E* _{nn}, S_{nn}: NP-NP; *E* _{pp}, S_{pp}: polymer-polymer; *E* _{np}, S_{np}: NP-polymer. *E* = *E* _{nn} + *E* _{pp} + *E* _{np}. The symbols are the calculated values and lines serve as a guide to the eye. Closed symbols correspond to energy values and open symbols correspond to excess entropy values. The error bars are of the order of symbol sizes.

The 2nd virial coefficient (B_{2}), excess entropy (S_{nn}), potential energy (E), and KBI (G_{nn}), as a function of ɛ_{ np }, are shown for *m* = 1, 30, and 50. The symbols are the calculated values and lines serve as a guide to the eye. The error bars are of the order of symbol sizes.

The 2nd virial coefficient (B_{2}), excess entropy (S_{nn}), potential energy (E), and KBI (G_{nn}), as a function of ɛ_{ np }, are shown for *m* = 1, 30, and 50. The symbols are the calculated values and lines serve as a guide to the eye. The error bars are of the order of symbol sizes.

Potential of mean force (PMF) as a function of inter-particle distance for *m* = 1, 30, and 50. Top panel corresponds to ɛ_{ np } = 0.1, middle panel corresponds to ɛ_{ np } = 1.0, and the bottom panel corresponds to ɛ_{ np } = 5.0.

Potential of mean force (PMF) as a function of inter-particle distance for *m* = 1, 30, and 50. Top panel corresponds to ɛ_{ np } = 0.1, middle panel corresponds to ɛ_{ np } = 1.0, and the bottom panel corresponds to ɛ_{ np } = 5.0.

Two branch structures of NPs at different state points as mentioned in the figure. The connectivity between adjacent beads of chains is not shown for clarity.

Two branch structures of NPs at different state points as mentioned in the figure. The connectivity between adjacent beads of chains is not shown for clarity.

Predicted ɛ_{ np } vs. *m* phase diagram of NP-polymer system. Symbols represent the phases as follows: (■) NPs and polymer melt phase separate, (●) NPs disperse in the polymer melt, (▲) string-like assembly of NPs, (★) branch structure of NPs, (♦) spherical shape assemblies of NPs.

Predicted ɛ_{ np } vs. *m* phase diagram of NP-polymer system. Symbols represent the phases as follows: (■) NPs and polymer melt phase separate, (●) NPs disperse in the polymer melt, (▲) string-like assembly of NPs, (★) branch structure of NPs, (♦) spherical shape assemblies of NPs.

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