^{1,a)}and Gary S. Grest

^{1}

### Abstract

Large-scale molecular dynamics simulations are used to simulate a layer of nanoparticles floating on the surface of a liquid. Both a low viscosity liquid, represented by Lennard-Jones monomers, and a high viscosity liquid, represented by linear homopolymers, are studied. The organization and diffusion of the nanoparticles are analyzed as the nanoparticle density and the contact angle between the nanoparticles and liquid are varied. When the interaction between the nanoparticles and liquid is reduced the contact angle increases and the nanoparticles ride higher on the liquid surface, which enables them to diffuse faster. In this case the short-range order is also reduced as seen in the pair correlation function. For the polymeric liquids, the out-of-layer fluctuation is suppressed and the short-range order is slightly enhanced. However, the diffusion becomes much slower and the mean square displacement even shows sub-linear time dependence at large times. The relation between diffusion coefficient and viscosity is found to deviate from that in bulk diffusion. Results are compared to simulations of the identical nanoparticles in 2-dimensions.

This research used resources of the National Energy Research Scientific Computing Center (NERSC), which is supported by the Office of Science of the United States Department of Energy under Contract No. DE-AC02-05CH11231, and the Oak Ridge Leadership Computing Facility located in the National Center for Computational Sciences at Oak Ridge National Laboratory, which is supported by the Office of Science of the United States Department of Energy under Contract No. DE-AC05-00OR22725. These resources were obtained through the Advanced Scientific Computing Research (ASCR) Leadership Computing Challenge (ALCC). This work was supported by the Laboratory Directed Research and Development program at Sandia National Laboratories. 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 under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION

II. SIMULATION METHODOLOGY

III. RESULTS AND DISCUSSION

A. Out-of-plane fluctuations

B. In-plane structure

C. Orientational order

D. Diffusion coefficient

IV. CONCLUSIONS

### Key Topics

- Nanoparticles
- 148.0
- Polymers
- 37.0
- Gas liquid interfaces
- 24.0
- Diffusion
- 13.0
- Interface diffusion
- 13.0

## Figures

Snapshots of nanoparticles floating at liquid/vapor interfaces: (a) monatomic LJ liquid and θ_{ c } = 137°; (b) monatomic LJ liquid and θ_{ c } = 29°; (c) 100-bead chain polymeric liquid and θ_{ c } = 93°. Only a small portion (100σ × 100σ × 100σ) of the simulation cell is shown in each snapshot.

Snapshots of nanoparticles floating at liquid/vapor interfaces: (a) monatomic LJ liquid and θ_{ c } = 137°; (b) monatomic LJ liquid and θ_{ c } = 29°; (c) 100-bead chain polymeric liquid and θ_{ c } = 93°. Only a small portion (100σ × 100σ × 100σ) of the simulation cell is shown in each snapshot.

Contact angle θ_{ c } vs. nanoparticle-liquid interaction strength *A* _{ns} for different liquids: monatomic LJ liquid (circles), 10-bead chain polymeric liquid (triangles), and 100-bead chain polymeric liquid (squares). Uncertainties in θ_{ c } are ∼1° − 2°, smaller than the symbol size. Lines are guides to the eye.

Contact angle θ_{ c } vs. nanoparticle-liquid interaction strength *A* _{ns} for different liquids: monatomic LJ liquid (circles), 10-bead chain polymeric liquid (triangles), and 100-bead chain polymeric liquid (squares). Uncertainties in θ_{ c } are ∼1° − 2°, smaller than the symbol size. Lines are guides to the eye.

Probability density distribution *P*(δ*z* _{ p }) of nanoparticle positions in the *z*-direction, where δ*z* _{ p } is the deviation of the position of a nanoparticle from the instantaneous mean of all nanoparticles. Data are for ϕ = 0.54, θ_{ c } = 93°, and various liquids: monatomic LJ liquid (circles), 10-bead chain polymeric liquid (triangles), and 100-bead chain polymeric liquid (squares). Lines are the corresponding Gaussian fits.

Probability density distribution *P*(δ*z* _{ p }) of nanoparticle positions in the *z*-direction, where δ*z* _{ p } is the deviation of the position of a nanoparticle from the instantaneous mean of all nanoparticles. Data are for ϕ = 0.54, θ_{ c } = 93°, and various liquids: monatomic LJ liquid (circles), 10-bead chain polymeric liquid (triangles), and 100-bead chain polymeric liquid (squares). Lines are the corresponding Gaussian fits.

The pair distribution function *g*(*r*). Lines in all main panels are for the monatomic LJ liquid: (a) θ_{ c } = 93° and ϕ = 0.45 (solid), 0.54 (dashed), 0.73 (dotted); (b) ϕ = 0.73 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted); (c) ϕ = 0.54 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted); (d) ϕ = 0.45 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted). Inset of (c): The dashed line is for the monatomic LJ liquid and the dashed-dotted line is for the polymeric liquid consisting of 100-bead chains; for both lines ϕ = 0.54 and θ_{ c } = 93°.

The pair distribution function *g*(*r*). Lines in all main panels are for the monatomic LJ liquid: (a) θ_{ c } = 93° and ϕ = 0.45 (solid), 0.54 (dashed), 0.73 (dotted); (b) ϕ = 0.73 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted); (c) ϕ = 0.54 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted); (d) ϕ = 0.45 and θ_{ c } = 137° (solid), 93° (dashed), 51° (dotted). Inset of (c): The dashed line is for the monatomic LJ liquid and the dashed-dotted line is for the polymeric liquid consisting of 100-bead chains; for both lines ϕ = 0.54 and θ_{ c } = 93°.

The pair distribution function *g*(*r*) for nanoparticles at the liquid/vapor interface for the LJ monatomic liquid at θ_{ c } = 93° (dashed lines) compared to results of 2D simulations (solid lines) for ϕ = 0.45 (red, with the lowest peaks), 0.54 (green, with the medium peaks), and 0.73 (blue, with the highest peaks).

The pair distribution function *g*(*r*) for nanoparticles at the liquid/vapor interface for the LJ monatomic liquid at θ_{ c } = 93° (dashed lines) compared to results of 2D simulations (solid lines) for ϕ = 0.45 (red, with the lowest peaks), 0.54 (green, with the medium peaks), and 0.73 (blue, with the highest peaks).

Density plots of *S*(**q**) in the *q* _{ x }-*q* _{ y } plane for θ_{ c } = 93° and the monatomic LJ liquid at ϕ = 0.54 (left) and 0.73 (middle), and the 100-bead chain polymeric liquid at ϕ = 0.54 (right).

Density plots of *S*(**q**) in the *q* _{ x }-*q* _{ y } plane for θ_{ c } = 93° and the monatomic LJ liquid at ϕ = 0.54 (left) and 0.73 (middle), and the 100-bead chain polymeric liquid at ϕ = 0.54 (right).

(a) The fraction of nanoparticles, *f* _{6}, having exactly six neighbors and (b) the N-H order parameter *m* _{6} vs. the nanoparticle density ϕ for nanoparticles on the surface of the monatomic LJ liquid at various contact angles: θ_{ c } = 137° (circles); θ_{ c } = 93° (triangles); θ_{ c } = 51° (squares). Results of 2D simulations are shown with diamonds. Lines are guides to the eye.

(a) The fraction of nanoparticles, *f* _{6}, having exactly six neighbors and (b) the N-H order parameter *m* _{6} vs. the nanoparticle density ϕ for nanoparticles on the surface of the monatomic LJ liquid at various contact angles: θ_{ c } = 137° (circles); θ_{ c } = 93° (triangles); θ_{ c } = 51° (squares). Results of 2D simulations are shown with diamonds. Lines are guides to the eye.

(a) Mean square displacement ⟨Δ*r* ^{2}⟩ vs. time *t*. The top three datasets are for the monatomic LJ liquid at ϕ = 0.45 and θ_{ c } = 137°; ϕ = 0.54 and θ_{ c } = 137°; ϕ = 0.54 and θ_{ c } = 93°. The second to bottom (bottommost) dataset is for the polymeric liquid consisting of 10-bead (100-bead) chains at ϕ = 0.54 and θ_{ c } = 93°.

(a) Mean square displacement ⟨Δ*r* ^{2}⟩ vs. time *t*. The top three datasets are for the monatomic LJ liquid at ϕ = 0.45 and θ_{ c } = 137°; ϕ = 0.54 and θ_{ c } = 137°; ϕ = 0.54 and θ_{ c } = 93°. The second to bottom (bottommost) dataset is for the polymeric liquid consisting of 10-bead (100-bead) chains at ϕ = 0.54 and θ_{ c } = 93°.

(a) Diffusion coefficient *D* and (b) ballistic time *t* _{ b } vs. the true nanoparticle density for nanoparticles floating on the surface of the monatomic LJ liquid at various contact angles: θ_{ c } = 137° (circles), θ_{ c } = 93° (triangles), θ_{ c } = 51° (squares), θ_{ c } = 29° (diamonds). is the same as ϕ except for θ_{ c } = 29°, where . Lines are guides to the eye.

(a) Diffusion coefficient *D* and (b) ballistic time *t* _{ b } vs. the true nanoparticle density for nanoparticles floating on the surface of the monatomic LJ liquid at various contact angles: θ_{ c } = 137° (circles), θ_{ c } = 93° (triangles), θ_{ c } = 51° (squares), θ_{ c } = 29° (diamonds). is the same as ϕ except for θ_{ c } = 29°, where . Lines are guides to the eye.

## Tables

Comparison of values of contact angle (θ_{ c }), viscosity (η), and diffusion coefficient (*D*) for nanoparticles floating on the surface of the monatomic LJ and 10-bead polymeric liquid at ϕ = 0.54.

Comparison of values of contact angle (θ_{ c }), viscosity (η), and diffusion coefficient (*D*) for nanoparticles floating on the surface of the monatomic LJ and 10-bead polymeric liquid at ϕ = 0.54.

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