^{1,a)}and Gerhard Hummer

^{2,b)}

### Abstract

We calculate the pair diffusion coefficient *D*(*r*) as a function of the distance *r* between two hard sphere particles in a dense monodisperse fluid. The distance-dependent pair diffusion coefficient describes the hydrodynamic interactions between particles in a fluid that are central to theories of polymer and colloid dynamics. We determine *D*(*r*) from the propagators (Green's functions) of particle pairs obtained from molecular dynamics simulations. At distances exceeding ∼3 molecular diameters, the calculated pair diffusion coefficients are in excellent agreement with predictions from exact macroscopic hydrodynamictheory for large Brownian particles suspended in a solvent bath, as well as the Oseen approximation. However, the asymptotic 1/*r* distance dependence of *D*(*r*) associated with hydrodynamic effects emerges only after the pair distance dynamics has been followed for relatively long times, indicating non-negligible memory effects in the pair diffusion at short times. Deviations of the calculated *D*(*r*) from the hydrodynamic models at short distances *r* reflect the underlying many-body fluid structure, and are found to be correlated to differences in the local available volume. The procedure used here to determine the pair diffusion coefficients can also be used for single-particle diffusion in confinement with spherical symmetry.

We thank Dr. Attila Szabo for many helpful discussions. This research was supported by the Intramural Research Program of the National Institutes of Health (NIH), NIDDK, and utilized the high-performance computational capabilities of the Biowulf PC/Linux cluster at the National Institutes of Health, Bethesda, MD (http://biowulf.nih.gov).

I. INTRODUCTION

II. METHODS

A. Theory

B. Algorithm to determine pair diffusion coefficient

III. SIMULATIONS

IV. RESULTS

V. CONCLUDING REMARKS

### Key Topics

- Diffusion
- 54.0
- Hydrodynamics
- 25.0
- Many body problems
- 13.0
- Green's function methods
- 11.0
- Rheology and fluid dynamics
- 9.0

##### B01J13/00

## Figures

Pair diffusion coefficient *D*(*r*) for two freely diffusing Brownian particles of diameter 1 with periodic boundary conditions. Results for different grid sizes Δ*r* = 0.004 (plus), 0.032 (cross), 0.024 (star), 0.016 (square) are compared to the exact value 2*D* _{0} = 0.1 (horizontal line). The vertical solid line marks the contact distance *r* = 1. To assess artifacts from periodic boundary conditions, the vertical dashed lines mark distances *r* = *L*/2, , and , where centered spheres touch the faces, edges, and corners of the cubic simulation box, respectively.

Pair diffusion coefficient *D*(*r*) for two freely diffusing Brownian particles of diameter 1 with periodic boundary conditions. Results for different grid sizes Δ*r* = 0.004 (plus), 0.032 (cross), 0.024 (star), 0.016 (square) are compared to the exact value 2*D* _{0} = 0.1 (horizontal line). The vertical solid line marks the contact distance *r* = 1. To assess artifacts from periodic boundary conditions, the vertical dashed lines mark distances *r* = *L*/2, , and , where centered spheres touch the faces, edges, and corners of the cubic simulation box, respectively.

Green's functions *G*(*r*, *t*|*r* ^{′}, 0) from simulations (symbols) and diffusion model (lines). *G*(*r*, *t*|*r* ^{′}, 0) is shown as a function of the pair distance *r* at packing fraction ϕ = 0.325. We use an observation time of *t* = 20 to obtain diffusion model parameters, combining results for lag times Δ*t* = 1, 2, …, 20. The arrow in the top panel reflects increasing *r* ^{′} = 1, 2, …, 7. Same color scheme is used for middle and bottom panels.

Green's functions *G*(*r*, *t*|*r* ^{′}, 0) from simulations (symbols) and diffusion model (lines). *G*(*r*, *t*|*r* ^{′}, 0) is shown as a function of the pair distance *r* at packing fraction ϕ = 0.325. We use an observation time of *t* = 20 to obtain diffusion model parameters, combining results for lag times Δ*t* = 1, 2, …, 20. The arrow in the top panel reflects increasing *r* ^{′} = 1, 2, …, 7. Same color scheme is used for middle and bottom panels.

Dependence of *D*(*r*) on diffusion model parameters. Pair diffusion coefficient *D*(*r*) versus distance *r* for a hard sphere fluid at packing fraction ϕ = 0.35 obtained for different (top) grid sizes Δ*r* (with fixed observation time *t* = 20) and (bottom) observation times *t* (with fixed grid size Δ*r* = 0.1). The lag time is Δ*t* = 1 consistently.

Dependence of *D*(*r*) on diffusion model parameters. Pair diffusion coefficient *D*(*r*) versus distance *r* for a hard sphere fluid at packing fraction ϕ = 0.35 obtained for different (top) grid sizes Δ*r* (with fixed observation time *t* = 20) and (bottom) observation times *t* (with fixed grid size Δ*r* = 0.1). The lag time is Δ*t* = 1 consistently.

Pair diffusion for a hard-sphere fluid. (Top) Calculated pair diffusion coefficient *D*(*r*) versus distance *r* with increasing packing fraction ϕ (symbols). Lines are the predictions of hydrodynamic theory (see text). (Bottom) Normalized pair diffusion coefficient *D*(*r*)/2*D* _{0}, where *D* _{0} is the self-diffusivity for a given ϕ. Symbols are our calculations, the thick line is the exact hydrodynamic theory,^{25,26} and the dashed line is the Oseen approximation.

Pair diffusion for a hard-sphere fluid. (Top) Calculated pair diffusion coefficient *D*(*r*) versus distance *r* with increasing packing fraction ϕ (symbols). Lines are the predictions of hydrodynamic theory (see text). (Bottom) Normalized pair diffusion coefficient *D*(*r*)/2*D* _{0}, where *D* _{0} is the self-diffusivity for a given ϕ. Symbols are our calculations, the thick line is the exact hydrodynamic theory,^{25,26} and the dashed line is the Oseen approximation.

Relation between fluid structure and dynamics. (Top) Pair diffusion coefficient *D*(*r*) (symbols connected by lines) and scaled pair correlation function *g* _{ a }(*r*) = *g*(*r*)/*a* (lines) versus distance *r* where *a* is an arbitrary scaling factor used to match *D*(*r*) and *g* _{ a }(*r*) at large *r*. (Bottom) *D*(*r*) as a function of the local fractional available volume *P* _{0}(*r*) (symbols; increasing packing fractions from right to left). The line is 2*D* _{0} versus *P* _{0} averaged over the entire system.

Relation between fluid structure and dynamics. (Top) Pair diffusion coefficient *D*(*r*) (symbols connected by lines) and scaled pair correlation function *g* _{ a }(*r*) = *g*(*r*)/*a* (lines) versus distance *r* where *a* is an arbitrary scaling factor used to match *D*(*r*) and *g* _{ a }(*r*) at large *r*. (Bottom) *D*(*r*) as a function of the local fractional available volume *P* _{0}(*r*) (symbols; increasing packing fractions from right to left). The line is 2*D* _{0} versus *P* _{0} averaged over the entire system.

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