^{1}and Kenneth S. Schweizer

^{2,3,a)}

### Abstract

We formulate and apply a microscopic statistical-mechanical theory for the non-hydrodynamic relative diffusion coefficient of a pair of spherical nanoparticles in entangled polymer melts based on a combination of Brownian motion, mode-coupling, and polymer physics ideas. The focus is on the mesoscopic regime where particles are larger than the entanglement spacing. The dependence of the non-hydrodynamic friction on interparticle separation, degree of entanglement, and tube diameter is systematically studied. The overall magnitude of the relative diffusivity is controlled by the ratio of the particle to tube diameter and the number of entanglements in a manner reminiscent of single-particle self-diffusion and Stokes-Einstein violations. A rich spatial separation dependence of mobility enhancement relative to the hydrodynamic behavior is predicted even for very large particles, and the asymptotic dependence is derived analytically in the small and large separation limits. Particle separations in excess of 100 nm are sometimes required to recover the hydrodynamic limit. The effects of local polymer-particle packing correlations are found to be weak, and the non-hydrodynamic effects are also small for unentangled melts.

We acknowledge financial support from Michelin-France, and stimulating and helpful discussions with Dr. Marc Couty.

I. INTRODUCTION

II. BACKGROUND: NANOCOMPOSITE MODEL AND SINGLE-PARTICLE DIFFUSIONTHEORY

A. Model

B. Self-diffusion

III. THEORY OF TWO-PARTICLE RELATIVE DIFFUSION

A. General approach and hydrodynamic limit

B. Non-hydrodynamic pair diffusiontheory

C. Limiting analytic analysis

IV. NUMERICAL RESULTS

A. Athermal limit model calculations

1. Effect of particle size

2. Effect of interparticle separation: Cross diffusivity

3. Deviations from hydrodynamics

B. Effect of molecular-level packing and interfacial adsorption

V. SUMMARY AND DISCUSSION

### Key Topics

- Polymers
- 49.0
- Nanoparticles
- 36.0
- Hydrodynamics
- 29.0
- Diffusion
- 19.0
- Polymer melts
- 17.0

##### B82B1/00

## Figures

Theoretical prediction for the Stokes-Einstein violation ratio as given by Eq. (10) using the structural continuum model with S 0 = 0.25. 12 The main frame shows results for N/N e = 1 (solid line), 2 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). (Inset) Stokes-Einstein violation ratio as a function of N/N e for (from top to bottom): 2R/d T = 1 (solid line), 2 (dashed line), 8 (short-dashed line), 32 (short-dotted line), and 256 (dashed-dotted line).

Theoretical prediction for the Stokes-Einstein violation ratio as given by Eq. (10) using the structural continuum model with S 0 = 0.25. 12 The main frame shows results for N/N e = 1 (solid line), 2 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). (Inset) Stokes-Einstein violation ratio as a function of N/N e for (from top to bottom): 2R/d T = 1 (solid line), 2 (dashed line), 8 (short-dashed line), 32 (short-dotted line), and 256 (dashed-dotted line).

Illustration of the studied two-particle system and schematic of the force correlation pathway that enters the dynamical vertex in Eq. (15) . Particle diameter (2R) and interparticle surface-to-surface separation (h) are defined by arrows.

Illustration of the studied two-particle system and schematic of the force correlation pathway that enters the dynamical vertex in Eq. (15) . Particle diameter (2R) and interparticle surface-to-surface separation (h) are defined by arrows.

Relative diffusivity normalized by the single-particle Stokes-Einstein self-diffusion coefficient as a function of h/2R based on the structural continuum model. Calculations are presented for 2R/d T = 10 and entangled melts with N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). The hydrodynamic result (solid curve) is also shown as a reference. (Inset) Same as main frame for larger particle size (2R/d T = 40).

Relative diffusivity normalized by the single-particle Stokes-Einstein self-diffusion coefficient as a function of h/2R based on the structural continuum model. Calculations are presented for 2R/d T = 10 and entangled melts with N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). The hydrodynamic result (solid curve) is also shown as a reference. (Inset) Same as main frame for larger particle size (2R/d T = 40).

Normalized non-hydrodynamic cross-diffusivity as a function of h/2R. Calculations are shown for one particle size (2R/d T = 10), the structural continuum model, and N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). Solid curve is the hydrodynamic result. (Inset) Non-hydrodynamic cross-diffusivity normalized by ΔD * ≡ ΔD (rel)(h/2R = 1/8) for N/N e = 128 and 2R/d T = 10 (solid line), 40 (dashed line), 200 (short-dashed line), and 400 (short-dotted line). A horizontal line of 1/e is shown as a guide.

Normalized non-hydrodynamic cross-diffusivity as a function of h/2R. Calculations are shown for one particle size (2R/d T = 10), the structural continuum model, and N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). Solid curve is the hydrodynamic result. (Inset) Non-hydrodynamic cross-diffusivity normalized by ΔD * ≡ ΔD (rel)(h/2R = 1/8) for N/N e = 128 and 2R/d T = 10 (solid line), 40 (dashed line), 200 (short-dashed line), and 400 (short-dotted line). A horizontal line of 1/e is shown as a guide.

Reduced relative diffusivity as defined in Eq. (30) as a function of h/d T based on the structural continuum model. Results for 2R/d T = 10 and N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line) are displayed along with the hydrodynamic result (solid). (Inset) Same as main frame for 2R/d T = 40.

Reduced relative diffusivity as defined in Eq. (30) as a function of h/d T based on the structural continuum model. Results for 2R/d T = 10 and N/N e = 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line) are displayed along with the hydrodynamic result (solid). (Inset) Same as main frame for 2R/d T = 40.

Ratio of the reduced relative diffusivity and its hydrodynamic counterpart as a function of N/N e for selected values of h (from top to bottom): h/d T = 0.25 (solid line), 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). All calculations employ 2R/d T = 10, N/N e = 16, and the structural continuum model.

Ratio of the reduced relative diffusivity and its hydrodynamic counterpart as a function of N/N e for selected values of h (from top to bottom): h/d T = 0.25 (solid line), 1 (dashed line), 4 (short-dashed line), 16 (short-dotted line), and 128 (dashed-dotted line). All calculations employ 2R/d T = 10, N/N e = 16, and the structural continuum model.

Re-plot of Fig. 6 as a function of h/d T for entangled melts where (from top to bottom): N/N e = 1 (solid line), 4 (dashed line), 16 (short-dashed line), and 128 (short-dotted line). (Inset) Ratio of the reduced diffusivity with N/N e = 128 and (from bottom to top) 2R/d T = 4 (solid line), 10 (dashed line), 40 (short-dashed line), 200 (short-dotted line), and 400 (dashed-dotted line).

Re-plot of Fig. 6 as a function of h/d T for entangled melts where (from top to bottom): N/N e = 1 (solid line), 4 (dashed line), 16 (short-dashed line), and 128 (short-dotted line). (Inset) Ratio of the reduced diffusivity with N/N e = 128 and (from bottom to top) 2R/d T = 4 (solid line), 10 (dashed line), 40 (short-dashed line), 200 (short-dotted line), and 400 (dashed-dotted line).

Reduced relative diffusivity as a function of h/d T calculated with PRISM theory structural input at ε/k B T = 0 (dashed line), 2 (short-dashed line), and 4 (short-dotted line). The structural continuum model result (solid line) is shown for comparison. Entanglement parameters are fixed to N/N e = 16 and d T /σ = 4 while particle sizes are: 2R/σ = 10 (main frame) and 12 (inset).

Reduced relative diffusivity as a function of h/d T calculated with PRISM theory structural input at ε/k B T = 0 (dashed line), 2 (short-dashed line), and 4 (short-dotted line). The structural continuum model result (solid line) is shown for comparison. Entanglement parameters are fixed to N/N e = 16 and d T /σ = 4 while particle sizes are: 2R/σ = 10 (main frame) and 12 (inset).

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