^{1}and Kenneth S. Schweizer

^{2,3,a)}

### Abstract

We propose a statistical dynamical theory for the violation of the hydrodynamic Stokes-Einstein (SE) diffusion law for a spherical nanoparticle in entangled and unentangled polymer melts based on a combination of mode coupling, Brownian motion, and polymer physics ideas. The non-hydrodynamic friction coefficient is related to microscopic equilibrium structure and the length-scale-dependent polymer melt collective density fluctuation relaxation time. When local packing correlations are neglected, analytic scaling laws (with numerical prefactors) in various regimes are derived for the non-hydrodynamic diffusivity as a function of particle size, polymer radius-of-gyration, tube diameter, degree of entanglement, melt density, and temperature. Entanglement effects are the origin of large SE violations (orders of magnitude mobility enhancement) which smoothly increase as the ratio of particle radius to tube diameter decreases. Various crossover conditions for the recovery of the SE law are derived, which are qualitatively distinct for unentangled and entangled melts. The dynamical influence of packing correlations due to both repulsive and interfacial attractive forces is investigated. A central finding is that melt packing fraction, temperature, and interfacial attraction strength all influence the SE violation in qualitatively different directions depending on whether the polymers are entangled or not. Entangled systems exhibit seemingly anomalous trends as a function of these variables as a consequence of the non-diffusive nature of collective density fluctuation relaxation and the different response of polymer-particle structural correlations to adsorption on the mesoscopic entanglement length scale. The theory is in surprisingly good agreement with recent melt experiments, and new parametric studies are suggested.

This work was supported by the Department of Energy Basic Energy Sciences via Oak Ridge National Laboratory. We thank Michael Mackay for discussion of his experimental results, and Venkat Ganesan for alerting us to Ref. 16.

I. INTRODUCTION

II. MODEL AND THEORY

A. Model

B. Dynamical theory

III. ANALYTIC ANALYSIS OF THE ATHERMAL CONTINUUM STRUCTURE MODEL

A. General results

B. Unentangled melts

C. Entangled melts

D. Unentangled vs. entangled friction crossover

IV. ATHERMAL LIMIT: MODEL CALCULATIONS

A. Continuum structure and entangled systems

B. Role of local packing structure

V. COMPARISON TO EXPERIMENTS

A. Theory versus measurement

B. Caveats and suggestions for new experiments

VI. ROLE OF INTERFACIAL ATTRACTION

VII. DISCUSSION

### Key Topics

- Polymers
- 67.0
- Nanoparticles
- 56.0
- Friction
- 37.0
- Polymer melts
- 31.0
- Polymer structure
- 23.0

##### B82B1/00

## Figures

An example of the amplitude of the unentangled friction contribution (solid, left axis) and the entangled friction contribution (dashed, right axis) as a function of wavevector for parameters relevant to nanoparticles in polystyrene^{5} (see text), a continuum athermal liquid structure, and *R* = 3 nm. (Inset) The *k*-space entanglement friction contribution defined in Eq. (16b). Here, *S* _{0} = 0.25, and calculations for a tube diameter (in nm) of 9 (dashed-dotted), 8 (dotted), 7 (dashed), and 6 (solid) are shown.

An example of the amplitude of the unentangled friction contribution (solid, left axis) and the entangled friction contribution (dashed, right axis) as a function of wavevector for parameters relevant to nanoparticles in polystyrene^{5} (see text), a continuum athermal liquid structure, and *R* = 3 nm. (Inset) The *k*-space entanglement friction contribution defined in Eq. (16b). Here, *S* _{0} = 0.25, and calculations for a tube diameter (in nm) of 9 (dashed-dotted), 8 (dotted), 7 (dashed), and 6 (solid) are shown.

Stokes-Einstein violation ratio as a function of degree of entanglement as computed from Eq. (22) for (top to bottom): *R*/*d* _{ T } = 0.25 (solid), 0.5 (dashed), 1 (dashed-dotted), 4 (dashed-dotted-dotted), 16 (short dashed), 64 (short dashed-dotted), at fixed *S* _{0} = 0.25. The values of *R*/*d* _{ T } need to be interpreted in terms of *R* _{ g } in the unentangled regime (*n* ≪ 1).

Stokes-Einstein violation ratio as a function of degree of entanglement as computed from Eq. (22) for (top to bottom): *R*/*d* _{ T } = 0.25 (solid), 0.5 (dashed), 1 (dashed-dotted), 4 (dashed-dotted-dotted), 16 (short dashed), 64 (short dashed-dotted), at fixed *S* _{0} = 0.25. The values of *R*/*d* _{ T } need to be interpreted in terms of *R* _{ g } in the unentangled regime (*n* ≪ 1).

Stokes-Einstein violation ratio as a function of *R*/*d* _{ T } for entangled melts with (bottom to top): *n* = 1 (solid), 2 (dashed), 4 (dashed-dotted), 8 (dashed-dotted-dotted), 16 (short dashed). *S* _{0} = 0.25 in all cases. Lines of slope –2 and –5 are indicated for reference. (Inset) Same as main frame for different values of *S* _{0} = 0.10 (solid), 0.25(dashed), and 0.40 (dashed-dotted) at fixed *n* = 16 which is representative of the limiting heavily entangled behavior.

Stokes-Einstein violation ratio as a function of *R*/*d* _{ T } for entangled melts with (bottom to top): *n* = 1 (solid), 2 (dashed), 4 (dashed-dotted), 8 (dashed-dotted-dotted), 16 (short dashed). *S* _{0} = 0.25 in all cases. Lines of slope –2 and –5 are indicated for reference. (Inset) Same as main frame for different values of *S* _{0} = 0.10 (solid), 0.25(dashed), and 0.40 (dashed-dotted) at fixed *n* = 16 which is representative of the limiting heavily entangled behavior.

The nanoparticle-polymer site-site pair correlation function with *R* = 3 nm and ε = 0 (solid), 1 (dashed), 2 (dotted), and 0 (dashed-dotted). The polymer melt packing fraction is η_{ t } = 0.40 for all cases except for the dashed-dotted curve for which η_{ t } = 0.35. (Inset) Fourier-transformed total correlation function, , obtained from PRISM theory for *R* = 3 nm and ɛ = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted), with η_{ t } = 0.40. The inset shows the surface excess as a function of interfacial attraction strength for *R* = 3 nm (solid) and 6 nm (dashed) at η_{ t } = 0.40.

The nanoparticle-polymer site-site pair correlation function with *R* = 3 nm and ε = 0 (solid), 1 (dashed), 2 (dotted), and 0 (dashed-dotted). The polymer melt packing fraction is η_{ t } = 0.40 for all cases except for the dashed-dotted curve for which η_{ t } = 0.35. (Inset) Fourier-transformed total correlation function, , obtained from PRISM theory for *R* = 3 nm and ɛ = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted), with η_{ t } = 0.40. The inset shows the surface excess as a function of interfacial attraction strength for *R* = 3 nm (solid) and 6 nm (dashed) at η_{ t } = 0.40.

Stokes-Einstein violation ratio calculated using PRISM structural input at η_{ t } = 0.40 (solid) and 0.35 (dashed-dotted), with *n* = 12 and *d* _{ T } = 8 nm. Analogous *D*/*D* _{SE} (dashed and dashed-dotted-dotted) calculations using the athermal continuum structure model with distance of closest approach modified from *R* to *R* + *d*/2 and *S* _{0} = are also shown. A line of slope of –3.5 is indicated which matches rather well our calculations at lower *R*, though this an apparent power law indicative of a crossover regime. (Inset) The same plot for an unentangled case (*n* = 1/4).

Stokes-Einstein violation ratio calculated using PRISM structural input at η_{ t } = 0.40 (solid) and 0.35 (dashed-dotted), with *n* = 12 and *d* _{ T } = 8 nm. Analogous *D*/*D* _{SE} (dashed and dashed-dotted-dotted) calculations using the athermal continuum structure model with distance of closest approach modified from *R* to *R* + *d*/2 and *S* _{0} = are also shown. A line of slope of –3.5 is indicated which matches rather well our calculations at lower *R*, though this an apparent power law indicative of a crossover regime. (Inset) The same plot for an unentangled case (*n* = 1/4).

Experimental^{5} (squares) and theoretical values of *D*/*D* _{SE}. In the theory, the polystyrene (PS) experimental values of *d* _{ T } = 8 nm, *n* = 12, and *S* _{0} are employed. Calculations for *R* = 3 nm (dashed) and 3.5 nm (dotted) are reasonably lower and an upper estimates of the effective radius of the CdSe nanoparticles in PS melts. (Inset) The same experimental data (squares) plotted as a function of with a linear fit; *S* _{0} is determined from *T* based on Eq. (21).

Experimental^{5} (squares) and theoretical values of *D*/*D* _{SE}. In the theory, the polystyrene (PS) experimental values of *d* _{ T } = 8 nm, *n* = 12, and *S* _{0} are employed. Calculations for *R* = 3 nm (dashed) and 3.5 nm (dotted) are reasonably lower and an upper estimates of the effective radius of the CdSe nanoparticles in PS melts. (Inset) The same experimental data (squares) plotted as a function of with a linear fit; *S* _{0} is determined from *T* based on Eq. (21).

Stokes-Einstein violation ratio as a function of polymer-particle attraction strength ε, with *R* = 3 nm,*d* _{ T } = 8 nm, and η_{ t } = 0.40 at (bottom to top): *n* = 1/4 (solid), 1 (dashed), 4 (dotted), 16 (dashed-dotted), and 100 (dashed-dotted-dotted).

Stokes-Einstein violation ratio as a function of polymer-particle attraction strength ε, with *R* = 3 nm,*d* _{ T } = 8 nm, and η_{ t } = 0.40 at (bottom to top): *n* = 1/4 (solid), 1 (dashed), 4 (dotted), 16 (dashed-dotted), and 100 (dashed-dotted-dotted).

The amplitude of the unentangled friction of Eq. (16a) for *R* = 3 nm, η_{ t } = 0.40 and (top to bottom at low k): ε = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted). (Inset) The amplitude of the entanglement friction, , for the same parameter set and labeling scheme.

The amplitude of the unentangled friction of Eq. (16a) for *R* = 3 nm, η_{ t } = 0.40 and (top to bottom at low k): ε = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted). (Inset) The amplitude of the entanglement friction, , for the same parameter set and labeling scheme.

Stokes-Einstein violation ratio as a function of attraction strength at fixed *n* = 12, *d* _{ T } = 8 nm, η_{ t } = 0.40 and various nanoparticle radii in nm (top to bottom): *R* = 2 (solid), 3 (dashed), 4 (dotted), 5 (dashed-dotted), and 6 (dashed-dotted-dotted).

Stokes-Einstein violation ratio as a function of attraction strength at fixed *n* = 12, *d* _{ T } = 8 nm, η_{ t } = 0.40 and various nanoparticle radii in nm (top to bottom): *R* = 2 (solid), 3 (dashed), 4 (dotted), 5 (dashed-dotted), and 6 (dashed-dotted-dotted).

Wavevector-resolved entanglement friction contribution of Eq. (16b) for *d* _{ T } = 8 nm and *S* _{0} = 0.25 for (top to bottom): ε = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted). Note that *k* _{cut} ≃ 0.4 nm^{−1}. (Inset) The same plot with *d* _{ T } = 3 nm.

Wavevector-resolved entanglement friction contribution of Eq. (16b) for *d* _{ T } = 8 nm and *S* _{0} = 0.25 for (top to bottom): ε = 0 (solid), 1 (dashed), 2 (dotted), 3 (dashed-dotted), 4 (dashed-dotted-dotted). Note that *k* _{cut} ≃ 0.4 nm^{−1}. (Inset) The same plot with *d* _{ T } = 3 nm.

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