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 polystyrene5 (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 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.
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).
Experimental5 (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).
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).
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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