Analytical center-of-mass velocity correlation functions . (a) Transverse [Eq. (36) ] (solid lines) and magnitudes of longitudinal [Eq. (41) ] (dashed lines) correlation functions. The dashed-dotted lines represent negative parts. (b) Total correlation functions (solid lines) and the contributions of the transverse parts (dashed lines). The magenta line indicates the fluid long-time tail according to Eq. (38) . The polymer lengths are L p /l = 10, 102, 103, 104, and 105 (top to bottom).
Analytical center-of-mass velocity correlation functions of the polymer of length L p /l = 103 and the collision time steps (red), 0.01 (blue), 0.03 (green), and 0.1 (black). These values correspond to the kinematic viscosities , 8.2, 2.8, and 0.9.
Polymer center-of-mass velocity autocorrelation functions. (a) The polymer length is N m = 160 and the collision time step ( ). The inset shows the data in semilogarithmic representation. (b) The polymer length is N m = 80 and ( ). The negative parts of C v are shown by dashed lines. The simulation results are displayed by red lines, the analytical results (33) by black lines, and the transverse contributions by green lines. The blue line in (a) indicates the correlation function of MPC particles. 65 The maximum mode numbers are (a) n m = 33 and (b) n m = 25 [cf. Eq. (60) ].
(a) Simulation results for polymer center-of-mass velocity autocorrelation functions of Gaussian polymers of lengths N m = 40, 80, 160, 320, 640, and 1280 (top to bottom), and (b) self-avoiding polymers of lengths N m = 40, 80, 160, 320, and 640 (top to bottom). The black lines correspond to the analytical approximation (33) with the maximum mode numbers (a) n m = 15, 25, 33, 50, and 40 for the two longer polymers, and (b) n m = 27, 43, 57, and 50 for the longer ones, respectively. The straight lines indicate the long-time tail, and the magenta lines, for the longest polymers, the correlation functions for infinite systems.
Time integrals of the center-of-mass velocity autocorrelation functions (61) of Gaussian polymers of length N m = 80 for the collision time steps step (blue), 0.05 (green), 0.02 (red), and 0.004 (black) (bottom to top). The corresponding kinematic viscosities are , 1.67, 4.12, and 20.54, respectively. The product μD(t) is scaled by the kinematic viscosity and the diffusion coefficient for the collision time step .
(a) Center-of-mass velocity correlation functions of polymers of length N m = 80. The lengths of the simulation boxes are L/a = 40 (green) and 140 (blue). At short times the two correlations are indistinguishable. The black and light blue lines are the corresponding theoretical results. The dashed line is the infinite system limit. (b) Integrated correlation functions. The same color code is applied as in (a). The black line follows as integral over the correlation function of the simulations up to and the theoretical correlation C v (t) beyond that time. The asymptotic value of the diffusion coefficient of the infinite system is .
(a) Means quare displacements of monomers (solid lines) and of polymer centers-of-mass (dashed-dotted lines) for Gaussian polymers. (b) Monomer MSDs in the center-of-mass reference frame . (c) Local slopes [Eq. (66) ] of the MSDs of (a) and (b): ζ m (t) (squares), ζ cm (diamonds), and ζ t (bullets). The polymer lengths are N m = 80 (red), 160 (blue), 320 (purple), 640 (light-blue), 1280 (with ) (orange), and 1280 (with ) (black). The dark-green curves are theoretical results following from Eqs. (50)–(52) . Inset in (b): Polymer-length dependence of the relaxation times. The solid line shows the power-law .
(a) Mean square displacements of self-avoiding polymers: Monomer MSDs in the center-of-mass reference frame (solid lines), total monomer MSDs (dashed lines), and center-of mass MSDs (dashed-dotted lines). The polymer lengths are N m = 40 (green), 80 (red), 160 (blue), 320 (purple), and 640 (light-blue). The dark-green curves are theoretical results following from Eqs. (50)–(52) . (Inset) Polymer-length dependence of the relaxation times. The solid line shows the power-law , with ν = 0.6. (b) Local slopes [Eq. (66) ] of the MSDs of (a): ζ m (t) (squares), ζ cm (diamonds), and ζ t (bullets).
Simulation parameters and results for Gaussian phantom chains. N m denotes the number of monomers, L is the length of the simulation box, R g is the radius of gyration, and τ r is the end-to-end vector relaxation time.
Simulation parameters and results for self-avoiding polymers. N m denotes the number of monomers, L is the length of the simulation box, R g is the radius of gyration, and τ r is the end-to-end vector relaxation time.
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