^{1,a)}, Gerhard Gompper

^{1,a)}and Roland G. Winkler

^{1,a)}

### Abstract

We analyze the effect of time-dependent hydrodynamic interactions on the dynamics of flexible polymers in dilute solution. In analytical calculations, the fluctuating hydrodynamics approach is adopted to describe the fluid, and a Gaussian model to represented the polymer. Simulations are performed exploiting the multiparticle collision dynamics approach, a mesoscale hydrodynamic simulation technique, to explicitly describe the fluid. Polymer center-of-mass velocity correlation functions are calculated for various polymer lengths. Similarly, segment mean square displacements are discussed and polymer diffusion coefficients are determined. Particular attention is paid to the influence of sound propagation on the various properties. The simulations reveal a strong effect of hydrodynamic interactions. Specifically, the time dependence of the center-of-mass velocity correlation functions is determined by polymer properties over a length-dependent time window, but are asymptotically solely governed by fluid correlations, with a long-time tail decaying as *t* ^{−3/2}. The correlation functions are heavily influenced by sound modes for short polymers, an effect which gradually disappears with increasing polymer length. We find excellent agreement between analytical and simulation results. This allows us to provide a theory-based asymptotic value for the polymer diffusion coefficient in the limit of large system sizes, which is based on a single finite-system-size simulation.

Financial support by the German Research Foundation (DFG) within SFB TR6 and the European Union through FP7-Infrastructures ESMI (Grant No. 262348) is gratefully acknowledged. We are grateful to the Jülich Supercomputer Centre (JSC) for allocation of a CPU-time grant.

I. INTRODUCTION

II. FLUCTUATING HYDRODYNAMICS

III. POLYMERDYNAMICS

A. Model

B. Center-of-mass velocity correlation function

1. Transverse center-of-mass velocity correlation function

2. Longitudinal center-of-mass velocity correlation function

C. Diffusion coefficient

D. Mean square displacement

E. Discussion

IV. SIMULATION MODELS

A. Polymer model

B. MPC fluid

C. Simulation parameter

V. SIMULATION RESULTS

A. Center-of-mass velocity-autocorrelation function

B. Diffusion coefficient

C. Mean square displacement

1. Gaussian polymer

2. Self-avoiding polymer

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Polymers
- 145.0
- Correlation functions
- 81.0
- Hydrodynamics
- 31.0
- Rheology and fluid dynamics
- 28.0
- Diffusion
- 21.0

## Figures

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, 10^{2}, 10^{3}, 10^{4}, and 10^{5} (top to bottom).

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, 10^{2}, 10^{3}, 10^{4}, and 10^{5} (top to bottom).

Analytical longitudinal center-of-mass velocity correlation functions for the polymer lengths *L* _{ p }/*l* = 10, 10^{3}, and 10^{5} (top to bottom). The solid lines represent the full expression (41) and the dashed lines the approximation (44) .

Analytical center-of-mass velocity correlation functions of the polymer of length *L* _{ p }/*l* = 10^{3} 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.

Analytical center-of-mass velocity correlation functions of the polymer of length *L* _{ p }/*l* = 10^{3} 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) ].

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.

(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 μ*D*(*t*) is scaled by the kinematic viscosity and the diffusion coefficient for the collision time step .

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 μ*D*(*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) 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) 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).

(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).

## Tables

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 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.

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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