^{1,a)}and Kostas Ch. Daoulas

^{1}

### Abstract

We investigate the ability of Monte-Carlo algorithms to describe the single-chain dynamics in a dense homogeneous melt and a lamellar phase of a symmetric diblock copolymer. A minimal, coarse-grained model is employed that describes connectivity of effective segments by harmonic springs and where segments interact via soft potentials, which do not enforce noncrossability of the chain molecules. Studying the mean-square displacements, the dynamic structure factor, and the stress relaxation, we show that local, unconstraint displacements of segments via a Smart Monte Carlo algorithm give rise to Rouse dynamics for all but the first Monte Carlo steps. Using the slithering-snake algorithm, we observe a dynamics that is compatible with the predictions of the tube model of entangled melts for long times, but the dynamics inside the tube cannot be resolved. Using a slip-link model, we can describe the effect of entanglements and follow the different regimes of the single-chain dynamics over seven decades in time. Applications of this simulation scheme to spatially inhomogeneous systems are illustrated by studying the lamellar phase of a symmetric diblock copolymer. For the local, unconstraint dynamics, the single-chain motions parallel and perpendicular to the interfaces decouples; the perpendicular dynamics is slowed down but the parallel dynamics is identical to that in a homogeneous melt. Both the slithering-snake dynamics and the slip-link dynamics give rise to a coupling of parallel and perpendicular directions and a significant slowing down of the dynamics in the lamellar phase.

It is a great pleasure to thank A. E. Likhtman and J. D. Schieber for helpful discussions. We gratefully acknowledge funding by the VW-foundation and the DFG Priority Programme, Polymer solid contacts: interfaces and interphases, under Grant No. Mu 1674/9. Ample computing time that was provided by the GWDG Göttingen, the HLRN Hannover and the JSC Jülich.

I. INTRODUCTION

II. MODEL AND TECHNIQUES

A. Coarse-grained model

B. Single-chain-in-mean-field simulations

C. Smart Monte Carlo algorithm for local, unconstraint dynamics

D. Reptation dynamics

1. Slithering-snake dynamics

2. Slip-link model

III. RESULTS

A. Single-chain dynamics in the homogeneous system

B. Stress relaxation in the homogeneous system

C. Single-chain dynamics in the lamellar phase

IV. CONCLUSION AND OUTLOOK

### Key Topics

- Polymers
- 26.0
- Brownian dynamics
- 21.0
- Stress relaxation
- 16.0
- Conformational dynamics
- 15.0
- Friction
- 15.0

## Figures

Mean-square end-to-end distance, , in units of , and radius of gyration, , in units of as a function of time increment, , in Brownian dynamics simulations and SMC simulations. , , and . The data are compared to the exact result (horizontal line) obtained without quasi-instantaneous field approximation. The gray vertical lines indicate the time step used in the previous and present studies.

Mean-square end-to-end distance, , in units of , and radius of gyration, , in units of as a function of time increment, , in Brownian dynamics simulations and SMC simulations. , , and . The data are compared to the exact result (horizontal line) obtained without quasi-instantaneous field approximation. The gray vertical lines indicate the time step used in the previous and present studies.

(a) Mean-square center-of-mass displacement, , per SMC step as a function of the time increment for various chain discretizations, , 32, 64, and 128 as indicated in the key (at fixed polymer number density, ). The dashed line according to Eq. (9) depicts the Brownian dynamics behavior. The inset presents the acceptance ratio of SMC moves as a function of the time step for the smallest and the largest chain length investigated. The gray vertical lines indicate the time step used in the previous (Ref. 41) and present studies. (b) Ratio of the mean-square displacement of the chain ends, , and all segments, , vs time measured in units of . Lines show the results of the Rouse model according to Eq. (13) for . Symbols show results of the local, unconstraint dynamics. Different data sets correspond to SMC simulations with different values of as given in the key. The inset depicts as a function of . Lines correspond to the prediction of the Rouse model while symbols correspond to SMC simulations. The arrow marks the time scale .

(a) Mean-square center-of-mass displacement, , per SMC step as a function of the time increment for various chain discretizations, , 32, 64, and 128 as indicated in the key (at fixed polymer number density, ). The dashed line according to Eq. (9) depicts the Brownian dynamics behavior. The inset presents the acceptance ratio of SMC moves as a function of the time step for the smallest and the largest chain length investigated. The gray vertical lines indicate the time step used in the previous (Ref. 41) and present studies. (b) Ratio of the mean-square displacement of the chain ends, , and all segments, , vs time measured in units of . Lines show the results of the Rouse model according to Eq. (13) for . Symbols show results of the local, unconstraint dynamics. Different data sets correspond to SMC simulations with different values of as given in the key. The inset depicts as a function of . Lines correspond to the prediction of the Rouse model while symbols correspond to SMC simulations. The arrow marks the time scale .

Sketch of the slip-link model for a diblock copolymer with chain discretization . Only one diblock copolymer out of 1444 is shown. The -block is depicted red, the -block is shown in yellow. slip links are attached to the backbone and they are represented by black lines. The anchor points, , are indicated blue caps of the black cylinders.

Sketch of the slip-link model for a diblock copolymer with chain discretization . Only one diblock copolymer out of 1444 is shown. The -block is depicted red, the -block is shown in yellow. slip links are attached to the backbone and they are represented by black lines. The anchor points, , are indicated blue caps of the black cylinders.

Mean-square displacements of segments, , and mean-square displacements of the center of mass, , as a function of time in the homogeneous melt . Time is measured in units of the longest single-chain relaxation time, . The graph represents the data for local, unconstraint dynamics generated by SMC moves, slithering-snake, and slip-link dynamics. Data for local, unconstraint dynamics (lines) and slithering-snake dynamics (circles) are hardly distinguishable in this representation. The predictions of the Rouse model (Ref. 18) and the tube model (Ref. 17) are indicated by gray lines. The mean-square displacements, of segments exhibit a -behavior at short times and a diffusive behavior at long times. The data for the slip-link model show a sequence of power laws according to Eqs. (21) and (22). The -behavior of at intermediate times is indicated in the graph. The inset depicts the local, unconstraint dynamics for different chain discretizations, , at fixed . The left inset presents mean-square displacements of the local, unconstraint dynamics for different chain discretizations, , while the right inset presents the simulation data for the slithering-snake dynamics for .

Mean-square displacements of segments, , and mean-square displacements of the center of mass, , as a function of time in the homogeneous melt . Time is measured in units of the longest single-chain relaxation time, . The graph represents the data for local, unconstraint dynamics generated by SMC moves, slithering-snake, and slip-link dynamics. Data for local, unconstraint dynamics (lines) and slithering-snake dynamics (circles) are hardly distinguishable in this representation. The predictions of the Rouse model (Ref. 18) and the tube model (Ref. 17) are indicated by gray lines. The mean-square displacements, of segments exhibit a -behavior at short times and a diffusive behavior at long times. The data for the slip-link model show a sequence of power laws according to Eqs. (21) and (22). The -behavior of at intermediate times is indicated in the graph. The inset depicts the local, unconstraint dynamics for different chain discretizations, , at fixed . The left inset presents mean-square displacements of the local, unconstraint dynamics for different chain discretizations, , while the right inset presents the simulation data for the slithering-snake dynamics for .

Comparison of the mean-square displacements in the simulations to the predictions of Rouse model (gray dashed lines) and tube model from Eqs. (21) and (22) depicted as blue solid lines. The mobility and the entanglement length are identified by matching the center-of-mass diffusion coefficient to the predictions. The symbols mark the simulation data at MCS. Arrows on top mark the entanglement time, , the Rouse time, , and the disengagement time, using the value extracted from the self-diffusion coefficients. The inset compares the mean-square displacements of our single-chain implementation of constraint release and the model by Likhtman (Ref. 57).

Comparison of the mean-square displacements in the simulations to the predictions of Rouse model (gray dashed lines) and tube model from Eqs. (21) and (22) depicted as blue solid lines. The mobility and the entanglement length are identified by matching the center-of-mass diffusion coefficient to the predictions. The symbols mark the simulation data at MCS. Arrows on top mark the entanglement time, , the Rouse time, , and the disengagement time, using the value extracted from the self-diffusion coefficients. The inset compares the mean-square displacements of our single-chain implementation of constraint release and the model by Likhtman (Ref. 57).

Single-chain dynamic structure factor, , for (a) local, unconstraint dynamics, (b) slithering-snake dynamics, and (c) slip-link dynamics. In panel (a) the predictions of the Rouse model (25) are represented by solid lines and the simulation results by symbols. Dashed lines mark asymptotic predictions. The inset compares the measured single-chain structure factor, , for the wave vectors ,4,8,16, and 32 with the Debye function. In panels (b) and (c) the data are compared to Eqs. (26) and (27), respectively.

Single-chain dynamic structure factor, , for (a) local, unconstraint dynamics, (b) slithering-snake dynamics, and (c) slip-link dynamics. In panel (a) the predictions of the Rouse model (25) are represented by solid lines and the simulation results by symbols. Dashed lines mark asymptotic predictions. The inset compares the measured single-chain structure factor, , for the wave vectors ,4,8,16, and 32 with the Debye function. In panels (b) and (c) the data are compared to Eqs. (26) and (27), respectively.

Stress relaxation for local unconstraint, slithering-snake, and slip-link dynamics in a homogeneous melt . Circles indicate the times MCS for unconstraint dynamics. The gray line represents the stress relaxation in the unconstraint model [cf. Eq. (33)] with . The dashed line with slope marks a power-law decay according to the unconstraint model [see Eq. (32)]. The arrow on the right hand side indicates the prediction of the tube model for the plateau modulus, . The point is also marked by a cross. The inset displays the data for local unconstraint and slip-link dynamics as a function of time measured in units of MCS. Lines correspond to the stress calculated from the multichain system while symbols represent the data extracted from the single-chain stress, .

Stress relaxation for local unconstraint, slithering-snake, and slip-link dynamics in a homogeneous melt . Circles indicate the times MCS for unconstraint dynamics. The gray line represents the stress relaxation in the unconstraint model [cf. Eq. (33)] with . The dashed line with slope marks a power-law decay according to the unconstraint model [see Eq. (32)]. The arrow on the right hand side indicates the prediction of the tube model for the plateau modulus, . The point is also marked by a cross. The inset displays the data for local unconstraint and slip-link dynamics as a function of time measured in units of MCS. Lines correspond to the stress calculated from the multichain system while symbols represent the data extracted from the single-chain stress, .

Complex moduli, and , in the homogeneous melt as obtained for local unconstraint, slithering-snake, and slip-link dynamics. The gray lines correspond to the predictions of the Rouse model. The low-frequency power laws for the storage and loss moduli, and , are indicated by short solid lines. The error bar indicates the uncertainty of the slip-link data due to inaccuracy of at long times.

Complex moduli, and , in the homogeneous melt as obtained for local unconstraint, slithering-snake, and slip-link dynamics. The gray lines correspond to the predictions of the Rouse model. The low-frequency power laws for the storage and loss moduli, and , are indicated by short solid lines. The error bar indicates the uncertainty of the slip-link data due to inaccuracy of at long times.

Configuration snapshots of one chain in the lamellar phase at different times. (a) Local, unconstraint dynamics . (b) Slithering-snake dynamics . (c) Slip-link dynamics .

Configuration snapshots of one chain in the lamellar phase at different times. (a) Local, unconstraint dynamics . (b) Slithering-snake dynamics . (c) Slip-link dynamics .

Mean-square displacements of segments, , and mean-square displacements of the center of mass, , as a function of time in the lamellar phase for (a) local, unconstraint dynamics, (b) slithering-snake dynamics, and (c) slip-link dynamics. Parallel and perpendicular displacements are shown and the results of the disordered phase (, cf. Fig. 4) are depicted for comparison.

Mean-square displacements of segments, , and mean-square displacements of the center of mass, , as a function of time in the lamellar phase for (a) local, unconstraint dynamics, (b) slithering-snake dynamics, and (c) slip-link dynamics. Parallel and perpendicular displacements are shown and the results of the disordered phase (, cf. Fig. 4) are depicted for comparison.

Mean-square displacements of segments, and of the center of mass, , in a homogeneous melt as a function of chain discretization, , and polymer density, , for slip-link dynamics. The first data set corresponds to Fig. 4 with . The second data set employs the same chain discretization, , and number of slip links but the lower polymer density corresponds to . The third data set corresponds to but uses a larger chain discretization and a larger number of slip links . The insets present measured in units of . The relaxation times for the three systems, (), (), and () are , , and in units of , respectively.

Mean-square displacements of segments, and of the center of mass, , in a homogeneous melt as a function of chain discretization, , and polymer density, , for slip-link dynamics. The first data set corresponds to Fig. 4 with . The second data set employs the same chain discretization, , and number of slip links but the lower polymer density corresponds to . The third data set corresponds to but uses a larger chain discretization and a larger number of slip links . The insets present measured in units of . The relaxation times for the three systems, (), (), and () are , , and in units of , respectively.

## Tables

Largest single-chain relaxation time, , calculated from the self-diffusion coefficient, , and normalized viscosity, calculated from the time integral of the stress relaxation function for the different single-chain dynamics. Entanglement time, , Rouse time, , and disengagement time, , are listed for the appropriate dynamics.

Largest single-chain relaxation time, , calculated from the self-diffusion coefficient, , and normalized viscosity, calculated from the time integral of the stress relaxation function for the different single-chain dynamics. Entanglement time, , Rouse time, , and disengagement time, , are listed for the appropriate dynamics.

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