^{1}and M. G. Guenza

^{1,a)}

### Abstract

The theory to reconstruct the atomistic-level chain diffusion from the accelerated dynamics that is measured in mesoscale simulations of the coarse-grained system, is applied here to the dynamics of *cis*-1,4-polybutadiene melts where each chain is described as a soft interacting colloidal particle. The rescaling formalism accounts for the corrections in the dynamics due to the change in entropy and the change in friction that are a consequence of the coarse-graining procedure. By including these two corrections the dynamics is rescaled to reproduce the realistic dynamics of the system described at the atomistic level. The rescaled diffusion coefficient obtained from mesoscale simulations of coarse-grained *cis*-1,4-polybutadiene melts shows good agreement with data from united atom simulations performed by Tsolou et al. [Macromolecules38, 1478 (Year: 2005)]10.1021/ma0491210. The derived monomer friction coefficient is used as an input to the theory for cooperative dynamics that describes the internal dynamics of a polymer moving in a transient regions of slow cooperative motion in a liquid of macromolecules. Theoretically predicted time correlation functions show good agreement with simulations in the whole range of length and time scales in which data are available.

We acknowledge support from the National Science Foundation (Grant No. DMR-0804145). Computational resources were provided by Trestles through the XSEDE project supported by NSF. We are grateful to V. G. Mavrantzas for the informative discussion of his paper.

I. INTRODUCTION

II. RESCALING OF THE FREE-ENERGY AND RESCALING OF THE FRICTION

III. THEORETICAL PREDICTIONS OF CENTER-OF-MASS DIFFUSION

IV. INTERNAL DYNAMICS

V. THEORETICAL PREDICTIONS USING THE FREELY ROTATING CHAIN MODEL

VI. CONCLUSIONS

### Key Topics

- Polymers
- 73.0
- Diffusion
- 49.0
- Friction
- 28.0
- Molecular dynamics
- 22.0
- Macromolecules
- 14.0

##### C08

## Figures

Dimensionless monomer friction coefficient as a function of the hard sphere diameter, based on Eq. (6) . (a) *cis*-1,4-polybutadiene samples with N = 32, 56, 128, 320 (solid, dashed, dotted-dashed, and dotted lines correspondingly) and parameters as reported in Table I and Ref. ^{ 15 } . (b) Polyethylene samples with N = 30, 44, 96, 270 (solid, dashed, dotted-dashed, and dotted lines correspondingly) and parameters as reported in Refs. ^{ 5 } and ^{ 6 } . Horizontal lines represent 1/*N* values, following the diffusion coefficient for unentangled chains, *D*βζ_{ m } = 1/*N*.

Dimensionless monomer friction coefficient as a function of the hard sphere diameter, based on Eq. (6) . (a) *cis*-1,4-polybutadiene samples with N = 32, 56, 128, 320 (solid, dashed, dotted-dashed, and dotted lines correspondingly) and parameters as reported in Table I and Ref. ^{ 15 } . (b) Polyethylene samples with N = 30, 44, 96, 270 (solid, dashed, dotted-dashed, and dotted lines correspondingly) and parameters as reported in Refs. ^{ 5 } and ^{ 6 } . Horizontal lines represent 1/*N* values, following the diffusion coefficient for unentangled chains, *D*βζ_{ m } = 1/*N*.

Diffusion coefficient predicted from the rescaled MS MD, when different values of the hard-sphere diameter are chosen as an input to the reconstruction procedure. Assuming different values of *d*, calculated by enforcing Rouse diffusive behavior for N = 56 (d = 1.4672), 64 (d = 1.4051), or 80 (d = 1.374) (circles, diamond, triangles correspondingly) leads to diffusion coefficients in good agreement with the UA MD simulation data (filled squares).

Diffusion coefficient predicted from the rescaled MS MD, when different values of the hard-sphere diameter are chosen as an input to the reconstruction procedure. Assuming different values of *d*, calculated by enforcing Rouse diffusive behavior for N = 56 (d = 1.4672), 64 (d = 1.4051), or 80 (d = 1.374) (circles, diamond, triangles correspondingly) leads to diffusion coefficients in good agreement with the UA MD simulation data (filled squares).

Center of mass self-diffusion coefficient as a function of degree of polymerization, N, for *cis*-1,4-polybutadiene melts with parameters defined in Table I . Diffusion coefficients reconstructed from MS MD by applying our procedure (open symbol) are compared against UA MD data (filled symbol) from Ref. ^{ 15 } . In analogy with the figure from the source, three scaling regimes in terms of power law dependence of *D* ∝ *N* ^{ b } are shown as dashed (*b* > 1), solid (*b* ≈ 1), and dotted-dotted-dashed (*b* ≈ 2) lines.

Center of mass self-diffusion coefficient as a function of degree of polymerization, N, for *cis*-1,4-polybutadiene melts with parameters defined in Table I . Diffusion coefficients reconstructed from MS MD by applying our procedure (open symbol) are compared against UA MD data (filled symbol) from Ref. ^{ 15 } . In analogy with the figure from the source, three scaling regimes in terms of power law dependence of *D* ∝ *N* ^{ b } are shown as dashed (*b* > 1), solid (*b* ≈ 1), and dotted-dotted-dashed (*b* ≈ 2) lines.

Density as a function of degree of polymerization, N, for *cis*-1,4-polybutadiene samples reported in Table I .

Density as a function of degree of polymerization, N, for *cis*-1,4-polybutadiene samples reported in Table I .

End-to-end vector time decorrelation function for *cis*-1,4-polybutadiene with N = 112. Data from UA MD simulations (symbols) are compared with predictions of the theory for cooperative dynamics (solid line) where the number of correlated chains is set to *n* = 7 and the monomer friction coefficient is reconstructed from MS MD simulations, using the procedure described in this paper.

End-to-end vector time decorrelation function for *cis*-1,4-polybutadiene with N = 112. Data from UA MD simulations (symbols) are compared with predictions of the theory for cooperative dynamics (solid line) where the number of correlated chains is set to *n* = 7 and the monomer friction coefficient is reconstructed from MS MD simulations, using the procedure described in this paper.

Center of mass mean square displacement as a function of time for *cis*-1,4-polybutadiene samples with N = 240 (circle), 320 (square), and 400 (triangle). Predictions of the theory for cooperative dynamics (solid lines) are compared against the UA MD simulations (symbols). Also shown are the purely diffusive slopes obtained from the rescaled MS MD simulations (dashed lines). Inset illustrates how the uncertainty in the radius-of-gyration affects the mean-square-displacement for the N = 240 sample: the upper and lower values, reported as dot-dashed lines, corresponds to the upper and lower values of *R* _{ g } as reported in Table I .

Center of mass mean square displacement as a function of time for *cis*-1,4-polybutadiene samples with N = 240 (circle), 320 (square), and 400 (triangle). Predictions of the theory for cooperative dynamics (solid lines) are compared against the UA MD simulations (symbols). Also shown are the purely diffusive slopes obtained from the rescaled MS MD simulations (dashed lines). Inset illustrates how the uncertainty in the radius-of-gyration affects the mean-square-displacement for the N = 240 sample: the upper and lower values, reported as dot-dashed lines, corresponds to the upper and lower values of *R* _{ g } as reported in Table I .

Normalized dynamic structure factor for *cis*-1,4-polybutadiene with N = 96 and q = 0.04 (squares), 0.1 (circles), 0.2 (triangles), 0.3 (diamonds) Å^{−1}. The data from UA MD simulations (symbols) are compared against the cooperative dynamics theory (solid lines) where the number of correlated chains is set to *n* = 15 and the monomer friction coefficient is reconstructed from MS MD simulations, using the procedure described in this paper.

Normalized dynamic structure factor for *cis*-1,4-polybutadiene with N = 96 and q = 0.04 (squares), 0.1 (circles), 0.2 (triangles), 0.3 (diamonds) Å^{−1}. The data from UA MD simulations (symbols) are compared against the cooperative dynamics theory (solid lines) where the number of correlated chains is set to *n* = 15 and the monomer friction coefficient is reconstructed from MS MD simulations, using the procedure described in this paper.

Normalized dynamic structure factor for *cis*-1,4-polybutadiene with N = 400 and q = 0.04 (squares), 0.1 (circles), 0.2 (triangles), 0.3 (diamonds) Å^{−1}. The data from UA MD simulations (symbols) are compared against the theory for cooperative dynamics (solid lines) where the number of correlated chains is set to *n* = 12 and the monomer friction coefficient is reconstructed from the MS MD simulations, using the procedure described in this paper.

Normalized dynamic structure factor for *cis*-1,4-polybutadiene with N = 400 and q = 0.04 (squares), 0.1 (circles), 0.2 (triangles), 0.3 (diamonds) Å^{−1}. The data from UA MD simulations (symbols) are compared against the theory for cooperative dynamics (solid lines) where the number of correlated chains is set to *n* = 12 and the monomer friction coefficient is reconstructed from the MS MD simulations, using the procedure described in this paper.

Monomer mean square displacement, averaged over the innermost chain segments, as a function of time for *cis*-1,4-polybutadiene with N = 112. Predictions from the theory of cooperative dynamics (solid line) are compared against UA MD simulations (circles). The slope obtained from the rescaled MS MD simulations is shown as well (dotted-dashed line).

Monomer mean square displacement, averaged over the innermost chain segments, as a function of time for *cis*-1,4-polybutadiene with N = 112. Predictions from the theory of cooperative dynamics (solid line) are compared against UA MD simulations (circles). The slope obtained from the rescaled MS MD simulations is shown as well (dotted-dashed line).

Density dependence of the semiflexibility parameter *g* calculated from Eq. (19) using values of the radius-of-gyration *R* _{ g } measured in UA MD and reported in Table III .

Radius of gyration squared over degree of polymerization as a function of *N*. UA MD data (circles), with statistical error, are compared against data calculated with the Freely Rotating Chain model using an averaged semiflexibility parameter, *g* = 0.6564 (solid line).

Radius of gyration squared over degree of polymerization as a function of *N*. UA MD data (circles), with statistical error, are compared against data calculated with the Freely Rotating Chain model using an averaged semiflexibility parameter, *g* = 0.6564 (solid line).

Diffusion coefficients reconstructed from MS MD simulations using the radius-of-gyration calculated with the freely rotating chain model (open circles) are compared against UA MD data (filled squares). Predictions for new systems with N =180, 280, 360, and 440 are shown as well (open diamonds).

Diffusion coefficients reconstructed from MS MD simulations using the radius-of-gyration calculated with the freely rotating chain model (open circles) are compared against UA MD data (filled squares). Predictions for new systems with N =180, 280, 360, and 440 are shown as well (open diamonds).

## Tables

Simulation parameters for 1,4-*cis*-PB chains of increasing lengths.

Simulation parameters for 1,4-*cis*-PB chains of increasing lengths.

Diffusion coefficient reconstructed from MS MD simulation compared against UA MD simulations.

Diffusion coefficient reconstructed from MS MD simulation compared against UA MD simulations.

Semiflexibility parameter *g* calculated from FRC expression.

Semiflexibility parameter *g* calculated from FRC expression.

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