^{1,a)}and Elsebeth Schröder

^{1}

### Abstract

We study the mutual interactions of simple parallel polymers within the framework of density-functional theory(DFT). As the conventional implementations of DFT do not treat the long-range dispersion [van der Waals (vdW)] interactions, we develop a systematic correction scheme for the nonlocal energy contribution of the polymer interaction at the intermediate to the asymptotic separations. We primarily focus on the three polymers, polyethylene, isotactic polypropylene, and isotactic polyvinylchloride, but the scheme presented applies also more generally to other simple polymers. From first-principle calculations we extract the geometrical and electronic structures of the polymers and the local part of their interaction energy, as well as the static electric response. The dynamic electrodynamic response is modeled on the basis of these static calculations, from which the nonlocal vdW interaction of the polymers is extracted.

This work was partly supported by the Swedish Research Council (VR), the Swedish National Graduate School in Materials Science, the EU Human potential research training network ATOMCAD under Contract No. HPRN-CT-1999-00048, and the Swedish Foundation for Strategic Research (SSF) through the consortium ATOMICS.

I. INTRODUCTION

II. DFT RESULTS AND METHOD

A. Structure determination

B. Electron density

III. THE POLYMER ELECTRODYNAMIC RESPONSE

IV. THE POLYMER-POLYMER van der Waals INTERACTION

V. NUMERICAL RESULTS AND DISCUSSION

A. Cylindrical approximation

B. Length averaging approximation

VI. COMPARISON WITH OTHER METHODS

VII. CONCLUSION

### Key Topics

- Polymers
- 80.0
- Density functional theory
- 24.0
- Polarizability
- 8.0
- Polarization
- 6.0
- Electric fields
- 5.0

## Figures

The three polymers shown from the top and from the side. The hydrogen atoms are not shown in order to make the helical structure more apparent. the gray spheres are carbon atoms and the dark gray spheres are chlorine atoms. The bottom figures show the corresponding contour plots of the valence electron density averaged along the chain. The contour lines are equally spaced by .

The three polymers shown from the top and from the side. The hydrogen atoms are not shown in order to make the helical structure more apparent. the gray spheres are carbon atoms and the dark gray spheres are chlorine atoms. The bottom figures show the corresponding contour plots of the valence electron density averaged along the chain. The contour lines are equally spaced by .

The calculated static susceptibility of a segment of PE for different angles . The orientation corresponds to the situation shown in Fig. 1 when the electric field is applied in the direction. The diamonds are values obtained from DFT calculations and the full line is obtained by Eq. (6). The dashed line is found using the plasmon-pole approximation and solving the electrodynamic equations with finite element method (FEM) techniques. The inset shows the graph in full scale.

The calculated static susceptibility of a segment of PE for different angles . The orientation corresponds to the situation shown in Fig. 1 when the electric field is applied in the direction. The diamonds are values obtained from DFT calculations and the full line is obtained by Eq. (6). The dashed line is found using the plasmon-pole approximation and solving the electrodynamic equations with finite element method (FEM) techniques. The inset shows the graph in full scale.

The orientation-dependent expansion coefficients and for PE, shown as equidistant contours. is the rotation angle of polymer about its center of mass. The initial positions of the polymers are indicated by the contours of the averaged charge densities in Fig. 1. The panels show (a) and (b) , respectively.

The orientation-dependent expansion coefficients and for PE, shown as equidistant contours. is the rotation angle of polymer about its center of mass. The initial positions of the polymers are indicated by the contours of the averaged charge densities in Fig. 1. The panels show (a) and (b) , respectively.

The PE-PE van der Waals interaction energy as a function of separation for the two orientations schematically shown in the inset, corresponding to the minimum and maximum interactions, respectively. The full lines are the results of the 2D numerical interaction integral, while the dotted lines show the results for the seventh order expansion in .

The PE-PE van der Waals interaction energy as a function of separation for the two orientations schematically shown in the inset, corresponding to the minimum and maximum interactions, respectively. The full lines are the results of the 2D numerical interaction integral, while the dotted lines show the results for the seventh order expansion in .

The PP-PP van der Waals interaction energy as a function of separation in different approximations. The full line corresponds to the evaluation of the 2D-interaction integral while the dotted line shows the expansion of the energy up to the seventh order in inverse distance. The inset shows the PE-PE, PP-PP, and PVC-PVC van der Waals interaction energies as a function of separation . All the polymers are oriented as shown by the density contours in Fig. 1 and separated in the direction.

The PP-PP van der Waals interaction energy as a function of separation in different approximations. The full line corresponds to the evaluation of the 2D-interaction integral while the dotted line shows the expansion of the energy up to the seventh order in inverse distance. The inset shows the PE-PE, PP-PP, and PVC-PVC van der Waals interaction energies as a function of separation . All the polymers are oriented as shown by the density contours in Fig. 1 and separated in the direction.

## Tables

Geometric data characterizing the polymers: the repetition length (unit-cell length), the average separation of atoms and , and the angle formed by the carbon atoms in the main chain. denotes a carbon atom with an attached subgroup or Cl, stands for or Cl.

Geometric data characterizing the polymers: the repetition length (unit-cell length), the average separation of atoms and , and the angle formed by the carbon atoms in the main chain. denotes a carbon atom with an attached subgroup or Cl, stands for or Cl.

Comparison between the expansion coefficients for the full 2D evaluation and the expansion coefficients where cylindrical symmetry is enforced.

Comparison between the expansion coefficients for the full 2D evaluation and the expansion coefficients where cylindrical symmetry is enforced.

Comparison of the coefficients for the PE-PE interaction obtained by other methods with those obtained by our method. Underlined quantities are estimated using Eq. (10).

Comparison of the coefficients for the PE-PE interaction obtained by other methods with those obtained by our method. Underlined quantities are estimated using Eq. (10).

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