^{1}, Joanne Budzien

^{2}, Francisco Avila

^{3}, Taylor C. Dotson

^{3}, Victoria J. Aston

^{3}, John D. McCoy

^{3,a)}and Douglas B. Adolf

^{4}

### Abstract

The rotational dynamics of chemically similar systems based on freely jointed and freely rotating chains are studied. The second Legendre polynomial of vectors along chain backbones is used to investigate the rotational dynamics at different length scales. In a previous study, it was demonstrated that the additional bond-angle constraint in the freely rotating case noticeably perturbs the character of the translational relaxation away from that of the freely jointed system. Here, it is shown that differences are also apparent in the two systems’ rotational dynamics. The relaxation of the end-to-end vector is found to display a long time, single-exponential tail and a stretched exponential region at intermediate times. The stretching exponents are found to be for the freely jointed case and for the freely rotating case. For both system types, time-packing-fraction superposition is seen to hold on the end-to-end length scale. In addition, for both systems, the rotational relaxation times are shown to be proportional to the translational relaxation times, demonstrating that the Debye-Stokes-Einstein law holds. The second Legendre polynomial of the bond vector is used to probe relaxation behavior at short length scales. For the freely rotating case, the end-to-end relaxation times scale differently than the bondrelaxation times, implying that the behavior is non-Stokes-Einstein, and that time-packing-fraction superposition does not hold across length scales for this system. For the freely jointed case, end-to-endrelaxation times do scale with bondrelaxation times, and both Stokes-Einstein and time-packing-fraction-across-length-scales superposition are obeyed.

Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION

II. ANALYSIS OF THE ROTATIONAL RELAXATION FUNCTIONS

III. RESULTS

IV. DISCUSSION

V. CONCLUSIONS

### Key Topics

- Relaxation times
- 48.0
- Chemical bonds
- 8.0
- Diffusion
- 7.0
- Dielectric relaxation
- 6.0
- Permittivity
- 5.0

## Figures

Dimensionless diffusion coefficients are plotted in a number of typical manners for the chain center of mass of freely jointed chains. The squares are for repulsive sites and the inverted triangles are for attractive sites. In (A), (B), and (C), the diffusion coefficient is reduced by the site mass , Lennard-Jones well depth , and Lennard-Jones site diameter . In (D), the diffusion coefficient is reduced by , the chain length , the kinetic energy , and the effective hard site diameter ; where is the Boltzmann constant and is the temperature. In (A) the reduced diffusion coefficient is plotted as a function of inverse temperature; in (B), as a function of the volume per site (where is the inverse density); in (C), as a function of the packing fraction ; and, in (D), as a function of reduced packing fraction, where is the location of the ideal glass transition. Error bars are smaller than symbol size. Logarithms are base 10 in all figures.

Dimensionless diffusion coefficients are plotted in a number of typical manners for the chain center of mass of freely jointed chains. The squares are for repulsive sites and the inverted triangles are for attractive sites. In (A), (B), and (C), the diffusion coefficient is reduced by the site mass , Lennard-Jones well depth , and Lennard-Jones site diameter . In (D), the diffusion coefficient is reduced by , the chain length , the kinetic energy , and the effective hard site diameter ; where is the Boltzmann constant and is the temperature. In (A) the reduced diffusion coefficient is plotted as a function of inverse temperature; in (B), as a function of the volume per site (where is the inverse density); in (C), as a function of the packing fraction ; and, in (D), as a function of reduced packing fraction, where is the location of the ideal glass transition. Error bars are smaller than symbol size. Logarithms are base 10 in all figures.

is plotted in a number of manners for a high packing fraction case in the FJ system. The solid lines in (A) and (C) are for the single-exponential tail and the solid lines in (B) and (D) are for the Kohlrausch function with . (, , .

is plotted in a number of manners for a high packing fraction case in the FJ system. The solid lines in (A) and (C) are for the single-exponential tail and the solid lines in (B) and (D) are for the Kohlrausch function with . (, , .

for a low and a high packing fraction repulsive-FR systems is fitted by KWW functions. In addition to the modified LP fit as in Fig. 2(D), a global KWW fit is done. The packing fraction in (A) and (C) is 0.456, while that in (B) and (D) is 0.552. The for the LP fit is constant while the global decreased.

for a low and a high packing fraction repulsive-FR systems is fitted by KWW functions. In addition to the modified LP fit as in Fig. 2(D), a global KWW fit is done. The packing fraction in (A) and (C) is 0.456, while that in (B) and (D) is 0.552. The for the LP fit is constant while the global decreased.

The rotational relaxation times from as a function of translational relaxation time for the chain center of mass. In (A). and are plotted against for FJ chains. In (B), this is repeated for FR chains. In (C), the ratio of to is plotted as a function of for both FJ and FR chains. In (D), the ratio of to is plotted as a function of for both FJ and FR chains. In (A) and (B) circles are for repulsive ; the squares, for repulsive ; the inverted triangles, for attractive ; and the triangles, for attractive . In (C) and (D) circles are for attractive FJ; the squares, for repulsive FJ; the triangles, for attractive FR; and the inverted triangles, for repulsive FR. Lines have slopes of 1 and are for reference.

The rotational relaxation times from as a function of translational relaxation time for the chain center of mass. In (A). and are plotted against for FJ chains. In (B), this is repeated for FR chains. In (C), the ratio of to is plotted as a function of for both FJ and FR chains. In (D), the ratio of to is plotted as a function of for both FJ and FR chains. In (A) and (B) circles are for repulsive ; the squares, for repulsive ; the inverted triangles, for attractive ; and the triangles, for attractive . In (C) and (D) circles are for attractive FJ; the squares, for repulsive FJ; the triangles, for attractive FR; and the inverted triangles, for repulsive FR. Lines have slopes of 1 and are for reference.

Modified Lindsey-Patterson of representative systems for both FJ and FR cases. This entails plotting against . When plotted as such, the KWW function is linear with a slope of . In (A), is analyzed for attractive FJ and, in (B), for attractive-FR systems. Results have been shifted both horizontally and vertically for clarity. In (C), has been analyzed for both FJ and FR systems. In (D), has been analyzed for both FJ and FR systems. In both (C) and (D), the times have been reduced by the translational relaxation times for the chain center of mass, and the FR results have all been shifted down in the vertical direction by 1 for clarity. Unlike in (A) and (B), there are no other shifts of the data. The FJ packing fractions in (A) are, from upper curve to lower, 0.429, 0.537, 0.566, 0.6178, and 0.630. The FR packing fractions in (B) are, from upper curve to lower, 0.494, 0.504, 0.517, 0.538, and 0.552. These packing fraction are also used in (C) and (D). In (A), all lines have slopes of 0.75 except the uppermost, which has a slope of 0.68. In (B), all lines have slopes of 0.68 except the lowestmost, which has a slope of 0.75.

Modified Lindsey-Patterson of representative systems for both FJ and FR cases. This entails plotting against . When plotted as such, the KWW function is linear with a slope of . In (A), is analyzed for attractive FJ and, in (B), for attractive-FR systems. Results have been shifted both horizontally and vertically for clarity. In (C), has been analyzed for both FJ and FR systems. In (D), has been analyzed for both FJ and FR systems. In both (C) and (D), the times have been reduced by the translational relaxation times for the chain center of mass, and the FR results have all been shifted down in the vertical direction by 1 for clarity. Unlike in (A) and (B), there are no other shifts of the data. The FJ packing fractions in (A) are, from upper curve to lower, 0.429, 0.537, 0.566, 0.6178, and 0.630. The FR packing fractions in (B) are, from upper curve to lower, 0.494, 0.504, 0.517, 0.538, and 0.552. These packing fraction are also used in (C) and (D). In (A), all lines have slopes of 0.75 except the uppermost, which has a slope of 0.68. In (B), all lines have slopes of 0.68 except the lowestmost, which has a slope of 0.75.

(A) and (B) are linear-log plots of the data in Fig. 5. (A) is for FJ systems, and (B) for FR systems. In order to show the degree of collapse, only curves through the data points are shown. In (C) are the translational relaxation times used to reduce the time scales in (A) and (B) as well as in Fig. 5. In (D) are the ratios of relaxation time for E to that of B for both FJ and FR systems. The circles are for attractive FJ; the squares, for repulsive FJ; the inverted triangles, for attractive FR; and the triangles, for repulsive FR.

(A) and (B) are linear-log plots of the data in Fig. 5. (A) is for FJ systems, and (B) for FR systems. In order to show the degree of collapse, only curves through the data points are shown. In (C) are the translational relaxation times used to reduce the time scales in (A) and (B) as well as in Fig. 5. In (D) are the ratios of relaxation time for E to that of B for both FJ and FR systems. The circles are for attractive FJ; the squares, for repulsive FJ; the inverted triangles, for attractive FR; and the triangles, for repulsive FR.

## Tables

State points simulated. Temperature in units of . Density in units of . Pressure in units of .

State points simulated. Temperature in units of . Density in units of . Pressure in units of .

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