^{1}, Julieanne V. Heffernan

^{1}, Joanne Budzien

^{2}, Keenan T. Dotson

^{1}, Francisco Avila

^{1}, David T. Limmer

^{1}, Daniel T. McCoy

^{1}, John D. McCoy

^{1,a)}and Douglas B. Adolf

^{2}

### Abstract

Dynamical properties of short freely jointed and freely rotating chains are studied using molecular dynamics simulations. These results are combined with those of previous studies, and the degree of rheological complexity of the two models is assessed. New results are based on an improved analysis procedure of the rotational relaxation of the second Legendre polynomials of the end-to-end vector in terms of the Kohlrausch–Williams–Watts (KWW) function. Increased accuracy permits the variation of the KWW stretching exponent to be tracked over a wide range of state points. The smoothness of as a function of packing fraction is a testimony both to the accuracy of the analytical methods and the appropriateness of as a measure of the distance to the ideal glass transition at . Relatively direct comparison is made with experiment by viewing as a function of the KWW relaxation time. The simulation results are found to be typical of small molecular glass formers. Several manifestations of rheological complexity are considered. First, the proportionality of -relaxation times is explored by the comparison of translational to rotational motion (i.e., the Debye–Stokes–Einstein relation), of motion on different length scales (i.e., the Stokes–Einstein relation), and of rotational motion at intermediate times to that at long time. Second, the range of time-temperature superposition master curve behavior is assessed. Third, the variation of across state points is tracked. Although no particulate model of a liquid is rigorously rheologically simple, we find freely jointed chains closely approximated this idealization, while freely rotating chains display distinctly complex dynamical features.

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. J.D.M. and T.C.D. thank Brian Borchers for useful discussions. We also thank the Materials Engineering Department at New Mexico Tech for partial support of the undergraduate research assistants (K.T.D., F.A., D.T.L., and D.T.M.).

I. INTRODUCTION

II. BACKGROUND AND METHODS

III. RESULTS

IV. DISCUSSION

### Key Topics

- Relaxation times
- 24.0
- Glass transitions
- 20.0
- Diffusion
- 7.0
- Dielectric relaxation
- 5.0
- Rheometry
- 4.0

## Figures

Analysis of the second Legendre polynomial of the end-to-end vector for ten-site FR-R at , , and . (A) vs ln(t). The lower thick line is really the overlapping points of the raw data which would obscure the curve passing through them. This curve is reproduced shifted to the right where the solid line is the Kohlrausch function; and the dashed, the single-exponential tail. (B) The Lindsey–Patterson plot. The points are the data. The dashed line is the single exponential tail; and the solid line has a slope of . (C) The modified Lindsey–Patterson plot. The points are the data. The dashed line shows a slope of one; and the solid, a slope of . (D) The Lindsey and Patterson cross-plot. The points are the data. The solid line has a slope of one and an intercept of with . In all cases, the vertical dashed lines denote the intermediate time regime. Logarithms are base .

Analysis of the second Legendre polynomial of the end-to-end vector for ten-site FR-R at , , and . (A) vs ln(t). The lower thick line is really the overlapping points of the raw data which would obscure the curve passing through them. This curve is reproduced shifted to the right where the solid line is the Kohlrausch function; and the dashed, the single-exponential tail. (B) The Lindsey–Patterson plot. The points are the data. The dashed line is the single exponential tail; and the solid line has a slope of . (C) The modified Lindsey–Patterson plot. The points are the data. The dashed line shows a slope of one; and the solid, a slope of . (D) The Lindsey and Patterson cross-plot. The points are the data. The solid line has a slope of one and an intercept of with . In all cases, the vertical dashed lines denote the intermediate time regime. Logarithms are base .

Examples of the determination of the KWW stretching exponent for a number of packing fractions for FR chains. [The are from top to bottom (1.6, 0.612, 0.318, 0.758)-R; (2.2, 0.944, 0.471, 0.681)-R; (2.0, 1.06, 0.536, 0.660)-R; (1.6, 1.06, 0.551, 0.656)-R; (1.6,1.06, 0.563, 0.650)-A]. In (A) are shifted Lindsey and Patterson cross-plots with lines of a slope of 1. In (B) are unshifted, modified Lindsey–Patterson plots with lines of slope .

Examples of the determination of the KWW stretching exponent for a number of packing fractions for FR chains. [The are from top to bottom (1.6, 0.612, 0.318, 0.758)-R; (2.2, 0.944, 0.471, 0.681)-R; (2.0, 1.06, 0.536, 0.660)-R; (1.6, 1.06, 0.551, 0.656)-R; (1.6,1.06, 0.563, 0.650)-A]. In (A) are shifted Lindsey and Patterson cross-plots with lines of a slope of 1. In (B) are unshifted, modified Lindsey–Patterson plots with lines of slope .

Chain center of mass diffusion coefficient for freely jointed chains [in (A) and (C)] and for freely rotating chains [in (B) and (D)]. The squares are for attractive FJ chains and the circles are for repulsive FJ chains. The triangles are for attractive FR chains and the inverted triangles are for repulsive FR chains. In (A) and (B), in Lennard–Jones units is plotted in Arrhenius fashion against inverse temperature. In (C) and (D), the reduced diffusion coefficient, , is plotted against packing fraction . In (C) and (D), the repulsive and attractive results have been shifted, as indicated by the arrows. The lines are powerlaw fits^{9} to : For FJ, ; for FR, . Logarithms are base 10.

Chain center of mass diffusion coefficient for freely jointed chains [in (A) and (C)] and for freely rotating chains [in (B) and (D)]. The squares are for attractive FJ chains and the circles are for repulsive FJ chains. The triangles are for attractive FR chains and the inverted triangles are for repulsive FR chains. In (A) and (B), in Lennard–Jones units is plotted in Arrhenius fashion against inverse temperature. In (C) and (D), the reduced diffusion coefficient, , is plotted against packing fraction . In (C) and (D), the repulsive and attractive results have been shifted, as indicated by the arrows. The lines are powerlaw fits^{9} to : For FJ, ; for FR, . Logarithms are base 10.

Variation of the KWW stretching exponent as a function of packing fraction in (A) and as a function of the inverse of the logarithm of the KWW relaxation time in (B) and (C). The circles correspond to FJ-R, the squares to FJ-A; the inverted triangles to FR-R, and the triangles to FR-A. In (A), a representative error bar of is shown to the left and vertical lines at the location of the packing fractions at the ideal glass transition are shown to the right [ and ]. In (B) is shown the results of Paluch *et al.* [Ref. 39(c)] for the fragile glass former [poly(bisphenol A-co-epichlorohydrin), glycidyl end capped] at a variety of pressures (symbols representing different pressures) and the line is a guide to the eye. The boxed area in (B) is reproduced in (C); the line is duplicated and the simulation results are plotted instead of experimental results. The relaxation times used to reduce the data in (B) and (C) are for experiment, and in Lennard–Jones time units, for the FR and for the FJ simulations. Logarithms are base 10.

Variation of the KWW stretching exponent as a function of packing fraction in (A) and as a function of the inverse of the logarithm of the KWW relaxation time in (B) and (C). The circles correspond to FJ-R, the squares to FJ-A; the inverted triangles to FR-R, and the triangles to FR-A. In (A), a representative error bar of is shown to the left and vertical lines at the location of the packing fractions at the ideal glass transition are shown to the right [ and ]. In (B) is shown the results of Paluch *et al.* [Ref. 39(c)] for the fragile glass former [poly(bisphenol A-co-epichlorohydrin), glycidyl end capped] at a variety of pressures (symbols representing different pressures) and the line is a guide to the eye. The boxed area in (B) is reproduced in (C); the line is duplicated and the simulation results are plotted instead of experimental results. The relaxation times used to reduce the data in (B) and (C) are for experiment, and in Lennard–Jones time units, for the FR and for the FJ simulations. Logarithms are base 10.

Variation of relaxation times. In (A), both and for FJ chains are plotted against the distance to the glass transition where . Circles are -R; squares, -R; triangles, -A; and inverted triangles, -R. The line has a slope of 2.2, In (B) the ratio of to is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. (C) is identical to (A) except for FR chains, , and the line has slope 3.0. In (D) is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. Logarithms are base 10.

Variation of relaxation times. In (A), both and for FJ chains are plotted against the distance to the glass transition where . Circles are -R; squares, -R; triangles, -A; and inverted triangles, -R. The line has a slope of 2.2, In (B) the ratio of to is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. (C) is identical to (A) except for FR chains, , and the line has slope 3.0. In (D) is plotted against . Circles are FJ-R; squares, FJ-A; triangles, FR-A; and inverted triangle, FR-R. Logarithms are base 10.

## 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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