^{1,a)}, W. Paul

^{2}and K. Binder

^{2}

### Abstract

Molecular dynamics simulations of a system of short bead-spring chains containing an additional dumbbell are presented and analyzed. This system represents a coarse-grained model for a melt of short, flexible polymers containing fluorescent probe molecules at very dilute concentration. It is shown that such a system is very well suited to study aspects of the glass transition of the undercooled polymer melt via single molecule spectroscopy, which are not easily accessed by other methods. Such aspects include data which can be extracted from a study of fluctuations along a trajectory of the single molecule, probing the rugged energy landscape of the glass-forming liquid and transitions from one metabasin of this energy landscape to the next one. Such an information can be inferred from “distance maps” constructed from trajectories characterizing the translational and orientational motion of the probe. At the same time, determining autocorrelation functions along such trajectories, it is shown for several types of probes (differing in their size and/or mass within reasonable limits) that this time-averaged information of the probe is fully compatible with ensemble averaged information on the relaxation of the glass-forming matrix, accessible from bulk measurements. The analyzed quantities include the fluorescence lifetime, linear dichroism, and also various orientational correlation functions of the probe, in order to provide guidance to experimental work. Similar to earlier findings from simulations of bulk molecular fluids, deviations from the Stokes-Einstein and Stokes-Einstein-Debye relations are observed.

One of the authors (R.V.) thanks the Fonds voor Wetenschappelijk Onderzoek Vlaanderen for a postdoctoral fellowship and a grant for a “study” stay abroad in the group of another author (K.B). Partial support from Sonderforschungsbereich 625/A3 of the German National Science Foundation and the EU network of excellence SOFTCOMP is also acknowledged.

I. INTRODUCTION

II. METHODS

III. RESULTS

A. Probe and matrix

B. Probe: Single molecule lifetime trajectories

C. Probe: Single molecule rotational trajectories

D. Probe: Correlations functions of the various trajectories

E. Probe: Relaxation times of the various observables—connections to the glass transition theories

IV. DISCUSSION AND CONCLUDING REMARKS

### Key Topics

- Polymers
- 41.0
- Relaxation times
- 32.0
- Fluorescence
- 25.0
- Glass transitions
- 25.0
- Correlation functions
- 23.0

## Figures

Self-incoherent scattering functions of the dumbbell (dash), the polymer chains surrounding it (solid), and of the model system simulated in the absence of the probe (open circles). The probe is either a dimer of the surrounding mers (, , top) or a larger probe (, , bottom). , 0.55, 0.55, 0.6, 0.7, 1.0, and 2.0 from right to left, respectively.

Self-incoherent scattering functions of the dumbbell (dash), the polymer chains surrounding it (solid), and of the model system simulated in the absence of the probe (open circles). The probe is either a dimer of the surrounding mers (, , top) or a larger probe (, , bottom). , 0.55, 0.55, 0.6, 0.7, 1.0, and 2.0 from right to left, respectively.

(Color online) Reduced radiative lifetime (full curves, left ordinate scale) and translational average square displacement (broken curve, right ordinate scale), trajectories of a probe (, ) in a polymer matrix for .

(Color online) Reduced radiative lifetime (full curves, left ordinate scale) and translational average square displacement (broken curve, right ordinate scale), trajectories of a probe (, ) in a polymer matrix for .

(Color) Distance matrix for translational DM (top) and rotational RDM (bottom) diffusion of a single probe (, , cf. Fig. 2) in the considered system at temperature . The gray scale indicates the values of (color online).

(Color) Distance matrix for translational DM (top) and rotational RDM (bottom) diffusion of a single probe (, , cf. Fig. 2) in the considered system at temperature . The gray scale indicates the values of (color online).

(Color) Legendre polynomials (solid) and (dashed) trajectories of a probe (, ) in a polymer matrix for (top) and (bottom). Corresponding translational (solid) and rotational (dashed) average square displacement trajectories ( in the case of the probe in the system at and for ).

(Color) Legendre polynomials (solid) and (dashed) trajectories of a probe (, ) in a polymer matrix for (top) and (bottom). Corresponding translational (solid) and rotational (dashed) average square displacement trajectories ( in the case of the probe in the system at and for ).

(Color online) Fluorescence lifetime time correlation functions for the dumbbell (, ) in the considered system at temperatures (squares), (circles), (diamonds) and (stars). The error bars are estimated by the Jackknife approach (Refs. 63 and 64).

(Color online) Fluorescence lifetime time correlation functions for the dumbbell (, ) in the considered system at temperatures (squares), (circles), (diamonds) and (stars). The error bars are estimated by the Jackknife approach (Refs. 63 and 64).

(Color) Fluorescence lifetime time correlation functions for the small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system for two temperatures: and . The error bars are estimated by the Jackknife approach (Refs. 63 and 64).

(Color) Fluorescence lifetime time correlation functions for the small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system for two temperatures: and . The error bars are estimated by the Jackknife approach (Refs. 63 and 64).

(Color online) Orientational time correlation functions of order (solid), 2 (dash), and 4 (dot) and (open diamonds) for the dumbbell (, ) in the considered system at temperatures (top), (middle) and (bottom). The error bars are estimated by the Jackknife approach (Refs. 63 and 64). Black lines are the best stretched exponential fits performed in the relaxation zone of the decays , with and values given in Table I.

(Color online) Orientational time correlation functions of order (solid), 2 (dash), and 4 (dot) and (open diamonds) for the dumbbell (, ) in the considered system at temperatures (top), (middle) and (bottom). The error bars are estimated by the Jackknife approach (Refs. 63 and 64). Black lines are the best stretched exponential fits performed in the relaxation zone of the decays , with and values given in Table I.

(Color online) (a) Angell plot of the relaxation times of (stars), (large full diamonds), (large filled circles), (small open diamonds), (small open circles), and (small balls) for the dumbbell (, ) in the considered system. (b) Angell plot of the relaxation times of (full symbols) and (open symbols) for a small (, , square; , , circle) or large (, , diamond; , , star) dumbbell in the considered system. Note that, in the standard Angell plot, is normalized by the glass transition temperature , while we normalize here by the critical temperature of mode coupling theory.

(Color online) (a) Angell plot of the relaxation times of (stars), (large full diamonds), (large filled circles), (small open diamonds), (small open circles), and (small balls) for the dumbbell (, ) in the considered system. (b) Angell plot of the relaxation times of (full symbols) and (open symbols) for a small (, , square; , , circle) or large (, , diamond; , , star) dumbbell in the considered system. Note that, in the standard Angell plot, is normalized by the glass transition temperature , while we normalize here by the critical temperature of mode coupling theory.

(Color online) Vogel-Fulcher plots of the relaxation times [ (stars), (full diamonds), (full circles)] (open diamonds) and (open circles) for the dumbbell (, ) in the considered system. The values of the Vogel temperature and of the “fragility parameter” are presented in Table II.

(Color online) Vogel-Fulcher plots of the relaxation times [ (stars), (full diamonds), (full circles)] (open diamonds) and (open circles) for the dumbbell (, ) in the considered system. The values of the Vogel temperature and of the “fragility parameter” are presented in Table II.

(Color online) Vogel-Fulcher plots of the relaxation times for a small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system. The values of the Vogel temperature and of the “fragility parameter” are presented in Table III.

(Color online) Vogel-Fulcher plots of the relaxation times for a small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system. The values of the Vogel temperature and of the “fragility parameter” are presented in Table III.

(Color online) Log-log plot of the relaxation times for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. The parameters and obtained from the fits to a power law (see text) are given in Table IV.

(Color online) Log-log plot of the relaxation times for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. The parameters and obtained from the fits to a power law (see text) are given in Table IV.

(Color) Translational (top) and rotational (bottom) mean square displacement curves for the small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system for two temperatures: and .

(Color) Translational (top) and rotational (bottom) mean square displacement curves for the small (, , squares; , , circles) or large (, , diamonds; , , stars) dumbbell in the considered system for two temperatures: and .

Log-log plot of the relaxation times obtained from the incoherent scattering functions at (a) and translational (b) and rotational (c) mean square displacement curves for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. The parameters and obtained from the fits to a power law (see text) are given in Table V.

Log-log plot of the relaxation times obtained from the incoherent scattering functions at (a) and translational (b) and rotational (c) mean square displacement curves for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. The parameters and obtained from the fits to a power law (see text) are given in Table V.

Power law fits of translational (a) and rotational (b) diffusivities and as functions of (, upper part) and (lower part); and for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. Results for the exponents are collected in Table VI.

Power law fits of translational (a) and rotational (b) diffusivities and as functions of (, upper part) and (lower part); and for the small (, , squares; , , diamonds) or large (, , circles; , , stars) dumbbell in the considered system. Results for the exponents are collected in Table VI.

## Tables

Amplitudes, relaxation times , and stretching parameters of the OTCFs for , 2, and 4 at various temperatures for the large and heavy dumbbell , , determined by fitting the KWW function to the curves in the relaxation zone. Three values are indicated in most columns for each line, which concern fits starting at , (0.5), and [0.4], respectively. In each case, the amplitude has only been given for the best fit (shown on Fig. 7). The errors determined by the fitting procedure, using a Levenberg-Marquadt algorithm with a least square minimization method, are also indicated for and . The quality of the fits has been judged on the base on the usual criterion.

Amplitudes, relaxation times , and stretching parameters of the OTCFs for , 2, and 4 at various temperatures for the large and heavy dumbbell , , determined by fitting the KWW function to the curves in the relaxation zone. Three values are indicated in most columns for each line, which concern fits starting at , (0.5), and [0.4], respectively. In each case, the amplitude has only been given for the best fit (shown on Fig. 7). The errors determined by the fitting procedure, using a Levenberg-Marquadt algorithm with a least square minimization method, are also indicated for and . The quality of the fits has been judged on the base on the usual criterion.

Parameters obtained by fitting the Vogel Fulcher law to the various relaxation times of the small and light dumbbell. Results of fits obtained by fixing the Vogel temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 9). Results of the “best” fits obtained by varying both Vogel temperature and the parameter are given in columns 4 and 5.

Parameters obtained by fitting the Vogel Fulcher law to the various relaxation times of the small and light dumbbell. Results of fits obtained by fixing the Vogel temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 9). Results of the “best” fits obtained by varying both Vogel temperature and the parameter are given in columns 4 and 5.

Parameters obtained by fitting the Vogal Fulcher law to the fluorescence lifetime relaxation time for the four types of dumbbell. Results of fits obtained by fixing the Vogel temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 10). Results of the “best” fits obtained by varying both the Vogel temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the Vogal Fulcher law to the fluorescence lifetime relaxation time for the four types of dumbbell. Results of fits obtained by fixing the Vogel temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 10). Results of the “best” fits obtained by varying both the Vogel temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the MCT law to the fluorescence lifetime relaxation time for the four types of dumbbell. Results of fits obtained by fixing the critical temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 11). Results of the “best” fits obtained by varying both the critical temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the MCT law to the fluorescence lifetime relaxation time for the four types of dumbbell. Results of fits obtained by fixing the critical temperature to the known value for this polymer model are given in column 3 (Data and fits are shown in Fig. 11). Results of the “best” fits obtained by varying both the critical temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the MCT law to the fluorescence lifetime relaxation time and the translational. and rotational diffusion constants for the four types of dumbbell. Results of fits obtained by fixing the critical temperature to the known value for this polymer model are given in column 4 (Data and fits are shown in Fig. 13). Results of the “best” fits obtained by varying both the critical temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the MCT law to the fluorescence lifetime relaxation time and the translational. and rotational diffusion constants for the four types of dumbbell. Results of fits obtained by fixing the critical temperature to the known value for this polymer model are given in column 4 (Data and fits are shown in Fig. 13). Results of the “best” fits obtained by varying both the critical temperature and the parameter are given in column 4 and 5.

Parameters obtained by fitting the fractional functional forms [Eq. (16)] to the relaxation times and for the four types of dumbbell (data and fits are shown in Fig. 14).

Parameters obtained by fitting the fractional functional forms [Eq. (16)] to the relaxation times and for the four types of dumbbell (data and fits are shown in Fig. 14).

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