^{1}and Laura J. Kaufman

^{1,a)}

### Abstract

The rotational dynamics of three perylene diimide dyes are studied on the single molecule (SM) level in *ortho*-terphenyl (OTP) near the glass transition temperature (T_{g}). At all temperatures probed, spanning 1.03–1.06 T_{g}, each of the three probes exhibits rotational correlation times, τ_{c}, that span more than a decade, consistent with the presence of spatially heterogeneous dynamics in OTP. No trend is found as a function of temperature, but a trend as a function of probe is observed: Average probe rotational correlation time scales inversely with breadth of SM τ_{c} distribution, with faster probes exhibiting broader τ_{c} distributions. This implies that dynamic exchange occurs on and below time scales associated with probe rotation. Extrapolating FWHM of rotational relaxation times to the structural relaxation time of the host shows that the τ_{c} distribution would span nearly two decades in the limit of no probe temporal averaging. Comparison with SM measurements in glycerol suggests that OTP demonstrates a greater degree of spatially heterogeneous dynamics in this temperature range than does glycerol.

This research was supported by the National Science Foundation (NSF) under Grant Nos. CHE 0744322, DGE 0801530, and an NSF GRF for L.M.L. We thank L. Campos and co-workers for supplying egPDI. We thank S. Snyder for assistance with ChemBio3D. We thank Keewook Paeng for valuable discussions.

I. INTRODUCTION

II. EXPERIMENTAL

A. Sample preparation

B. Optical setup

C. Data analysis

D. Heating correction

E. Simulations

III. RESULTS

A. Median rotational relaxation times

B. Degree of spatially heterogeneous dynamics

1. Breadth of relaxation times

2. Evaluation of stretching exponent

IV. DISCUSSION

A. Rotational relaxation rates

B. Reports of heterogeneous dynamics

1. Breadth of relaxation times

2. Evaluation of stretching exponent

V. CONCLUSIONS

### Key Topics

- Relaxation times
- 23.0
- Rotational correlation time
- 21.0
- Hydrodynamics
- 10.0
- Temperature measurement
- 10.0
- Spatial dimensions
- 8.0

##### C09B

## Figures

Schematic diagram (not to scale) of the epi-fluorescence microscope. BS = beam splitter, F = filter, M = mirror, PEB = piezoelectric buzzer, L = lens.

Schematic diagram (not to scale) of the epi-fluorescence microscope. BS = beam splitter, F = filter, M = mirror, PEB = piezoelectric buzzer, L = lens.

Rotational relaxation times vs. temperature for tbPDI (blue triangles), dpPDI (red circles), and egPDI (black squares) together with each dye's molecular structure outlined in the same color. Each point represents the heat corrected τ_{c,med} value for a single movie. Lines represent the best-fit DSE fit for each PDI dye. Extracted hydrodynamic radii are V_{h} = 1.87 nm^{3} (tbPDI), V_{h} = 1.17 nm^{3} (dpPDI), and V_{h} = 0.80 nm^{3} (egPDI). The structural relaxation of OTP as measured by dielectric spectroscopy ^{ 30 } is plot as a function of temperature (green line).

Rotational relaxation times vs. temperature for tbPDI (blue triangles), dpPDI (red circles), and egPDI (black squares) together with each dye's molecular structure outlined in the same color. Each point represents the heat corrected τ_{c,med} value for a single movie. Lines represent the best-fit DSE fit for each PDI dye. Extracted hydrodynamic radii are V_{h} = 1.87 nm^{3} (tbPDI), V_{h} = 1.17 nm^{3} (dpPDI), and V_{h} = 0.80 nm^{3} (egPDI). The structural relaxation of OTP as measured by dielectric spectroscopy ^{ 30 } is plot as a function of temperature (green line).

(Left) Distribution of SM τ_{c} values for (a) tbPDI, (b) dpPDI, and (c) egPDI in OTP at 258 K (black), 257 K (red), 256 K (green), 255 K (blue), 254 K (magenta), 253 K (wine), 252 K (olive), 251 K (orange), and 250 K (cyan). All histograms are normalized to the maximum number of occurrences. Each histogram is taken from two data sets for each dye, with the histograms across data sets normalized to the median τ_{c} value of one of the data sets for that particular PDI dye. This alleviates any potential widening of the distribution arising from differing thermal contact of the sample and stage between data sets. (Right) SM data from all temperatures normalized by τ_{c,med} and combined to form a single histogram for (d) tbPDI, (e) dpPDI, and (f) egPDI. Each histogram is fit with a Gaussian function (black line). Histogram of τ_{c} distribution of simulations of homogeneous rotational diffusion with trajectory length for each simulation tuned to match average experimental trajectory length as described in the text is shown by the red lines.

(Left) Distribution of SM τ_{c} values for (a) tbPDI, (b) dpPDI, and (c) egPDI in OTP at 258 K (black), 257 K (red), 256 K (green), 255 K (blue), 254 K (magenta), 253 K (wine), 252 K (olive), 251 K (orange), and 250 K (cyan). All histograms are normalized to the maximum number of occurrences. Each histogram is taken from two data sets for each dye, with the histograms across data sets normalized to the median τ_{c} value of one of the data sets for that particular PDI dye. This alleviates any potential widening of the distribution arising from differing thermal contact of the sample and stage between data sets. (Right) SM data from all temperatures normalized by τ_{c,med} and combined to form a single histogram for (d) tbPDI, (e) dpPDI, and (f) egPDI. Each histogram is fit with a Gaussian function (black line). Histogram of τ_{c} distribution of simulations of homogeneous rotational diffusion with trajectory length for each simulation tuned to match average experimental trajectory length as described in the text is shown by the red lines.

Distributions of β values from individual SM ACF fits for (a) tbPDI, (b) dpPDI, and (c) egPDI. The solid black lines indicate the median value, β_{med}, of these distributions. The red curve represents distributions from simulations of homogeneous rotational diffusion with trajectory length set to match average experimental trajectory length. For all experimental and simulated data, β is allowed to float from 0.3 to 2.0 when each ACF is fit with a stretched exponential as described in the text. (d) β_{QE} values for each probe for each movie collected as a function of true temperature for tbPDI (black squares), dpPDI (red circles), and egPDI (blue triangles). The dashed lines of corresponding color represent the mean β_{QE} value for each probe for all temperatures studied. The solid lines of corresponding color represent the β_{QE} values from the corresponding simulations.

Distributions of β values from individual SM ACF fits for (a) tbPDI, (b) dpPDI, and (c) egPDI. The solid black lines indicate the median value, β_{med}, of these distributions. The red curve represents distributions from simulations of homogeneous rotational diffusion with trajectory length set to match average experimental trajectory length. For all experimental and simulated data, β is allowed to float from 0.3 to 2.0 when each ACF is fit with a stretched exponential as described in the text. (d) β_{QE} values for each probe for each movie collected as a function of true temperature for tbPDI (black squares), dpPDI (red circles), and egPDI (blue triangles). The dashed lines of corresponding color represent the mean β_{QE} value for each probe for all temperatures studied. The solid lines of corresponding color represent the β_{QE} values from the corresponding simulations.

SM data from all temperatures normalized and combined to form a single histogram for (a) each PDI in OTP: tbPDI (blue), dpPDI (red), and egPDI (black) and (b) each PDI in glycerol: dpPDI (red), dapPDI (green), and tbPDI (blue). All histograms are normalized to their maximum number of occurrences. Each PDI in OTP histogram is fit with a fixed height Gaussian and each PDI in glycerol histogram is fit with a fixed-height Lorentzian (lines of solid colors corresponding to histogram colors). The histogram for egPDI in OTP is additionally fit with a fixed-height Lorentzian (black dashed line in a) and tbPDI in glycerol is fit with a fixed-height Gaussian (blue dashed line in b). (c) τ_{c,med} from each movie plot with respect to T_{g}/T for egPDI (black squares), dpPDI (red circles), and tbPDI (blue triangles) in OTP and tbPDI (open blue triangles), dapPDI (open green sideways triangle), and dpPDI (open red circles) in glycerol. Structural relaxation data for OTP (dashed blue line) ^{ 30 } and glycerol (dashed black line) ^{ 29 } are plot with respect to T_{g}/T. (d) FWHM from Gaussian (squares) and Lorentzian (circles) fits to each of the OTP (blue) and glycerol (black) histograms pictured in (a) and (b) plot as a function log(τ_{c}/τ_{α}). FWHM from Gaussian fits of OTP (blue squares) and from Lorentzian fits of glycerol (black circles) τ_{c} distributions vs. log(τ_{c}/τ_{α}) are fit to lines. y-intercepts correspond to FWHM for τ_{c}/τ_{α} = 1.

SM data from all temperatures normalized and combined to form a single histogram for (a) each PDI in OTP: tbPDI (blue), dpPDI (red), and egPDI (black) and (b) each PDI in glycerol: dpPDI (red), dapPDI (green), and tbPDI (blue). All histograms are normalized to their maximum number of occurrences. Each PDI in OTP histogram is fit with a fixed height Gaussian and each PDI in glycerol histogram is fit with a fixed-height Lorentzian (lines of solid colors corresponding to histogram colors). The histogram for egPDI in OTP is additionally fit with a fixed-height Lorentzian (black dashed line in a) and tbPDI in glycerol is fit with a fixed-height Gaussian (blue dashed line in b). (c) τ_{c,med} from each movie plot with respect to T_{g}/T for egPDI (black squares), dpPDI (red circles), and tbPDI (blue triangles) in OTP and tbPDI (open blue triangles), dapPDI (open green sideways triangle), and dpPDI (open red circles) in glycerol. Structural relaxation data for OTP (dashed blue line) ^{ 30 } and glycerol (dashed black line) ^{ 29 } are plot with respect to T_{g}/T. (d) FWHM from Gaussian (squares) and Lorentzian (circles) fits to each of the OTP (blue) and glycerol (black) histograms pictured in (a) and (b) plot as a function log(τ_{c}/τ_{α}). FWHM from Gaussian fits of OTP (blue squares) and from Lorentzian fits of glycerol (black circles) τ_{c} distributions vs. log(τ_{c}/τ_{α}) are fit to lines. y-intercepts correspond to FWHM for τ_{c}/τ_{α} = 1.

(a) β_{QE} with respect to median trajectory length in terms of τ_{c,med} for each measurement for egPDI (black squares), dpPDI (red circles), and tbPDI (blue triangles) in OTP and tbPDI (open blue triangles), dapPDI (open green sideways triangles), and dpPDI (open red circles) in glycerol. Inset shows the same data with trajectory length plot on a log scale. (b) Average β_{QE} for each PDI studied in OTP (blue) or glycerol (black) with respect to FWHM from Gaussian (OTP) or Lorentzian (glycerol) fits to each histogram pictured in Figs. 5(a) and 5(b) as well as best-fit lines to the data.

(a) β_{QE} with respect to median trajectory length in terms of τ_{c,med} for each measurement for egPDI (black squares), dpPDI (red circles), and tbPDI (blue triangles) in OTP and tbPDI (open blue triangles), dapPDI (open green sideways triangles), and dpPDI (open red circles) in glycerol. Inset shows the same data with trajectory length plot on a log scale. (b) Average β_{QE} for each PDI studied in OTP (blue) or glycerol (black) with respect to FWHM from Gaussian (OTP) or Lorentzian (glycerol) fits to each histogram pictured in Figs. 5(a) and 5(b) as well as best-fit lines to the data.

## Tables

Number of molecules and FWHM values of best-fit Gaussian distributions of log(τ_{c}) values for all SM data shown in Fig. 3 , left panel. Average FWHM over all temperatures as a function of probe as well as the FWHM values for the combined histograms shown in the right panel of Fig. 3 are also given.

Number of molecules and FWHM values of best-fit Gaussian distributions of log(τ_{c}) values for all SM data shown in Fig. 3 , left panel. Average FWHM over all temperatures as a function of probe as well as the FWHM values for the combined histograms shown in the right panel of Fig. 3 are also given.

Various quantities for experiments and simulations for the three employed probes in OTP. Lifetime/τ_{c} is determined for each movie as the average time each SM is “on” (typically the time until photobleaching) divided by the average τ_{c} value for that movie. These quantities are then averaged. FWHM, β_{med}, and β_{QE} are experimental quantities as described in the text and shown in Figs. 3 and 4 . FWHM_{sim}, β_{med,sim}, and β_{QE,sim} are quantities obtained from simulation as described in the text and shown in Figs. 3 and 4 .

Various quantities for experiments and simulations for the three employed probes in OTP. Lifetime/τ_{c} is determined for each movie as the average time each SM is “on” (typically the time until photobleaching) divided by the average τ_{c} value for that movie. These quantities are then averaged. FWHM, β_{med}, and β_{QE} are experimental quantities as described in the text and shown in Figs. 3 and 4 . FWHM_{sim}, β_{med,sim}, and β_{QE,sim} are quantities obtained from simulation as described in the text and shown in Figs. 3 and 4 .

Molecular weight (MW), extracted hydrodynamic volumes (V_{h}), van der Waals volume (V_{v}), V_{h}/V_{v} ratios, and τ_{c}/τ_{α} values as described in the text for the three probes investigated in OTP and in glycerol. ^{ 14 }

Molecular weight (MW), extracted hydrodynamic volumes (V_{h}), van der Waals volume (V_{v}), V_{h}/V_{v} ratios, and τ_{c}/τ_{α} values as described in the text for the three probes investigated in OTP and in glycerol. ^{ 14 }

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