Nonexponential Relaxation of Poly(cyclohexyl acrylate

Nonexponential Relaxation of Poly(cyclohexyl acrylate): Comparison of Single-Molecule and Ensemble Fluorescence Studies. Chia-Yin Joyce Wei and David ...
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J. Phys. Chem. B 2009, 113, 2253–2261

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Nonexponential Relaxation of Poly(cyclohexyl acrylate): Comparison of Single-Molecule and Ensemble Fluorescence Studies Chia-Yin Joyce Wei† and David A. Vanden Bout*,†,‡ Department of Chemistry and Biochemistry and Center for Nano and Molecular Science and Technology, The UniVersity of Texas at Austin, 1 UniVersity Station, A5300, Austin, Texas 78712 ReceiVed: July 16, 2008; ReVised Manuscript ReceiVed: NoVember 7, 2008

Rotational motions of probe molecules in poly(cyclohexyl acrylate) (PCA) were investigated by both ensemble fluorescence recovery after photobleaching (FRAP) and single-molecule spectroscopy at temperatures near the glass transition temperature (Tg) of the polymer host. FRAP measurements of the ensemble anisotropy decay show a nonexponential decay with β values of 0.5-0.6 when fit by a stretched exponential function. The relationship between the average relaxation time and temperature follows the Williams-Landel-Ferry equation, whereas β shows no temperature dependence over this range. The same system was also studied by single-molecule spectroscopy at 2 °C above the Tg of PCA. The rotational dynamics of the probe molecule can be measured by the autocorrelation function of the linear dichroism signals. Each single-molecule correlation function was fit to the stretched exponential function. The results from all single-molecule data yield broad distributions of both the correlation times (τ) and β values. The average of the single-molecule correlation times agrees with the ensemble relaxation time, and the sum of all single correlation functions has a nonexponential decay that is almost identical to the ensemble anisotropy decay. The ensemble β values are smaller than the average β values in the single-molecule experiments, demonstrating that the system exhibits heterogeneous dynamics. However, the dynamics are not described by an ensemble of molecules that all have single-exponential correlation functions with different time constants. I. Introduction Many disordered systems, such as viscous liquids, polymers, and glassy solids, exhibit nonexponential relaxations upon perturbation.1-6 This nonexponential decay has been observed by different means including dielectric relaxation,7,8 multidimensional NMR spectroscopy,9,10 optical anisotropic recovery,11-13 and single-molecule spectroscopy,14-19 among others. The relaxation time scales increase by several orders of magnitude as the temperature is lowered toward the glass transition temperature, and eventually, the molecular dynamics slow until the system becomes frozen in the glassy state. These amorphous materials are used in a wide range of applications and have very interesting properties that are of importance to physical scientists and engineers. Intensive studies have been done throughout the past few decades to account for the nonexponential nature of these glass-forming materials, but this still remains a challenging area of active research.1,2,20 When probing a system using ensemble techniques, the observed dynamics exhibit a nonexponential relaxation, which indicates that the dynamics are not governed by a single time scale, as one would expect for diffusive motion. This can be described by two possible underlying schemes: One is a heterogeneous scheme, in which all molecules relax exponentially with different time constants. The resulting average over these molecules with different dynamics creates a nonexponential relaxation for the ensemble. The second scheme is homogeneous, where all molecules relax by identical decays. In this * Corresponding author. E-mail: [email protected]. Phone: (512) 232-2824. † Department of Chemistry and Biochemistry. ‡ Center for Nano and Molecular Science and Technology.

case, the nonexponential decay for the ensemble can result only if each molecule exhibits the same nonexponential relaxation. This nonexponential relaxation has been addressed in many early theories of glass transitions,21-27 and many experiments have madestrongargumentsforspatiallyheterogeneousdynamics.10,14,15,19,28-37 The results of a probe rotation experiment always depend on the magnitude of the time scale of the rotational motion compared to that of the lifetime of any heterogeneity. Experiments using a large, slowly rotating molecule as a probe can exhibit homogeneous dynamics even if the host material itself is heterogeneous.13 This results from the fact that the underlying heterogeneity is fluctuating faster than the time scale of the probe. As such, the observed dynamics could lie at any continuum between these two extremes depending on the relative time scale of the dynamics. To explore which scheme can best describe the dynamics, a single-molecule technique would be an ideal tool because it can probe individual molecular behaviors and properties that might be obscured in the ensemble average.15,19,38-44 In this article, the relaxation of poly(cyclohexyl acrylate) is studied via an ensemble photobleaching technique that measures the reorientation of molecular probes doped into the polymer. At all temperatures, the relaxation of the probe results in nonexponential decays. To further investigate the underlying molecular motions that contribute to the observed dynamics, single-molecule fluorescence spectroscopy is utilized to study the same polymer and obtain individual transient data for statistical analysis. Comparisons between ensemble and singlemolecule results are made, and the statistical errors associated with analyzing single-molecule data are also considered.45

10.1021/jp806293x CCC: $40.75  2009 American Chemical Society Published on Web 01/30/2009

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SCHEME 1: Schematic Representation of the Ensemble FRAP Experimental Setupa

a The laser polarization is modulated by the electro-optic modulator,58 whose frequency is controlled by the function generator. The solenoid holding and the neutral density filter (ND) are used to switch between the high-intensity bleaching beam and the low-intensity probe beam. The configuration of the half-wave plate at 22.5° and detection at 90° angle with a polarizer at 35.3° allows for probing intensities of the absorption dipoles moments. Data collection and integration of all the devices are controlled by the LabView program (National Instruments) on a computer.

II. Experimental Section 1. Ensemble Photobleaching Experiments. Sample Preparation. Poly(cyclohexyl acrylate) (PCA) was purchased from Scientific Polymers as a solution in toluene with an approximate molecular weight of 150000 Da. The glass transition temperature, Tg, for the polymer was measured by differential scanning calorimetry by the manufacturer and reported to be 19 °C, which is in agreement with literature values.46 PCA was precipitated by adding cold methanol to the toluene solution. Rhodamine 6G dye obtained from Spectra-Physics was dissolved in dichloromethane or tetrahydrofuran/methanol. The polymer was doped with dye by adding a small aliquot of the stock dye solution to the polymer solution. The concentration of R6G in the polymer was estimated to be in the range between 10 and 100 ppm by weight. The mixture was first dried in a hood under flowing air and then put under vacuum to remove residual solvent. Dried samples were then transferred to a 4-mm-diameter cuvette and heated under vacuum to 110 °C for several days until a homogeneous polymer melt formed. Temperature was controlled by either a water chiller or a Janus ST-100 cryostat with a LakeShore temperature controller. Photobleaching. The fluorescence recovery after photobleaching (FRAP) technique can be used to measure the rotational dynamics of fluorescent probe molecules.12,47,48 The FRAP measurement uses a high-intensity polarized laser beam to selectively photobleach probe molecules that have absorption dipoles oriented along the excitation polarization, thus creating an anisotropic distribution of fluorescent molecules in the sample. A weaker beam polarized either parallel or perpendicular to the bleaching beam is then used to excite the fluorescence in the sample. The intensities of the fluorescence from both parallel and perpendicular excitation are measured as a function of time after the initial bleaching. As the molecules rotate in the polymer, the initial anisotropy created by the photobleaching event is lost, and the system reverts to an isotropic distribution. Scheme 1 shows the ensemble FRAP experimental setup that was used in this work. The excitation source was a continuouswave 532-nm diode laser made by Power Technology. An

electro-optic modulator (EOM) manufactured by FastPulse Technology was used to modulate the beam polarization between 0° and 90° at 10-Hz frequency. A half-wave plate was placed after the EOM to rotate the polarization by 45° to produce a probing beam modulated (45° from vertical. Fluorescence was detected at 90° through a polarizer at the “magic angle” of 35.3° from the vertical. This detection configuration allowed for the observation of the absorption intensities proportional to the probing beam polarization.47 Photobleaching was accomplished by exciting the sample with a high-intensity bleaching beam polarized at 45°. This was accomplished by removing a neutral density filter from the excitation source with a solenoid and simultaneously disabling the EOM. After a fixed bleaching duration (typically a few seconds), the neutral density was placed back into the beam to attenuate the laser intensity, and polarization modulation by the EOM was resumed. By varying bleaching power and duration, different bleach depths were achieved. Typically, the bleaching power ranged from 0.5 to 3.5 mW, and the reading power ranged from 0.5 to 3.5 µW. The bleaching duration was between 1 and 6 s, and the bleach depth was between 10% and about 40% of the original fluorescence intensity. The fluorescence signal was continuously monitored by a Hamamatsu photomultiplier tube, and the raw data were sorted into separate fluorescence signals denoted as parallel (I|) and perpendicular (I⊥) based on the modulator signal. The time-dependent anisotropy of the unbleached molecules was calculated as

r(t) )

∆I|(t) - ∆I⊥(t) ∆I|(t) + ∆2I⊥(t)

(1)

where ∆I(t) is the difference in the fluorescence signal in each polarization channel before and after photobleaching. The anisotropy decay was fit to the Kohlrausch-Williams-Watts (KWW) stretched exponential function

Nonexponential Relaxation of Poly(cyclohexyl acrylate)

exp[-(t/τ)β]

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(2)

to obtain the rotational time constant, τ, and the stretching exponent, β. From τ and β, the average correlation time, τc, can be defined as

τc )

τ 1 Γ β β

()

(3)

2. Single-Molecule Experiments. To ensure that the probe molecules were well-isolated, a dilute solution containing 0.1 nM R6G and 4 wt % PCA was prepared. The solution was spincast onto glass coverslips to make thin films that were dried in a hood prior to observation. The single-molecule experiments were performed using a laboratory-built microscope, as shown in Scheme 2. Transient data were taken at room temperature of 21 °C, so no cryostat was used. A 532-nm diode laser by Compass was focused onto the sample by an oil-immersion objective with a numerical aperture (NA) of 1.2. A quarterwave plate was placed before the objective to change the polarization of the laser from linearly polarized to circularly polarized, so that R6G molecules could be excited independent of orientation. Emission from the dye molecule was collected and collimated by the same objective and separated from the excitation wavelength by a dichroic mirror, a notch filter, and several long-pass filters. The signal was split by a cube beam splitter into two orthogonal polarizations, designated as Is and Ip, on the plane perpendicular to the objective axis. Signals were collected on two Perkin-Elmer avalanche photodiodes. The sample ws mounted on an X-Y piezo-scanning stage manufactured by Queensgate Instruments. An image scan was first taken to locate a single molecule; then, the target molecule was moved into the laser spot for excitation. The power was attenuated 100-1000-fold (∼0.5 µW) for recording transients. The fluorescent signals were collected continuously until the molecule had photobleached. The reduced linear dichroism, A(t), which is a measure of the orientation of the dipole moment, was calculated as

A(t) )

Is(t) - Ip(t) Is(t) + Ip(t)

(4)

As the dipole rotates in space, A(t) fluctuates, and a rotational time constant can be characterized from this timed signal by taking the autocorrelation function of A(t)49 T-t′-1

CA(t) ) 〈A(t′) A(t + t′)〉 )

∑

t′)0 T-1

A(t′) A(t′+t)

∑ A(t′) A(t′)

(5)

t′)0

The autocorrelation function is a measure of the similarity between the signals at two times, t′ and t + t′. As t′ increases, the difference between A(t′) and A(t′ + t) increases, and CA(t) becomes small. In general, the autocorrelation function will resemble an exponential form decaying from 1 to 050,51 and thus can also be fit with the KWW stretched exponential function to obtain τ and β.

III. Results and Discussion 1. Ensemble Experiments. An example of the ensemble experiment data is shown in Figure 1. After photobleaching, ∆I| is larger than ∆I⊥ because probe molecules with dipole moments parallel to the bleaching polarization are depleted more efficiently than those with other dipole orientations. As the molecules reorient, the ensemble becomes more isotropic; ∆I| becomes equal to ∆I⊥; and the anisotropy value, r(t), eventually decays to 0 for an equal distribution of intensities. The anisotropic relaxation is a nonexponential decay that can be readily fit by the KWW equation. All of the ensemble anisotropy measurements had decays with βens ≈ 0.5-0.6 (henceforth, the subscript ens is used to represent fitted results of the ensemble anisotropy relaxation). This is clearly nonexponential, and one explanation for this fact is that the dynamics is heterogeneous.4 The magnitude of β describes the extent of heterogeneity: the smaller the value of β, the more nonexponential the decay, and the more heterogeneous the system. From this perspective, it should be expected that β would be small at low temperatures and approach closer to 1 at higher temperatures, where one might expect the heterogeneity to vanish. The current experiments did not demonstrate a strong correlation between the magnitude of β and temperature. However, one might not expect a large variation in β over the limited temperature range that was probed. The parameter τc corresponds to the average rotation time for the unbleached R6G molecules in PCA. The probe dynamics are very sensitive to the local environment; therefore, the probe responds to minor variations in the host polymer structure. Figure 2 is a plot of temperature dependence of the measured average relaxation time for the ensemble of probe dyes, τens, which can be well described by the Williams-Landel-Ferry equation

log(τens) ) A -

C1(T - Tr) (C2 + T - Tr)

(6)

where A, C1, and C2 are empirical coefficients and Tr is a reference temperature (typically chosen to be Tg). The WLF equation is mathematically equivalent to the frequently applied Vogel-Fulcher-Tamman equation and many other empirical equations that are used to describe polymer viscosity and relaxation. This equation implies that rotational dynamics become orders of magnitude slower as the temperature approaches Tg and that, at C2 degrees below the reference temperature, the dynamics diverge.2 The fit with Tg chosen as the reference temperature yields C1 ) 12 and C2 ) 85. This shows a slightly shallower temperature dependence than has previously been reported for PCA.52 Nonetheless, the fact that the rotational FRAP results show a trend consistent with the WLF equation confirms that the rotation of the probe molecules is dependent on the dynamics of the host polymer matrix. The anisotropy measurements were repeated for three PCA/ R6G samples made at different times with different dye concentrations. For each of these samples, there was a slight shift in the temperature dependence of the dynamics. As such, the temperature for each sample was normalized to yield a consistent WLF trend for each sample. This difference might have arisen from the individual sample histories as a result of the preparation procedure and the time for which the sample had been exposed to the ambient environment. Trace amounts of solvent trapped in the melt could also result in faster dynamics. Even though the same preparation procedure was

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SCHEME 2: Schematic Representation of the Instrumental Setup for Single-Molecule Experimentsa

a The quarter-wave plate is used to change the linearly polarized laser beam to circular polarization. The objective focuses the excitation light onto the sample and also collects fluorescent photons from the sample. The X-Y piezo stage allows for image scans across the sample area to locate single-molecule positions. The dichroic mirror and notch filter are used to separate excitation photons from the fluorescence signal. The CCD camera is confocal with the sample to help with initial alignment of the laser focus. The fluorescent signal is split by a polarizing beamplitter onto two avalanche photodiodes (APDs).

Figure 2. Anisotropic relaxation times of poly(cyclohexyl acrylate) as a function of temperature. The line is the fit to the WLF equation with Tg ) 292 K.

Figure 1. Example of ensemble FRAP experimental raw data. ∆I denotes the difference between signals measured before and after photobleaching in the parallel and perpendicular directions, which are emissions from the unbleached molecules. Anisotropy was calculated as r(t) ) ∆I| - ∆I⊥/∆I| + 2∆I⊥. The dashed line is the fit to the KWW stretched exponential function.

followed in each case, the conditions varied slightly. The WLF trend is used to compare rotational times for the various samples. For example, at Tg + 2 °C, the measured rotational times are 124 s for PCA sample 1, 116 s for PCA sample 2, and 157 s for PCA sample 3. Whereas the statistical errors for the time constant that result from the stretched exponential fit are quite small, the time constant has a large covariance with the value of β. The correlation times are reported to the nearest 1 s, but they could have errors as large as (10 s if their corresponding β values are allowed to change. It is evident that the rotational times measured have the same order of magnitude at Tg + 2 °C across different samples, and this also shows that the polymer dynamics measurements are not strongly affected by the concentration of the probe molecules.

Previous rotational FRAP experiments have displayed a dependence of the relaxation time on the bleaching depth.29,36 This was the result of a preferential photobleaching of molecules with faster rotation times. As a result, the deeper the bleach, the slower the resulting dynamics. In the current study, a wide range of bleaching depths from 10% to 40% was studied. No trend between bleaching depth and the resulting rotation times was observed. This is likely the result of a difference in the photobleaching mechanism between R6G and the small aromatic molecules used in the previous studies.29,36 2. Single-Molecule Experiments. As discussed in the previous section, the ensemble experiments revealed a nonexponential anisotropic relaxation and measured the average rotational times for the system at various temperatures. To gain further understanding of the origins of the inhomogeneous dynamics, single-molecule spectroscopy was utilized to study the rotational motions of single R6G molecules in PCA. With the experimental design, fluctuations in the orientation of the emission transition dipole of a molecule are reflected in the reduced linear dichroism, which fluctuates between +1 and -1 as the molecule rotates. As a result of the finite numerical aperture (NA) of the optics, the fluorescence signal collected was never 0 on one

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Figure 3. Example of single transient data: (a) orthogonal fluorescent signals, Is and Ip; (b) reduced linear dichroism signal, A(t); (c) autocorrelation function, CA(t), of the dichroism (plotted on a logarithmic scale for clarity).

detector or the other. Therefore, the dichroism never reached either limit of (1.16 Because A(t) is a fluctuating signal that varies around a well-defined mean, we can extract the periodic component associated with the rotational dynamics by taking the autocorrelation function of A(t). Theoretically, for isotropic diffusion, CA(t) can be expressed in terms of spherical harmonic functions

CA(t) ) ΣalCl(t) ) l

∑ ale-l(l+1)Dt

(7)

l

where l labels a Legendre polynomial, Pl, and its corresponding coefficient, al, and the rotational diffusion constant is D.49,53 When taking into account the effects from using a high-NA objective, as in a single-molecule experiment, the resulting correlation function, CA(t), is dominated by the second harmonic term, C2 ∼ exp(-6Dt).54 Thus, the observed correlation function in the single-molecule experiments would be essentially a singleexponential decay if the system exhibited only diffusive motion.16 The l ) 2 correlation function, C2 ) exp(-6Dt), measured in the single-molecule experiments is exactly the same molecular correlation function as is measured in the FRAP experiments.11,47 Therefore, the result for the single-molecule correlation function, τsm, should be directly comparable to τens, the rotation constant calculated from the ensemble measurements.

Figure 3 shows an example of the single-molecule transient data (Figure 3a), the dichroism signals (Figure 3b), and the correlation function CA(t) (Figure 3c). The correlation function was fit with the KWW stretched exponential function to obtain βsm and τsm. It should be noted that the fitting result is sensitive to fitting criteria and can vary slightly when given different constraints. When the KWW function is used to characterize dynamics, β is usually fit in the range of 0 < β e 1. Statistically speaking, there is no need to constrain β within unity. Having β > 1 is not a physical result, but could arise as a result of statistical fluctuations from estimating the correlation function.55 The result of confining β e 1 is forcing all β values greater than 1 to exactly equal 1. As such, for the least-squares fitting, β was fit without constraint so that the distributions of β would accurately reflect the statistical fluctuations. In addition, because of the finite length of the transient, the calculated autocorrelation function was essentially zero after a certain time lag q, beyond which time the standard errors were larger than the value of the correlation function. When fitting each transient, the fitting region was restricted to times between t ) 0 and t ) q rather than the full time range.50 The transient shown in Figure 3 yields a correlation function that is fit up to its statistically significant lag time of 500 s. The decay is well represented by the stretched exponential fit with τsm ) 542 s and βsm )0.50. A set of 58 single transients of R6G in PCA film were collected at room temperature of 21 °C, which is 2 °C above

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Figure 4. Select single-molecule rotational correlation functions. Note the logarithmic scale for the time axis. All transients have different correlation functions with different decay times. None are well fit by a single-exponential decay.

Figure 5. Distributions of fitted τsm/〈τsm〉 and βsm values for a set of 58 single-molecule transients (bar). Average values are 〈τsm〉 ) 149.22 s, 〈βsm〉 ) 0.78. The overlapping lines are the distributions from simulated transients of T ) 12.75τ1 for comparison.

Tg of PCA. Figure 4 shows a typical collection of such correlation functions. Each transient has a correlation function that has a distinguishable rotational time. In addition, they are not all single-exponential functions. When the correlation functions of all molecules are compared, a wide distribution for βsm and τsm is observed, as shown in Figure 5. Note that τsm is normalized to the average, 〈τsm〉. Referring back to the previous discussion, there are two possible underlying schemes that can lead to the same final result of nonexponential decay in an ensemble measurement: the heterogeneous case, composed of pure diffusions on different time scales, and the homogeneous case, where all microenvironments have identical nonexponential relaxations. From the distribution of single-molecule rotational correlation functions that were obtained, the system can be described as inhomogeneous because each of the molecules is distinct; however, the correlation functions for the individual molecules are not single exponentials. This is very similar to the observation by Schob et al. for poly(methylacrylate) in which

a distribution of stretching exponents was found for the rotational correlation functions of all of the single molecules measured.19 3. Comparison of Single-Molecule and Ensemble Results. To make comparisons between single-molecule and ensemble experiments, it is critical to relate the time scales measured in each experiment back to the molecular motions. As previously described, the high NA of the single-molecule collection optics creates a situation in which the single-molecule correlation functions can be estimated by the l ) 2 term of the full rotational correlation function, C2 ∼ exp(-6Dt). The decay measured in the bulk anisotropy experiments is also given the by C2(t). Therefore, the single-molecule and ensemble FRAP decays can be compared directly, as they reflect the exact same molecular correlation function. For the single-molecule experiments, the average values for the 58 transients were calculated from individual fits of the correlation function. The averages of the individual times and

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Figure 6. Comparison of ensemble anisotropy decay at Tg + 2 °C (solid line) and the sum of all single-molecule correlation functions (dashed line). When fit with the KWW stretched exponential function, the ensemble decay yields τen,sum ) 132.69 s, βen,sum ) 0.58; for the single molecules, τsm,sum ) 84.66 s, βsm,sum ) 0.51.

β values yielded 〈τsm〉 ) 149 s and 〈βsm〉 ) 0.78, respectively. The ensemble average of the correlation time is 〈τc,sm〉 ) 221 s. When β was constrained by 0 < β e 1, the fitting results were very similar, with 〈τsm〉 )151 s, 〈βsm〉 ) 0.76, and 〈τc,sm〉 ) 230 s. The difference is trivial and affects only the shape of the β distributions, in that all values of βsm > 1 collapse to βsm ) 1 when the fitting is constrained. The average ensemble relaxation time for the PCA samples measured in the FRAP experiments at Tg + 2 °C yielded an average time constant of 〈τc,ens〉 ) 219 s, which agrees with the average of all single molecules, 〈τc,sm〉 ) 221 s, especially given that the error is likely to be (10 s and the temperature of the cryostat is reproducible to only (0.5 K. The average β value from the ensemble experiments is 〈βens〉 ) 0.56, which is smaller than the single-molecule result, 〈βsm〉 ) 0.78. The discrepancy could be due to the fitting of the single-molecule correlation functions, where statistical errors can bias the fitting results when transient lengths are not sufficient (as discussed in the next section). Nevertheless, the fact that β from both experiments does not equal 1 implies that the dynamics are not homogeneous in the polymer. Moreover, the smaller β value for the ensembleaverage data is evidence of heterogeneous dynamics. If the distribution of β arose simply from statistical fluctuations, one would expect the ensemble average to have a β value closer to 1. The opposite behavior is observed. By averaging all of the single-molecule correlation functions together, one obtains an “ensemble” of single molecules that can be compared to the dynamics measured by the FRAP experiments. As depicted in Figure 6, the single-molecule plot (dashed line) is the sum of all single-molecule correlation functions. The ensemble decay (solid line) is the average of the FRAP anisotropy decays from three samples at Tg + 2 °C. The two decays nearly overlap, and when fit to the stretched exponential function, the sum of single-molecule correlation functions, C(t)sum, yields τsm,sum ) 85 s and βsm,sum ) 0.51 (τc,sm,sum ) 163 s); the sum of ensemble anisotropy decays, r(t)sum, yields τens,sum ) 133 s and βens,sum ) 0.58 (τC,ens,sum ) 209 s). Because of the similarity of the two decays, it is clear that the sum of all single molecules creates an ensemble-average result that is nearly identical to the ensemble relaxation measured in the FRAP experiments. The comparable decays observed using the two methods demonstrate that the same dynamics are being measured in both experiments and further confirms that the single-molecule experiments probe a subset of molecules that are representative of the full ensemble. Although the singlemolecule ensemble decay and the FRAP ensemble decay yield very similar results, as shown previously, if one averages the

decay times and β values for the single molecules (rather than the correlation functions), the results are very different. Fitting C(t)sum gives τsm,sum and βsm,sum values that are smaller than the averages of independently fitted trajectories, 〈τsm〉 and 〈βsm〉. This effect can be understood by the fact that the summation of a series of single-exponential functions (β ) 1) with different decay times (τ) does not yield a single exponential. Because the single-molecule transients all have different τ and β values, it can be expected that the sum of all correlation functions will be more “stretched” than the individual decays. Conversely, if the distributions of correlation functions were purely a result of statistical fluctuations, then one would expect the correlation functions to be normally distributed about the true correlation function. The effect of averaging many samples would be that the average correlation function would converge toward the true correlation time and β value. In the case that this correlation function is purely exponential, the averaged correlation functions should become less, rather than, more stretched. 4. Statistical Errors Associated with Rotational Correlation Function Analysis. Even though single-molecule experiments reveal a broad distribution of measured properties, the question of whether the system is heterogeneous is not readily answered. It has been shown using simulated data that, for a pure rotation system, if the observed single transients have very short lengths, the correlation functions will give distributions of τ and β values even though there is only one true rotational constant, and all correlation functions should be exponential. Moreover, these distributions broaden as the transient length shortens.55 This is an inherent statistical error associated with finite sampling, which will propagate to the other parameters when characterizing the correlation function. Thus, one needs to be careful when analyzing single-molecule correlation functions; to properly determine whether the system is heterogeneous, the quality of the data needs to be evaluated. The width of the distribution of values measured should be compared to the natural width arising purely from statistics to determine whether the heterogeneity truly exists. To compare distributions, a χ2 test is performed to see if the standard deviation of experimental data is the same as that of the simulated data. If the χ2 test value is greater than the upper critical value χR2 or smaller than the lower critical value χ1-R2 at significance level R, then the standard deviation of the experimental data is greater that of the simulation. If the test value falls between the upper and lower critical values, then the two standard deviations are not statistically different from each other.56 The simulation has only one rotational constant, τ1, and yields a single-exponential correlation function for an

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infinitely long trajectory. A trajectory of 106 time points was calculated and broken into shorter lengths. The correlation function for each of these segments was calculated, and each decay was fit to a stretched exponential. The average ratio of the transient length, T, to τsm was found to be 12.75 for the set of single-molecule data; hence, the statistics arising from simulations with transient of length T ) 12.75τ1 was used for comparison. These segments were calculated to determine the statistical distributions of τ and β that would arise from trajectories of the length measured in the experiments.55 To directly compare the resulting values to the simulation, the single-molecule data were fit with no restrictions on the maximum value of β. Figure 5 shows the distribution of time constants for the single-molecule correlation functions with the decay times normalized to their ensemble average, τsm/〈τsm〉. This distribution has a standard deviation of S(τsm) ) 1.0801. The distribution of time constants calculated from the diffusion simulation is narrower than the experimental result, with a standard deviation of S(τsm) ) 0.823. The calculated one-way χ2 test value for τ is

[ ]

P(τ) ) (N - 1)

S(τsm) S(τsim)

2

) 98.157

(8)

where N - 1 is the number of degrees of freedom. The upper critical χ2 value for the 0.1% significance level is χ0.0012 ) 95.751. Therefore, the single-molecule data have a different distribution of rotational constants than the simulated distributions. The calculated χ2 test value for β yields P(β)) 20.425, which falls below the lower critical value of χ0.9992 ) 29.592. Therefore, the distribution of βsm is not the same as the simulated distribution either. A plot of the distributions of simulated transients along with the experimental data is shown as the dark lines in Figure 5. The differences are immediately apparent. The distribution of τsm for the single-molecule data is peaked at very small times and tails off at a larger extreme than the simulated data. The distribution of βsm for the single-molecule data is not centered at 1, whereas the simulated transients have a broader range of β values that is slightly peaked at β > 1. It should be mentioned that, for simulated transients, despite the trajectory lengths and the broad distribution, the average β value is always ∼1 (for these simulated T ) 12.75τ1 transients, 〈β〉 ) 1.030); for the experimental transients, 〈βsm〉 ) 0.780. This basic comparison demonstrates that the single-molecule results are very different from pure diffusion. However, the transient lengths are only on the order of 10 times the average rotational time, which unfortunately results in inherently greater statistical errors than the ideal case of an infinitely long trajectory. In practice, fluorescent probe molecules cannot be observed infinitely because of irreversible photobleaching. Hence, it is very difficult to obtain transients that have sufficient lengths (at least 100 times longer than the true rotational time)55 to reduce the impact of statistical errors. Previously, efforts have been made to directly characterize the exchange time from single-molecule transients.14 This was done by examining the length of time that the system displayed a consistent rotation time as judged by the distribution of angular jumps. However, in the current data, it is not possible to distinguish such discrete changes in dynamics. Moreover, such changes appear even in the simulation of homogeneous diffusion. This is essentially the same problem as the transient length. Rigorous establishment of a consistent rotational diffusion constant from a distribution of angular jumps requires that the

exchange time be very long compared to the rotational time. It is not currently possible to distinguish between the data and the simulation in a robust fashion using the angular changes as a measure of the exchange time of the heterogeneous environments. Recently, a new statistical method was published that analyzes finite trajectories in both the time and frequency domains, and it could be very useful in determing the exchange time despite the limitations in trajectory length.57 IV. Conclusions This article compares polymer rotational dynamics using an ensemble FRAP technique and single-molecule spectroscopy. The two types of experiments yield similar reorientation times for an R6G dye probe in a matrix of poly(cyclohexyl acrylate) at 2 °C above its glass transition temperature. The ensemble FRAP measurements show the rotational correlation function to be strongly exponential, which suggests that the polymer dynamics are not homogeneous. The single-molecule studies reveal that the rotational correlation function for each probed molecule is distinct. Rotational times vary from molecule to molecule, and the single-molecule decays are also nonexponential with a distribution of β values. This is in contrast to recent single-molecule results for supercooled glycerol, in which nearly all of the single-molecule correlation functions were found to be single exponentials (although with a much larger probe molecule).15 In the PCA system, all single-molecule trajectories are nonexponential decays. The single-molecule and ensemble FRAP experiments yield very similar results. The average of all rotational constants in the single-molecule experiments is the same as that measured in the ensemble FRAP experiments. However, the β values for the individual singlemolecule correlation functions are always closer to unity than the ensemble value, suggesting that the system is heterogeneous. This conclusion is supported by the fact that the average of the single-molecule correlation functions yields an ensemble correlation function that is nearly identical to that measured in the FRAP experiments. The β value for this ensemble-average single-molecule correlation function is also 0.55, suggesting that a portion of the distribution of time scales observed in the ensemble experiments results from a distribution of molecular environments in the polymer. It is also shown by comparison with a pure diffusion simulation that the distribution of time constants and β values is not the result of statistical fluctuations that arise from estimating the correlation function from finite trajectories. The rotational dynamics in the system appear to be heterogeneous, but the underlying dynamics for each single molecule do not appear to be simple diffusion. Acknowledgment. This work was supported by the National Science Foundation (CHE-0316088) and the Welch Foundation (F-1377). References and Notes (1) Ediger, M. D.; Angell, C. A.; Nagel, S. R. J. Phys. Chem. 1996, 100, 13200. (2) Angell, C. A.; Ngai, K. L.; McKenna, G. B.; McMillan, P. F.; Martin, S. W. J. Appl. Phys. 2000, 88, 3113. (3) Phillips, J. Rep. Prog. Phys. 1996, 59, 1133. (4) Sillescu, H. J. Non-Cryst. Solids 1999, 243, 81. (5) Ediger, M. D. Annu. ReV. Phys. Chem. 2000, 51, 99. (6) Lubchenko, V.; Wolynes, P. G. Annu. ReV. Phys. Chem. 2007, 58, 235. (7) Stickel, F.; Fischer, E. W.; Richert, R. J. Chem. Phys. 1995, 102, 6251. (8) Stickel, F.; Fischer, E. W.; Richert, R. J. Chem. Phys. 1996, 104, 2043.

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