Translational and Rotational Motion of Probes in Supercooled 1,3,5

Nov 14, 1996 - Connection of translational and rotational dynamical heterogeneities with the breakdown of the Stokes-Einstein and Stokes-Einstein-Deby...
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J. Phys. Chem. 1996, 100, 18249-18257

18249

Translational and Rotational Motion of Probes in Supercooled 1,3,5-Tris(naphthyl)benzene F. R. Blackburn, Chia-Ying Wang, and M. D. Ediger* Department of Chemistry, UniVersity of Wisconsin-Madison, 1101 UniVersity AVenue, Madison, Wisconsin 53706 ReceiVed: July 22, 1996; In Final Form: September 17, 1996X

Measurements of the rotational and translational motion of several probe molecules in supercooled 1,3-bis(1-naphthyl)-5-(2-naphthyl)benzene (RRβ-TNB) are reported. Rotational correlation times from 10-10 to 103 s were measured in the temperature range from 341 to 515 K (Tg ) 343.5 K). Translational diffusion coefficients ranged from 10-6 to 10-15 cm2/s over a similar temperature range. The rotational correlation times of all probes have essentially the same temperature dependence as the viscosity of TNB. In contrast, the translational diffusion of the smallest probe, tetracene, is observed to have a significantly weaker temperature dependence. At Tg, the translational diffusion coefficient of tetracene is 300 times larger than expected based on its rotational correlation time. Similar enhancements of translational diffusion near Tg have recently been observed in other glass-forming liquids. These observations suggest that dynamics in TNB and other supercooled liquids are spatially heterogeneous. The probes used in this study were tetracene, rubrene, 9,10-diphenylanthracene, and 9,10-bis(phenylethynyl)anthracene.

Introduction While systematic studies of the glass transition and the properties of supercooled liquids began at least seven decades ago,1 the last decade has seen a resurgence of interest in these problems.2 Relaxation processes in fragile supercooled liquids are characterized by non-Arrhenius temperature dependences and nonexponential relaxation functions.2 Despite vigorous effort, the fundamental origin of these and other features of fragile liquids is still in dispute. For example, nonexponential relaxation functions can be interpreted in two different ways. One can imagine that a heterogeneous set of environments exists in a supercooled liquid; relaxation in a given environment is nearly exponential but the relaxation time varies significantly among environments. Alternatively, one can imagine that supercooled liquids are homogeneous and that each molecule relaxes nearly identically in an intrinsically nonexponential manner. Of course, these two viewpoints are extreme positions, and it is possible that elements of both pictures are applicable. We report here measurements of the rotational and translational motion of several probe molecules in supercooled 1,3bis(1-naphthyl)-5-(2-naphthyl)benzene (RRβ-TNB, Tg ) 343.5 K). As explained below, these measurements suggest that nonexponential relaxation functions in fragile supercooled liquids are at least partially the result of a heterogeneous set of environments. Rotational correlation times from 10-10 to 103 s are reported here for probe molecules in liquid and supercooled TNB; these data were acquired between 341 and 515 K. We also report translational diffusion coefficients from 10-6 to 10-15 cm2/s for some of the same probe molecules over a similar temperature range. The rotational correlation times of all probes have essentially the same temperature dependence as the viscosity of TNB. In contrast, the translational diffusion of the smallest probe, tetracene, is observed to have a significantly weaker temperature dependence than the viscosity. At Tg, the translational diffusion coefficient of tetracene is 300 times larger than expected on the basis of its rotational correlation time. X

Abstract published in AdVance ACS Abstracts, November 1, 1996.

S0022-3654(96)02204-6 CCC: $12.00

Background. Molecular motion in liquids is often compared to the predictions of the Debye-Stokes-Einstein3 (DSE) and Stokes-Einstein4 (SE) equations. These equations describe the rotational and translation motion of a sphere of radius rs in a hydrodynamic continuum with viscosity η and temperature T. The DSE equation predicts the rotational correlation time τc to be

τc )

4πηrs3 3kT

(1)

Similarly, the SE equation predicts that the translational diffusion coefficient is

DT )

kT 6πηrs

(2)

These relationships are remarkably useful for describing the motions of molecules surrounded by other molecules of similar or smaller size. In particular, experimental results5 for translation and rotation in liquids above their melting points are usually in good agreement with the temperature dependences predicted by eqs 1 and 2, i.e., τc ∝ η/T and DT ∝ T/η. In 1992, Sillescu and co-workers reported a striking observation concerning rotational and translational motion in supercooled o-terphenyl (OTP).6 Rotation of neat OTP followed the temperature dependence of the DSE equation as the viscosity was changed by 12 orders of magnitude. Over the same temperature range, translational diffusion of a probe similar in size to OTP showed more than 2 orders of magnitude deviation from the temperature dependence of the SE equation. Translation measurements on neat OTP could only be performed over a more limited range, but over this range agreed with the probe translation measurements and showed enhancement of a factor of 4. Recently Cicerone and Ediger extended these results, comparing the rotational and translational motion of four probes in OTP.7 This work allowed the translation and rotation of the same molecules to be compared near Tg. These results confirmed the dramatic enhancement of translational motion © 1996 American Chemical Society

18250 J. Phys. Chem., Vol. 100, No. 46, 1996 inferred by Sillescu and co-workers and showed that this effect was quite sensitive to probe size. Is enhanced translation near Tg unique to OTP? Or is this a general feature of low molecular weight glass-formers, fragile glass-formers, or even perhaps all glass-formers? Recent work has confirmed this effect in two fragile polymeric glassformers.8,9 Indirect evidence for this effect in salol, a low molecular weight fragile glass-former, has also been presented.10 To provide an unambiguous test for enhanced translation in a low molecular weight glass-former other than OTP, we have performed rotation and translation experiments on probe molecules in supercooled RRβ-TNB. TNB is a fragile liquid and has been studied extensively.11-21 We observe that probe translation and rotation in TNB are qualitatively similar to the results reported for OTP. For example, in each matrix, the translational diffusion coefficient of tetracene is about 300 times larger at Tg than expected on the basis of the rotational correlation time. These results, when combined with previous work, indicate that enhanced translational diffusion may be a common or universal feature of fragile glass-formers. Enhanced translational diffusion is most reasonably explained if dynamics in these systems are spatially heterogeneous and if this heterogeneity becomes more prominent as the temperature is lowered toward Tg. Thus we envision that near Tg rotational and translational motion might be significantly faster in one region of space than in another region perhaps 5 nm away. In such a system, the different temperature dependences for probe translation and rotation arise because the translation and rotation experiments average dynamics in very different ways. A strong argument in favor of this interpretation is the correlation observed between the enhancement of translational motion and the degree of nonexponentiality observed in the rotational correlation function. On the basis of these results, we suggest that spatially heterogeneous dynamics play a significant role in many glass-forming materials near Tg. Experimental Section Samples. The structures of the probe molecules used in this study are shown in Figure 1. Rubrene, tetracene, and 9,10diphenylanthracene (DPA) were purchased from Aldrich. 9,10-Bis(phenylethynyl)anthracene (BPEA) was received from 3M. All probes were used without further purification. The host material used in this study was 1,3-bis(1-naphthyl)5-(2-naphthyl)benzene (RRβ-TNB). Samples of RRβ-TNB were generously provided by Dr. J. H. MaGill13 (University of Pittsburgh) and by Dr. R. J. McMahon22 (University of Wisconsin). The two samples were synthesized via different pathways, but NMR studies have shown them to be the same isomer.22 Previous papers published on 1,3,5-tris(naphthyl)benzene have identified the isomer as 1,3,5-tri-R-naphthylbenzene (a.k.a. 1,3,5-tris(1-naphthyl)benzene or RRR-TNB).11-18 Whitaker and McMahon22 have determined that the TNB used in many of these studies was RRβ-TNB and not RRR-TNB. It has not been possible to test the TNB used in the viscosity measurements of MaGill and Plazek11-15 since none of this material remains. Based on Tg and Tm values for the TNB isomers, it seems likely that the TNB used in the viscosity measurements was also the RRβ isomer. We will assume in this paper that the isomer used for the viscosity measurements and all previous studies was RRβ-TNB. Below we will refer to this compound as simply TNB. The glass transition temperature Tg of TNB was measured to be 343.5 ( 0.5 K using differential scanning calorimetry (midpoint convention, heating rate of 10 K/min).23 This is

Blackburn et al.

Figure 1. Structures of RRβ-TNB and the probe molecules used in this study. The double-headed arrows represent the electronic transition dipoles of the probe molecules.

consistent with values reported in the literature.13,16 The melting point was determined to be 465 ( 1 K.24 Samples were prepared by adding one of the probes dissolved in benzene to a 0.8 or 3.0 mm cuvette. The benzene was removed under a partial vacuum, leaving probe molecules on the cuvette surface. Next, powdered TNB was added to the cuvette, and the sample was placed under vacuum for about 1 h at room temperature. Still under vacuum, the sample was then heated to 483 K for ∼4 h to allow the probe molecules to dissolve and diffuse into the TNB. Using this method, we could reliably produce samples of good optical quality with optical densities ranging from 0.05 to 0.3. Probe concentrations in TNB were 10-80 ppm. We verified that these low concentrations did not affect the measured Tg. Because data were acquired near and below the calorimetric Tg, aging of the sample was a serious consideration. The data reported in this paper were collected on samples in the equilibrium supercooled liquid; that is, experiments were repeated until results no longer depended upon aging time. At the lowest temperature, this required about one day. Temperature control of the sample was typically (0.05 K with absolute temperature known to within 0.3 K. Rotation Measurements: Time-correlated single-photon counting (TCSPC) and a photobleaching method were used to measure probe reorientation in TNB. The two techniques are conceptually similar and have been described in detail elsewhere.25,26 Both techniques use polarized laser light to photoselect an anisotropic subset of probe molecules in the equilibrium supercooled liquid. The orientational randomization of this subset is followed using either absorption or the combination of absorption and fluorescence. The data from both techniques can be used to calculate an anisotropy function, r(t). The anisotropy functions are directly proportional to the orientation correlation functions CF(t). The correlation function measured by TCSPC is27

CF(t) ) r(t)/r(0) ) 〈P2[µˆ a(0)‚µˆ et]〉

(3)

Here P2 is the second Legendre polynomial, and µˆ is the

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Figure 2. Typical anisotropy decay functions from the TCSPC experiment (left curve, bottom axis) and from the photobleaching experiment (right curve, upper axis). The solid curves are KWW fits to the data.

transition dipole moment (subscripts “a” and “e” refer to absorption and emission). The two transition dipoles are almost parallel, and hence the measured correlation function is essentially an autocorrelation. The orientation of the probe molecules’ transition dipole moments are shown in Figure 1. For the photobleaching experiments, µˆ e in eq 3 is replaced with µˆ a. In all cases, we fit the data with the Kohlrausch-WilliamsWatts (KWW) or stretched exponential function:

CF(t) ) e-(t/τ)

β

(4)

The KWW function is used because it fits the data well and has been used extensively for describing relaxation processes in supercooled liquids. The integral of the correlation function provides a model independent reorientation time, τc. This correlation time is given by ∞ τ τc ) ∫0 CF(t) dt ) Γ(1/β) β

(5)

Error bars for log(τc) values reported here are less than (0.05 for tetracene and rubrene and less than (0.08 for DPA and BPEA. TCSPC allows the measurement of rotational correlation times from 100 ps to 10 ns while the photobleaching method measures correlation times from 100 ms to >104 s. Figure 2 shows typical anisotropy decay curves with fits to the KWW equation. The BPEA data on the left (bottom time axis) were acquired with TCSPC, while the rubrene data on the right (upper time axis) were acquired with the photobleaching method. In the photobleaching experiment, the excitation wavelengths used were 488 nm for rubrene and 476 nm for tetracene. Typical laser powers used for photobleaching ranged from 0.1 to 2.0 W/mm2, with bleaching times ranging from 0.02 to 1 s. Excitation and emission wavelengths used for the TCSPC experiments were as follows: DPA (394 nm, 414 nm); BPEA (404 nm, 480 nm); and rubrene (575 nm, 600 nm). Translation Measurements. Translational diffusion coefficients of probe molecules were measured with a holographic fluorescence recovery after photobleaching technique (holographic FRAP).8,28 Two laser beams (writing beams) are crossed in the sample, creating a sinusoidal intensity pattern. Probes in regions of high intensity are preferentially destroyed (about 30% of the probe molecules are bleached in this step). The same two laser beams, attenuated by ∼106, are used to detect translational motion. The phase of one of these reading

Figure 3. Translational relaxation function for tetracene in TNB: T ) 375.4 K, d ) 3.60 µm. The inset demonstrates that the relaxation function decays exponentially. The solid curve is an exponential fit with τT ) 131 s.

beams is modulated relative to the other so that a reading grating is swept across the written grating at a constant velocity. Each time the reading grating is out of phase with the written grating the fluorescence signal due to unbleached probes will be maximized. Conversely, when the reading grating is in phase with the written grating, the fluorescence signal will be a minimum. This modulation of the fluorescence signal eventually decays due to translational motion of the unbleached probe molecules. Because this technique monitors the probe concentration grating with fluorescence, and since only unbleached probes absorb the laser light and fluoresce, the holographic FRAP experiment measures the translation of only the unbleached probe molecules. (Similarly, photobleaching measurements of rotation only detect reorientation of unbleached probes.)29 The experimental observable, c(q,t), is proportional to the spatial Fourier transform of the single-particle van Hove function:

c(q,t) ∞ ) ∫-∞Gs(x,t)eiqx dx ≡ c(t) c(q,0)

(6)

Here q ) 2π/d is the grating wavevector, and d is the grating period. When translational motion is diffusive, i.e., the mean squared displacement of probe molecules is a linear function of time, Gs(x,t) is a Gaussian, and the relaxation function, c(t), decays exponentially:

c(t) ) e-t/τT

(7)

c(t) decays with a characteristic relaxation time τT ) (d2/4π2DT), where DT is the diffusion coefficient. The grating period depends upon the vacuum wavelength λ and the angle θ at which the beams are crossed:

d)

λ 2sin(θ/2)

(8)

This equation applies to copropagating beams. For counterpropagating beams, λ must be divided by the refractive index of the host material.30 In this study, d was varied from 0.138 to 350 µm. Figure 3 shows typical data for the translation of tetracene in TNB. The solid curve is a single-exponential fit. There are

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Figure 4. Translational relaxation times as a function of grating period squared for tetracene in TNB at T ) 375.4 K (DT ) 2.3 × 10-11 cm2/ s). The solid line is a linear least-squares fit to the data. The proportionality of τT and d2 is expected for diffusive transport.

two tests that can be used to determine whether probe translation is diffusive. Diffusive transport leads to an exponential decay of c(t). The inset of Figure 3, which plots log c(t) versus time, indicates that this is the case at this temperature. Diffusive transport also implies that the translational relaxation time τT should be a linear function of the square of the grating period. Figure 4 shows that this is the case at this temperature. Since c(t) was observed to decay exponentially at all temperatures,31 and since a linear relation between τT versus d2 was observed at several temperatures, we are confident that all translation observed in these experiments is diffusive in nature. For experiments performed in the counterpropagating geometry (θ ) 180°) a refractive index of 1.72 was used in order to calculate d.32 The translation measurements used excitation wavelengths of 488 nm for rubrene and 476 nm for tetracene. Typical bleaching powers used in these experiments ranged from 0.1 to 2.0 W/mm2. Bleaching times ranged from 0.02 to 5 s. For rubrene and tetracene, photobleaching is believed to occur through an excited state reaction with O2.26 The error bars for log(DT) values reported here are (0.08 for tetracene and (0.12 for rubrene. Sample Degradation. TNB degradation occurred during both rotation and translation measurements after prolonged exposure to high temperatures, leading to an increase in background fluorescence. Surprisingly, this background signal was polarized and thus had the potential of significantly influencing our rotation measurements.33 This problem was circumvented by higher probe concentrations and more highly purified TNB (obtained from McMahon). The error bars noted above account for any inaccuracy associated with sample degradation. The impurity molecules were also observed to translate during holographic FRAP experiments. For samples of tetracene in TNB, we sometimes observed a distinct biexponential decay. This decay was easily fit to extract the translational diffusion coefficient for tetracene. Several checks ensured the accuracy of this procedure. In newly prepared tetracene/TNB samples, the translational relaxation times were observed to decay exponentially; these relaxation times agreed with those extracted from the biexponential fits at the same temperature. We were also able to observe exponential decays in samples that had previously given biexponential decays by increasing the probe concentration significantly. The uncertainty in tetracene diffusion coefficients determined from biexponential fits was about 10%, well within the reported error bars. It was more difficult to determine rubrene diffusion coefficients because these were

Blackburn et al. very similar to those of the impurity; consequently, the error bars on these data are larger. Reversible Bleaching. Photobleaching is usually observed to be irreversible in our experiments; that is, the signal destroyed by the initial bleach does not return. At high temperatures, Tg + 100 K, both tetracene and rubrene exhibited some reversible bleaching. This represents a potential problem since, under these conditions, it is no longer true that the translational motion of the bleached probes is irrelevant. In TNB and some other systems, we have observed that increasing the oxygen concentration in the matrix reduces reversible bleaching. In the few cases where the reversible bleaching could not be eliminated, we fit our data in a different fashion (see below). We are confident that any error associated with reversible bleaching in these TNB measurements is smaller than the reported error bars. When a sample exhibits only permanent bleaching, the signal obtained from the translation experiment depends upon the difference between the fluorescence intensity when the reading grating is out-of-phase, Io(t), and in-phase, Ii(t), with respect to the written grating: ∆I(t) ) Io(t) - Ii(t). The average fluorescence intensity is simultaneously measured, A(t) ) [Io(t) + Ii(t)]/2. In the absence of reversible bleaching, c(t) is proportional to ∆I(t)/A(t).34 If reversible bleaching is contributing significantly to the grating relaxation, both the in-phase and the out-of-phase components of the fluorescence are affected. To approximately compensate for this effect, we subtract the average fluorescence intensity before the bleach, Ao, from both components. The resultant equation for c(q,t) is

c(q,t) )

∆I(t) [A(t) - A0]

(9)

Our kinetic analysis indicates that this expression is exact if (1) residual bleaching can be ignored and (2) the bleached and unbleached probes have identical diffusion coefficients. Results and Discussion A. Rotation in TNB. Figure 2 shows the general features of our experimental results. Here anisotropy decays for rubrene near Tg (upper axis) and BPEA far above Tg (lower axis) are shown. The dark curves are fits to the KWW function with β values of 0.74 and 0.90 for rubrene and BPEA, respectively. The KWW function provided a reasonable fit to the data for all probes at all temperatures. Temperature Dependence of Correlation Times. Figure 5 shows the rotational correlation times τc for tetracene and rubrene near Tg. The solid curve is the temperature dependence of η/T,11-13,15 vertically shifted to overlap with the rubrene reorientation data. The rotational correlation times of these probes follow the temperature dependence of η/T (as predicted by the DSE equation) very well over this temperature range. Some previous studies in low molecular weight glass-formers suggested that probes of different sizes rotate with the same relaxation times near Tg.35 Here, we observe that the larger probe, rubrene, rotates approximately 0.8 decades slower than tetracene over the entire range studied. Also shown in Figure 5 are the results of 2D solid state 13C NMR16 and photon correlation spectroscopy17 (PCS) measurements made on neat TNB. The NMR experiment measures the reorientation of C-H bond vectors; in principle, both overall molecular reorientation and the internal rotation of naphthalene groups contribute to this signal. PCS measures a collective relaxation time associated with molecular reorientation. As is the case in the photobleaching experiments, the NMR and PCS

Supercooled 1,3,5-Tris(naphthyl)benzene

Figure 5. Rotational correlation times for rubrene and tetracene in TNB. The solid curve is log η/T vertically shifted to coincide with the rubrene data. Error bars are smaller than the symbol size. NMR and PCS relaxation times for neat TNB are also shown.

Figure 6. Rotational correlation times for probe molecules in TNB. The low temperature data have been reduced to a representative number of data points for clarity. The solid curve is the temperature dependence of η/T vertically shifted to coincide with the low-temperature rubrene data. Rubrene rotation follows η/T for over 11 decades in time.

measurements near Tg follow the temperature dependence of η/T fairly well. Correlation times for TNB from the NMR measurement and those for rubrene from the photobleaching measurement are similar. This is expected since these two molecules are similar in size (456 g/mol (TNB) versus 532 g/mol (rubrene)). The slightly longer relaxation times for TNB should not be overinterpreted given that the temperature stability cited in ref 16 for the NMR measurements is (1 K. A temperature difference of 1 K would account for the differences shown in Figure 5. The satisfactory agreement between these correlation times supports our view that probe/TNB interactions are very similar to the interactions among TNB molecules. Figure 6 shows selected data from the previous figure plus high temperature rotation data for DPA, BPEA, and rubrene. Also included are the complete NMR and PCS results. The solid curve is again the temperature dependence of η/T, vertically shifted to coincide with the low-temperature rubrene data. Both DPA and BPEA rotation follow η/T at high temperatures. Comparing the high temperature rubrene data point to the low temperature rubrene data, we see a deviation from an η/T dependence of only 0.15 decade. Considering that the viscosity changes by more than 11 decades, this deviation is rather small. Although we do not have high-temperature data for tetracene, we believe it is reasonable that the temperature

J. Phys. Chem., Vol. 100, No. 46, 1996 18253 dependence of rotation for tetracene will also follow the temperature dependence of η/T. In support of this argument, rotational correlation times of tetracene and rubrene in OTP follow the temperature dependence of η/T over a similarly large viscosity range.36 In OTP, the correlation times for these two probes differed by 1.1 decades near Tg, very similar to the 0.8 decade difference observed in TNB. The data discussed in this section are all consistent with the hypothesis that the reorientation of probes in TNB and the reorientation of TNB itself follows the temperature dependence of η/T from Tg - 2 K to Tg + 170 K. β Values. Many relaxation processes in glassy systems are nonexponential and can be fit to the empirical KWW expression (eq 4). This nonexponential behavior implies that these processes are either inherently nonexponential or that different regions of the sample have different relaxation times. The rotation measurements exhibited the following β values: 0.47 ( 0.08 for tetracene, 0.74 ( 0.1 for rubrene at low temperatures, 0.83 ( 0.15 for rubrene at high temperature, 0.90 ( 0.05 for DPA, and 0.88 ( 0.04 for BPEA. At high temperatures, nearly exponential relaxation is observed for rubrene, DPA, and BPEA. Clearly, tetracene and rubrene relax nonexponentially near Tg. Probe rotation in OTP is also observed to be nearly exponential at high temperature and quite nonexponential near Tg.36 Zemke et al.16 fit their NMR data to a log Gaussian distribution and provided KWW β parameters for comparison to other work. A β of ∼0.6 was calculated. Because TNB and rubrene are similar in size, it is expected that their β parameters would be similar; they are fairly similar (0.6 versus 0.74 ( 0.1). The sensitivity of the NMR measurements to internal motion of the naphthalene groups may be at least partially responsible for the difference. The β value obtained from the PCS experiments is 0.55.17 Because the PCS experiment measures a collective orientational relaxation process, it is difficult to compare the PCS results directly to the photobleaching results. Values of r(0) observed in the photobleaching measurements were in the range 0.1-0.2. Since this is significantly less than theoretical maximum of 0.4, it is likely that partial reorientation of rubrene and tetracene occurs on time scales shorter than our bleaching times. Thus the actual relaxation time distribution for probe rotation may be somewhat broader than that implied by the KWW β parameters obtained from fitting the observed anisotropy decays. Values of r(0) observed in the TCSPC measurements were in the range of 0.2-0.32. B. Translation in TNB. Figures 3 and 4 show the general features of the experimental results. Exponential translational relaxation functions were observed for both rubrene and tetracene in TNB, except for a few cases in which sample degradation occurred (see above). This, and the linear relationships observed between translational relaxation times and the grating spacing squared, shows that the observed translational motion is diffusive. Temperature Dependence of Diffusion Coefficients. Figure 7 shows translational diffusion coefficients for tetracene and rubrene in TNB as a function of temperature. Note that -log DT is plotted in order to emphasize the comparison with Figure 6. The solid curve is the temperature dependence of η/T vertically shifted to coincide with the high-temperature rubrene data. Clearly, the translational diffusion of both probes deviates from this temperature dependence. The inset in Figure 7 allows this deviation to be seen more clearly. The product DTη/T is predicted to be independent of temperature by the SE equation. Even at the highest temperatures, DT values for rubrene and tetracene do not follow the temperature dependence of T/η. Over

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Figure 7. Translational diffusion coefficients for rubrene, tetracene and TTI (2,2′-bis(4,4-dimethylthiolan-3-one)). The solid curve is the temperature dependence of η/T, vertically shifted to coincide with the high-temperature rubrene data. The inset tests how well probe translation follows the SE equation. DTη/T (cm2 P (sK)-1) is predicted to be a temperature independent quantity. Clearly, tetracene and TTI do not follow this prediction. Error bars for rubrene and tetracene are smaller than the symbol size.

the temperature range of our measurements, DTη/T varies by 2.5 decades for tetracene and 0.6 decade for rubrene. We have also plotted translational diffusion coefficients of the probe TTI (2,2′-bis(4,4-dimethylthiolan-3-one)) in both parts of the figure. DT values for TTI were measured by Ehlich and Sillescu using forced Rayleigh scattering (FRS).19 The behavior of TTI and tetracene are quite similar, as expected on the basis of their similar sizes (256 g/mol (TTI) versus 228 g/mol (tetracene)). The observation that probe translation in TNB has a weaker temperature dependence than T/η is consistent with studies of probe translation in OTP6,7 and also with measurements on neat OTP.6 Enhanced Translation. Since probe translation has a weaker temperature dependence than probe rotation, probes on aVerage translate farther and farther per rotational correlation time as Tg is approached from aboVe. For example, tetracene translates a root-mean-squared displacement of 1000 Å at high temperatures (Tg + 110 K) in about 80 000 rotational correlation times. At Tg, only about 370 rotational correlation times are required to traverse the same distance. If we assume that the diffusion equation is applicable on molecular length scales, we can calculate that tetracene translates 3.5 Å per rotational correlation time at Tg + 110 K and 52 Å per rotational correlation time at Tg. Thus translational diffusion at Tg is enhanced. Figure 8 is a convenient format for examining the temperature dependence of this enhanced translation. The horizontal axis is inverse temperature normalized to Tg. The vertical axis is the logarithm of the product DTτc normalized to the value of this product predicted by the SE and DSE equations ((DTτc)SE,DSE ) 2rs2/9). The Stokes radius, rs, could be estimated in a number of ways; the choice of method does not have a significant effect on Figure 8. We used values derived from high temperature rotation measurements in o-terphenyl (OTP).36 To obtain rotational correlation times for Tg/T values less than 0.97, we assumed that rotation at these higher temperatures continues to have an η/T dependence. This assumption was justified above and is quite reasonable considering that rubrene rotation is observed to follow the temperature dependence of η/T quite closely over 12 decades. If the SE and DSE equations were obeyed, DTτc would be a constant over the entire temperature range studied. As normalized in this figure, results for all probes would fall on the solid line. Rubrene deviates from this behavior systematically, but

Blackburn et al.

Figure 8. DTτc for rubrene and tetracene in TNB and OTP plotted versus temperature. As the temperature is lowered toward Tg, translational diffusion is enhanced. The dashed and dotted curves are drawn to guide the eye.

only by 0.5 decade. Tetracene, on the other hand, deviates significantly from SE behavior over the entire temperature range. Near Tg translational diffusion of tetracene is enhanced by more than 2.5 decades. Enhanced translation of probe molecules has also been observed in OTP.6,7 DTτc values for rubrene and tetracene in OTP have been reproduced in Figure 8. In both OTP and TNB, tetracene translation is enhanced about 2.5 decades near Tg. Rubrene, on the other hand, shows little enhanced translation in either matrix. C. Interpretation. We begin by summarizing the key observations of our rotation and translation experiments in TNB: (1) Rotational relaxation near Tg occurs nonexponentially; (2) near Tg, the larger probe molecule rotates more exponentially than the smaller probe; (3) probe rotation follows η/T for over 12 decades in viscosity; (4) probe translation does not follow T/η and is enhanced at lower temperatures; and (5) the translation of the larger probe is not enhanced as much as that of the smaller probe. The fourth observation above is the most striking. We know of only two possible explanations for this. One possibility is that probe translation becomes increasingly anisotropic as Tg is approached; at Tg, individual tetracene molecules would have to translate four times their length while reorienting on average only about 45°. The other possibility is that dynamics in TNB are spatially heterogeneous, i.e., that the rotation and translation of probes in some regions of the sample are significantly faster than the motion of probes in other regions perhaps 5 nm away.37 The five observations above have also been made for probes in OTP.6,7,36 In that case, four probes were studied and enhanced translation was observed to be correlated with probe size and not probe shape.7 This observation argues against the anisotropic translation explanation. We assume that the same mechanism is responsible for enhanced translation in both OTP and TNB. A detailed (if still qualitative) explanation of the five observations above in terms of spatially heterogeneous dynamics has been presented in reference 7. Here we summarize the major features of this explanation as it applies to probe motion in TNB. Other related explanations have been presented elsewhere.10,38-41 How can spatially heterogeneous dynamics explain the different temperature dependences for probe translation and rotation? We envision a system in which the dynamics in some regions of the sample are faster than in others. The idea that dynamics in deeply supercooled liquids are spatially heterogeneous has been around for some time, and recently a number

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of experiments have provided independent evidence for its validity.42-46 We assume that the DSE and SE relations are at least roughly obeyed locally (within a given region); that is, a region in which molecules rotate unusually fast is also a region where translation is unusually fast. We also assume the chemical potential for the probe molecule is uniform throughout the system; that is, the fraction of probes in a particular type of region is proportional to the volume fraction of that type of region. In a spatially heterogeneous system, the translation and rotation experiments each average over the heterogeneity in their own way. As explained briefly below, the rotation experiment emphasizes the dynamics in the slowest regions, while the translation experiment emphasizes the dynamics in the most mobile regions. This allows the two measurements to have different temperature dependences. Rotation Experiments. In a heterogeneous system, the orientation correlation function (eq 3) is a superposition of the orientation relaxation functions for the different regions of the sample. Molecules in more mobile regions relax quickly and are responsible for the fast initial decay in CF(t). Molecules in less mobile regions give rise to a long tail in this function. Clearly CF(t) is expected to be nonexponential in a heterogeneous system, in qualitative agreement with the experimental results presented here. Since τc is the integral of the correlation function, it weights regions of slower mobility to a much greater extent than the regions of higher mobility. It is easy to make this argument quantitative if we make the simplifying assumption that rotational relaxation within one region is exponential. Let F(τ) be the normalized distribution of rotation times. In this case the correlation function and correlation time can be written as:

CF(t) ) ∫0 dτ F(τ)e-t/τ ∞

(10)

τc ) ∫0 dt CF(t) ) ∫0 dτ F(τ)∫0 dt e-t/τ ) ∞

∞

∞

∫0∞dττ F(τ) ) 〈τ〉

(11)

If F(τ) is broad, τc is almost completely determined by the longest τ values. How do we understand the observation that larger probes rotate more exponentially than smaller probes? If the characteristic size of a region with similar dynamics is comparable to the size of the larger probe, this probe may simultaneously experience more than one local environment. Larger probes may also rotate slowly enough that the dynamics of the environment might change before reorientation is complete. Either of these mechanisms average over the heterogeneity of the host matrix and would give rise to a more exponential relaxation function. Translation Experiments. The translational diffusion experiment measures the average time required for a probe molecule to traverse a distance comparable to the grating period. If the grating period were comparable to the characteristic domain size, then transport would not be expected to be diffusive. In this case, the translational relaxation function would be nonexponential. Since diffusive transport is observed in our experiments, we infer that probe molecules must visit many domains of differing dynamics before they translate a distance equal to the grating spacing. Thus the grating spacing must be large compared to the characteristic size of the dynamic heterogeneity. How is the long time diffusion coefficient measured in our experiments related to the local diffusion coefficients of individual domains? It is the regions of faster mobility that

allow probes to translate the farthest distance in a given time period and thus contribute most to the relaxation of the written grating. Consequently, translational diffusion measurements emphasize the regions of faster mobility. Two arguments can be made in favor of this conclusion. A two-state model for a heterogeneous medium developed by Zwanzig47 predicts this result. The following heuristic argument, based on an analogy with electrical resistance, indicates that this conclusion is also valid in a system with a continuous distribution of local relaxation times. The time required to traverse a fixed distance is proportional to the overall resistance to transport, Roverall. In three dimensions, this quantity is determined by the local resistance Ri of various regions as follows:

Roverall ∝

1 ∑1/Ri

(12)

Since the local resistance is proportional to local relaxation time τi, we can write

DT ∝

〈〉

1 1 1 ∝ ∝ Roverall ∑Ri τ

(13)

If we combine eqs 11 and 13, and supply the proportionality constant, we find that

DTτc )

(92r )〈τ〉〈1τ〉 2

s

(14)

Equation 14 is in exact agreement with results based on Zwanzig’s two-state model if the domains are large and the time needed to randomize domain dynamics is short. In the limit that F(τ) is a delta function, 〈τ〉〈1/τ〉 is 1 and the prediction of the SE and DSE equations is recovered. If F(τ) is broad, DTτc is larger than the SE/DSE result, indicating that translation is enhanced relative to rotation. If F(τ) is a log-Gaussian with a fwhm of 2.5 decades, DTτc is (380)(2rs2/ 9). Thus log(DTτc/(DTτc)SE/DSE) would be 2.6, comparable to the value observed for tetracene in TNB at Tg. Of course, this result is only illustrative since the derivation of eq 14 is not rigorous. Experimentally, we observe that translational motion has a weaker temperature dependence than rotation and DTτc increases with decreasing temperature. These observations are consistent with eq 14 if F(τ) broadens as the temperature is lowered. As in the case of rotation, we argue that larger probes experience less heterogeneity as a result of their size and/or sluggishness; that is, F(τ) is less broad. Thus we observe that DTτc for rubrene in TNB is closer to the prediction of the SE and DSE equations than is the tetracene result. D. Comparison to Work on Other Glass-Formers. Correlation between Enhanced Translation and Nonexponential Reorientation. If dynamics in TNB and OTP are spatially heterogeneous, and if this heterogeneity is responsible both for nonexponential probe rotation and for enhanced translation, then a direct correlation should exist between these quantities. Figure 9 tests this correlation at Tg for rubrene and tetracene in TNB and OTP,7 and for these same probes in two polymer systems (polystyrene8 and polysulfone9). Data for two other probes in OTP are also included. KWW β values have been used to characterize the nonexponentiality of the rotational correlation functions. Figure 9 shows a strong correlation between the DTτc and the KWW β parameter in all four matrices. Probe molecules whose rotational correlation times decay nearly exponentially

18256 J. Phys. Chem., Vol. 100, No. 46, 1996

Blackburn et al. of subensembles with distinct dynamics on the time scale of the rotational correlation time.45 NMR experiments on poly(vinyl acetate) by Heuer et al.46 and on polycarbonate by Li et al. have been interpreted similarly.48 Chemical kinetics in glassy polymer matrices often show evidence for sites of varying molecular mobility.49-51 Summary

Figure 9. Correlation of enhanced translation (DTτc) with the KWW β parameter at Tg: tetracene/TNB (1); rubrene/TNB (b); tetracene/ OTP (3); rubrene/OTP (O); anthracene/OTP (]); BPEA/OTP (0); tetracene/PS (×); rubrene/PS (/); tetracene/PSF (unfilled hourglass); rubrene/PSF (filled hourglass). This correlation is expected if the dynamics in these systems are spatially heterogeneous.

exhibit very little enhanced translational diffusion (e.g., rubrene/ OTP). On the other hand, probe molecules that experience a heterogeneous environment, as evidenced by a nonexponential orientation correlation function, also exhibit greatly enhanced translational diffusion (e.g., tetracene/TNB). This correlation supports the idea that spatially heterogeneous dynamics are responsible for enhanced translational diffusion near Tg in all of these systems. The enhanced translation observed in OTP and TNB is different in at least one regard; that is, we speculate that selfdiffusion at Tg would be significantly enhanced in OTP but not in TNB. Tetracene in OTP shows dramatically enhanced translation. Tetracene is similar in size to OTP, and these two molecules in OTP have very similar rotational and translational motion over a considerable range of temperature.7,36 Selfdiffusion in OTP has not been measured near Tg, but on the basis of behavior of tetracene, we expect that self-diffusion in OTP would show a 2 decade enhancement at Tg. Rubrene is similar in size to TNB, and these two molecules have similar rotation times near Tg. Since rubrene does not show significantly enhanced translation in TNB at Tg, we anticipate that self-diffusion in TNB at Tg would also not show much enhancement. Other EVidence for Spatially Heterogeneous Dynamics. A number of recent experiments on OTP have been interpreted as indicating spatially heterogeneous dynamics. Recently, Cicerone and Ediger performed photobleaching rotation experiments in which the dynamics of a slow subensemble of probe molecules were selected.43 The fact that this selection is possible indicates that different regions of the sample have different dynamics. The dynamics of this slow subensemble were observed to relax very slowly back to the average dynamics for OTP at that temperature. The time required for reequilibration of local environments was 100-1000 times the rotational correlation time at Tg. Conceptually similar experiments on OTP using multidimensional NMR were recently reported by Bo¨hmer et al.44 They found that the time required for re-equilibration of local environments was similar to the rotational correlation time at Tg + 10 K. Earlier 2H NMR T1 measurements by Schnauss et al. also indicated motional heterogeneity on a time scale at least comparable to the rotational correlation time near Tg.42 Very recent dielectric hole burning experiments by Schiener et al. on propylene carbonate near Tg have also been interpreted as indicating the existence

In this paper, we have described the rotational and translational motion of probe molecules in TNB. For both tetracene and rubrene, translational diffusion has a weaker temperature dependence than reorientation. Thus, as the temperature is lowered toward Tg, on average probe molecules travel further and further per rotational correlation time. The most striking result is that translational diffusion of tetracene in TNB at Tg is enhanced by about 2.5 decades. The similar rotational correlation times for rubrene and TNB argue that rubrene and tetracene act as unbiased reporters of the dynamics of the TNB matrix. These results for probe rotation and translation in TNB are qualitatively similar to recently reported measurements of probe rotation and translation in o-terphenyl.7,36 Enhanced translation has also been observed over a narrower temperature range in neat OTP.6 This supports our position that the enhanced translation observed in TNB is not an artifact of measurements involving probes but an observation fundamentally related to the dynamics of supercooled TNB. The results for TNB and OTP, taken together with evidence for even more dramatically enhanced translational diffusion in polystyrene and polysulfone, indicate that this phenomenon may be generally important in fragile supercooled liquids near Tg. In our understanding, the enhanced translation observed in these materials near Tg must be a result of either extremely anisotropic probe translation or spatially heterogeneous dynamics. Having reasonable arguments for rejecting anisotropic translation and independent evidence for the existence of spatially heterogeneous dynamics in related systems, we concentrate on the second explanation. We envision that the dynamics of some regions of the TNB sample are significantly faster than the dynamics in other regions perhaps 5 nm away. Rotation and translation measurements average over this heterogeneity in different ways. While the rotation measurements emphasize dynamics in the slowest regions, translation measurements are most sensitive to dynamics in the most mobile regions. If the matrix dynamics become more heterogeneous as the temperature is lowered, translational diffusion will have a weaker temperature dependence than rotational motion and be enhanced at low temperature. The existence of spatially heterogeneous dynamics qualitatiVely explains all the experimental results for TNB presented here and also explains enhanced translational diffusion in three other glass-forming systems. One quantitative test of this explanation awaits an accurate description of translational motion in a medium with a broad range of local diffusivities. Simulations of this problem are currently in progress. Acknowledgment. We gratefully acknowledge the receipt of TNB samples from J. MaGill (University of Pittsburgh) and R. McMahon and C. Whitaker (University of WisconsinMadison). We thank Hans Sillescu for information on the refractive index of TNB, and Neil Moe for performing some NMR measurements. Christopher Grayce and Thatcher Root are acknowledged for helpful ideas regarding the consequences of spatially heterogeneous dynamics. The authors thank the NSF (CHE-9322838) for funding.

Supercooled 1,3,5-Tris(naphthyl)benzene References and Notes (1) Kauzmann, W. Chem. ReV. 1948, 43, 219. (2) Ediger, M. D.; Angell, C. A.; Nagel, S. R. J. Phys. Chem. 1996, 100, 13200. (3) Debye, P. Polar Molecules; Dover: New York, 1929; pp 72-85. (4) Einstein, A. InVestigations on the Theory of Brownian Motion; Dover: New York, 1956. (5) See for example: Ravi, R.; Ben-Amotz, D. Chem. Phys. 1994, 183, 385; Artaki, I.; Jonas, J. J. Chem. Phys. 1985, 82, 3360. Bauer, D. R.; Alms, G. R.; Brauman, J. I.; Pecora, R. Ibid. 1974, 61, 2255. Chuang, J.; Eisenthal, K. B. Chem. Phys. Lett. 1971, 11, 368. Paul, E.; Mazo, R. M. J. Chem. Phys. 1968, 48, 1405. (6) Fujara, F.; Geil, B.; Sillescu, H.; Fleischer, G. Z. Phys. B. 1992, 88, 195. (7) Cicerone, M. T.; Ediger, M. D. J. Chem. Phys. 1996, 104, 7210. (8) Cicerone, M. T.; Blackburn, F. R.; Ediger, M. D. Macromolecules 1995, 28, 8224. (9) Hwang, Y.; Ediger, M. D. J. Polym. Sci., Polym. Phys. Ed., in press. (10) Chang, I.; Fujara, F.; Geil, B.; Heuberger, G.; Mangel, T.; Sillescu, H. J. Non-Cryst. Solids 1994, 172-174, 248. (11) MaGill, J. H.; Ubbelohde, A. R. Trans. Faraday Soc. 1958, 54, 1811. (12) MaGill, J. H.; Plazek, D. J. J. Chem. Phys. 1967, 46 (10), 3757. (13) Plazek, D. J.; MaGill, J. H. J. Chem. Phys. 1966, 45 (8), 3038. (14) MaGill, J. H. J. Chem. Phys. 1967, 47 (8), 2802. (15) Plazek, D. J.; MaGill, J. H. J. Chem. Phys. 1968, 49 (8), 3678. (16) Zemke, K.; Schmidt-Rohr, K.; MaGill, J. H.; Sillescu, H.; Spiess, H. W. Mol. Phys. 1993, 80 (6), 1317. On the basis of the work of Whitaker and McMahon (ref 22), we presume that the isomer used in this work was RRβ-TNB and not RRR-TNB. (17) Zhu, X. R.; Wang, C. H. J. Chem. Phys. 1986, 84 (11), 6086. (18) Ma, R. J.; He, T. J.; Wang, C. H. J. Chem. Phys., 1988, 88 (30), 1497. (19) Ehlich, D.; Sillescu, H. Macromolecules 1990, 23 (6), 1600. (20) Fujara, F.; Petry, W. Europhys. Lett. 1987, 4 (8), 921. (21) Bartsch, E.; Debus, O.; Fujara, F.; Kiebel, M.; Petry, W.; Sillescu, H.; MaGill, J. H. Physica B 1992, 180&181, 808. (22) Whitaker, C. M.; McMahon, R. J. J. Phys. Chem. 1996, 100, 1081. (23) Initially a Tg of 345.2 K was measured for samples obtained from both MaGill and McMahon. After heating under vacuum to 423 K for 1 h, Tg was determined to be 343.5 K. Additional heating did not yield further changes in Tg. 13C NMR measurements on the heat-treated samples confirmed that no chemical reaction had taken place; that is, only the RRβ isomer was present. (24) The melting curve for TNB was qualitatively different than those obtained for indium and zinc standards. The onset of melting for TNB was very gradual, covering a temperature range of about 15 K before the heat flow began to rise sharply.

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