Self-Diffusion of Supercooled o-Terphenyl near the Glass Transition

Dec 14, 2005 - Self-diffusion coefficients for the low molecular weight glass former o-terphenyl have been measured near Tg by isothermally desorbing ...
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J. Phys. Chem. B 2006, 110, 507-511

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Self-Diffusion of Supercooled o-Terphenyl near the Glass Transition Temperature Marie K. Mapes, Stephen F. Swallen, and M. D. Ediger* Department of Chemistry, UniVersity of Wisconsin-Madison, Madison, Wisconsin 53706 ReceiVed: September 30, 2005; In Final Form: October 13, 2005

Self-diffusion coefficients for the low molecular weight glass former o-terphenyl have been measured near Tg by isothermally desorbing thin film bilayers of deuterio and protio o-terphenyl in a vacuum chamber. We observe translational diffusion that is about 100 times faster at Tg + 3 K than the Stokes-Einstein prediction. Predictions from random first order transition theory and a dynamic facilitation approach are in reasonable agreement with our results; in these approaches, enhanced translational diffusion is associated with spatially heterogeneous dynamics. Self-diffusion controls crystallization in o-terphenyl for most of the supercooled liquid regime, but at temperatures below Tg + 10 K, the reported crystallization rate increases suddenly while the self-diffusion coefficient does not. This work and previous work on trisnaphthylbenzene both find a self-diffusion-controlled crystal growth regime and an enhancement in self-diffusion near Tg, suggesting that these phenomena are general characteristics of fragile low molecular weight glass formers. We discuss the width of the relaxation time distributions of o-terphenyl and trisnaphthylbenzene as they relate to the observation of enhanced translational diffusion.

I. Introduction The glass transition, which transforms an equilibrium liquid into an amorphous solid, has been the subject of longstanding investigation. The glass transition observed in a typical laboratory experiment is not a thermodynamic phase transition but rather is a kinetic event that occurs when molecular motions slow to the time scale of seconds. As such, our deepest insights into this phenomenon have come from studies of the dynamics of molecules in liquids that are near the glass transition temperature (Tg). Small molecules that easily supercool and form glasses are valuable model systems for the investigation of dynamics near Tg. Dynamics in these systems have been investigated by dielectric relaxation,1-4 dynamic heat capacity measurements,5 light 6,7 and neutron scattering,8,9 single molecule spectroscopy,10 and other techniques.11,12 Interestingly, near Tg very little is known about what is arguably the most fundamental measure of molecular motion, the self-diffusion coefficient. Sillescu et al. used field-gradient NMR to measure self-diffusion of supercooled o-terphenyl13 and several other glass formers,14 but the NMR measurements are limited to above 1.15Tg. The only measurements of selfdiffusion near Tg on a low molecular weight glass former, using forward recoil spectrometry, were published by Swallen et al.12 They found that self-diffusion at Tg for trisnaphthylbenzene was 400 times faster than that predicted by the Stokes-Einstein equation. These results, and previous results for probe diffusion in supercooled liquids,13,15 indicate that self-diffusion coefficients are not easily predicted from other observables but rather provide important new insights into the nature of supercooled liquids. In addition, self-diffusion has practical links to predicting the rate of crystal growth in supercooled liquids, a process with many important implications in material16 and pharmaceutical sciences.17 Here we report self-diffusion coefficients near Tg for o-terphenyl, which is perhaps the best characterized fragile glassforming material.18 Below about 1.2Tg, self-diffusion in o-terphenyl has a substantially weaker temperature dependence than does the viscosity. Our results show enhancement of self-

diffusion in o-terphenyl by a factor of about 100 at Tg + 3 K relative to that expected on the basis of viscosity or molecular reorientation measurements. We compare our results for selfdiffusion in o-terphenyl with the predictions of recent theoretical efforts based upon the random first-order transition theory19,20 and upon a dynamic facilitation approach.21 While both approaches can be viewed as consistent with the experimental results, there are uncertainties in using either of these approaches in their current state of development to predict self-diffusion. The enhancement in self-diffusion reported here for o-terphenyl is similar to that reported earlier for trisnaphthylbenzene when compared at the same T/Tg. For both of these liquids, we review experimental measurements of the temperature dependence of the width of the relaxation time distribution. In contrast to expectations based upon earlier work, significant evidence indicates that a weak temperature dependence for translational diffusion can occur in systems where the relaxation time distribution shows little temperature dependence. We also compare self-diffusion coefficients for o-terphenyl to crystallization kinetics. We find that self-diffusion controls crystallization in o-terphenyl for most of the supercooled liquid regime, but at temperatures below Tg + 10 K, the reported crystallization rate increases suddenly while the self-diffusion coefficient does not. This work and previous work on trisnaphthylbenzene both find a self-diffusion-controlled crystal growth regime and an enhancement in self-diffusion near Tg, suggesting that these phenomena are general characteristics of fragile low molecular weight glass formers. II. Experimental Technique A full description of our experiments is provided elsewhere.22 Self-diffusion coefficients are found by isothermally desorbing thin film bilayers of protio and perdeuterated o-terphenyl in a vacuum chamber. Our approach is based upon the desorption technique developed by Kay and co-workers23 to measure selfdiffusion in amorphous solid water. Detection of desorbing molecules from the surface of a film can be used to measure

10.1021/jp0555955 CCC: $33.50 © 2006 American Chemical Society Published on Web 12/14/2005

508 J. Phys. Chem. B, Vol. 110, No. 1, 2006

Figure 1. Isothermal desorption data from the mass spectrometer, plotted as normalized concentration of protio (b) and deuterio (O) o-terphenyl versus time as the sample desorbs at 253 K (Tg + 10 K). For clarity, every 10th data point collected is plotted in this graph, except at the beginning and end of the experiment. The thin film bilayer sample is vapor-deposited on a polystyrene-coated silicon wafer, then isothermally desorbed as diffusion occurs simultaneously. The bilayer in this experiment consists of a 214 nm protio film and a 158 nm deuterio film. Above the curves are schematic illustrations of the sample’s composition and thickness at various times during the desorption experiment.

diffusion if these two processes occur on similar time scales. A typical desorption experiment for o-terphenyl is shown in Figure 1. Bilayer films are initially produced by sequential vapor deposition of deuterio and protio o-terphenyl. The initial thickness of the films, on the order of tens to hundreds of nanometers, is determined by a quartz crystal microbalance. During desorption of the bilayer, a mass spectrometer records a characteristic m/z peak for a fragment of protio (101 amu) and for deuterio (120 amu) o-terphenyl. The normalized traces of these signals give the concentration of material evaporating from the surface as a function of time. Using the known film thickness, the length of time to desorb the sample, and the timedependent concentration, we fit the traces with a Fickian model of diffusion and calculate a self-diffusion coefficient. Three types of control experiments have been conducted to check the consistency of our findings.22 First, we checked that changing the order of deposition of protio and deuterio o-terphenyl did not change the observed self-diffusion coefficient. Second, “reduced contrast” samples were prepared by depositing one layer of a bilayer from a premixed deuterio and protio o-terphenyl source. Reduced contrast experiments produced the same self-diffusion coefficients as pure protio and deuterio bilayers. These two types of experiments show no detectable isotope effect, confirming a basic assumption of our approach: interdiffusion of deuterio and protio o-terphenyl is indistinguishable from self-diffusion of protio o-terphenyl. In a third control experiment, films that varied by a factor of 3 in thickness produced the same value of D, verifying that we are in the Fickian regime. The self-diffusion coefficients that we report here are independent of the thermal history of the samples and thus characterize molecular motion in the equilibrium supercooled liquid.22 III. Results and Discussion Figure 2 presents self-diffusion coefficients of o-terphenyl from 246 to 265 K (Tg + 3 K to Tg + 22 K) measured with

Mapes et al.

Figure 2. Self-diffusion coefficients of o-terphenyl compared to viscosity (η).48 The left axis is log D for diffusion coefficients determined by NMR (b)13 and isothermal desorption (9, this work). The right axis is T/η (solid line), which has been shifted to overlay the values of high-temperature self-diffusion data. The dotted line is η-0.80, vertically shifted. The structure of o-terphenyl is shown as an inset.

isothermal desorption; the uncertainty in log D is 0.10 decades above 250 K and 0.15 below 250 K. Our results are plotted with high-temperature NMR field gradient measurements of selfdiffusion by Fujara et al.13 As seen in the figure, the two data sets fit together smoothly. Our new measurements expand the range of known self-diffusion coefficients of o-terphenyl by 6 decades in log D. A. Comparison to Viscosity and Rotational Correlation Times. Figure 2 shows a comparison between o-terphenyl selfdiffusion coefficients and the viscosity (η);48 the quantity Tη-1 is shown as a solid curve. At high temperatures, the two observables share the same temperature dependence, as predicted by the Stokes-Einstein relation (D ∼ Tη-1) and observed in many liquids. However, at lower temperatures, Figure 2 clearly shows that self-diffusion has a weaker temperature dependence than viscosity; the dashed line in the figure represents η-0.80 and serves only as a guide to the eye. In the figure, Tη-1 has been vertically shifted to coincide with the high-temperature diffusion data. We regard this curve as the Stokes-Einstein expectation for diffusion at lower temperatures. Compared to this expectation, self-diffusion is enhanced by a factor of 100 at Tg + 3 K. The self-diffusion coefficients are plotted in a different manner in Figure 3 to compare to theoretical predictions. The product of the self-diffusion coefficient and the rotational correlation time, Dτc, is a quantity that is independent of η and T for a large object moving freely in a viscous continuum. The τc values needed to construct this plot were obtained from Fujara et al.;13 these τc values have the temperature dependence of ηT-1. As temperature decreases toward Tg, Dτc for o-terphenyl increases by a factor of 100. This decoupling can be viewed as more fundamental than the breakdown of the Stokes-Einstein equation shown in Figure 2. One can argue that the StokesEinstein equation (which describes the motion of an object large compared to the surrounding particles) should not apply to selfdiffusion. In contrast, Dτc is the product of two single particle observables. Changes in this product as a function of temperature unambiguously indicate a change in some mechanism underlying rotational and/or translational motion. It has been previously argued that an increase in Dτc as the liquid is cooled toward Tg is due to the increasing importance of spatially heterogeneous dynamics.25-27 In a spatially heterogeneous system, some

Self-Diffusion of Supercooled o-Terphenyl

Figure 3. Deviation from the Stokes-Einstein/Debye-StokesEinstein relation for o-terphenyl as a function of temperature. The product of the self-diffusion coefficient and rotational correlation time13 is normalized to its high-temperature value. Comparisons to two theoretical predictions are also shown. The predictions of the dynamic facilitation approach have been shifted vertically to coincide with the data.

molecules find paths through the material that allow them to translate many times their own diameter while during the same time period other molecules remain fixed in space. B. Comparison to Theoretical Predictions. Figure 3 compares the predictions of the random first-order transition (RFOT) theory19 and the dynamic facilitation approach21 (East model) to our experimental data. In both of these approaches, spatially heterogeneous dynamics are responsible for the enhancement of translational diffusion. Xia and Wolynes developed the diffusion predictions19 for the random first-order transition theory,20 which attributes characteristic properties of supercooled liquids to underlying thermodynamic changes through the molecular rearrangements that occur in “entropic droplets”. Within this approach, the barrier for structural relaxation in supercooled liquids is calculated by using the value of the heat capacity jump at Tg as an input. This approach predicts that fragile glass formers, like o-terphenyl, have a large Stokes-Einstein deviation and strong glass formers (e.g., SiO2) have a small deviation. The published prediction of Xia et al.19 for o-terphenyl (bold line in Figure 3) is calculated based upon the fragility of o-terphenyl; a fragility D ) 8 was used to produce the curve shown. Clearly this prediction captures the qualitative trend of the data. In our view, some ambiguity exists in comparing the theory to our data, considering that fits to viscosity or dielectric data produce different values for fragility depending upon the temperature range fitted. As a specific example, a fragility D ) 21 is obtained from fitting over the temperature range from Tg to the point where self-diffusion is first observed to deviate from the Stokes-Einstein relation (243-300 K). With use of this value of D, the RFOT prediction for log(Dτc) at Tg is 1.9, which would be in nearly perfect agreement with the experimental data in Figure 3. Recently barrier softening effects have been incorporated into the RFOT theory.28 If these calculations are extended to predict diffusion, any ambiguity regarding the fragility parameter would no longer influence the RFOT prediction. Garrahan, Chandler, Berthier, and co-workers have recently developed a theoretical framework for supercooled liquid dynamics based upon dynamic facilitation.21,29 They view viscous slowdown and heterogeneous dynamics as a consequence of kinetic constraints rather than thermodynamics. The dynamic facilitation prediction in Figure 3 is based upon the East model,21 a lattice model in which dynamic facilitation has directional persistence. In the 1D East model, a change in mobility is only allowed if the neighbor to the left is in a mobile state. The

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Figure 4. Deviation from the Stokes-Einstein/Debye-StokesEinstein relation for o-terphenyl (Tg ) 243 K) and trisnaphthylbenzene12 (Tg ) 342 K) as a function of reduced temperature.

prediction in Figure 3 is based upon the diffusion of a probe particle on the lattice that is allowed to jump into mobile sites. While Jung et al. consider the East model to provide predictions that are valid for all fragile supercooled liquids, they specify a procedure to relate the model to any particular liquid.21 This procedure does not produce a high-temperature prediction of Dτc and thus the vertical position of the dashed line in Figure 3 is arbitrary. We have chosen to overlay the prediction with the experimental data. Thus one should only conclude that the dynamic facilitation approach and the experimental data show similar temperature dependences. C. Comparison of Self-Diffusion in o-Terphenyl and Trisnaphthylbenzene. In Figure 4, the product Dτc for o-terphenyl is compared to the same quantity for trisnaphthylbenzene.12 Both compounds are considered typical fragile glass formers. When m is used to characterize fragility [m ) d(log τ)/d(Tg/T)|Tg], o-terphenyl and trisnaphthylbenzene have m ) 78 and 91, respectively.30,31 As shown in Figure 4, the values of Dτc for trisnaphthylbenzene and o-terphenyl are similar when compared on a T/Tg scale, although o-terphenyl does show a more rapid rise in Dτc near 1.03Tg. We do not know a priori that the T/Tg scale is the most appropriate manner in which to compare different liquids. The data presented in Figure 4 are consistent with the hypothesis that similar fragilities lead to similar enhancements in translational diffusion as Tg is approached from above. We are currently working to produce self-diffusion measurements on glass formers with a wide range of fragilities. D. Relaxation Time Distributions of o-Terphenyl and Trisnaphthylbenzene. In early models of spatially heterogeneous dynamics, an enhancement of translational diffusion was associated with an increase in the breadth of the relaxation time distribution (e.g., a decreasing βKWW value) with decreasing temperature.25-27 In these models, translational motion is more heavily influenced by the fast regions (since motion through such regions dominates the mean-square displacement) while the slow regions have a greater influence on rotation (since the rotational correlation function cannot decay until the slowest molecules reorient). Within this framework, the product Dτc will increase as temperature decreases if the relaxation time distribution broadens. The RFOT predictions for diffusion in o-terphenyl,19 and recent work by Merabia and Long,32 are also associated with a strong temperature dependence of the relaxation time distribution. In light of these explanations for enhanced translational diffusion, it is interesting to note that not all experimental

510 J. Phys. Chem. B, Vol. 110, No. 1, 2006 determinations of the temperature dependence of βKWW for o-terphenyl are consistent with this explanation. Some studies show very little change in the relaxation time distribution in the temperature range where Dτc changes significantly. For o-terphenyl, a recent evaluation of the dielectric relaxation spectra from T/Tg ) 1.02-1.17 is consistent with a constant value of βKWW ()0.50( 0.04).30 βKWW measured from photon correlation spectroscopy is roughly consistent with the dielectric relaxation measurements, with a change from 0.55 to 0.62 over a range of T/Tg ) 1.01-1.12. 33 In contrast, 2H NMR measurements of βKWW for partially deuterated o-terphenyl show a strong temperature dependence, with βKWW increasing from 0.5 to 0.7 over T/Tg ) 1.04-1.08.13,34 Dynamic heat capacity measurements have also been performed on an o-terphenyl mixture (9% o-phenylphenol).35 In this case, βKWW also increases considerably (0.61-0.73) from T/Tg ) 1.02-1.10; it is unclear what influence the addition of 9% o-phenylphenol had on the reported values. The results described in the previous paragraph are difficult to reconcile although it is not surprising that different experiments obtain somewhat different values of βKWW. Some observables are related to correlation functions of the first Legendre polynomial P1 (dielectric) while others (NMR, photon correlation spectroscopy) are related to P2.36 The NMR experiments measure a single particle correlation function while all other techniques mentioned measure a collective observable.14 For this comparison with translational diffusion, the single particle observable seems the most relevant. On the other hand, in the dielectric and photon correlation measurements, βKWW is obtained more directly from the data and over a wider temperature range. In contrast to o-terphenyl, measurements of βKWW for trisnaphthylbenzene are quite consistent with each other and all indicate a temperature-independent distribution of relaxation times. Dielectric relaxation measurements of trisnaphthylbenzene from T/Tg ) 1.01-1.22 show no change in the shape of the loss spectra; the data are consistent with βKWW equal to 0.50.31 Photon correlation spectroscopy measurements report that βKWW ) 0.55 ( 0.01 from T/Tg ) 1.02-1.12.37 NMR measurements over the range T/Tg ) 1.01-1.09 are consistent with a constant βKWW ) 0.5.38 Is the experimental eVidence consistent with the explanation that a strong increase in Dτc is associated with an increase in the breadth of the relaxation time distribution? For trisnaphthylbenzene, our answer at present is “no” although it must be added that trisnaphthylbenzene has not been studied as thoroughly as has o-terphenyl. For o-terphenyl, there is active disagreement on this point among those working in the field. The NMR and heat capacity spectroscopy show a sufficiently strong temperature dependence of βKWW to be consistent with the RFOT prediction19 and the other approaches.27 The dielectric relaxation and photon correlation spectroscopy results imply a relaxation time distribution whose shape is too temperature independent to cause the increase in Dτc, at least within the framework of current approaches. If one (provisionally) accepts that Dτc can increase significantly without an associated increase in the breadth of the relaxation time distribution, does this mean that the increase in Dτc is not associated with spatially heterogeneous dynamics? No. The failure of one class of spatially heterogeneous models would not mean that all explanations based on spatial heterogeneity must fail. For example, an increase in a length scale associated with heterogeneity might produce an increase in Dτc even for a constant relaxation time distribution. Alternately, an

Mapes et al.

Figure 5. Self-diffusion (D) for o-terphenyl, compared with the kinetic part of the crystallization rate (Gkin) and viscosity (T/η). The two sources of crystallization rate data, Magill et al.43 and Hikima et al.,44 are vertically shifted on the log D axis to overlay the self-diffusion data near 255 K. The right axis is T/η, which has been shifted to overlay the high-temperature self-diffusion data.

increasing barrier height might create an increasing dispersion of translational jump times even with constant βKWW and a constant length scale.39 As a third possibility, Berthier et al. argue that in dynamic facilitation models the breadth of the relaxation time distribution need not be correlated with the increase in Dτc.40 Finally, we note that a recent simulation of polydisperse hard spheres shows a substantial enhancement of translational motion as the volume fraction increases while minimal changes occur in the shape of the relaxation time distribution (as obtained from the intermediate scattering function).41 In this simulation, the enhanced translation is unambiguously associated with spatially heterogeneous dynamics. E. Crystallization in o-Terphenyl and Trisnaphthylbenzene. Self-diffusion has an important impact on crystallization kinetics. Using probe diffusion data in o-terphenyl and trisnaphthylbenzene, Ngai et al. argued that the kinetic portion of the crystal growth rate Gkin has the same temperature dependence as diffusion.42 On this basis, they concluded that diffusion (not viscosity) controls the rate of crystallization. With the measurements reported here for o-terphenyl, we can extend this comparison to self-diffusion. In Figure 5 we have plotted selfdiffusion coefficients and Gkin for o-terphenyl crystals grown from the supercooled liquid.43,44 Gkin is calculated from the observed isothermal growth rate by dividing out the thermodynamic nucleation term determined by Magill and Li.43 Figure 6 presents the analogous comparison between crystal growth kinetics and self-diffusion for trisnaphthylbenzene.12 Figure 5 shows that crystal growth in o-terphenyl is controlled by self-diffusion from 253 K to the melting point. Below 253 K, a jump in crystal growth rate indicates that additional factors influence crystal growth. This phenomenon, which has been called homogeneous nucleation based growth,44 is characterized by a new crystal shape produced by an accumulation of crystal nuclei and has been seen in other fragile supercooled liquids.45 Figure 6 shows that trisnaphthylbenzene self-diffusion controls crystal growth at least down to Tg + 26 K, which is the limit of reported crystal growth measurements.12,24,46,47 The existence of a wide temperature range where selfdiffusion controls crystal growth for both o-terphenyl and trisnaphthylbenzene suggests that this is a common feature of

Self-Diffusion of Supercooled o-Terphenyl

Figure 6. Self-diffusion (D) for trisnaphthylbenzene,12 compared with the kinetic part of the crystallization rate (Gkin)46 and viscosity (T/η).24 The crystallization rate data have been vertically shifted on the log D axis to overlay the self-diffusion data at high temperatures.

fragile low molecular weight glass-forming materials. This further supports the suggestion42 that crystal growth rate data may be used to extend high-temperature self-diffusion coefficients to lower temperatures in cases where direct measurements are unavailable. This is a connection that deserves to be more thoroughly explored. IV. Conclusion In summary, we have reported self-diffusion coefficients for o-terphenyl down to Tg + 3 K, extending the range of measured diffusion coefficients by 6 decades. Self-diffusion of o-terphenyl at Tg + 3 K is 100 times faster than expected based upon viscosity data and the Stokes-Einstein equation. While selfdiffusion controls the rate of crystal growth above 253 K, at lower temperatures additional factors must play an important role. Enhancement in self-diffusion near Tg and a self-diffusion controlled crystal growth regime seem to be general characteristics of fragile small molecule glass formers. The temperature dependence of the enhanced translational motions is consistent with both the random first-order transition theory and dynamic facilitation, although neither approach in its present form provides an unambiguous prediction. Acknowledgment. We thank Ranko Richert, Peter Wolynes, Hans Sillescu, Gregor Diezemann, Lian Yu, Bruce Kay, and Jim Skinner for helpful discussions. We acknowledge support for this work from NSF Chemistry (NSF-CHE 0245674). References and Notes (1) Williams, G.; Cook, M.; Hains, P. J. J. Chem. Soc., Faraday Trans. 2 1972, 68, 1045-1050. (2) Lunkenheimer, P.; Schneider, U.; Brand, R.; Loidl, A. Contemp. Phys. 2000, 41, 15-36. (3) Wagner, H.; Richert, R. J. Phys. Chem. B 1999, 103, 4071-4077. (4) Stickel, F.; Kremer, F.; Fischer, E. W. Physica A (Amsterdam) 1993, 201, 318-321. (5) Wu, L.; Dixon, P. K.; Nagel, S. R.; Williams, B. D.; Carini, J. P. J. Non-Cryst. Solids 1991, 131-133, 32-36.

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