J. Phys. Chem. B 2008, 112, 9909–9913
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The Physics of Heating by Time-Dependent Fields: Microwaves and Water Revisited Wei Huang and Ranko Richert* Department of Chemistry and Biochemistry, Arizona State UniVersity, Tempe, Arizona 85287 ReceiVed: April 30, 2008; ReVised Manuscript ReceiVed: May 30, 2008
Heating samples by microwave radiation is a particular example of the more general phenomenon where materials absorb energy from an external time-dependent field of an electric, magnetic, or mechanical nature. How this compares with conventional heating is a question of continued interest. Here, we show that the origin of the absorptivity determines whether energy accumulates in the slower configurational degrees of freedom or transfers rapidly to the phonon bath, where only the latter situation is equivalent to conventional heating. Based upon time-resolved measurements of the configurational temperatures, evidence is provided for simple liquids displaying nonthermal behavior if heated by external fields, with molecules being more mobile than expected on the basis of the actual temperature. However, water and related materials are the exception regarding absorptive heating, because energy is transferred to the phonons more rapidly than it is absorbed from the field, and nonthermal effects thus remain absent. Introduction The widely applied use of microwave radiation for the purpose of heating water, aqueous materials, and food stuff has been scrutinized regarding possible differences from conventional heating, with the generally accepted result that microwave heating produces only thermal effects that are equivalent to conventional heating. For bulk water, it is the prominent Debyetype dielectric loss peak positioned at a frequency of ν ≈ 16 GHz1 that is the main source of absorptivity at the typical (household) microwave frequency of 2.45 GHz.2 The scientific interest in absorptive heating has experienced a more recent surge through the field known as “microwave chemistry”, that is, the enhancement of chemical reaction rates by microwave instead of conventional heating of solvents.3,4 Again, the general view tends toward the absence of microwave-specific or nonthermal effects regarding the chemical reaction rates in aqueous solutions,5,6 and the advantage of microwave chemistry thus reduces to the more effective use of energy and to the higher rates of temperature changes. The uptake of energy from an external field with ν e 100 GHz requires “slow” modes that match that frequency ν, because phonon modes will not absorb in this range. The principle of microwave (MW) heating is commonly described as follows: Any significant fluctuation of charge in a material couples to the electric field component of an external source of electromagnetic radiation. In many situations, charge displacement does not follow the field instantaneously, and the friction responsible for this time lag is the source of heat.3,5 Although the MW energy will enter via the slow degrees of freedom, there is little reason to suspect effects that could differ from conventional heating if the transfer of the energy to the phonon bath is practically instantaneous. In the case of absorption from an external field, it is the dielectric loss ε′′(ω) at the frequency ω of the electric field E(t) ) E0 sin(ωt) that determines the amount of energy Q ) πε0E20ε′′(ω)V a sample of volume V absorbs each period.7 Apart from dc-conductivity, the origin of significant dielectric loss ε′′ in a typical polar liquid is a reorientational mode of permanent dipoles, with corresponding signatures in * Corresponding author. E-mail:
[email protected].
both dielectric relaxation and heat capacity measurements that occur on a common time scale τR of the structural (R-) relaxation.8 Therefore, the configurational modes of the liquid absorb the energy from the field, and the details of how the microwave energy is eventually transferred to the phonon bath are often disregarded. Experimentally, the very short time scale involved in the case of water is a major obstacle to monitoring such effects for MW heating. However, this can be overcome by studying glass-forming liquids of sufficiently high polarity at much lower frequencies, as outlined below. For generic liquids, details of the energy absorption from an external electric field and the subsequent energy flow have been studied in detail, and the situation differs qualitatively from conventional heating.9,10 From differential scanning calorimetry (DSC)11 and in more detail from dynamic heat capacity experiments,12–15 it is known that only one-half of the heat capacity of a typical liquid is supplied by vibrational modes, while the remainder originates from slow degrees of freedom. Structural relaxation in supercooled liquids involves broad distributions of relaxation times, and energy absorption by these slow configurational modes is thus spatially heterogeneous.16,17 This dynamic heterogeneity is experimentally well established and implies that the relaxation time dispersion of viscous materials originates from a spatial distribution of exponentially relaxing domains. This situation is outlined schematically in the upper part of Figure 1, where the existence of multiple relaxation times is represented by five modes of different time constants (τ1...τ5). These domains retain their time constants τ for times τex that are comparable to the longest structural relaxation time of the system,18 and possibly much longer for temperatures very close to Tg.19 The spatial dimensions ξ of these domains have been estimated at 1-3 nm.20 Therefore, these domains are essentially independent and exchange energy only with the phonon reservoir, not among themselves. As a result, a liquid will generally not absorb energy from an external monochromatic field in a spatially homogeneous manner. For each domain, the transfer or delocalization of the absorbed energy into the phonon modes and participation in the heat conductivity κ of the material occurs only after its relaxation time τi has elapsed;21,22 that is, it is only for times in excess of
10.1021/jp8038187 CCC: $40.75 2008 American Chemical Society Published on Web 07/23/2008
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Figure 1. Schematic representation of five independent slow modes with different relaxation times (τi) and how they contribute to heat capacity Cp (top) and to the dielectric loss spectrum ε′′(ω) (bottom). The shaded arrows indicate the difference in where energy enters via conventional heating (“heat”) and via absorption from an external field (“ext. field”). Each mode contributes with a Debye peak to the entire loss spectrum, shown as overall solid line in the ε′′ versus log ω graph. The arrows at the individual loss peaks indicate the shift in relaxation time that originates from the increased configurational temperatures after applying a large electric field at frequency ω0. The net effect of these shifts is an increase of the loss at ω0, represented by the dot. As has been observed experimentally, the dashed line would result from measuring ε′′ at large fields across the entire spectrum.
Huang and Richert
Figure 2. Experimental results (symbols) for the frequency-resolved dielectric loss, ε′′, and its relative field-induced change, ∆ ln ε′′, of propylene carbonate at T ) 166 K. (a) The loss curves at high (E0 ) 177 kV/cm) and low (E0 ) 14 kV/cm) fields, together with an HN fit to the low field data (line). (b) The relative change of the dielectric loss, ∆ ln ε′′, for a field of E0 ) 177 kV/cm. The dashed lines reflect the calculation following eq 1, using ∆cp ) 0.47 J K-1 cm-3 (ref 20). The relative signal increase reaches 20% at E0 ) 177 kV/cm.
τR that a single temperature is sufficient for describing the system. Introducing energy via the configurational modes (arrow “ext. field”) has the potential of generating effects other than conventional heating (arrow “heat”), unless a rapid transfer of the absorbed energy into heat occurs. Such nonthermal effects contrast what is assumed for the heating of aqueous materials by microwaves. On the other hand, microwave-specific effects have been shown to occur in other heterogeneous materials such as ceramics.23 This Article clarifies this situation by providing evidence for water (and related materials) again being the exception rather than the rule for how energy is transferred from an external electric field to the phonon bath. To this end, viscous liquids are exposed to large electric fields, where the relaxation times are sufficiently long so that the energy flow and resulting changes in configurational temperatures can be quantified and resolved in time.
by a factor of 100 with a Trek PZD-700 voltage amplifier. A typical pattern would be a ν ) 1 kHz sine wave with an amplitude Vrms ) 30 V for the first 8 periods, followed by Vrms ) 200 V for the remaining 24 periods. Such a complete signal with duration 32 ms is triggered with a repetition rate as low as 1 Hz to allow the sample to cool for a sufficiently long time. (In cases where the 16 000 point waveform length was insufficient, the voltage for the internal sine wave was switched from low to high to low under computer control for approximately 0.3 s, without repetition.) The signal is applied to the stainless steel sample disk capacitor with diameter 14 mm and electrode separation 10 µm. The current is measured as voltage drop across a resistor (typically 1 kΩ) using a 50 kHz low pass filter. Voltage and current signals are recorded with a Nicolet Sigma 100 storage scope, using a sampling rate of 1 Ms/s at 14 bit resolution and averaging over up to 5000 trigger events. The traces are subject to period-by-period Fourier analysis at the frequency of interest, and the phase difference ∆φ is used to determine the tangent of the loss angle via tan δ ) ε′′/ε′ ) tan(π/2 - ∆φ). The glass-forming liquids used in this study are propylene carbonate (PC, 99.7%, anhydrous) and 2-methyl-3-pentanol (2M3P, 99%). The materials have been purchased from Aldrich and are used as received.
2. Experimental Section
3. Results and Discussion
We measure the configurational temperature Tcfg of a mode via its relaxation time τ*, and define Tcfg by the temperature that is required to establish the same time constant τ* of that mode in equilibrium. Impedance measurements provide access to these dielectric relaxation times with high resolution, but usually without the time dependence desired for the present study. Combining a resolution of 5 × 10-5 regarding tan δ at ν ) 1 kHz with period-by-period time resolution is achieved by the following novel technique. The arbitrary waveform feature (e16 000 points, 12 bit) of a Stanford Research DS 345 synthesized function generator is used to provide a sequence of sinusoidal cycles, with the output voltage being increased
Propylene carbonate with a static dielectric constant near 100 is a good candidate for demonstrating the above technique. The frequency dependence of the field-induced changes is shown in Figure 2 for at a temperature of T ) 166 K, at which the peak of the loss profile is positioned at a frequency of νmax ) 20 Hz. Significant field effects are observed for frequencies ν g νmax, while the low frequency wing remains field invariant.22 The time-resolved effects for PC at T ) 166 K measured with ν0 ) 1 kHz are compiled in Figure 3. Following the application of the high field at time t ) 0, it is seen that tan δ rises by 5-15% over a time of approximately 50 cycles, with the amplitude of the effect scaling with E02; that is, the plateau
The Physics of Heating by Time-Dependent Fields
Figure 3. Experimental results for the time-resolved field-induced relative change of tan δ measured at ν0 ) 1 kHz for propylene carbonate at T ) 166 K. The curves are recorded at the different field strengths of 122, 142, and 162 kV/cm, as indicated. The high field is applied for times between 0 and 0.32 s; the field is significantly lower elsewhere. As expected, the plateau heights increase quadratic with the field. The temperature axis indicates the added configurational temperature ∆T for the dominant mode. The inset displays the first 50 periods for the 162 kV/cm case (corrected for a constant Tbath), together with the time dependence predicted from eq 1.
height of ∆ ln(tan δ) is linear in the amount of energy absorbed. The constant slope observed for times 0.1 s e t e 0.3 s is the result of residual phonon heating. The rationale for the above observation is as follows. The dynamic heterogeneity established for supercooled liquids justifies assigning the spectrum of relaxation time constants to independent modes;16,17 that is, the system is considered to consist of dynamically distinct domains, and in each domain the dielectric (ε) and the thermal (Cp) relaxations proceed exponentially with identical values for τ.9,21 As we will focus on the effects at frequencies exceeding 1/τR, the particular value of the exchange time τex is of no concern in what follows. The offset of the configurational temperature, ∆T ) Tcfg- Tbath, is governed by a gain term +p/cp (p being the power absorbed from the field) and a loss term -∆T/τ (with τ ) τT being the thermal relaxation time).21 For an electric field represented by E0 sin(ωt) for times t g 0 and zero field for t < 0, the change in ∆T can be obtained from: 2 d∆T(t) ε0E0∆ε ω2τ2 ∆T(t) λ(t) ) , ∆T(0) ) 0 (1) dt 2F∆Cpτ 1 + ω2τ2 τ
as long as the changes are linear in the energy. Here, ε0 is the permittivity of vacuum, F is the density, ∆Cp is the configurational (specific) heat capacity, and ∆ε ) εs - ε∞ is the dielectric relaxation strength. The term λ(t) ) 2(ωτ sin ωt + cos ωt e-t/τ)2/(1 + ω2τ2) accounts for transient effects,24 and typically oscillates between 0 and 2 at a frequency of 2ω with an average of unity for times t . τ. An increase of the configurational temperature by ∆T translates into a reduction of the relaxation time via ln τ* ) ln τ - TA∆T/T2, with EA ) kBTA being the activation energy. The final step is to evaluate the current density associated with each mode and subject their sum to a periodby-period Fourier transform, which provides the time-resolved tan δ as in the experiment. The steady-state case (d∆T/dt ) 0 and λ(t) ≡ 1) of eq 1 is represented as a dashed line in Figure 2b and agrees well with the observation.22 Clearly, any such offset between configurational and phonon temperature establishes a nonthermal effect because fluctuations are faster than expected on the basis of the phonon temperature.
J. Phys. Chem. B, Vol. 112, No. 32, 2008 9911 It is important to realize the asymmetry regarding the position of τ relative to the frequency of the external field, ω0, in Figure 1. The two modes labeled “τ3” and “τ5” display the same loss ε′′ at ω0 and thus absorb the same amount of energy (per volume). However, the rates of energy transfer to the phonon bath differ by the ratio τ3/τ5, and, therefore, ∆T for the slower mode is considerably higher, as indicated by the arrows representing ∆ ln τ for each mode. The net result regarding ε′′(ω0) is an increase, as indicated by the dot in Figure 1 above the solid line representing the low field limit for ε′′(ω). The dashed line in Figure 1 represents the result of measuring the entire loss spectrum at high fields, consistent with the observation for PC in Figure 2 and for other liquids.21,22 Toward ωτ . 1, the steady-state effect saturates at a level of ∆T ) 1/2ε0E20∆ε/ ∆cp, while for ωτ , 1 the extent of ∆T remains negligible. Examples of time-resolved configurational temperature changes observed at ν0 ) 1 kHz (see arrow in Figure 2a) are displayed in Figure 3 for PC at T ) 166 K, where the peak loss frequency is at νmax ) 20 Hz and the dielectric relaxation strength is ∆ε ≈ 100. The magnitude of the relative change in the loss tangent, ∆ ln(tan δ), reaches 10% already and a field of E0 ) 142 kV/ cm and shows the expected quadratic field dependence; that is, the change is linear in the energy absorbed and very large as compared to other nonlinear dielectric effects.25 More importantly, the total ∆T approaches 0.35 K slowly over a time of ∼50 periods ()50 ms) and fades on the same time scale when the high field is removed at t ≈ 0.32 s. The residual slope for times 0.1 s e t e 0.3 s results from gradually heating Tbath with a power of 0.89 W for the E0 ) 162 kV/cm case. Although the metal electrodes serve as effective heat sinks, isothermal conditions for the bath are not maintained because the thermal wavelength λ ) κ/(ω0Fcp) ≈ 4.7 µm is not much larger than the sample thickness of 10 µm.26 The inset of Figure 3 shows how well the model of eq 1 reproduces the time dependence of ∆ ln(tan δ) without adjustable parameters. In conclusion, this type of “heating” is nonthermal as it can increase the configurational temperature (and thus the mobility of molecules) beyond the bath temperature, and in general this ∆T offset will be as spatially heterogeneous as the liquid dynamics unless averaged over a sufficient time scale. Now we ask what these results imply for the heating via microwave radiation, for example, for bulk water and related materials. Typical household MW ovens operate at 2.45 GHz, thus providing much more power at lower fields (the above case of using p ) 0.89 W for PC at 1 kHz would become p > 2 MW at 2.45 GHz for identical conditions regarding E0 and ε′′). The position of the 2.45 GHz frequency within the absorption band of bulk water at 19 °C is shown in Figure 4a, and the loss ε′′ (ν ) 2.45 GHz) ) 10 determines the efficiency for energy absorption. While ice has an even higher dielectric constant, the loss is shifted to lower frequencies by 7 decades, as shown in Figure 4b. The lack of overlap of the loss of ice with the MW band leads to the absence of significant heating (as was easily confirmed in a household MW unit), whereas strong fields in the kHz range can heat ice effectively. Associating the prominent 10 ps relaxation mode of liquid water with the structural relaxation peak implies that the model of eq 1 should apply here. By analogy to the field effect of PC shown in Figure 2b, exciting on the low frequency side of the peak positioned at νmax ) 15.6 GHz would not lead to any considerable increase of the configurational temperature. However, this is different for the 95 GHz mark included in Figure 4a, which is used by “active denial systems”. Based upon the steady state limit of eq 1 that describes the PC experiment of Figure 2, this latter
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Figure 4. (a) Dielectric loss data (symbols) for water at 19 °C compiled by Kaatze (ref 30), including the decomposition into two Debye-type contributions separated in frequency by a factor of ∼80 (dashed lines) and their sum (solid line). Open symbols display the case of n-propanol at 117 K (ref 26), but shifted up in frequency by 6.5 × 109 to match the water peak position. (b) Dielectric loss (symbols) for ice at -10.8 °C taken from the work of Auty and Cole (ref 29). Note the similar loss amplitude but lower peak frequency (by 7 decades) relative to the liquid counterpart.
Figure 5. (a) Low field dielectric loss spectrum of 2-methyl-3-pentanol at T ) 191 K, showing the prominent Debye contribution (“D”) and the primary structural peak (“R”). The frequency ν0 used for the high field experiment is fast as compared to the Debye mode, but slow relative to the R-process. (b) Time-resolved field-induced change of tan δ for the same sample observed by increasing the field from 39 to 193 kV/cm at time zero. The solid line reflects the model result if it is assumed that the thermal relaxation τT is equal to the dielectric time constant τD, whereas the dashed line is obtained for the case τT e 10-2 τD.
case would generate a configurational temperature which is higher than Tbath by ∆T ) 1/2ε0E20∆ε/∆cp. As this estimate of ∆T does not bear a direct dependence on frequency, ∆T for MW heated water is not expected to reach 1 K. However, this conclusion holds only if there is no additional bottleneck involved in the energy transfer from configurational to phonon modes, and a longer thermal relaxation time could increase ∆T considerably. Additionally, powers so significantly exceeding the ones of our experiments could enter the regime in which the effects are no longer linear in the energy and where the present model requires modification. While the above considerations do apply to generic molecular liquids (where the primary structural mode absorbs the energy), there are reasons to question that the mechanism of energy flow is the same for water and other highly associating liquids such as monohydroxy alcohols. The main source of suspicion is the lacking connection between thermal (or calorimetric) modes and the strong dielectric relaxations that provide the high values of the loss ε′′ of many alcohols.11,27 For monohydroxy alcohols, it has been demonstrated that the calorimetric and structural relaxation time (τR) is orders of magnitudes shorter than that of the prominent Debye-type dielectric polarization (τD).28–31 The high loss of such dielectric modes can absorb substantial amounts of energy from the field, but the transfer of energy to the phonon bath is practically instantaneous because it will proceed on the time scale of τR (or faster) rather than as slow as τD.
The frequency dependence for such systems resembles the case of PC, that is, no effect on the low frequency side and a near constant relative increase of the loss for frequencies exceeding the peak value. The time-resolved observation of this effect is shown in Figure 5 for a typical case, 2-methyl-3pentanol, where τR , τD. As in the above case of PC, the frequency of the external field is ν0 ) 1 kHz, that is, positioned within the high frequency wing of the main loss peak (see Figure 5a). In this 2M3P case, the total change is as small as tan δ ) 0.0012 (relative change of 0.85%), equivalent to ∆T ≈ 30 mK. The main observation, however, is the very fast thermal relaxation time τT involved in the rise of tan δ when the field is increased, shown in Figure 5b (the fall of tan δ is not shown, but the curves are symmetric as in the PC case). The solid line in Figure 5b is based upon τT ) τD, analogous to the case of PC in Figure 3, while the dashed curve is obtained for τT e τD/100. The time dependence of tan δ that occurs every time the field amplitude is changed originates from the transient polarization response, not from a lag of the fictive temperature. The fast and efficient transfer of energy to the phonons is thus observed very directly and greatly reduces the energy offset that can be sustained in these modes by an external field. At the same time, the heat capacity of these modes is only a very small fraction of the total Cp,27 and therefore the changes of the dielectric behavior become measurable. The same can be expected for energy absorption via dc-conductivity, where the frequency ω0 of the field is small relative to the reciprocal
The Physics of Heating by Time-Dependent Fields thermal time constant 1/τT. In these cases, there is practically no difference between conventional heating and energy absorption from an external field, because energy is transferred from the absorbing modes to the phonons more rapidly than the power associated with the absorption process. What are the implications of the above alcohol case for water? The prominent Debye-type dielectric relaxations of both ice (Ih)32 and water33 are shown in Figure 4, indicating a 7 decade shift regarding their time constant τD but very similar amplitudes ∆ε. Apparently, restricting the structural relaxation by freezing the water does not reduce the large amplitude of polarization fluctuations. Additionally, the absorption spectrum of liquid water is highly reminiscent of the typical monohydroxy alcohol behavior, where a prominent Debye peak with high Kirkwood correlation factor gK is accompanied by a further process that appears at higher frequencies with relatively small amplitude. This is demonstrated by including the case of n-propanol (shifted along log ν scale) in Figure 4a, where the small higher frequency (R-) peak is dispersive rather than Debye like,29,34 as expected for any supercooled liquid. A similar comparison for n-hexanol at 20 °C and water is provided by Kaatze.33 Because only the smaller and faster dielectric process of many alcohols coincides with the calorimetric and mechanical modes,28–31,35 it is identified as the dielectric signature of structural relaxation. In particular, a study of numerous alcohols suggests that the time constant ratio τD/τR approaches ∼100 in the limit of high temperatures or short relaxation times,30 in agreement with the observation of τ1/τ2 ) 78 in Figure 4a. Therefore, we concur with Angell’s conclusion36 that the structural relaxation of water should be faster than the main dielectric mode by at least a factor of 10, and suggest that the 0.13 ps mode of water at 19 °C be understood as the primary structural relaxation, that is, τR ) τT ) 0.13 ps. For microwave heating of water, this implies that the absorbed energy is surrendered to the phonon system at a rate that exceeds the power of the external source and that practically no difference to conventional phonon heating should be expected. Because of the situation that τR , τD, the above notion holds irrespective of the position of the radiation frequency within the main absorption band and equally for water and monohydroxy alcohols in their liquid regime. 4. Summary and Conclusion In summary, we have measured the rates at which energy is transferred from the absorbing modes to the phonon system. For generic molecular liquids, the slowest configurational degrees of freedom determine the time scale at which energy can be accumulated as excess configurational temperature and at which it will be released to the phonon bath. It has been demonstrated that it can take as long as 50 periods of the applied frequency to saturate the energy stored in these slow modes, and the energy will remain localized for the same time. The extent of such nonthermal effects will depend on the position of the frequency within the absorption band. With the tunable subradio frequencies used here, spatially targeted energy deposition involving several watts of power into dielectrically inhomogeneous materials is an interesting application beyond the simple liquids studied thus far. While the importance of dynamic heterogeneity tends to vanish in the more fluid regime of a liquid, all other aspects of the model outlined in eq 1 should apply irrespective of the frequency, as long as the configurational modes are responsible for the absorption process. In the cases of monohydroxy alcohols, some secondary amides, and most likely water, the frequencies ν where the dielectric loss achieves significant values are low as compared
J. Phys. Chem. B, Vol. 112, No. 32, 2008 9913 to the reciprocal thermal relaxation times τT, that is, ν , νT ) (2πτT)-1. As a result, the rate of energy transfer from the absorbing mode to the phonons is fast as compared to the power of the absorption process, and the system will remain near thermal equilibrium even for high powers. The same argument for the absence of nonthermal effects holds for absorption via the dc-conductivity, for example, from ionic species, where again ν , νT applies. An interesting and perhaps intermediate case is the class of room-temperature ionic liquids, which display dispersive loss signals with high amplitude,37 where dc- and ac-conductivities show significant overlap.38 It is expected that measurements of the steady-state absorption profiles ε′′(ω) in the GHz range at high fields (analogous to the results of Figure 2) will provide more direct information on the details of heating in the microwave range of frequencies. Acknowledgment. We acknowledge the donors of the American Chemical Society Petroleum Research Fund (ACSPRF) for support of this research under Grant No. 42364-AC7. References and Notes (1) von Hippel, A. IEEE Trans. Electron Insul. 1988, 23, 801. (2) Gabriel, C.; Gabriel, S.; Grant, E. H.; Halstead, B. S. J.; Mingos, D. M. P. Chem. Soc. ReV. 1998, 27, 213. (3) Dallinger, D.; Kappe, C. O. Chem. ReV. 2007, 107, 2563. (4) de la Hoz, A.; Dı´az-Ortiz, A.; Moreno, A. Chem. Soc. ReV. 2005, 34, 164. (5) Galema, S. A. Chem. Soc. ReV. 1997, 26, 233. (6) Laurent, R.; Laporterie, A.; Dubac, J.; Berlan, J.; Lefeuvre, S.; Audhuy, M. J. Org. Chem. 1992, 57, 7099. (7) Fro¨hlich, H. Theory of Dielectrics; Clarendon: Oxford, 1958. (8) Schro¨ter, K.; Donth, E. J. Chem. Phys. 2000, 113, 9101. (9) Schiener, B.; Bo¨hmer, R.; Loidl, A.; Chamberlin, R. V. Science 1996, 274, 752. (10) Jeffrey, K. R.; Richert, R.; Duvvuri, K. J. Chem. Phys. 2003, 119, 6150. (11) Kauzmann, W. Chem. ReV. 1948, 43, 219. (12) Birge, N. O.; Nagel, S. R. Phys. ReV. Lett. 1985, 54, 2674. (13) Birge, N. O. Phys. ReV. B 1986, 34, 1631. (14) Minakov, A. A.; Adamovsky, S. A.; Schick, C. Thermochim. Acta 2003, 403, 89. (15) Christensen, T.; Olsen, N. B.; Dyre, J. C. Phys. ReV. E 2007, 75, 041502. (16) Ediger, M. D. Annu. ReV. Phys. Chem. 2000, 51, 99. (17) Richert, R. J. Phys.: Condens. Matter 2002, 14, R703. (18) Huang, W.; Richert, R. J. Chem. Phys. 2006, 124, 164510. (19) Cicerone, M. T.; Ediger, M. D. J. Chem. Phys. 1995, 103, 5684. (20) Tracht, U.; Wilhelm, M.; Heuer, A.; Feng, H.; Schmidt-Rohr, K.; Spiess, H. W. Phys. ReV. Lett. 1998, 81, 2727. (21) Richert, R.; Weinstein, S. Phys. ReV. Lett. 2006, 97, 095703. (22) Wang, L.-M.; Richert, R. Phys. ReV. Lett. 2007, 99, 185701. (23) Freeman, S. A.; Booske, J. H.; Cooper, R. F. J. Appl. Phys. 1998, 83, 5761. (24) Richert, R. Physica A 2003, 322, 143. (25) Weinstein, S.; Richert, R. Phys. ReV. B 2007, 75, 064302. (26) The thermal conductivity κ ) 0.177 W m-1 K-1 and density F ) 1.35 g cm-3 for T ) 166 K are extrapolated using the respective data from: (a) Tuliszka, M.; Jaroszyk, F.; Portalski, M. Int. J. Thermophys. 1991, 12, 791. (b) Simeral, L.; Amey, R. L. J. Phys. Chem. 1970, 74, 1443. For the heat capacity, we use Cp,∞ ) 0.953 J g-1 K-1. The thermal wavelength λ ) 4.7 µm is stated for a frequency of ω0 ) 2π × 103 Hz. (27) Huth, H.; Wang, L.-M.; Schick, C.; Richert, R. J. Chem. Phys. 2007, 126, 104503. (28) Murthy, S. S. N. J. Phys. Chem. 1996, 100, 8508. (29) Hansen, C.; Stickel, F.; Berger, T.; Richert, R.; Fischer, E. W. J. Chem. Phys. 1997, 107, 1086. (30) Wang, L.-M.; Richert, R. J. Chem. Phys. 2004, 121, 11170. (31) Wang, L.-M.; Tian, Y.; Liu, R.; Richert, R. J. Chem. Phys. 2008, 128, 084503. (32) Auty, R. P.; Cole, R. H. J. Chem. Phys. 1952, 20, 1309. (33) Kaatze, U.; Behrends, R.; Pottel, R. J. Non-Cryst. Solids 2002, 305, 19. (34) Cole, R. H.; Davidson, D. W. J. Chem. Phys. 1952, 20, 1389. (35) Litovitz, T. A.; McDuffie, G. E. J. Chem. Phys. 1963, 39, 729. (36) Angell, C. A. Annu. ReV. Phys. Chem. 1983, 34, 593. (37) Daguenet, C.; Dyson, P. J.; Krossing, I.; Oleinikova, A.; Slattery, J.; Wakai, C.; Weinga¨rtner, H. J. Phys. Chem. B 2006, 110, 12682. (38) Ito, N.; Richert, R. J. Phys. Chem. B 2007, 111, 5016.
JP8038187
