Temperature Dependence of the Segmental Relaxation Time of

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Temperature Dependence of the Segmental Relaxation Time of Polymers Revisited B. Schmidtke, M. Hofmann, A. Lichtinger, and E. A. Rössler* Experimentalphysik II, Universität Bayreuth, D-95444 Bayreuth, Germany ABSTRACT: We analyze the temperature dependence of the segmental relaxation time τ of several low-Tg polymers with varying molar masses (M) as obtained from field-cycling 1H NMR relaxometry and dielectric spectroscopy. They are compared with those of molecular liquids (ML). Time constants in the range 3 × 10−12 s−1000 s, i.e., between Tg and 413 K, are covered. Describing τ(T) by the Vogel−Fulcher−Tammann (VFT) eq a systematic difference with respect to ML is found. While VFT fails for the latter it works well for polymers. The apparent activation energy at high temperatures shows a trend toward a temperature independent value E∞. For polymers, its Mdependence follows that of Tg(M), thus E∞(M) can be described by a Fox−Flory equation. Attempting to understand the difference among the two classes of liquids, we take recourse to our approach first applied to ML [J. Chem. Phys. 2013, 139, 084504]; i.e., we decompose the temperature-dependent activation energy E(T) controlling τ(T) in a constant high-temperature value E∞(M) and a “cooperative part” Ecoop(T). The latter turns out to depend exponentially on temperature, as in ML. Introducing a plot in terms of Ecoop(T)/E∞ vs T/E∞, a master curve for each polymer series is revealed. Taking averaged parameters for all polymers a three-parameter fit well interpolates τ(T) up to highest temperatures. Describing molecular and polymer liquids within the same approach, the difference lies in the fact that the ratio E∞/Ecoop(Tg) is systematically higher for polymers; i.e., τ(T) displays an Arrhenius behavior extending over a larger temperature range.

1. INTRODUCTION One of the most intriguing features of supercooled liquids and polymer melts is the non-Arrhenius temperature dependence of their transport coefficients such as viscosity, diffusion coefficient, or relaxation (or correlation) time τ, the latter characterizing reorientational (or local segmental) dynamics.1−5Actually, in the case of simple (molecular) liquids this behavior sets in already above the melting point Tm.6−10 Often, the temperature dependence of the transport coefficients close to the glass transition temperature Tg is described by the Vogel−Fulcher−Tammann (VFT) equation. While it works reasonable well close to Tg, it is known to fail at high temperatures.6−10 In particular, the usually anticipated crossover from VFT to an Arrhenius law is not accounted for even by most of the approaches introduced to improve the VFT description. For example, the recently discussed approach by Mauro et al. becomes inappropriate when high temperature data are included.9,10 As full understanding of the dynamics in (supercooled) liquids is still missing, most of the offered descriptions, including the VFT equation, are purely phenomenological, and none is broadly accepted. Moreover, the corresponding parameters depend on the temperature range covered by the experiments. Here one has to emphasize that experimental data of molecular glass formers at high temperatures reaching the boiling point Tb are still rare. One reason is that the standard technique applied to investigate the relaxation in such liquids, namely dielectric spectroscopy (DS), usually covers only frequencies, say, below 1 GHz7,11−14 (see, however, ref 15). © XXXX American Chemical Society

Yet, reorientational spectra can now routinely be measured up to Tb by applying depolarized light scattering (DLS) using tandem-Fabry−Perot interferometry and double monochromator spectroscopy.16−19 Likewise, the optical Kerr effect can be applied for providing reorientational time constants.20−22 Still, reaching such high temperatures requires special experimental efforts. For instance, in the case of o-terphenyl (Tg = 245 K), one of the most studied molecular liquids, temperatures up to Tb = 605 K have to be covered. Recently, a series of low-Tg molecular liquids were studied from Tg up to Tb.8,9,19 Two facts emerged from these studies. “Glassy dynamics” in terms of bimodal and long-time stretched correlation functions as well as super-Arrhenius temperature dependence of τ(T) establish well above Tm. The crossover to monomodal correlation functions and an Arrhenius behavior of τ(T) is only observed when approaching Tb. Thus, glassy dynamics occurs already in the thermodynamically stable state of a liquid; it is a central feature of dense fluids. However, it is difficult to identify a clear-cut onset temperature below which glassy dynamics becomes dominant. In the case of polymers, again the VFT equation often in form of the Williams−Landel−Ferry (WLF)23,24 equation is well established to interpolate τ(T). With respect to simple liquids, the experimental situation here is even worse as most polymers cannot be studied up to high temperatures due to Received: January 30, 2015 Revised: April 16, 2015

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DOI: 10.1021/acs.macromol.5b00204 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules thermal degradation. Moreover, many low-Tg polymers, the most appropriate ones to test temperature laws, exhibit a rather small dipole moment as well as weak electronic polarizability making DS and DLS measurements challenging. Here fieldcycling 1H NMR relaxometry (FC 1H NMR) has changed the situation.25−29 Typically, the technique measures the frequency dependence (dispersion) of the spin−lattice relaxation rate R1 = 1/T1 in a frequency range of 10 kHz−20 MHz. Applying earth field compensation even frequencies down to 100 Hz may be covered.30−33 In order to enlarge the effective frequency window, frequency−temperature superposition (FTS) is applied, and usually a range of 10−12 < τ/s < 10−7 is covered. In an Arrhenius representation, Figure 1 presents segmental time constants compiled for six low-Tg polymers, namely

Table 1. Parameters of the Investigated Polymers: Polymer Type, Molar Masses M, Glass Transition Temperature Tg, High-Temperature Activation Energy E∞ from Interpolation along Eq 3, Fragility (Steepness Index) m, and Generalized Fragility Parameter μ as Resulting from Fits with Eq 2a

Figure 1. Segmental time constants obtained from FC 1H NMR relaxometry and dielectric spectroscopy as a function of reciprocal temperature.34−38 Color indicates data sets of one polymer with various M (cf. Table 1). For PB a single data point from rheological experiments,41 for PDMS two data points from neutron scattering experiments42 are included (open symbols). For comparison, time constants of the molecular liquids o-terphenyl (OTP)8 and ethylbenzene (EB), obtained by depolarized light scattering (DLS), dielectric spectroscopy (DS),39 and 1H FC NMR,9 are included. Dashed line represents an Arrhenius law.

polymer

M [g/mol]

Tg [K]

E∞ [K]

m

μ

PI PI PI PI PI PI PI PI PI PI PI PPG PPG PPG PDMS PDMS PDMS PDMS PDMS PDMS PDMS PDMS PDMS PB PB PB PB PB PB PB PEP PPS OTP EB

1040 1370 1920 2390 4470 9910 13500 21200 47300 110000 157000 134 192 18000 860 1600 2490 5940 11000 21600 41400 128000 232000 355 777 1450 2020 2760 4600 87000 200000 44000 230 106

185 190 195 193 201 204 204 203 205 207 207 195 191 202 133 138 140 143 144 144 145 145 145 141 166 171 174 174 174 174 205 229 245 115

3246 3328 3402 3439 3508 3552 3561 3571 3580 3584 3585 3032 3116 3306 2177 2257 2291 2326 2337 2344 2347 2350 2350 2654 3011 3151 3196 3227 3261 3309 3374 3600 2630 1229

80 77 79 81 78 82 76 76 77 81 84 71 79 105 100 105 115 118 121 120 121 126 123 67 84 85 81 92 88 90 89 116 82 86

111 103 105 106 107 111 110 109 110 110 108 73 82 126 154 183 177 166 167 171 172 174 174 87 103 121 114 132 130 127 90 119 79 74

a Data for the two molecular liquids o-terphenyl (OTP) and ethylbenzene (EB) are included.8

poly(butadiene) (PB),34 poly(dimethylsiloxane) (PDMS),35 poly(isoprene) (PI),36 poly(propylene glycol) (PPG),37 poly(propylene sulfide) (PPS),38 and poly(propylene-alt-ethylene) (PEP)38 combining FC NMR and DS39 data. Here we note that the present list of low-Tg polymers is rather limited and actually their fragility does not vary strongly (cf. below). Correlation times in the range 3 × 10−12 s < τ < 103 s are covered, and temperatures up to 413 K are reached for the polymers. Molar masses (M) in a large range are studied (cf. Table 1). For comparison, we added data for ethylbenzene (EB, Tg = 115 K) and o-terphenyl (OTP, Tg = 245 K).8,40 For PB a time constant obtained by rheological experiments41 and for PDMS two data points reported by neutron scattering (NS)42 are included (open symbols). The perfect matching with the FC 1H NMR data demonstrates the applicability of FTS in these cases (see also data for EB). A crossover to a hightemperature Arrhenius law is well documented for the low-Tg liquid ethylbenzene,9 and it may also be anticipated for polymersas will be shown. In contrast to simple liquids, for which mainly packing effects control τ(T), as assumed, e.g., in mode coupling theory,43 in polymers one expects that conformational dynamics due to the connectivity of the

monomers provide an additional contribution to the temperature dependence of τ(T); more precisely, intramolecular torsional barriers may change the dynamics in polymers.44,45 Minding the structural differences between simple liquids and polymers, it appears paradox that both systems are usually described by the VFT equation. Some while ago, searching for a universal representation of transport coefficients, we compared simple liquids and polymers and concluded that both types show significantly different temperature dependence; we introduced the terms “elementary” and polymer glasses.46 The difference was discovered by mapping transport data to a rescaled temperature axis of the kind F·(Tg/T − 1), where F is proportional to the fragility index m = [(∂ log τ)/∂(Tg/T)]|T=Tg. Using selected data of Figure 1, we show such a plot in Figure 2, but with the abscissa m(T/Tg − 1). This representation of the data is convenient as the VFT equation can be recast in a form including m, Tg, and τ∞ instead of the usual parameters T0, D, and τ∞; explicitly, the VFT equation B


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temperature dependence in many cases. The idea behind the approach of Kivelson et al. was to find a clear-cut determination of an “onset temperature” defining the crossover from simple liquid to glassy dynamics guided by their frustration-based theory, an attempt not fully compelling so far.52,54 We also note that incorporating the crossover to a high-temperature Arrhenius law was already suggested rather early. 6,48 Furthermore, we introduced a “generalized Angell plot” displaying Ecoop(T)/E∞ vs T/E∞. Looking for a minimal number of system-specific parameters in the range 10−12 s < τ < 102 s (or between Tb and Tg), three parameters turn out to be sufficient for describing τ(T).8,9 This description was inspired by the correlation between Tg and E∞, pointing to the possibility that high- and low-temperature dynamics in (glass forming) liquids are interrelated. Thus, instead of Tg the energy scale of the glass transition phenomenon might be set by the experimentally accessible high-temperature quantity E∞. We emphasize that the activation energy E∞ must not be identified with some well-defined barrier in a simple one-particle picture. Here we note that a “two-barrier model” is also the idea behind the recently developed “elastic cooperative activated barrier hopping theory” by Schweizer and co-workers.55−57 In this approach, which was extended to polymers very recently,50 the primary relaxation is a process of mixed spatial character, involving local and collective barriers. Yet, both barriers are interrelated. One barrier reflects the local cage formed in a dense liquid while the second is due to the elastic-like response of the surrounding liquid to accommodate the local motion on the cage scale. This quite appealing approach allows interpolating the transport coefficients of glass formers up to highest temperatures. In the present contribution we apply our two-barrier model to polymers; i.e., we analyze the data in Figure 1 and attempt to decompose the activation energy E(T) along the lines sketched above. Our results (i) explain the difference of the temperature dependence of τ(T) between polymers and simple liquids as a consequence of E∞ being comparatively high in polymers. (ii) E∞ depends on M but saturates at high M just like Tg(M), while it appears to continuously grow for simple liquids. (iii) As in molecular systems the activation energy Ecoop(T) shows an exponential increase with lowering temperature. (iv) A plot Ecoop(T)/E∞ vs T/E∞ collapses the data for a given polymer series. As a note of caution, even with the high-temperature FC 1 H NMR data at hand, we emphasize that the Arrhenius law may not always be fully reached. This is also the case for many molecular liquids with Tg values, say, above 200 K.9 Thus, our analysis is still preliminary.

Figure 2. Reorientational correlation times of selected high-M polymers (data from Figure 1) in comparison to molecular liquids8 plotted as a function of a reduced variable (cf. eq 1) which leads at Tg to a collapse of the data as well as to a common slope; m is the fragility index; solid lines: VFT interpolations; dashed straight line: common slope −1 at Tg.

log

τ D = τ∞ T − T0

transforms to47 2

log

K0 τ = τ∞ m(T /Tg − 1) + K 0

(1)

Here K0 = log(τg/τ∞) and m = TgD/(Tg − T0) holds. Following eq 1, in Figure 2 the (rescaled) reduced temperature T/Tg is shifted to provide a slope of −1 for the data at Tg, i.e., at T/Tg − 1 = 0. In order to describe the temperature dependence of the data with the VFT equation, in such representation the only free parameter is τ∞. For all polymer data displayed, the VFT equations with log(τ∞/s) = −12.5 works reasonably well even up to highest temperatures, and its application is well established in polymer science.45,48−50 In the case of molecular liquids a master curve is approximately found as well, yet the VFT description clearly fails at high temperatures, as said, a fact well-known for long.6,7 Here, we emphasize that most of the high-temperature data for τ < 10−10 s in Figure 2 were only recently compiled.8,9 It seems that two different approaches are needed for describing τ(T) of molecular liquids and polymers, respectively. We also mention that the fragility parameter m of polymers often exceeds that of molecular liquids.49−51 Yet, as we will demonstrate, the description of the temperature dependence τ(T) of both classes of liquids may be put on the same base. We will choose an approach which for the case of molecular liquids was recently introduced to describe τ(T) from Tb down to Tg. In particular, the crossover to the hightemperature Arrhenius law is clearly identified in molecular liquids.8,9 Following ideas of Kivelson et al.52 and Sastry,53 we assume an Arrhenius-law for τ(T) to hold for all temperatures and the temperature-dependent activation energy E(T) (cf. eq 2) is decomposed into a temperature-independent part E∞ describing the high-temperature regime well above Tm and a quantity Ecoop(T) ≡ E(T) − E∞ reflecting ”cooperative dynamics” dominating in the low-temperature regime close to Tg.8,9 In molecular liquids Ecoop(T) appears to follow an exponential 2

2. ANALYSIS Figure 3 displays the apparent activation energy (given in K) defined by Ea = (d ln τ)/(d(1/T)) for one polymer of each series in Figure 1 as a function of temperature. As in the case of molecular liquids (examples are included in Figure 3), always a trend to a high-temperature Arrhenius law is observed; i.e., Ea exhibits a trend to become temperature independent at high temperatures. Again, as in the case of molecular liquids, the high-temperature limit of Ea, i.e. E∞, appears to be the higher the lower is Tg. Yet, comparing both classes of liquids E∞ is higher in polymers. In any case, the trend to a crossover to Arrhenius behavior at high temperature makes us confident to attempt a decomposition of E(T) along E(T) = E∞ + Ecoop(T) also for polymers. C


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the quantity E∞ grows with M until saturation is reached, yet some scatter is still observed. In order to reduce this, we interpolate E∞(M) by ∞ E∞(M ) = E∞ − ΔE /M

(3)

where ΔE denotes some constant characteristic of a polymer series. The interpolation in Figure 4a well reproduces the essential features of E∞(M). Repeating a three-parameter fit of the data in Figure 1 along eq 2, and taking the interpolated values for E∞(M) in Figure 4, no systematic deviations from the mean value log(τ∞/s) = −14 are found for τ∞(M) as is demonstrated in Figure 5. Thus, we

Figure 3. Apparent activation energy Ea as a function of temperature for the data of some selected polymers and molecular liquids.8,9

Next we apply a four-parameter fit to the data in Figure 1 employing the following formulas: ln τ /τ∞ = E(T )/T E(T ) = E∞ + Ecoop(T ) Ecoop(T ) = E∞ exp(−μ(T /E∞ − b))

(2)

In the case of molecular liquids the parameter b = TA/E∞ turns out to vary weakly. Moreover, the existence of a common intersection point of the Ecoop(T)/E∞ vs T/E∞ lines at b with Ecoop(TA/E∞) ≅ E∞ is suggested (cf. below).8,9 For a given series of polymers, the scatter of the fit parameters of eq 2, in particular E∞ and the attempt time τ∞, is still large, and as expected, the variation of the two is correlated. In the case of PDMS, for which the largest temperature range is covered since Tg is lowest, the τ∞ values scatter around 10−14 s. As demonstrated in Figure 2, no large variation of τ∞ is expected for polymers. Thus, we fixed log(τ∞/s) = −14 for all polymers. In molecular liquids we find −13 > log(τ∞/s) > −14.8,9 Changing from log(τ∞/s) > −14 to log(τ∞/s) > −13.5 leads to a systematic decrease of the E∞ values by about 17%; the error of E∞ due to the scatter of the experimental data may reach five percent (cf. error bars). Repeating the fit procedure by eq 2 with fixed τ∞, one obtains E∞(M) as shown in Figure 4a. In each polymer series,

Figure 5. Attempt time τ∞ for the different polymers of Figure 1 assuming smoothened values of the high-temperature activation energy E∞(M) (cf. Figure 4a).

assume the such obtained E∞(M) values are reliable to attempt the decomposition of the activation energy E(T). Before doing this, we plot E∞ versus Tg in Figure 6. As in the case of molecular liquids (also included) proportionality is found approximately, yet the slope is larger than in the case of simple liquids; a ratio of E∞/Tg ≅ 11 for simple liquids and E∞/Tg ≅ 16 for polymers is found. Thus, in polymers the hightemperature activation energy E∞ is larger compared to molecular systems; i.e., the Arrhenius law is more pronounced, and this might already explain the difference between “elementary” and polymer glass-formers. We will come back to this point below.

Figure 4. (a) High-temperature activation energy E∞ as a function of molar mass M for the different polymers. The polymer data E∞(M) are interpolated by eq 3. (b) Glass transition temperature Tg of the polymers as a function of M interpolated by a Fox−Flory equation (eq 4). For comparison in each figure data of molecular liquids8,9 are included and as guide for the eye a power-law dependence with exponent α = 0.5 obtained from a survey of nonpolymer liquids including M up to 1000 g/mol (solid black lines) is shown.59 D


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some curvature of the data close to Tg. Here, we note that PB exhibits a strong secondary relaxation process, and disentangling of the main relaxation from the overall bimodal relaxation pattern close to Tg in the dielectric spectra depends on the assumed model of decomposing main and secondary relaxation.12 For all data sets scatter or systematic deviations increase when low values of Ecoop(T) are considered. This is expected as here the accuracy of E∞ becomes important as well as that of the experimental data itself. Also, the data points do not cover equally well all temperatures since the temperature range covered by FC 1H NMR and dielectric spectroscopy does not always overlap. Nevertheless, we assume for the present context that Ecoop(T) follows an exponential temperature dependence, yet keeping in mind that the exact behavior of Ecoop(T) depends on the correct determination of E∞. As discussed, the new approach suggests to construct a “generalized Angell plot” by rescaling Ecoop(T) as well as the temperature T by E∞,8,9 an approach also taken by Kivelson and co-workers.52 This is done in Figure 8. Indeed, the data for each polymer essentially collapse to a single line. While for a given polymer series the slope μ (cf. eq 2) in the semilogarithmic plot of the Figure 8a−d virtually does not change with M there are clear differences among the different polymers. Of course, as said before, the quality of such master curves or, more precisely, the extent to which the data actually follows a master curve depends on the correct determination of E∞. For example, allowing a change of E∞ by, say, 10% may change the picture. Also, giving up the route taken here to fix the attempt time τ∞ would change the results for E∞. In any case, given that not always enough high-temperature data are available, E∞ is expected to be overestimated. As already mentioned, in the case of molecular liquids a “coarse grained” inspection of the data Ecoop(T)/E∞ led us to

Figure 6. High-temperature activation energy E∞ plotted against Tg for the polymers discussed here and for molecular glass-formers.8,9

We note that eq 3 is mathematically identical with the Fox− Flory equation often applied to interpolate Tg(M),58 explicitly Tg(M ) = T g∞ − ΔTg /M

(4)

As demonstrated in Figure 4b, the Fox−Flory equation reproduces the salient features of Tg(M), a fact well-known. Here, we define Tg via τ(Tg) ≡ 100 s. For the present context we ignored a possible discontinuous evolution of Tg(M) as discussed previously.39 Taking the smoothened E∞(M) values from Figure 4a, we can extract Ecoop(T) along eq 2. The results are shown in Figure 7 in a semilogarithmic plot. Essentially, straight lines are observed, indicating an exponential temperature dependence of Ecoop(T), a trend also found for most molecular liquids.8,9 In the case of PB this behavior is less obvious; there is a trend to

Figure 7. Extracted Ecoop as a function of temperature T as obtained by a three-parameter fit with E∞(M) fixed according to the fits in Figure 4a: (a) PDMS, (b) PI, (c) PB, and (d) PPG, PEP, and PPS. E


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Figure 8. Generalized Angell plot: Ecoop(T) and temperature T are rescaled by the high-temperature activation energy E∞ (a) PI, (b) PDMS, (c) PB, and (d) PPG, PEP, and PPS.

there is a common intersection. In order to test this possibility we plot Ecoop(0)/E∞, the intercept of the straight lines in Figure 9 with the ordinate, against μ, the slope in the figure. The data of the two groups (polymers and molecular liquids) follow each a more or less linear behavior. The observed scatter of the points is actually smaller in the case of polymers. Accepting again that the scatter is not systematic but results from random errors, in particular of E∞, a linear interpolation allows to calculate the coordinates of the common point. A common point at Ecoop(T/E∞ = 0.066)/E∞ = 0.45 is suggested (cf. Figure 9). In Figure 9 we included the result of our previous global fit for molecular liquids (dashed line), which differs somewhat from the present approach. This is not unexpected as a different statistical weighting procedure was involved. We do not want to overstress this issue, whether there exists a common intersection for each, polymers as well as molecular liquids, too far, but introducing such a point is also a pragmatic way to reduce the description of τ(T) by eq 2 to a threeparameter formula. As previously shown for molecular liquids,8,9 such a three-parameter fit (with b = 0.104 and E∞, μ, and τ∞ free) is sufficient to allow a complete description of τ(T) between Tg and Tb for 18 systems. Another threeparameter fit with taking E∞(M) from the interpolation in Figure 4a is shown in Figure 1 for the polymers. Clearly, a very satisfying interpolation is achieved. Once again, given the current quality of the data we do not claim that such common intersection definitively exists. Clearly, our analysis reveals differences in the parameters describing τ(T) of polymers and simple liquids, respectively. As already mentioned, the influence of the Arrhenius law appears stronger in polymers. This is confirmed in Figure 11a where the ratio E∞ over Ecoop at Tg is displayed. It is significantly higher for polymers and does not dependent on M for a given series.

speculate whether there exists a common intersection of the straight lines in the generalized Angell plot, i.e., at a common rescaled temperature TA = bE∞ for which all the individual curves Ecoop(T)/E∞ vs T/E∞ might intersect.8,9 Specifically, within a global fit of the τ(T) data the quantities E∞ were slightly varied (about 10%) to induce such a common intersection of all data at b = 0.104. This was justified by the expected comparatively large error in determining E∞. A look at Figure 8d, where the data of PPG, PPS, and PEP are displayed, does not exclude such a possibility also in the case of polymers. In Figure 9, we show now all polymer data in a common plot with rescaled axes. Indeed, one may again speculate whether

Figure 9. Reduced quantities plotted against each other for all the polymers investigated. The point marked by the intersection of the dashed lines indicates a possible common intersection provided that one accepts some random error of the experimentally determined E∞ values. F


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Figure 10. Intersection with the ordinate axis plotted vs slope μ in the generalized Angell plot (Figure 9) for both polymer and molecular liquid data.8,9 Linear extrapolation and slope suggest identification of a common intersection provided one accepts random errors in E∞ (cf. Figure 9). In the case of the molecular liquids we added our previous results from a global fit strategy to “optimize” E∞.

Figure 12. Conventionally defined fragility m vs generalized fragility parameter μ. Some correlation is observed.

mers, a trend toward an Arrhenius temperature dependence is found as it is well established for molecular liquids. In the latter case very high T/Tg ratios can be reached experimentally what is usually impossible in the case of most polymers. Notwithstanding, attempting to isolate the high-temperature activation energy E∞ in the case of low-Tg polymers, our analysis reveals a systematic trend of E∞ with M following that of Tg(M). Yet, E∞ is systematically higher in polymers. This is reflected in the ratio E∞/Ecoop(Tg) which is larger in the case of polymers (cf. Figure 11a); also, the ratio E∞/Tg is larger. In other words, the thermally activated high-temperature dynamics controls τ(T) over a larger temperature range before Ecoop(T) takes over. The essential difference between polymers and molecular liquids appears to be in the fact that E∞(M) saturates while it continuously grows for molecular liquids along a power-law E∞(M) ∝ Mα with α ≅ 0.5 at least for M < 1000 g/mol and provided that proportionality between E∞ and Tg is assumed (cf. Figure 4a).59 We note that the relation Tg(M) ∝ M0.5 for simple liquids as well as the saturation of E∞(M) for polymers is forecast by the approach of Mirigian and Schweizer.50,56,57 As mentioned in the Introduction, one expects that in polymers the conformational barriers play an important role controlling the temperature dependence of the segmental dynamics. Yet, inspecting E∞(M) in the case of PI, for example, saturation only occurs around M ≅ 20 000, i.e., at N = 274 Kuhn elements.60 This is quite a high number of repeat units

In Figure 11b, the “steepness parameter” μ is shown as a function of M (cf. also Table 1). While μ is high for polymers it is significantly smaller for simple liquids. For the limit of low M the polymer data approaches that of molecular liquids. This is reflected in the finding that the fragility index m of polymers shows the trend to be rather highfor the correlation between m and μsee Figure 12. Actually, the variation appears to be larger in μ than in m. Finally, the most important difference between polymers and simple liquids is demonstrated in Figure 4a: While for polymers E∞(M) saturates, in the case of molecular liquids it continuously grows, just along with Tg(M). A recent survey of Tg(M) data of about 100 molecular liquids including molar masses up to 1000 g/mol and Tg values up to 500 K revealed a power-law trend Tg(M) ∝ Mα with α ≅ 0.5.59 Assuming still proportionality of E∞ with Tg the current E∞(M) essentially follows this behavior (cf. Figure 4a).

3. DISCUSSION Phenomenologically, the temperature dependence of the segmental relaxation time of polymers and molecular liquids turns out to be quite different. While in the case of polymers a VFT equation works well over a large temperature range, in the case of molecular liquids VFT fails already close to Tg. Investigating the high-temperature behavior of low-Tg poly-

Figure 11. (a) Ecoop(Tg)/E∞ vs molar mass M comparing the data for polymers and molecular liquids. (b) Generalized fragility parameter μ vs molar mass M. G


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Macromolecules suggesting that E∞ is not a local quantity as often claimed,61 and thus the torsional barrier is not the relevant contribution to the effective barrier. Moreover, in the limit of low-M the quantity E∞ approaches lower values typical of molecular liquids (Figure 4a). As demonstrated by molecular dynamics simulations, the dynamics is controlled by correlated jumps of consecutive conformational transitions, and it is impossible to explain the glassy slowdown on the basis of a single event.44 Thus, a direct correlation of E∞ and a torsion barrier appears not possible. According to the two-barrier model of Schweizer and co-workers,50,55−57 the high-temperature activation energy is actually a combined effect of an effective binary collision term and the first barrier contribution which reflects the local cage formed in a dense liquid. The second barrier contribution becoming relevant at low temperatures is due to the elastic-like response of the surrounding liquid to accommodate the local motion on the cage scale. Still, the magnitude of E∞ may be a measure of the flexibility of a polymer. For example, it is wellknown that PDMS forms highly flexible chains and the corresponding E∞ values are comparatively small (cf. Figure 4a). As in molecular liquids, Ecoop(T) follows an exponential temperature dependence in good approximation. Moreover, the proportionality between E∞ and Tg suggests that the energy scale of the glass transition is determined by E∞, and a generalized Angell plot using E∞ instead of the conventionally defined Tg may be more appropriate to compare different glassforming liquids. This suggests that an adequate measure of fragility is given by the parameter μ which defines the “steepness” of Ecoop(T)/E∞ vs T/E∞ (cf. ref 62). This “generalized fragility parameter” μ is high in polymers and small in molecular liquids in accordance with the finding that the conventionally defined fragility index m is rather high in polymers.48 Since in the generalized Angell plot the master curves for the different polymers come rather close, introducing some common intersection point, independent whether it indeed exists, may be a pragmatic way to reduce our approach to a three-parameter interpolation of τ(T), an approach being successful for molecular liquids.8,9 Admittedly, the present concept of describing τ(T) is difficult to be broadly validated as it can be tested only for low-Tg polymers, since only in this case sufficiently high temperatures or short correlation times can be reached in order to determine the high-temperature activation energy E∞. Nevertheless, our approach sets the description of τ(T) of polymers and molecular liquids on the same base which opens the possibility to achieve a common theoretical understanding of the glass phenomenon in both classes of disordered systems.

temperature-independent value E∞, a fact not reproduced by the VFT equation. The M dependence of E∞ follows that of Tg(M) and thus may be interpolated by a Fox−Flory equation. This opens the possibility to apply our recent approach for the description of τ(T) in molecular liquids also to polymers. The temperature-dependent activation energy E(T) assumed to control τ(T) is decomposed in a constant high-temperature value E∞ and a “cooperative part” Ecoop(T). The latter turns out to depend exponentially on temperature as in the case of molecular liquids. Introducing a “generalized Angell plot” in terms of Ecoop(T)/E∞ vs T/E∞, a master curve for each polymer series is revealed. Small differences in this plot distinguish the different polymers. Within our approach, the main difference between polymers and molecular liquids is given by the fact that the E∞ values are systematically higher than those for molecular liquids; specifically the ratio E∞/ Ecoop(Tg) is higher. That is the temperature dependence of the segmental time shows an Arrhenius behavior extending over a larger temperature range. In addition, while E∞(M) saturates for polymers it appears to continuously grow for molecular liquids.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (E.A.R.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support by the Deutsche Forschungsgemeinschaft (DFG) through the grants FU-308/14 and RO-907/17 is appreciated. REFERENCES

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4. CONCLUSION The temperature dependence of the segmental relaxation time τ of polymers like PDMS, PI, PB, and PPG with varying molar masses (M) as obtained from measurements by field-cycling 1H NMR relaxometry and dielectric spectroscopy are analyzed. Relaxation times in the range 3 × 10−12 s−100 s, i.e., in a broad interval between Tg and 413 K, are covered. Scaling out Tg as well as the fragility index m a master curve for τ(T) results which follows a VFT equation. This is in stark contrast to the case of molecular liquids for which VFT fails already close to Tg. Thus, the transport coefficients of polymer melts and molecular liquids appear to show distinct temperature dependence. Yet, as in the case of simple liquids, the apparent activation energy at high temperatures exhibits a trend toward a H


Article

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Temperature Dependence of the Segmental Relaxation Time of

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