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Temperature Dependence of the Structural Relaxation Time in Equilibrium below the Nominal Tg: Results from Freestanding Polymer Films K. L. Ngai,*,†,‡ Simone Capaccioli,†,‡ Marian Paluch,§ and Daniele Prevosto‡ †

Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, I-56127, Pisa, Italy CNR-IPCF, Institute for Chemical and Physical Processes, Largo B. Pontecorvo 3, I-56127, Pisa, Italy § Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland ‡

ABSTRACT: When the thickness is reduced to nanometer scale, freestanding high molecular weight polymer thin films undergo large reduction of degree of cooperativity and coupling parameter n in the Coupling Model (CM). The finite-size effect together with the surfaces with high mobility make the α-relaxation time of the polymer in bulk nanoconfinement, τnano α (T), much shorter than τα (T) in the bulk. The consequence is avoidance of vitrification at and below the bulk glass transition temperature, Tbulk g , on cooling, and the freestanding polymer thin film remains at thermodynamic equilibrium at temperatures below Tbulk g . Molecular dynamics simulations have shown that the specific volume of the freestanding film is the same as the bulk glass-former at equilibrium at the same temperatures. Extreme nanoconfinement renders total or almost total removal of cooperativity of the α-relaxation, and τnano α (T) becomes the same or almost the same as the JG β-relaxation time τbulk β (T) of the bulk glass-former at equilibrium and at bulk temperatures below Tbulk g . Taking advantage of being able to obtain τβ (T) at bulk equilibrium density below Tg by extreme nanoconfinement of the freestanding films, bulk and using the CM relation between τbulk α (T) and τβ (T), we conclude that the Vogel−Fulcher−Tammann−Hesse (VFTH) bulk dependence of τbulk (T) cannot hold for glass-formers in equilibrium at temperatures significantly below Tbulk α g . In addition, τα (T) does not diverge at the Vogel temperature, T0, as suggested by the VFTH-dependence and predicted by some theories of glass transition. Instead, τbulk α (T) of the glass-former at equilibrium has a much weaker temperature dependence than the VFTHdependence at temperature below Tbulk and even below T0. This conclusion from our analysis is consistent with the temperature g dependence of τbulk α (T) found experimentally in polymers aged long enough time to attain the equilibrium state at various temperatures below Tbulk g . divergence.8−14 In parallel, there are also theoretical models of the structural α-relaxation indicating that τα does not diverge at a finite temperature.15−19 The difficulty encountered in considering the VFTH dependence of τα in the experimental studies far below Tbulk is the immense time needed to attain g equilibrium by aging. Hence, the experimental studies of the equilibrium dynamics were limited to less than 16 K below Tg. Although deviation from the VFTH law was found, the size of the deviation is 4 decades or less.8−14 The issue has received impetus from the results of the study20 of an extremely stable (20 million year) Dominican fossil amber, which established upper bounds on the equilibrium dynamics of a glass-forming material at temperatures down to 43.6 K below Tbulk g . The results show that the upper bounds deviate strongly by many orders of magnitude from the VFTH dependence suggested by the classical1−6 and some modern theories,7 but are consistent with the models with no divergence of τα.15−19

1. INTRODUCTION The temperature dependence of the structural relaxation time, τbulk of bulk glass-forming systems on approaching the nominal α glass transition temperature Tbulk from above usually follows the g Vogel−Fulcher−Tammann−Hesse (VFTH) equation1−3 ⎡ B ⎤ τα(T ) = τ∞ exp⎢ ⎥ ⎣ T − T0 ⎦

(1)

where τ∞ is a prefactor, B is a material constant, and T0 is the temperature at which τα would become infinite. This VFTH law is the signature of τα of glass-forming systems in the equilibrium liquid state at temperature above Tg. Throughout the history of research on the glass transition problem, it has often been assumed that the VFTH law continues to describe the τα of the equilibrium state of the system at temperature below Tg, and diverges at T0. There are classical models4 based on either free volume,5 or configurational entropy,6 and some recent theoretical model7 of glass transition supporting this belief. However, evidence from experimental studies have surfaced contradicting the validity of the VFTH of the equilibrium state below Tgbulk, and the existence of a finite temperature © 2014 American Chemical Society

Received: March 21, 2014 Revised: May 2, 2014 Published: May 5, 2014 5608

dx.doi.org/10.1021/jp502846t | J. Phys. Chem. B 2014, 118, 5608−5614

The Journal of Physical Chemistry B

Article

The spectacular deviations from the VFTH dependence of τα of amber in the equilibrium state below Tbulk have motivated us g to provide another case to support the deviation. It is based on the strong connections between the properties of the structural α-relaxation and the secondary β-relaxation of a special kind,21−25 and in particular a relation between τα of the former and τβ of the latter. These connections were suggested by the Coupling Model26−31 from a similar relation between τα and the primitive relaxation time τ0. Subsequently the relation between τα and τβ has been found by experiments and simulations in many glass-forming systems with diverse physical structures and chemical compositions.31−38 To distinguish the secondary relaxation from other secondary relaxations not having these properties, the term Johari−Goldstein (JG) βrelaxation for them has been used.23,24 The universal existence of the JG β-relaxation also was verified through the analogy with the primitive relaxation in the CM, and the approximate relation between the primitive relaxation time, τ0, and τβ23,24,31 τβ ≈ τ0 (2)

coexists with the effect of reduction of the coupling parameter n. The much shorter τnano has been observed by spectroscoα py,40−42 or can be inferred by the tremendous reduction of the glass transition temperature from Tgbulk to Tgnano of the freestanding polymer.43−45 Due to the fact that Tnano is g significantly less than Tbulk , in the temperature range g Tgnano < T < Tgbulk

the nanoconfined polymer has not been vitrified, and it remains in equilibrium. Molecular dynamics simulations of freestanding polymer films have shown that density goes smoothly to zero at the surface and the entire film practically has density of the bulk polymer at various temperatures.46 Therefore, the values of τnano from experiments were obtained at the equilibrium density α of the bulk polymer at temperatures below the nominal Tbulk g . In the following sections we present the experimental data of τnano ≈ τbulk in the temperature range 5 obtained by the other α β workers in high molecular weight freestanding polymer films. By applying the relation 4 between τbulk and τbulk ≈ τnano α β α , the bulk approximate T-dependence of τα in the equilibrium state at temperatures below Tbulk is obtained. We shall show τbulk g α (T) of the polymers in equilibrium at temperatures way below the nominal Tbulk of the bulk polymer deviate substantially from the g extrapolated VFTH dependence of the values of the experimental τbulk at temperatures above Tbulk to below Tbulk α g g . Moreover, τbulk (T) does not diverge at the Vogel temperature α T0 as indicated by the VFTH-dependence and predicted by some theories. Thus, in support of the conclusions made by other workers, this paper provides additional and independent evidence of the deviation from the VFTH-dependence and bulk nondivergence of τbulk α (T) at temperatures below Tg .

In conjunction with the time honored CM relation26−31 τα(T ) = [tc−n(T )τ0(T )]1/(1 − n(T ))

(3)

relation 2 leads to another approximate relation τα(T ) ≈ [tc−n(T )τβ(T )]1/(1 − n(T ))

(5)

(4)

In eqs 3 and 4, n(T) is the coupling parameter, and its complement, 0 < 1 − n(T) ≡ βK(T) ≤ 1 is the fractional exponent of the Kohlrausch function used to fit the time correlation function of the α-relaxation. The plethora of experimental data in various glass-formers accumulated over the past years31 show beyond doubt that relation 4 exists between α-relaxation and the JG β-relaxation in the equilibrium liquid state above the glass transition temperature Tbulk g . This leads to the expectation that the same relation should hold in the equilibrium state at any temperature below Tbulk g , albeit the latter is hard to reach or impossible to reach in the laboratory if far below Tbulk g . This is because of the exceedingly long times needed to attain equilibrium, as well as τα becoming too long to be measured. On the other hand, τβ is much shorter than τα for many glass-formers and can be measured directly at temperatures way below Tbulk g . If there is some means of obtaining τβ(T) of the equilibrium state below Tgbulk, then relation 4 will also give approximately the corresponding τα(T) at equilibrium below Tbulk g . In this paper, we present such a possibility coming from the studies of the dynamics of high molecular weight freestanding polymer films with thickness reduced to the order of 10 nm. Different from the bulk is the presence of the two highly mobile free surfaces, the reduced size of the nanoconfined polymer becoming less than the radius of gyration of the polymer and the cooperative and heterogeneous length-scale of the structural relaxation of the bulk polymer at the bulk glass transition temperature, Tbulk g . The consequence is significant reduction of the coupling parameter n down to zero or nearly zero. It follows from eq 3 and relation 4 that τnano becomes the same (or nearly the same) α as τbulk 0 , and approximately the same (or approximately nearly the same) as τbulk of the bulk polymer. We hasten to point out β that experimentally the observed relaxation in freestanding polymer films has frequency dispersion much broader than exponential relaxation.40−42 This broadening originates from the intrinsic spatial heterogeneity of the freestanding film, and

2. EXPERIMENTAL EVIDENCE OF τNANO ≈ τBULK IN α β FREESTANDING POLYSTYRENE THIN FILMS In the following, the results are taken from experimental studies of high molecular weight freestanding polystyrene ultrathin films. The data are used to show that we can obtain approximately the JG β-relaxation time or the primitive relaxation time of the bulk glass-former, τbulk β , in the equilibrium state at temperatures way below the nominal glass transition temperature Tbulk of the bulk polystyrene. These results will be g utilized in section 3 to substantiate the conclusion of the paper. Isothermal photon correlation spectroscopy (PCS)40 and dielectric41,42 measurements had been made in ultrathin freestanding polystyrene (PS) films of high molecular weights. The dielectric experiment was carried out on freestanding films 40 nm thick of atactic polystyrene with MW = 932 000 g/mol, and Mw/Mn = 1.2. The PCS experiment was on 22 nm thick PS film with Mw = 767 000 g/mol and Mw/Mn = 1.11. The data of τnano α (T) from these measurements are plotted altogether in Figure 1 as well as the VFTH-dependence used to fit the dielectric τnano α (T) of the d = 40-nm-thick freestanding PS film, and the VFTH-dependence of τbulk in bulk PS. From the α extrapolation of the VFTH fit to lower temperatures, the dielectric Tnano is determined by either the definition that g nano τnano (T ) is equal to 100 or 1000 s. The τα(T) data of the α g freestanding 22 nm PS film from PCS are sparse, and no VFTH fit is made. Nonetheless, ellipsometry measurements43,45 had determined T gnano for freestanding PS thin films with comparable molecular weights as a function of d. The values of Tnano from ellipsometry with Mn = 820 000 g/mol are g represented by two points in Figure 1 with coordinates (1000/ 5609



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dynamics simulations46 of freestanding high molecular weight PS films that the density ρ is practically the same as in the bulk at all temperatures, increasing on cooling from the value of 0.85 at T = 1 to 1.013 at T = 0.44, where the units of all quantities are derived from the Lennard-Jones potential used in the simulations. Therefore, the specific volume of the freestanding PS film has the same value as the equilibrium state of the bulk polymer at any temperature within the range, Tnano < T < Tbulk g g . After combining this result with that in the previous section, the experimental α-relaxation time, τnano α (T), is either the same as or can be considered an upper bound of the JG β-relaxation time of the bulk polymer, τbulk β (T), in the equilibrium state of bulk PS at temperatures T significantly lower than the bulk Tbulk g . It is crucial to recognize not only that τbulk β (T) has been obtained by this method in ultrathin freestanding PS films at temperatures < T < Tbulk within the range, Tnano g g , but also that it is the JG βrelaxation time of the bulk polymer at equilibrium for these temperatures. With this essential result in hand, we ask the following questions. What is the corresponding τbulk α (T) of the bulk polymer in equilibrium at temperatures within the same bulk < T < Tbulk temperature range, Tnano g g ? Does τα (T) deviate from the VFTH dependence given by eq 1 at temperatures significantly below the conventional glass transition temperature Tbulk ? Does τbulk g α (T) diverge at T0 as required by the VFTH-dependence, and predicted by some theories? We answer these related questions by exploiting the connection bulk between τbulk α (T) and τβ (T) of glass-formers in general to deduce the T-dependence of τbulk α (T) in the equilibrium states far below Tbulk g . By now, a voluminous amount of experimental data has accumulated to show that the dynamics of the α-relaxation bear inseparable relation to that of the JG β-relaxation. A microscopic connection was established by spin−lattice relaxation weighted stimulated-echo spectroscopy. The technique had been used to find evidence for a strong connection between the α- and the JG β-relaxation above the glass transition temperature of ortho-terphenyl, D-sorbitol, and cresolphthalein-dimethyl ether.32,34 By suppressing the contributions of some subensembles of the JG β-relaxation in these glass-formers, it was found that the α-relaxation is modified. Hence the linkage between the two relaxations has been demonstrated microscopically. More connections in dynamic properties between the JG β-relaxation and the α-relaxation can be found in the literature.31 The most commonly found connection is the approximate agreement between the experimental τβbulk(T,P) and the bulk theoretical τbulk 0 (T,P) calculated from τα (T,P) and (1 − n(T,P)) by eq 3, after including the dependence on pressure P of the variables, in many glass-forming systems at Tg and above.35−39 Thermodynamic scaling of α-relaxation, or τα as a function of Tρ−γ where γ is a material specific constant,47−49 has been found to apply in general for glass-formers. Shown recently is that τβ also obeys thermodynamic scaling and is approximately another function of Tρ−γ with the same γ.50 For the present purpose and using current notations, these facts can be restated as follows. If τbulk β (T) and n(T) are known and used to evaluate the right-hand-side of relation 4, the result is approximately τbulk α (T). Since relation 4 is valid for many glassformers including amorphous polymers at equilibrium temper51 it is reasonable to assert that it continues atures above Tbulk g , bulk to hold below Tg , provided the glass-former is maintained at thermodynamic equilibrium. In Section 2, by the effect of

Figure 1. Segmental relaxation time, τnano α , measured by dielectric relaxation on a 40 nm freestanding PS film (red filled circles, data from Rotella et al.41), and by PCS on a 22 nm freestanding PS film (purple filled triangles, data from Forrest et al.40). The blue line is the VFTH fit to τα(T) of bulk PS with location of Tg and T0 indicated by the vertical lines. The open black circle and triangle are deduced from the glass transition temperature from ellipsometry measurements43,45 for film thickness that is approximately the same as h = 40 and 22 nm, of respectively. The red line is the VFTH fit to the dielectric τnano α Rotella et al. The black filled diamond is the primitive relaxation time, τbulk 0 (Tg), of bulk PS calculated by eq 3 with n = 0.64. The green line is PCS data together with τ0. The Arrhenius fit to the 22 nm film τnano α upper broken line and symbols on it are log10(τα) calculated from τnano α of the 22 nm film by taking it as τ0 of bulk PS (see text), and n = 0.64 of bulk PS. The black broken line indicates the values of the coupling parameter, n, needed to have τα calculated by eq 3 to be the same as the VFTH fit. 3 3 nano Tnano g (d = 40 nm), 10 s) and (1000/Tg (d = 20 nm), 10 s). nano It can be seen by inspection that the dielectric-Tg for d = 40 nm is not far from the ellipsometry-Tnano for roughly the same g film thickness. Also, the PCS τα(T) data of the 22 nm PS film seem to have the PCS-Tnano consistent with the ellipsometryg Tnano as well. g bulk bulk The primitive relaxation time, τbulk 0 (Tg ), of bulk PS at Tg = 373 K is shown in Figure 1. This value is calculated by eq 3 bulk 3 by taking τbulk α (Tg ) = 10 s, and (1 − nα) equal 0.36 from PCS measurement of bulk PS, and it is a good estimate of the shortest τnano that freestanding film may attain at the bulk Tbulk α g when the film becomes extremely thin. We can see in Figure 1 that the VFTH fit of the dielectric τα(T) of the 40 nm film is 2 bulk bulk or more decades longer than τbulk 0 (Tg ) at the bulk Tg . Presumably, this limit may be approached more closely by decreasing d further to 22 nm as in the case of the PCS experiment. Indeed the PCS and ellipsometry data of the 22 bulk nm freestanding film together with τbulk 0 (Tg ) can be fit reasonably well by Arrhenius dependence as shown in Figure 1. Thus, this Arrhenius τnano α (T) of the 22 nm freestanding film can be taken either the same as τbulk β (T) or an upper bound of τbulk (T) of the equilibrium state of bulk PS at temperature lower β than the bulk Tbulk g .

3. DEVIATION OF τBULK (T) FROM THE α VFTH-DEPENDENCE AT TEMPERATURES FAR OF THE BULK POLYMER AT BELOW THE BULK TBULK G EQUILIBRIUM What extreme nanoconfinement has done for us is to make bulk τnano α (T) much shorter than τα (T) and to delay vitrification on cooling to a lower temperature Tnano than Tbulk g g . Molecular 5610



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For the glass-formers where the FWHM or n(T) remains constant below some temperature equal to or less than Tbulk g , bulk nano the τbulk α (T) can be calculated from the known τβ (T) ≈ τα and the T-independent value of n by the CM relation 4. This operation was carried out and the results are shown in Figure 1. nano Because the temperature dependence of τbulk is β (T) ≈ τα nearly Arrhenius, and n is T-independent, the calculated bulk τbulk α (T) is also close to Arrhenius. This property of τα (T) at thermodynamic equilibrium at temperatures below Tbulk has g support from mechanical and calorimetric measurements after aging to the glass for a long time to attain equilibrium by McKenna and co-workers,8,10−12,20 although mention must be made of different results deduced from dielectric data by others.60−62 If the FWHM or n(T) is not totally constant but increases slightly and slowly with decreasing temperature from bulk n(Tbulk to n(T0) at T0, the calculated values of τbulk g ) at Tg α (T) will depart moderately from strictly Arrhenius, but lie within the bounds calculated with n(Tg) and n(T0). Moreover, via τnano α (T), in Figures.1 we have obtained either bulk τ0 (T) ≈ τbulk β (T), or their upper bound, at thermodynamic equilibrium at temperature even lower than T0. There, the VFTH-dependence give no clue to what the value of τbulk α (T) should be, although its counterpart, τbulk β (T), exists and is known. On the other hand, the value of τbulk α (T) below T0 can be obtained by the CM relation, as shown in Figure 1.

extreme nanoconfinement of high-molecular-weight PS ultrathin films, we have obtained approximately τbulk β (T) of the polymers at thermodynamic equilibrium below Tbulk from the g experimentally measured τnano α (T), which has been justified by the approximate equality, τnano ≈ τbulk ≈ τbulk α 0 β . Although n(T) is not known at temperatures way below Tbulk g , we still can inquire bulk what its values have to be in order for τnano α (T) ≈ τβ (T) to bulk satisfy relation 4 if τα (T) continues to follow the VFTHdependence of eq 1 at temperatures significantly below Tbulk g . This task has been carried out for the ultrathin freestanding PS films in Figure 1. The calculated values of n(T), indicated by the black broken line in this figure, show rapid increase on decreasing temperature below Tbulk g , reaching the limiting value of unity at the Vogel temperature T0. It is more illuminating to appreciate this increase of n(T) by its relation to the full-width at half-maximum (FWHM), W, of the dispersion of the αrelaxation as a function of log(frequency). An approximate relation W and n was given by Dixon.52 This relation rewritten in the form 1 − n = 0.047 − 1.047(W/1.144)−1 shows that W increases rapidly as n increases toward unity depicted in Figure 2. This relation is not accurate for values of n close to 1 because

4. DISCUSSION AND CONCLUSION There is strong connection in dynamic properties of the structural α-relaxation and the JG β-relaxation or the primitive relaxation of the coupling model (CM), as well as a relation bulk between their relaxation times, τbulk α (T) and τβ (T), predicted by the CM relation 4 and verified by experiments and molecular dynamics simulations in many glass-forming systems. The verifications by experiments and simulations are made in the thermodynamically equilibrium “liquid” state of the systems at temperatures above the laboratory or simulation glass transition temperature, Tbulk g . As a general property of the equilibrium state, we can expect that the CM relation also holds for the equilibrium state at temperatures below Tbulk g . In ultrathin freestanding PS films, finite size effect and the presence of free surfaces greatly reduce the dynamic heterogeneous length scale, degree of cooperativity, and the coupling parameter n of the CM. The effect is making the αrelaxation time of the polymer under this kind of nanoconfinement, ταnano(T), much shorter than in the bulk. Thus, vitrification at the bulk Tbulk on cooling is avoided. Taking g advantage of the density of freestanding polymer films has essentially the same density as the bulk polymer,46 the polymer thin film remains at thermodynamic equilibrium with its specific volume the same as that of the bulk polymer at temperatures lower than the nominal Tg. Extreme nanoconfinement renders total or almost total removal of cooperativity of the α-relaxation, and τnano α (T) becomes the same or almost the same as the JG β-relaxation time τbulk β (T) or the primitive relaxation time τbulk of the bulk polymer at equilibrium at 0 temperatures below the bulk Tbulk g . With the advantage of being bulk able to obtain τbulk (T) or τ at equilibrium below Tbulk β 0 g , and employing once more the CM relation, we find the VFTHdependence of τbulk α (T) cannot hold at temperatures significantly below Tbulk for the polymer at equilibrium, and the g divergence of τbulk (T) at the Vogel temperature, T0, cannot α occur. Instead, τbulk α (T) of the polymers at equilibrium has a weaker temperature dependence than the VFTH-dependence

Figure 2. Increase of the FWHM in decades with n appearing in the Kohlrausch stretched exponential correlation function of the αrelaxation, ϕ(t) = exp[−(t/τα)1−n].

for n exactly equal to one, the Kohlrausch function is replaced by a power law which has no time scale, or in other words, the width is infinite. Thus, by employing the CM relation 4 between τbulk α (T) and bulk τβ (T) ≈ τnano at temperatures in the range Tnano < T < Tbulk α g g , and if τbulk α (T) were continuing to obey the VFTH dependence all the way down to T0 where it diverges, then we have the following consequence. On decreasing temperature, concomitant with the increase of n(T) toward unity, the FWHM of the frequency dispersion of the α-relaxation becomes very broad, increases rapidly, and diverges to infinity at the Vogel temperature T0. There are experimental data of a number of glass-formers showing the width of the frequency dispersion of the α-relaxation increase initially on decreasing temperature, but remain constant or at best increasing very slowly after τbulk α (T) has reached a certain threshold value. Some examples can be found in refs 8,11,12,14, and 53−59. Therefore, the rapid increase of the FWHM toward infinity needing to be consistent with the VFTH-dependence is not possible or at least not plausible. Hence τbulk α (T) of glass-formers has to deviate from the VFTH-dependence and its predicted divergence at T0 does not exist. 5611



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for temperature below Tbulk and even below T0. This conclusion g is consistent with the temperature dependence of τbulk α (T) found experimentally in polymers at equilibrium state arrived at by long-term aging experiments,8,10−13 and in 20 million-yearold amber by thermal analysis.20 Before closing, we caution the reader not to confuse the VFTH-dependence considered in this paper with the temperature dependence of the aging time t∞ for the enthalpy to reach equilibrium. The results of t∞ obtained by various workers have been summarized in a recent publication by Koh and Simon.63 The plot of t∞ as a function of aging temperature in Figure 9 of this reference clearly shows a weaker T-dependence than the VFTH dependence of the equilibrium liquid state below the nominal Tg. Similar deviation from VFTH dependence was found for the temperature dependence of t∞ required for enthalpy and volume to reach equilibrium in polystyrene by Badrinaryananan and Simon,64 where a few works with different results are cited. Thus, the results from Simon and coworkers13,63,64 as well as from most other related studies are consistent with the conclusion by McKenna and co-workers.8,12,20,61 The presence of free surfaces has been generally recognized to be the principal cause of the large enhancement of mobility of the segmental α-relaxation of freestanding polymer films. Recently Chai et al.65 were able to measure the viscosity of surface localized flow of a low molecular weight PS with Mw = 3000 g/mol and polydispersity index of 1.09. The experiments provide a measure of surface PS chain mobility in a simple geometry where confinement and substrate effects are negligible. The measured viscosity changes from whole-film flow to surface localized flow over a narrow temperature region near the bulk Tg = 343 K. The lowest temperature of measurement at 314 K is 29 K below Tg, and the measured enhancement of viscosity than the bulk is about 4 orders of magnitude. This is much smaller enhancement than the high molecular weight freestanding film suggested by Figure 1 here for 29 °C below Tg. The disparity of enhancement may seem to suggest that there is a contradiction. However, it is important to recognize that the viscous flow at the surface observed by Chai et al. is transpired by the sub-Rouse modes, and not by the segmental α-relaxation for the low molecular weight PS studied by Chai et al.66,67 The chains of the low molecular weight PS are too short to support Rouse modes.67 The shift factors of sub-Rouse modes have weaker temperature dependence than the segmental α-relaxation, as can be seen from Figure 53 in ref 67 for PS labeled PC11 with Mw = 3600 g/mol and polydispersity similar to that of Chai et al. Moreover, the sub-Rouse modes have smaller coupling parameter nsR than nα of the segmental α-relaxation.68,69 Consequently the enhancement of mobility of the sub-Rouse modes is significantly less than that of the segmental α-relaxation in thin polymers films.69 Thus, it is not surprising that the enhancement of viscosity coming from sub-Rouse modes at the surface by Chai et al. is much less than that of the segmental α-relaxation indicated in Figure 1. The results from recent surface viscosity study of low molecular weight PS do not contradict the larger reduction of segmental α-relaxation time of freestanding high molecular weight PS films. It is noteworthy that the different enhancements in mobility of the sub-Rouse modes and the segmental α-relaxation are also the key to explaining69 the large reduction of plateau compliance observed in ultrathin nanobubble inflated thin films of PS and polycarbonate.70,71 As well, the sub-Rouse modes have been used to explain72 the upper of two transitions

found by ellipsometry in high molecular weight freestanding films.45

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AUTHOR INFORMATION

Corresponding Author

*Phone: +39-0502214322, e-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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