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Why is surface diffusion the same in ultrastable, ordinary, aged, and ultrathin molecular glasses?

K. L. Ngai *a, Marian Paluch ab and Cristian Rodríguez-Tinoco *ab
aSilesian Center for Education and Interdisciplinary Research, 75 Pulku Piechoty 1, 41-500 Chorzow, Poland. E-mail:;
bInstitute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland

Received 7th August 2017 , Accepted 13th September 2017

First published on 31st October 2017

Recently Fakhraai and coworkers measured surface diffusion in ultrastable glass produced by vapor deposition, ordinary glass with and without physical aging, and ultrathin films of the same molecular glass-former, N,N′-bis(3-methylphenyl)-N,N′-diphenylbenzidine (TPD). Diffusion on the surfaces of all these glasses is greatly enhanced compared with the bulk diffusion similar to that previously found by others, but remarkably the surface diffusion coefficients DS measured are practically the same. The observed independence of DS from changes of structural α-relaxation due to densification or finite-size effect has an impact on the current understanding of the physical origin of enhanced surface diffusion. We have demonstrated before and also here that the primitive relaxation time τ0 of the coupling model, or its analogue τβ, the Johari–Goldstein β-relaxation, can explain quantitatively the enhancement found in ordinary glasses. In this paper, we assemble together considerable experimental evidence to show that the changes in τβ and τ0 of ultrastable glasses, aged ordinary glasses, and ultrathin-films are all insignificant when compared with ordinary glasses. Thus, in the context of the explanation of the enhanced surface diffusion given by the coupling model, these collective experimental facts on τβ and τ0 further explain approximately the same DS in the different glasses of TPD as found by Fakhraai and coworkers.

1. Introduction

Surface diffusion coefficients DS(T) of several molecular glass-formers including indomethacin (IMC), nifedipine (NIF), and ortho-terphenyl (OTP),1–3 and a canonical metallic glass, Pd40Cu30Ni10P20 (Pd40)4 were measured at temperatures slightly above and mostly below the bulk glass transition temperature Tg using the method of surface-grating decay. The surface diffusion coefficients DS(T) are found to be many orders of magnitude larger than the bulk diffusion coefficients DV(T) at the same temperature. The size of the enhancement of DS over DV depends on the glass-former, as shown by nearly the two orders of magnitude greater enhancement in the case of OTP than that in IMC. In all these previous studies1–4 at temperatures below Tg the glasses were formed ordinarily by rate cooling or liquid quenching, and are referred to as ordinary glass (OG) to distinguish from a new class of ultrastable glasses (SG) produced by physical vapor deposition. Several theoretical models were proposed to address this huge enhancement of surface mobility of small molecule OGs5–7 and metallic glass.8

Recently Fakhraai and coworkers9–11 have investigated the effect of variations in bulk dynamics on the surface diffusion of the molecular glass, N,N′-bis(3-methylphenyl)-N,N′-diphenylbenzidine (TPD) with its ordinary Tg = 330 K. Using the tobacco mosaic virus as a probe particle, they measured DS(T) on glasses of the same composition but with a large difference in bulk relaxation dynamics and in the glass transition temperature. The glasses of TPD include the ordinary glass (OG) obtained by liquid quenching, annealed glass after physical aging at 0.9Tg for a week, ultrastable glass (SG) fabricated by physical vapor deposition at various substrate temperatures, and 12 to 30 nm thin films. The fictive temperature Tf is reduced in aged glass and much reduced in the SG. The onset temperature for the transformation from the stable glass to the supercooled liquid, Ton, is significantly higher than the Tg of the OG. These changes imply orders of magnitude increase of the structural α-relaxation time τα in the SG. It is generally believed that the high surface mobility or the much larger surface diffusion coefficients DS(T) rather than DV(T) plays a critical role in allowing the formation of highly stable glasses.12,13 Glasses of higher stability will be formed in systems where surface diffusion is faster. This is borne out by OTP2 having a larger DS(Tg) than IMC1 and also having formed a more stable glass than IMC.12,13 However, despite the large difference in the bulk dynamics of SG, OG, and annealed OG of TPD, the surface diffusion coefficients of these glasses measured9–11 turn out to be nearly identical at two temperatures below the Tg of bulk OG. These results have led Fakhraai and coworkers to suggest that surface diffusion has no dependence on the bulk relaxation dynamics when measured below Tg, and to question the validity of theories proposed to account for the size of the DS(T).

Furthermore, Fakhraai and coworkers11 measured the average α-relaxation dynamics in ultrathin films of molecular glass TPD supported on a silicon substrate with film thickness h in the range 12 nm < h < 53 nm. As the film thickness is decreased, Tg decreases rapidly, and the average relaxation time of the films is shorter by 6–14 orders of magnitude, depending on the measuring temperature below Tg of bulk TPD. The enhancement of relaxation in the TPD thin films also originates from the high mobility of the free surface, which induces faster dynamics of the film interior as concluded from studies of the reduction of the glass transition temperature of polymer thin films.14–20 Surface diffusion of the thin films was measured by Fakhraai and coworkers.11 Again, surprisingly they found that the surface diffusion coefficients DS(T) are approximately the same, in spite of the large variations of the averaged film relaxation dynamics for film thicknesses in the range of 12 nm < h < 400 nm. Even more surprising, the DS(T) of the thin films are practically the same as that of the SG and OG, indicating a complete decoupling of the average film relaxation and surface diffusion.11

Collectively the results from the studies of Fakhraai and coworkers have an impact on the research on surface diffusion, ultrastable glasses, and dynamics of thin films. The results also challenge theories of surface diffusion enhancement, and may yield insight into the α-relaxation dynamics of the ultrastable glass, which cannot be directly measured. The basic question is why is the surface diffusion coefficient DS(T) approximately the same despite widely different structural α-relaxation dynamics in the different cases? Part of the answer is at hand if a mechanism for the enhanced DS(T) can be found and is present in all cases. The answer is complete if the mechanism can give quantitatively the size of the enhanced DS(T) with approximately the same value for all cases. In this paper, we provide such a mechanism in the primitive relaxation of the coupling model21–25 or its analogue the Johari–Goldstein (JG) β-relaxation25–29 exemplified in experiment and simulations.21 It has been demonstrated before6,8 that the primitive relaxation time τ0 or the JG β-relaxation time τβ can account for the size of the enhanced DS(T). In this paper, we show directly from experiments that these relaxation times have approximately the same values in all cases, and thus provide an answer to the question why the measured DS(T) all have approximately the same value. Moreover, the change of τβ in ordinary glasses at ambient pressure and under high pressures up to 500 MPa is shown to be similar to that found between τβ in SG and τβ in OG. We use this result to support our explanation of the roughly invariant surface diffusion coefficient in SG, OG, and annealed OG as well as in nanometer thin films.

2. Evidence from experiments

The basic question from the findings of Fakhraai et al. is why the surface diffusion coefficient DS(T) is approximately the same in SG, OG, aged OG, and nanometer thin films, despite widely different bulk structural α-relaxation dynamics. One way to answer this question is to identify the mechanism that is omnipresent, and can account quantitatively for the size of the surface diffusion enhancement, DS(T)/DV(T), in all four cases. But before doing that, one must be able to have a mechanism working in the simpler case of bulk OG before showing it continues to work for the other three more sophisticated cases, particularly the SG. Here we consider the mechanism from the primitive relaxation of the coupling model (CM) and its analogue, the JG β-relaxation.


In our previous applications of the CM to surface diffusion in OG,6 it was assumed at the surface that intermolecular cooperativity of the α-relaxation is totally removed and the many-body relaxation is reduced to the primitive relaxation.21–25,28,29 Consequently, the α-relaxation time at the surface is given by the primitive relaxation time τ0(T), which can be many orders of magnitude shorter than τα(T) in the bulk according to the time-honored CM equation,
τα = [tcnτ0]1/(1−n),(1)
if n is not zero. In eqn (1)tc is the onset time of classical chaos21–24 and its magnitude depends on the interaction potential. Its value is ≈1 to 2 ps for soft matter including molecular glass-formers and polymers and was determined by quasielastic neutron scattering experiments and molecular dynamics simulations.21 The parameter (1 − n) is the fractional exponent of the Kohlrausch correlation function,
φ(t) = exp[−(t/τα)1−n].(2)

According to the CM, τ0 is the α-relaxation time that the structural relaxation would have if all the cooperativity associated with the many-body α-relaxation had been removed, and the coupling parameter n is reduced to zero. In fact from eqn (1), it is clear τα is reduced to τ0 when n becomes equal to zero. The JG β-relaxation is an analogue of the primitive relaxation,21,25,28,29 and their relaxation times are found in general to be approximately equal,

τβ(T) ≈ τ0(T)(3)
This relation has been verified multiple times in many glass-formers of different types since the first paper in 1998,25 and is justified by both the primitive relaxation and the JG β-relaxation being non-cooperative precursors of the structural α-relaxation and having similar properties.21

Free of neighboring molecules and totally free space to explore on one side, molecules diffusing on the surface are not slowed down by intermolecular coupling, provided there is no widespread hydrogen bonding and extensive penetration of the molecules into the interior to constrain and retard surface diffusion. Thus, at the surface the coupling parameter n can become zero or nearly zero. The α-relaxation time, τS(T), at the surface is obtained by substituting n = 0 into eqn (1), and therefore is the same or nearly the same as τ0(T),

τS(T) = τ0(T)(4)
From eqn (4) and the relation of τS(T) to the surface diffusion coefficient, DS(T), given by DS(T) = d2/4τS(T), the CM immediately predicts DS(T) quantitatively by
DS(T) = d2/4τ0(T)(5)
where d is the size of the molecule. Taking relation (3) into consideration, we have
τS(T) ≈ τβ(T),(6)
and alternatively, DS(T) can be determined from the JG β-relaxation time by
DS(T) ≈ d2/4τβ(T)(7)
Predicting DS(T) by eqn (5) requires calculating the primitive τ0(T) from eqn (1), which needs the input of τα(T) and n. By contrast, eqn (7) directly determines DS(T) from the experimental τβ(T) without performing any calculation. This is worth emphasizing at the outset because we shall apply it to surface diffusion in SG, OG, annealed OG, and thin films later on.

Although eqn (5) and (7) are straightforward, the magnitude of d is not exactly known. This indeterminate parameter is eliminated if the enhancement of surface diffusion given by the ratio, DS(T)/DV(T), is considered. Incidentally, the experimental values of this ratio at Tg for several molecular glass-formers are published in the literature.1–4,30 If DS(T)/DV(T) is the same as the ratio τα(T)/τS(T) or τα(T)/τ0(T) except for a factor 3/2 which is insignificant for the consideration herein and henceforth is neglected, then viaeqn (1)DS(T)/DV(T) is given by

image file: c7cp05357f-t1.tif(8)
Furthermore from eqn (6), we have approximately
image file: c7cp05357f-t2.tif(9)
For ordinary glasses, the values of τα(T) and n are known at temperatures above and below Tg from dielectric relaxation. For indomethacin (IMC), the value of n = 0.41 was determined.6 Using this value together with tc = 1 to 2 ps for soft matter and the choice of τα(Tg) = 102 or 103 s, the CM eqn (1) gives τ0(Tg) = 10−3.62 and 10−3.03 s respectively. Assuming d = 1 nm, eqn (5) predicts DS(Tg) = 10−15.16 and 10−15.75 m2 s−1 compared to the experimental value of DS(Tg) = 10−14.3 m2 s−1 (see inset of Fig. 1). The discrepancy can be due to the arbitrary choice of the value of d. On the other hand, after eliminating d, eqn (8) yields τα(Tg)/τS(Tg) ≈ 106 s, which is in order of magnitude agreement with the experimental value of DS(Tg)/DV(Tg).1 The temperature dependence of τ0(T) is also about the same as that of the DS(T) data below Tg = 314 K of OG, indicating τ0(T) continues to match τS(T) in value in the OG of IMC. The JG β-relaxation of IMC is not resolved at ambient pressure (albeit resolved at a high pressure of 400 MPa), and its relaxation time τβ(T) at ambient pressure cannot be determined exactly.

image file: c7cp05357f-f1.tif
Fig. 1 Green closed circles and squares are the α-relaxation times τα(T), and the primitive relaxation times τ0(T) (calculated), and the red closed squares are the β-relaxation times τβ(T) of bulk OTP. The blue closed diamonds represent −log[thin space (1/6-em)]DS(T) + C with C = −17.35 and the original data of DS(T) are from ref. 2. The inset showing DS(T) and DV(T) is reproduced from ref. 2. Added is the line, which is a fit of the DS(T) of the OG of TPD by the Arrhenius dependence.

In this paper we test the CM further by considering the new case of OTP, which had its JG β-relaxation resolved in the OG,27,31 and the experimental values of τβ(T) can be used to directly predict DS(T) viaeqn (7).

OTP has a larger value of n = 0.50 than IMC at a temperature near Tg.32 The dielectric τα(T) and a Vogel–Fulcher fit are shown in Fig. 1 together with τβ(T), the calculated τ0(T), and DS(T) after it was shifted by a constant to match roughly τ0(T) above Tg. Although τ0(T) can only be obtained above Tg, τβ(T) was determined experimentally below Tg. By extrapolating the Arrhenius temperature dependence of τβ(T) back to Tg the relationship τβ(T) ≈ τ0(T) is verified. Hence, from eqn (8) and (9), we have at Tg. the CM predicted value of

DS(Tg)/DV(Tg) = τα(Tg)/τS(Tg) = τα(Tg)/τ0(Tg) = 107.5,(10)
DS(Tg)/DV(Tg) = τα(Tg)/τS(Tg) ≈ τα(Tg)/τβ(Tg) ≈ 107.5(11)
This prediction of enhanced surface mobility based on τ0(T) and τβ(T) is to be compared with the experimental value of DS(Tg)/DV(Tg) ≈ 107.6 at Tg shown in the inset of Fig. 1 taken from the paper of Zhang et al.2 Deeper into the glassy state, the shifted DS(T) and τβ(T) are within half a decade apart and can be considered the same if the uncertainty of τβ(T) determined from the broad frequency dispersion of the JG β-relaxation is taken into consideration. Thus eqn (8) and (9) for TTg in OG of OTP is verified.

We can obtain DS(Tg) of OTP directly from the experimental value of τβ(Tg) by using eqn (7) and again assuming d = 1 nm. The Arrhenius extrapolation of τβ(T) in Fig. 1 determines τβ(Tg) = 10−5.56 s. Hence the predicted value of DS(Tg) is 10−13.22 m2 s−1, which is smaller than the experimental value of 10−12.1 m2 s−1. Assumed in eqn (5) and (7) is that τs(Tg) is the same as the CM primitive relaxation time τ0(Tg) ≈ τβ(Tg) in the OTP bulk. But the actual value of τ0(Tg) ≈ τβ(Tg) at the surface can be shorter, due to more free space, and hence the actual value of DS(Tg) = 10−13.22 m2 s−1 calculated by eqn (7) can underestimate the real enhancement. It can be seen from Fig. 1 that τβ(T) continues to describe well the temperature dependence of DS(T) deep into the glassy state of OTP. This means the DS(T) from eqn (7) not only agrees with the experimental value at Tg but also at temperatures below it.

The exercise presented in the above in the case of OTP demonstrates that the experimental data of τβ(T) can be used directly viaeqn (7) to determine DS(T). Therefore, if there is no or little change in τβ(T) in glasses of different origins, we can immediate conclude that the surface diffusion coefficients DS(T) measured in these glasses will be practically the same. This point is relevant for understanding the collection of studies of the surface diffusion of SG, OG, and annealed OG of TPD in the following subsections B and C.

Shown in the inset of Fig. 1 is the Arrhenius fit of the measured tracer surface diffusion coefficients of the OG of TPD.10 The enhancement DS(Tg)/DV(Tg) of TPD at Tg lies in between that of the surface self-diffusion of IMC and OTP. If the tracer surface diffusion of TPD measured is not too different from self-diffusion, the approximately same order of magnitude of the enhancement at Tg of TPD and OTP can be understood from the comparable value of n = 0.50 and 0.48 for TPD and OTP respectively.

B. Aged OG

From the previous Section A we have validated eqn (6) and (7) and established that the JG β-relaxation time τβ(T) is a quantitative indicator of the enhanced surface diffusion coefficient DS(T) for TTg in the OG. Now, if the OG is densified by physical aging over a long period of time, any change of DS(T) will be observed by the corresponding change in τβ(T). It is known from the classical study of physical aging of 43.3 mol% toluene in pyridine glass at several temperatures33 that there is practically no change of the β loss peak frequency, fβ = (1/2πτβ), except a reduction of the amplitude of the loss. A slight increase of the peak frequency was found on aging the JG β-relaxation of glassy polyvinylethylene.34 On the other hand, a small increase of peak frequency was found on aging of dipropyleneglycol dibenzoate.35 In all cases, including others not mentioned, there is either no change or a small change of τβ by physical aging. From this general aging property of τβ(T), it follows from eqn (6) and (7) that τS(T) and DS(T) of the aged TPD glass are nearly the same as that of OG, as found by Fakhraai and coworkers.


The key for continued use of eqn (6) and (7) to predict the enhancement of surface diffusion at Tg or Ton in SG is to have data on τβ(T) of SG. The onset temperature Ton is defined as the intersection of the extrapolated glassy line and the tangent of the transformation from the glassy state to the supercooled liquid state. Among the SGs produced by vapor deposition, only the SG of toluene36,37 has been studied by dielectric relaxation and reported.38 Toluene is a rigid molecule and its sole secondary relaxation is the JG β-relaxation.28 The relationship between the α- and the β-relaxations of toluene is in accord with the CM eqn (1)–(3) as shown by an example in the inset of Fig. 2. Shown there is the reported dielectric loss spectrum of toluene at 119 K with fα = 0.2 Hz39 and the Kohlrausch fit with exponent (1 − n) = 0.52. The calculated primitive frequency f0 = (1/2πτ0) is about half a decade higher than fβ = (1/2πτβ), verifying the approximate eqn (3).
image file: c7cp05357f-f2.tif
Fig. 2 The main figure is reproduced from Yu et al.38 It shows logarithmic relaxation times of α and β processes of ordinary glass (open symbols), and the relaxation times for the vapor-deposited samples (solid symbols) vs. reciprocal temperature. Added are the two vertical lines located at 1000/Tg and 1000/Ton. Inset: The dielectric loss spectrum of toluene, the fit by the Fourier transform of the Kohlrausch function with n = 0.48, and the calculated value of the primitive frequency f0.

The experimental data of τβ(T) of a toluene SG deposited at a substrate temperature 98 K (= 0.84Tg) from Yu et al.38 are reproduced in Fig. 2. The value of Ton of the SG is 123 K,36 and the corresponding 1000/Ton is located in Fig. 2 by the vertical black broken line. The other vertical broken line is at 1000/Tg with Tg = 117 K. Shown also are τβ(T) and τα(T) of the toluene OG from dielectric and NMR measurements.40,41 Remarkably, τβ(T) of the SG is longer than the OG by less than a decade at the same temperature, and the T-dependence of both are Arrhenius with the activation energies Eβ = 27 ± 3 and 25 ± 2 kJ mol−1 respectively. From these experimental facts, eqn (6) and (7) of the CM readily explain why the surface diffusion time τS(T) and coefficient DS(T) of the SG are practically the same as those of the OG.

We have extrapolated the Arrhenius T-dependence of τβ(T) in the SG to higher temperatures to intersect 1000/Ton. The intersection determines τβ(Ton) = 10−4.31 s for the SG, while a similar operation determines τβ(Tg) = 10−4.57 s for the OG. The nearly same value of τβ suggests also τS(T) of the SG at Ton is similar to τS(T) of the OG at Tg.

D. Nanometer thin films

Zhang and Fakhraai11 measured average α-relaxation time in ultrathin films of molecular glass TPD supported on a silicon substrate with film thicknesses ranging from 12 nm to 400 nm. The film Tg decreases rapidly with decreasing film thickness. They also measured the tracer surface diffusion coefficient DS(T) of all the films and found that the DS(T) are all enhanced at all temperatures by the same orders of magnitudes as the DS(T) of OG and SG, despite the fact that the average film α-relaxation time decreases significantly with decreasing film thickness.

It is generally agreed that the enhanced surface mobility in supported or freestanding polymer thin films is primarily the cause of the decrease of the film Tg as thickness is decreased. The enhanced surface mobility with the fast relaxation time τS(T) is transmitted into the interior of the film but the effect is attenuated layer by layer.19 The substrate may have an opposite effect on the mobility if physical or chemical interaction with the film exists at the interface. Consequently, the averaged α-relaxation of the film measured is slower than at the surface, and the effective relaxation time is longer than τS(T) as in the study of Zhang and Fakhraai. However, the JG β-relaxation is a local process and its relaxation time τβ(T) is not sensitive to changes of film thickness, as is supported by experiments. For example, dielectric measurements of poly(methyl methacrylate) (PMMA) thin films show little change of τβ(T) for thickness down to 20 nm. The τβ(T) of the 15 nm, and 9.5 nm thick films are only about a factor of 5 and 8 respectively shorter than the thick 1070 nm film, while the change of Tg is more than 20 K.42 Direct measurement of surface relaxation of syndiotactic (PMMA) films by lateral force microscopy also finds the τβ(T) at the surface shorter than the bulk by about 1 decade at higher temperatures of measurements.43 Thus, from eqn (6) and (7), the small difference between τβ(T) in thin films and in bulk explains the same for the surface diffusion DS(T) observed in glassy thin films of TPD compared to OG.

It is worthwhile mentioning a related experimental fact of the average segmental relaxation time, τnanoα, measured by dielectric relaxation on a 40 nm, and by PCS on a 22 nm freestanding polystyrene (PS) thin film. The values of τnanoα(T) were shown to match the primitive relaxation times τ0(T) in the glassy state of bulk PS.44 In these ultrathin freestanding films the high mobility of the two free surfaces dominate in determining τnanoα, and hence the finding of τnanoα(T) ≈ τ0(T) supports the fact that τS(T) is the same as τ0(T), or eqn (4).

The cooperative α-relaxation in nominal glass-formers such as OTP at 10 degrees above Tg has a length-scale of the order of 20 nm already.45 Therefore, when the film thickness is much reduced to the order of 1 nm, not only the surface mobility dominates the entire film but also the interior is also removed of the cooperativity provided that the substrate has no effect. The ideal situation is realized in the 1.5 nm thin films of poly(methylphenylsiloxane) confined in galleys of nanocomposites19,46 or poly(dimethylsiloxane) confined in 2 nm glass pores,47 and the calculated τ0(T) by eqn (1) from the bulk is in quantitative agreement with the experimental τS(T) as shown in ref. 19 (see Fig. S1 for the 1.5 nm thin film in the ESI). Thus eqn (4) is verified and, viaeqn (5), it supports the explanation of the nearly no change of DS(T) to change of film thickness.

3. Comparing SG at ambient pressure with OG at elevated pressures

Studies of the structure of vapor-deposited SG have shown the differences from OG, which include higher density, different local packing and orientations of the molecules.48,49 Since higher density is a primary cause of the stability of SG, it is profitable to consider the glasses formed by applying very high pressure to the liquid and comparing some properties. This task is relevant in view of a recent experiment,50 which compares the pressure dependence of the onset of devitrification, Ton, between the SG and OG of IMC. The dynamics and thermodynamic stabilities of the two glasses are very different at ambient pressure. Ultrastable glasses of IMC have densities up to 1.4% more than the liquid-cooled glass.13,37 The two glasses have different dTon/dP values at low pressures, but become nearly the same when Ton of the OG approaches that of the SG with increase of pressure to 300 MPa.50

At about 75 MPa, the increase of Ton of OG with pressure has caught up with the Ton of SG at ambient pressure. From this, we can expect at higher pressures, such as a few hundred MPa routinely reached in dielectric studies of the dynamics of glass-formers, that Ton as well as the density of the pressurized OG is significantly higher than the UG at ambient pressure. Thus, the difference in τβ(T) between the pressurized OG and OG at ambient pressure in the glass state should be larger than that between the SG and OG at ambient pressure. This can be verified by examining some experimental τβ(T) data of OG at ambient and elevated pressure such as poly(phenylglycidylether) (PPGE),51 diglycidyl ether of bisphenol-A with Mw = 380 g mol−1 (also known as EPON 828),52 and di-propylene glycol dibenzoate (DiPGDB).35 In Fig. S2–S4 (ESI) are the data of τβ and τα of the three glass-formers shown at a fixed temperature as a function of pressure, and at a fixed pressure as a function of temperature. From these plots, Tg changes by 54 K for EPON 828 with the increase in pressure of 400 MPa; by 43 K and 77 K for DiPGDB with increase of 268 and 530 MPa respectively; and by 17 and 64 K for PPGE with increase of 95.5 and 500 MPa respectively. The Tg change of 17 K at 95.5 MPa matches the difference between Tg = 242.3 K and Ton = 259 K for the most stable glass of OTP vapor-deposited at 0.85Tg.12 Even for the much larger increase of pressure up to 500 MPa, it only changes τβ by one to two orders of magnitude in the glassy state at a fixed temperature in all cases. The change of τβ is even less at pressures lower than 100 MPa to match the increase in density with that of the IMC SG at ambient pressure. This we can use as corroborative evidence to support the small difference in τβ between SG and OG at ambient pressure, and the use of the CM eqn (6) and (7) to explain the same surface diffusion coefficient for the two glasses. The properties of pressurized OG and SG produced by vapor deposition are the same as long as the pressure on the former is maintained, and the common cause is significant densification of the glass. By no means do the similar properties diminish the achievement of vapor-deposited SG in producing glassy materials with properties very different from OG at ambient pressure. By contrast, when the pressure is released, the pressurized glass returns to OG. Nevertheless, the physics of the pressurized glass is the same as SG as far as the comparison is confined to their glassy states. The analogy is no longer valid when the relationship of the glassy state to the equilibrium liquid is considered. For example, during transformation of the SG, the emerging liquid phase is at ambient pressure, and cannot be compared to the transformation of pressurized OG, where both the glass and liquid phases are under high pressure. Consequently, SG and pressurized OG do not exhibit the same transformation behavior. More importantly, the limiting fictive temperature (Tf′) of a SG is lower than the OG, while the Tf′ of the pressurized OG is higher than that of the OG at ambient pressure (note that this does not contradict the higher density of the pressurized OG than the ambient pressure OG). To be more specific, when calculating the Tf′ of a SG, one considers the corresponding liquid at ambient pressure, while the Tf′ of the pressurized OG refers to the liquid at the corresponding pressure. Again, since the definition of Tf′ of either glass involves the corresponding equilibrium liquid, the mentioned analogy is not applicable in this point.

It is worthwhile to point out from the data shown in Fig. S2–S4 (ESI) that τβ(T,P) has approximately the same value at any arbitrary chosen long time τα(T,P), independent of the choice of the fixed T as a function of P, or the choice of fixed P as a function of T.35,51,52 The relationship τβ(T,P) ≈ τ0(T,P) has also been verified.

4. Conclusions

The findings of practically no change of the surface diffusion coefficient DS(T) on ultrastable glass (SG), ordinary glass (OG) with and without physical aging, and nano-meter glassy thin films of TPD by Fakhraai and coworkers are interesting, and certainly have an impact on identifying the mechanism of enhanced surface diffusion. They suggest that fast surface diffusion is decoupled from the bulk dynamics, although the exact mechanism of this decoupling merits further investigation. We show the mechanism of decoupling is the removal of intermolecular coupling or cooperativity in relaxation and diffusion of molecules at the surface, resulting in the surface relaxation time τS(T) being the same as the primitive relaxation time τ0(T) of the coupling model (CM), and orders of magnitude shorter than the bulk α-relaxation time τα(T). The enhanced surface diffusion is given by DS(T) = d2/4τS(T) = d2/4τ0(T) where d is the size of the molecule. In the case of OG at Tg, the values of τα(Tg) and the coupling parameter n are known, and the calculated DS(Tg) are in semi-quantitative agreement such as in the case of IMC,4 OTP herein, and ethylcyclohexane (ECH).53DS(Tg) of ECH is several orders of magnitude less enhanced than that of IMC and OTP,53 but it is still consistent with the CM eqn (8) because ECH has a smaller n ≤ 0.32 than IMC and OTP. From the fact that the Johari–Goldstein β-relaxation is the analogue of the primitive relaxation, and its relaxation time τβ(T) is approximately the same as τ0(T) as proven in many materials, one can take the values of τβ(T) from experiment and directly calculate the enhanced surface diffusion in the glassy state by DS(T) ≈ d2/4τβ(T). The crux of our explanation of the approximate invariance of DS(T) in SG, OG with and without physical aging, and nano-meter glassy thin films of TPD is the corresponding approximate invariance of τβ(T), despite the huge differences in the bulk α-dynamics. This property of τβ(T) is demonstrated directly from experimental data in all these different glasses. It is perhaps unsurprising since both the primitive relaxation and the JG β-relaxation are local process without cooperativity, although a more in-depth study to substantiate it theoretically is needed and will be forthcoming. We compare also the change of τβ(T) in some other ordinary glasses by elevating the pressure up to 500 MPa. The advantage of studying OG under pressure is because τβ(T) is resolved in the glassy state of many glass-formers. Pressure below 100 MPa is able to densify the OG to the same level and have a similar Ton as the best SG deposited at 0.85Tg. The change of τβ(T) in the pressurized OG is found to be small. Much higher Ton is found in OG pressurized at higher levels above 100 MPa. Notwithstanding, the change of τβ(T) is still not large. This comparative study of τβ(T) in OG at elevated pressures lends support to the explanation for the ultrastable glass.

In this paper we use eqn (5)–(7) exclusively and directly from the experimental τβ or the theoretical τ0 to compare with the experimental value of DS(T) of SG like TPD. The CM eqn (1) need not be used to calculate τ0 from the structural α-relaxation time τα. The exception is the previous work on indomethacin (IMC)6 where τα is known from experiment above and below Tg, and OTP above Tg in Fig. 1 herein. This situation in IMC and OTP confers the benefit of predicting the size of the enhancement by the ratio, τα/τ0, and comparing with the experimental value of DS(T)/DV(T), as successfully done for IMC and OTP. In the case of SG prepared by vapor deposition, τα is not known and eqn (1) cannot and need not be used to calculate τ0 or τβ in the stable glass.

Conflicts of interest

There are no conflicts of interest to declare.


We thank Prof. Ranko Richert and Prof. Haibin Yu for sharing with us their data on the secondary relaxation in the ultrastable glass of toluene. C. R.-T. acknowledges support from the National Science Centre through the Polonez scheme (Grant No. DEC-2015/19/P/ST3/03540/2). This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement no. 665778.


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Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp05357f

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