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K. L.
Ngai
*^{a},
Marian
Paluch
^{ab} and
Cristian
Rodríguez-Tinoco
*^{ab}
^{a}Silesian Center for Education and Interdisciplinary Research, 75 Pulku Piechoty 1, 41-500 Chorzow, Poland. E-mail: kiangai@yahoo.com; cristian.rodriguez-tinoco@smcebi.edu.pl
^{b}Institute 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 D_{S} measured are practically the same. The observed independence of D_{S} 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 τ_{0} of the coupling model, or its analogue τ_{β}, 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 τ_{β} and τ_{0} 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 τ_{β} and τ_{0} further explain approximately the same D_{S} in the different glasses of TPD as found by Fakhraai and coworkers.

Recently Fakhraai and coworkers^{9–11} 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 T_{g} = 330 K. Using the tobacco mosaic virus as a probe particle, they measured D_{S}(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 0.9T_{g} 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 T_{f} 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, T_{on}, is significantly higher than the T_{g} of the OG. These changes imply orders of magnitude increase of the structural α-relaxation time τ_{α} in the SG. It is generally believed that the high surface mobility or the much larger surface diffusion coefficients D_{S}(T) rather than D_{V}(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 OTP^{2} having a larger D_{S}(T_{g}) than IMC^{1} 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 measured^{9–11} turn out to be nearly identical at two temperatures below the T_{g} 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 T_{g}, and to question the validity of theories proposed to account for the size of the D_{S}(T).

Furthermore, Fakhraai and coworkers^{11} 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, T_{g} decreases rapidly, and the average relaxation time of the films is shorter by 6–14 orders of magnitude, depending on the measuring temperature below T_{g} 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 D_{S}(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 D_{S}(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 D_{S}(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 D_{S}(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 D_{S}(T) with approximately the same value for all cases. In this paper, we provide such a mechanism in the primitive relaxation of the coupling model^{21–25} or its analogue the Johari–Goldstein (JG) β-relaxation^{25–29} exemplified in experiment and simulations.^{21} It has been demonstrated before^{6,8} that the primitive relaxation time τ_{0} or the JG β-relaxation time τ_{β} can account for the size of the enhanced D_{S}(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 D_{S}(T) all have approximately the same value. Moreover, the change of τ_{β} in ordinary glasses at ambient pressure and under high pressures up to 500 MPa is shown to be similar to that found between τ_{β} in SG and τ_{β} 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.

τ_{α} = [t_{c}^{−n}τ_{0}]^{1/(1−n)}, | (1) |

φ(t) = exp[−(t/τ_{α})^{1−n}]. | (2) |

According to the CM, τ_{0} 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 τ_{α} is reduced to τ_{0} 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,

τ_{β}(T) ≈ τ_{0}(T) | (3) |

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, τ_{S}(T), at the surface is obtained by substituting n = 0 into eqn (1), and therefore is the same or nearly the same as τ_{0}(T),

τ_{S}(T) = τ_{0}(T) | (4) |

D_{S}(T) = d^{2}/4τ_{0}(T) | (5) |

τ_{S}(T) ≈ τ_{β}(T), | (6) |

D_{S}(T) ≈ d^{2}/4τ_{β}(T) | (7) |

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, D_{S}(T)/D_{V}(T), is considered. Incidentally, the experimental values of this ratio at T_{g} for several molecular glass-formers are published in the literature.^{1–4,30} If D_{S}(T)/D_{V}(T) is the same as the ratio τ_{α}(T)/τ_{S}(T) or τ_{α}(T)/τ_{0}(T) except for a factor 3/2 which is insignificant for the consideration herein and henceforth is neglected, then viaeqn (1)D_{S}(T)/D_{V}(T) is given by

(8) |

(9) |

Fig. 1 Green closed circles and squares are the α-relaxation times τ_{α}(T), and the primitive relaxation times τ_{0}(T) (calculated), and the red closed squares are the β-relaxation times τ_{β}(T) of bulk OTP. The blue closed diamonds represent −logD_{S}(T) + C with C = −17.35 and the original data of D_{S}(T) are from ref. 2. The inset showing D_{S}(T) and D_{V}(T) is reproduced from ref. 2. Added is the line, which is a fit of the D_{S}(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 τ_{β}(T) can be used to directly predict D_{S}(T) viaeqn (7).

OTP has a larger value of n = 0.50 than IMC at a temperature near T_{g}.^{32} The dielectric τ_{α}(T) and a Vogel–Fulcher fit are shown in Fig. 1 together with τ_{β}(T), the calculated τ_{0}(T), and D_{S}(T) after it was shifted by a constant to match roughly τ_{0}(T) above T_{g}. Although τ_{0}(T) can only be obtained above T_{g}, τ_{β}(T) was determined experimentally below T_{g}. By extrapolating the Arrhenius temperature dependence of τ_{β}(T) back to T_{g} the relationship τ_{β}(T) ≈ τ_{0}(T) is verified. Hence, from eqn (8) and (9), we have at T_{g}. the CM predicted value of

D_{S}(T_{g})/D_{V}(T_{g}) = τ_{α}(T_{g})/τ_{S}(T_{g}) = τ_{α}(T_{g})/τ_{0}(T_{g}) = 10^{7.5}, | (10) |

D_{S}(T_{g})/D_{V}(T_{g}) = τ_{α}(T_{g})/τ_{S}(T_{g}) ≈ τ_{α}(T_{g})/τ_{β}(T_{g}) ≈ 10^{7.5} | (11) |

We can obtain D_{S}(T_{g}) of OTP directly from the experimental value of τ_{β}(T_{g}) by using eqn (7) and again assuming d = 1 nm. The Arrhenius extrapolation of τ_{β}(T) in Fig. 1 determines τ_{β}(T_{g}) = 10^{−5.56} s. Hence the predicted value of D_{S}(T_{g}) is 10^{−13.22} m^{2} s^{−1}, which is smaller than the experimental value of 10^{−12.1} m^{2} s^{−1}. Assumed in eqn (5) and (7) is that τ_{s}(T_{g}) is the same as the CM primitive relaxation time τ_{0}(T_{g}) ≈ τ_{β}(T_{g}) in the OTP bulk. But the actual value of τ_{0}(T_{g}) ≈ τ_{β}(T_{g}) at the surface can be shorter, due to more free space, and hence the actual value of D_{S}(T_{g}) = 10^{−13.22} m^{2} s^{−1} calculated by eqn (7) can underestimate the real enhancement. It can be seen from Fig. 1 that τ_{β}(T) continues to describe well the temperature dependence of D_{S}(T) deep into the glassy state of OTP. This means the D_{S}(T) from eqn (7) not only agrees with the experimental value at T_{g} but also at temperatures below it.

The exercise presented in the above in the case of OTP demonstrates that the experimental data of τ_{β}(T) can be used directly viaeqn (7) to determine D_{S}(T). Therefore, if there is no or little change in τ_{β}(T) in glasses of different origins, we can immediate conclude that the surface diffusion coefficients D_{S}(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 D_{S}(T_{g})/D_{V}(T_{g}) of TPD at T_{g} 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 T_{g} of TPD and OTP can be understood from the comparable value of n = 0.50 and 0.48 for TPD and OTP respectively.

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/T_{g} and 1000/T_{on}. 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 f_{0}. |

The experimental data of τ_{β}(T) of a toluene SG deposited at a substrate temperature 98 K (= 0.84T_{g}) from Yu et al.^{38} are reproduced in Fig. 2. The value of T_{on} of the SG is 123 K,^{36} and the corresponding 1000/T_{on} is located in Fig. 2 by the vertical black broken line. The other vertical broken line is at 1000/T_{g} with T_{g} = 117 K. Shown also are τ_{β}(T) and τ_{α}(T) of the toluene OG from dielectric and NMR measurements.^{40,41} Remarkably, τ_{β}(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 E_{β} = 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 τ_{S}(T) and coefficient D_{S}(T) of the SG are practically the same as those of the OG.

We have extrapolated the Arrhenius T-dependence of τ_{β}(T) in the SG to higher temperatures to intersect 1000/T_{on}. The intersection determines τ_{β}(T_{on}) = 10^{−4.31} s for the SG, while a similar operation determines τ_{β}(T_{g}) = 10^{−4.57} s for the OG. The nearly same value of τ_{β} suggests also τ_{S}(T) of the SG at T_{on} is similar to τ_{S}(T) of the OG at T_{g}.

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 T_{g} as thickness is decreased. The enhanced surface mobility with the fast relaxation time τ_{S}(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 τ_{S}(T) as in the study of Zhang and Fakhraai. However, the JG β-relaxation is a local process and its relaxation time τ_{β}(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 τ_{β}(T) for thickness down to 20 nm. The τ_{β}(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 T_{g} is more than 20 K.^{42} Direct measurement of surface relaxation of syndiotactic (PMMA) films by lateral force microscopy also finds the τ_{β}(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 τ_{β}(T) in thin films and in bulk explains the same for the surface diffusion D_{S}(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, τ^{nano}_{α}, measured by dielectric relaxation on a 40 nm, and by PCS on a 22 nm freestanding polystyrene (PS) thin film. The values of τ^{nano}_{α}(T) were shown to match the primitive relaxation times τ_{0}(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 τ^{nano}_{α}, and hence the finding of τ^{nano}_{α}(T) ≈ τ_{0}(T) supports the fact that τ_{S}(T) is the same as τ_{0}(T), or eqn (4).

The cooperative α-relaxation in nominal glass-formers such as OTP at 10 degrees above T_{g} 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 nanocomposites^{19,46} or poly(dimethylsiloxane) confined in 2 nm glass pores,^{47} and the calculated τ_{0}(T) by eqn (1) from the bulk is in quantitative agreement with the experimental τ_{S}(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 D_{S}(T) to change of film thickness.

At about 75 MPa, the increase of T_{on} of OG with pressure has caught up with the T_{on} 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 T_{on} as well as the density of the pressurized OG is significantly higher than the UG at ambient pressure. Thus, the difference in τ_{β}(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 τ_{β}(T) data of OG at ambient and elevated pressure such as poly(phenylglycidylether) (PPGE),^{51} diglycidyl ether of bisphenol-A with M_{w} = 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 τ_{β} and τ_{α} 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, T_{g} 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 T_{g} change of 17 K at 95.5 MPa matches the difference between T_{g} = 242.3 K and T_{on} = 259 K for the most stable glass of OTP vapor-deposited at 0.85T_{g}.^{12} Even for the much larger increase of pressure up to 500 MPa, it only changes τ_{β} by one to two orders of magnitude in the glassy state at a fixed temperature in all cases. The change of τ_{β} 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 τ_{β} 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 (T_{f}′) of a SG is lower than the OG, while the T_{f}′ 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 T_{f}′ of a SG, one considers the corresponding liquid at ambient pressure, while the T_{f}′ of the pressurized OG refers to the liquid at the corresponding pressure. Again, since the definition of T_{f}′ 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 τ_{β}(T,P) has approximately the same value at any arbitrary chosen long time τ_{α}(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 τ_{β}(T,P) ≈ τ_{0}(T,P) has also been verified.

In this paper we use eqn (5)–(7) exclusively and directly from the experimental τ_{β} or the theoretical τ_{0} to compare with the experimental value of D_{S}(T) of SG like TPD. The CM eqn (1) need not be used to calculate τ_{0} from the structural α-relaxation time τ_{α}. The exception is the previous work on indomethacin (IMC)^{6} where τ_{α} is known from experiment above and below T_{g}, and OTP above T_{g} in Fig. 1 herein. This situation in IMC and OTP confers the benefit of predicting the size of the enhancement by the ratio, τ_{α}/τ_{0}, and comparing with the experimental value of D_{S}(T)/D_{V}(T), as successfully done for IMC and OTP. In the case of SG prepared by vapor deposition, τ_{α} is not known and eqn (1) cannot and need not be used to calculate τ_{0} or τ_{β} in the stable glass.

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## Footnote |

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

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