Why is the change of the Johari–Goldstein β-relaxation time by densification in ultrastable glass minor?

Abstract

Ultrastable glasses (USG) formed by vapor deposition are considerably denser. The onset temperature of devitrification, T_on, is significantly higher than T_on or T_g of ordinary glass (OG) formed by cooling, which implies an increase of the structural α-relaxation time by many orders of magnitude in USG compared to that in OG at the same temperature. However, for a special type of secondary relaxation having properties strongly connected to those of the α-relaxation, called the Johari–Goldstein β-relaxation, its relaxation time in USG is about an order of magnitude slower than that in OG and it has nearly the same activation energy, E_β. The much smaller change in τ_β and practically no change in E_β by densification in USG are in stark contrast to the behavior of the α-relaxation. This cannot be explained by asserting that the Johari–Goldstein (JG) β-relaxation is insensitive to densification in USG, since the JG β-relaxation strength is significantly reduced in USG to such a level that it would require several thousands of years of aging for an OG to reach the same state, and therefore the JG β-relaxation does respond to densification in USG like the α-relaxation. Here, we provide an explanation based on two general properties established from the studies of glasses and liquids at elevated pressures and applied to USG. The increase in density of the glasses formed under high pressure can be even larger than that in USG. One property is the approximate invariance of the ratio τ_α(T_on)/τ_β(T_on) to density change at constant τ_α(T_on), and the other is the same ρ^γ/T-dependence of τ_β in USG and OG where ρ is the density and γ is a material constant. These two properties are derived using the Coupling Model, giving a theoretical explanation of the phenomena. The explanation is also relevant for a full understanding of the experimental result that approximately the same surface diffusion coefficient is found in USG and OG with and without physical aging, and ultrathin films of a molecular glass-former.

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Introduction

Glasses produced by vapor deposition on a substrate at an optimal temperature below the bulk glass transition temperature (∼0.85T_g, where T_g is the glass transition temperature of the material) have density considerably higher than ordinary glasses even after aging for a realistically long time.^1–7 These glasses are more stable than the ordinary glasses (OG), and hence are called ultrastable glasses (USG). Compared with OG, some dynamic and thermodynamic properties of USG are novel and challenging to explain.^8–11 One such property and its explanation are the focus of the present paper.

The stability of USG is measured by the onset temperature for the transformation into the supercooled liquid, T_on. Its value is significantly higher than T_on or T_g of the ordinary glass (OG) formed by cooling. These changes in T_on imply an increase of the structural α-relaxation time, τ_α, by many orders of magnitude in USG compared to that in OG at the same temperature. Often observed in OG and USG is not only the α-relaxation but also a secondary relaxation. Some, but not all, secondary relaxations have properties strongly connected or related to those of the α-relaxation, and hence they are fundamentally important.¹² These secondary relaxations are called the Johari–Goldstein (JG) β-relaxations in order to distinguish them from the other trivial ones.¹² In contrast to the α-relaxation time τ_α, the JG β-relaxation time, τ_β, in USG of toluene,¹³ etoricoxib¹⁴ and telmisartan¹⁵ is about an order of magnitude longer than that in OG and has nearly the same activation energy E_β. The data from these three cases are extracted from C. Rodríguez-Tinoco et al.¹⁵ and shown in Fig. 1. On the other hand, the secondary relaxations of maltose octa-acetate, carvedilol and celecoxib, also discussed in ref. 15, involve intramolecular degrees of freedom, bear no connection to the α-relaxation,¹⁵ do not belong to the class of JG β-relaxations, and are not considered in this paper.

Fig. 1 Adapted from ref. 15. Arrhenius plot of τ_β in USG and OG of (A) toluene, (B) etoricoxib, and (C) telmisartan. Black and red closed symbols correspond to the USG and the OG obtained after transformation of the USG and subsequent cooling down, respectively. The red open points and the orange VFT fit to τ_α in (B and C) are from the literature (etoricoxib,^14,16,17 telmisartan^18,19). All the data of toluene are extracted from Yu et al.¹³ The black and red circles indicate the value of τ_β at T_on. The horizontal blue lines in graphs a–c bring out the invariance of τ_β at T_on. The sketch of each molecule is also depicted.

On the other hand, the relaxation strength of the JG β-relaxation, Δε_β, is significantly reduced in USG in toluene to such a level that it would require several thousands of years of annealing for an ordinary glass to achieve the same state.^13–15 The reduction of Δε_β indicates that the JG β-relaxation does respond to densification in the USG like the α-relaxation. Notwithstanding, the much smaller changes of τ_β and E_β by densification in USG are in stark contrast to the behavior of the α-relaxation time τ_α, and this remarkable experimental observation deserves an explanation. In this paper, we provide an explanation based on two general properties established from the studies of glasses and liquids at elevated pressures^20–27 and applied it to USG. The increase in density of the glasses formed under high pressure can be even larger than that in USG, and hence the general properties should apply. One of the properties deduced and applied to USG is the approximate invariance of the ratio τ_α(T_on)/τ_β(T_on) to density change at constant τ_α(T_on), and the other is the same ρ^γ/T-dependence of τ_β in USG and OG where ρ is the density and γ is a material constant. Furthermore, we show how these two general properties can be derived theoretically by applying the Coupling Model,²⁸ and by this application we have completed our explanation of the observed minor changes of τ_β and E_β by densification in USG.

The explanation is also relevant in achieving a better understanding than before²⁹ of another remarkable experimental finding, which is about the same surface diffusion coefficient, D_S, in USG and OG with and without physical aging, and ultrathin films of the molecular glass, N,N′-bis(3-methylphenyl)-N,N′-diphenylbenzidine (TPD) by Fakhraai and coworkers.^30–32

1. Two general properties applied to USG

As mentioned in the Introduction, to explain the properties of the JG β-relaxation in USG compared with the OG we shall apply two general dynamic and thermodynamic properties of the α-relaxation and JG β-relaxation established before from studies at elevated pressures in molecular glass-formers.^20–27 The application is justified by the fact that density is increased at elevated pressure, and the increase can be comparable to or even larger than that in USG. Through the application, we are able to explain why the Johari–Goldstein (JG) β-relaxation time, τ_β, in USG of toluene,¹³ etoricoxib¹⁴ and telmisartan¹⁵ is about an order of magnitude longer than that in OG, and has nearly the same activation energy E_β. Later we shall show how the two general properties are derived from theoretical considerations. Thus, a theoretical explanation of the properties of the JG β-relaxation in USG is obtained.

A. General property leading to the same τ_β(T_on) for USG and OG

By elevating pressure P to hundreds of MPa or more, one can increase the glass transition temperature from T_g at ambient pressure, P₀ = 0.1 MPa, of many molecular glass-formers by several tens of degrees to T_gP and the density by a few per cent. These changes are comparable to or larger than those achieved by producing USG. Found in general in many molecular glass-formers^20–28 is the approximate invariance of τ_β(T,P) to variations of P and T provided τ_α(T,P) are kept constant. Examples of experimental evidence for this property in OG are shown in Fig. S1 and S2 in the ESI.† Since τ_α(T_gP,P) are approximately the same at T_gP for all P by definition of glass transition temperature, the property specified at T_gP states that τ_β(T_gP,P) is invariant to variations of P. In turn, this can be restated as approximate invariance of τ_β(T_gP,ρ_gP) to variations of ρ_gP in OG, where ρ_gP is the density at temperature equal to T_gP and pressure P determined from the equation of state connecting P, T, and specific volume V or density ρ of the glass-former. Pressure P is just a parameter used to vary T_gP and ρ_gP to bring out the property of the invariance of τ_β(T_gP,ρ_gP). It can be replaced by density ρ in some cases where ρ is changed directly by some operation instead of applied pressure. Producing USG by vapor deposition is such an operation whereby the density is increased compared to OG. The equivalence of effects due to change in density either by the route of USG or by application of pressure on OG can be appreciated from the fact that T_on of the USG of indomethacin is larger than that of the OG at ambient pressure, but T_on of the two glasses increases with pressure and becomes nearly the same upon reaching 300 MPa.³³ Thus, the approximate invariance of τ_β(T_gP,ρ_gP) to variations of ρ_gP translates to the approximate invariance of τ_β(T_on) with the change from OG to USG. It is expressed explicitly by

τ_β,USG(T_on,_USG) ≈ τ_β,OG(T_on,OG), (1)

after replacing T_gP by T_on,USG for USG and T_on,OG for OG. Thus, this property found in USG and OG of toluene,¹³ etoricoxib¹⁴ and telmisartan¹⁵ is a special application of the general property of the approximate invariance of τ_β(T_gP,P) to variations of P or equivalently the invariance of τ_β(T_gP,ρ_gP) to variations of ρ_gP in OG. We shall provide the theoretical justification of the general property in Section (C) after the presentation of the other general property in Section (B) from which we can rationalize why the Arrhenius temperature dependence of τ_β,USG in USG is not much different from that of τ_β,OG in OG.

B. General property leading to similar Arrhenius T-dependence of τ_β,USG and τ_β,OG

The structural α-relaxation time, τ_α, of non-associated small molecular and polymeric glass-formers obeys thermodynamic scaling, i.e. τ_α is a function of the product variable, ρ^γ/T, where γ is a material dependent parameter.^34–36 This property in many glass-formers was deduced from measurements of τ_α at ambient and elevated pressures analyzed in conjunction with P–V–T data. The property of the invariance of the ratio τ_α(T,P)/τ_β(T,P) to variations of T and P while keeping τ_α(T,P) constant^20–27 is key to arriving at the results in Section (A). It also has the immediate consequence that τ_β is also a function of ρ^γ/T, if τ_α is a function of ρ^γ/T, although the two functions τ_β = f_β(ρ^γ/T) and τ_α = F_α(ρ^γ/T) are different. This is an example of the dynamic and thermodynamics properties of the JG β-relaxation similar to those of the α-relaxation,^20–27 which follows from the approximate relation,^23,27, where t_c (∼1 ps) is a constant and β_KWW is the stretch exponent of the Kohlrausch–Williams–Watts correlation function for the α process. From this approximate relation and the fact that the JG β-relaxation occurs before the α-relaxation in time (i.e., its precursor), it follows that the ρ^γ/T-dependence of τ_α = F_α(ρ^γ/T) originates from τ_β = f_β(ρ^γ/T)²⁷ as required by causality. Another piece of evidence for ρ^γ/T-dependence originating from τ_β = f_β(ρ^γ/T) can be drawn from the remarkably small value of r_s at which the steepness of the repulsive part of the intermolecular potential U(r) determines the scaling exponent γ.²⁷ The short distance r_s was found by molecular dynamics simulations of several Lennard-Jones liquids³⁷ and in cis 1,4-polybutadiene as shown in Fig. S3 in the ESI,† ³⁸ and reviewed in ref. 27.

The result, τ_β = f_β(ρ^γ/T), is the general property we now use to deduce why the Arrhenius temperature dependence of τ_β in USG is not much different from that in OG. Although the ρ^γ/T-dependences of τ_α = F_α(ρ^γ/T) and τ_β = f_β(ρ^γ/T) are mostly considered in the liquid state, it is applicable also in the glassy state. An example demonstrating this is the study of diglycidyl ether of bisphenol-A (DGEBA) with M_w = 380 g mol⁻¹.²³ In the glassy state, the temperature dependence of f_β(ρ^γ/T) becomes effectively Arrhenius. The difference between the activation energy E_β,USG in USG and E_β,OG in OG comes from the factor ρ^γ and the difference between the density ρ_USG of USG and ρ_OG of OG. Hence the ratio E_β,USG/E_β,OG can be estimated from the value of (ρ_USG/ρ_OG)^γ. Ultrastable glasses of IMC have a density up to 1.4% more than that of the liquid-cooled OG,⁴ and a similar value is obtained for USG of TPD.³⁹ The ratio E_β,USG/E_β,OG is equal to (1.014)^γ. Typical values of γ fall within the range of 2 to 7, and toluene has γ = 7. The largest value of E_β,USG/E_β,OG = (1.014)^γ is 1.1 for γ = 7. In other words, E_β,USG is larger than E_β,OG by merely 10%, and therefore the small increase of activation energy in USG compared to OG is a consequence of the general property of the ρ^γ/T-dependence of τ_β.

There are other experimental results that prove the fact that the ρ^γ/T-dependence of τ_α and τ_β is the same in both USG and OG because the repulsive part of the intermolecular potential is unchanged. Wide-angle X-ray scattering (WAXS) experiments were performed on USG and OG of indomethacin (IMC)³ and etoricoxib.¹⁴ In both cases, the intensity as a function of the scattering wave vector q exhibits no difference at q values higher than that of the peak, indicating that the relation of g(r) with the repulsive part of the intermolecular potential is the same for USG and OG. Moreover, molecular dynamics simulations of USG^40,41 show that the changes of its g(r) from that of OG occur at larger values of r, which are irrelevant for the determination of γ from the slope of the repulsive part of the intermolecular potential. Therefore, from these two facts we conclude the same repulsive part of the potential, and hence the same ρ^γ/T-dependence of τ_β for USG and OG. In a recent report,⁷ the authors used the calorimetric transformation time of glasses with different thermodynamic stability to infer the structural relaxation times τ_α of the glassy systems. Interestingly, they found that for all the glasses, USG and OG, the relaxation data have the same ρ^γ/T-dependence, with a unique γ exponent (see Fig. S4 in ESI†). This is another indication that γ is practically the same in USG and OG.

C. Derivation of the two general properties to complete the explanation

The two general properties of the JG β-relaxation time τ_β and its relation to τ_α have been applied to compare USG with OG in Sections (A) and (B) with two results: (i) the value of τ_β,USG(T) at T = T_on,USG is the same as τ_β,OG(T) at T = T_on,OG, i.e., eqn (1), and (ii) the activation energy E_β,USG of the Arrhenius temperature dependence of τ_β,USG(T) for T < T_on,USG is only slightly increased from E_β,OG of τ_β,OG(T) for T < T_on,OG. The two results combined explain why the difference between τ_β,USG(T) and τ_β,OG(T) over the entire common temperature range below T_on,OG is minor as observed experimentally in toluene,¹³ etoricoxib,¹⁴ and telmisartan.¹⁵ However, the explanation is not complete until we have given the theoretical basis of the two general properties. This is our next task and we use the Coupling Model (CM).

The CM starts from the primitive relaxation which is independent and local. The time honored CM equation,^27,28

τ_α(T,P) = [t^−n(T,P)_cτ₀(T,P)]^{1/[1−n(T,P)]} (2)

links the primitive relaxation time τ₀(T,P) to the cooperative many-molecule α-relaxation. In eqn (2), t_c is the onset time of classical chaos²⁸ and its magnitude is about 1 to 2 ps for molecular glass-formers and polymers, and was directly determined by quasielastic neutron scattering experiments and molecular dynamics simulations.²⁸ The parameter n(T,P) is the fractional exponent of the Kohlrausch correlation function,

The independent and local nature of the primitive relaxation suggests that it is similar to the JG β-relaxation. The two are not identical because the JG β-relaxation is composed of a distribution of processes^42,43 and the primitive relaxation is only the leading part. Notwithstanding, approximate correspondence of the two relaxation times τ₀(T,P) and τ_β(T,P) is expected and predicted.^27,28,44 Written as

τ_β(T,P) ≈ τ₀(T,P), (4)

this relation has been verified in many glass-formers of different types,^20–28,44 where the JG β-relaxation is resolved and its loss peak frequency provides a direct and unequivocal determination of τ_β(T,P). In many of these cases, the value of τ_β(T,P) differs from τ₀(T,P) calculated by eqn (4) within one order of magnitude or less.

The CM is based on classical chaos engendered by the anharmonic intermolecular potential.^28,45 In the model, it is the intermolecular potential in conjunction with the primitive relaxation time τ₀(T,P) which exclusively determines φ_K(t). Upon varying P and T while keeping τ₀(T,P) the same, the intermolecular potential is also unchanged,^37,38,46 therefore φ_K(t) is the same, or both τ_α(T,P) and n(T,P) are the same. Therefore, the exact co-invariance to changes of P and T of the three quantities, τ₀(T,P), τ_α(T,P) and n(T,P), is immediately the consequence of the CM. Combining this property with τ_β(T,P) ≈ τ₀(T,P), we have derived the approximate co-invariance of τ_β(T,P), τ_α(T,P) and n(T,P). From this result, the general property used in Section (A) (i.e., approximate invariance of τ_β(T,P) to variations of P and T at constant τ_α(T,P)) is now derived from the CM. The reader may recall the application of this general property to USG and OG in showing that τ_β,USG(T_on,USG) is approximately equal to τ_β,OG(T_on,OG).^14,15

Derived also from the CM is the invariance of n(T,P) or the α-relaxation frequency dispersion to variations of P and T at constant τ_α(T,P), which has not been utilized in this paper in the relation between USG and OG. Nevertheless, it is worthwhile to point out that it is a remarkable prediction. Equally remarkable is that this property has been verified in so many glass-formers, molecular or polymeric.^47,48 The importance of the frequency dispersion is brought out by the property.

In Section (B) we use another property and give a reason to show that the Arrhenius activation energies E_β,USG of USG and E_β,OG of OG are not much different. The property is that τ_β is a function f(ρ^γ/T) of ρ^γ/T with the same γ as the function F(ρ^γ/T) of τ_α. The reason is that γ is determined by the repulsive part of the intermolecular potential which is the same in USG and OG, and hence the same f(ρ^γ/T) for USG and OG. We have demonstrated in Section (B) that the property, τ_β a function of ρ^γ/T, is a consequence of the approximate invariance of the ratio τ_α(T,P)/τ_β(T,P) to variations of T and P while keeping τ_α(T,P) constant. The latter is just part of the approximate co-invariance of τ_β(T,P), τ_α(T,P) and n(T,P) derived above from the CM, and hence the property is justified as well.

D. Accounting for the minor difference in τ_β(T) between USG and OG

Having derived in (C) the two properties in (A and B), we are ready to put them together to account for the minor change of τ_β(T) in USG compared to OG observed experimentally. The change is given by the ratio, τ_β,USG(T)/τ_β,OG(T), which we only need to calculate at T = T_on,OG because the activation energies E_β,USG and E_β,OG of the Arrhenius T-dependences of τ_β,USG(T) and τ_β,OG(T) in the glassy states are about the same. Since the ratio is the main interest, it is denoted by R̂ and written explicitly as

R̂ ≡ τ_β,USG(T_on,OG)/τ_β,OG(T_on,OG). (5)

Upon combining eqn (5) with τ_β,USG(T_on,USG) ≈ τ_β,OG(T_on,OG), i.e., eqn (1), we have the result R̂ ≈ [τ_β,USG(T_on,OG)/τ_β,USG(T_on,USG)], which can be calculated from the change of the factor, exp(E_β,USG/RT), of the temperature dependence of τ_β,USG(T). The final result iswhere E_β,USG/RT_on,OG ≡ Λ. In the case of toluene, by taking T_on,OG = 117 K, T_on,USG = 124 K,¹³ and Λ = 28.5, we have log₁₀(R̂) = 0.7. It is slightly less than a decade, and in approximate agreement with the experimental data of Yu et al.¹³

For etoricoxib, with E_β = 51 kJ mol⁻¹, T_on,OG = 327 K, T_on,USG = 360 K, and Λ = 18.76,¹⁴ we have log₁₀(R̂) = 0.75, which compares well with the experimental value of ∼0.84 at T = 293 K. The small value of Λ for etoricoxib is because the secondary relaxation is not the usual JG β-relaxation.^14,17 Notwithstanding, this secondary relaxation has some of the properties of the JG β-relaxation such as a significant dependence of its relaxation time on pressure. This is the reason why it is used for comparison between USG and OG since the usual JG β-relaxation of etoricoxib is not resolved.

The difference between T_on,USG of USG and T_on,OG of OG is 18 degrees for TPD.³⁹ Taking T_on,OG = 330 K and T_on,USG = 348 K, and assuming E_β/RT_on,OG ≡ Λ = 24, the predicted log₁₀(R̂) = 0.54. This small increase of τ_β,USG(T) in USG of TPD with reference to τ_β,OG(T) of OG is consistent with the CM explanation²⁹ of the invariant surface diffusion observed in the two glasses by Fakhraai and coworkers.^31,32

2. Discussion

The main objective of this paper is to explain why the differences in magnitude and temperature dependence of τ_β between USG and OG are minor compared with the huge change of τ_α. Our goal is accomplished by the theoretical treatment given in Section 1. The explanation is beneficial to complete a previous rationalization of another remarkable experimental finding,²⁹ which is approximately the same surface diffusion coefficient D_S in USG, OG, and nanometer thin films of the same glass-former,^30–32 while their structural α-relaxation times differ by many orders of magnitude. The rationalization is based on the relation D_S(T) ≈ d²/4τ(T), where d is the size of the molecule, used previously to account for the enhancement of the surface diffusion²⁹ (further support for the use of this expression is shown in the ESI,† Fig. S5), and the experimental evidence showing that τ_β(T) is approximately the same in USG, aged OG, OG, and ultrathin films. The development in the present paper justifies on theoretical grounds the rationalization given before.

Glass with density higher than USG can be realized by applying pressure P much higher than ambient pressure P₀ = 0.1 MPa. If the minor changes in the magnitude and activation energy of τ_β in USG compared to OG is general, it should be observed by elevating pressure from P₀ to P. The answer is positive from the results of studies at elevated pressures.^20–28 An example is the isobaric data of DGEBA (diglycidyl ether of bisphenol-A, M_w = 380 g mol⁻¹, also known as EPON 828). Shown in Fig. 2 are the α-relaxation times τ_α and JG β-relaxation times τ_β at ambient pressure P₀ of 0.1 MPa and at P = 400 MPa. Defining glass transition temperatures by τ_α = 100 s, we have T_g = 252.8 K at P₀ and T_gP = 306.7 K at P = 400 MPa. The increase of 54 K is significantly larger than that expected for USG. The value of τ_β(T_g,P₀) is about the same as τ_β(T_gP,P). The activation energy is E_β = 52.2 kJ mol⁻¹ at ambient pressure, and it is about the same as the value at P = 400 MPa. Thus, the relation of τ_β at high pressure to ambient pressure is the same as that of USG to OG. The ratio τ_β(T,P)/τ_β(T,P₀) ≈ 100 is larger, and this is consistent with the density increase by elevating pressure, although the value of γ of DGEBA is 3.5, which is a factor of 2 smaller than that of toluene.

Fig. 2 Logarithm of characteristic time of dielectric loss maximum of DGEBA (diglycidyl ether of bisphenol-A, M_w = 380 g mol⁻¹, also known as EPON 828) for the α-relaxation and JG β-relaxation under isobaric conditions versus reciprocal temperature. Red symbols, P = 400 MPa; blue symbols, P₀ = 0.1 MPa. The difference in T_g is 54 K, which is matched by the separation of the JG β-relaxation time as indicated by the dashed line.

In this paper we have exclusively considered toluene, etoricoxib, and telmisartan with regard to the change of their secondary relaxations upon densification in USG compared to OG. The properties of the secondary relaxation in these three glass-formers are connected to or correlated with that of the α-relaxation, as discussed before in ref. 15. They belong to the class of JG β-relaxations if the relaxation time τ_β(P,T) satisfies the criteria set forth in ref. 12 and 44, including the approximate relation (4), i.e. τ_β(P,T) ≈ τ₀(P,T). This criterion together with the CM eqn (2) led to the approximate relation,

τ_α(P,T) ≈ [t⁻ⁿ_cτ_β(P,T)]^1/(1−n). (7)

From this relation, another criterion for JG β-relaxation is that the P, T and ρ dependences of τ_β(P,T,ρ) are correlated with those of τ_α(P,T,ρ) approximately. The τ_β(P,T,ρ) of toluene, etoricoxib, and telmisartan increase upon densification in USG compared to OG in concert with τ_α(P,T,ρ), indicating that the secondary relaxations are the JG β-relaxations.

The secondary relaxation with properties uncorrelated with those of the α-relaxation and that does not satisfy the criteria is unimportant and is referred to as a non-JG relaxation. This is the case for the well-resolved secondary δ-relaxation of celecoxib, and γ-relaxations of D-maltose octa-acetate and carvedilol, discussed also in ref. 15. Their τ_γ(T,ρ) becomes faster while τ_α(T,ρ) becomes slower on densification of the glass by vapour deposition. Thus, the change of τ_γ(T,ρ) upon densification is opposite to that of τ_α(T), and hence the γ-relaxations in D-maltose octa-acetate and carvedilol are the non-JG relaxations. For celecoxib, it was shown before⁴⁹ that it is the fast δ-relaxation, and its relaxation time τ_δ is much shorter than the calculated primitive relaxation time τ₀ ≈ τ_β. The JG β-relaxation is present but it has low dielectric strength and is detected as a broad shoulder in the glassy state. Its relaxation time τ_β is in agreement with τ₀. It was not considered in ref. 15. For D-maltose octa-acetate and carvedilol, the non-JG nature of the resolved secondary relaxation is further justified by the fact that the observed relaxation time τ_γ(T) is much shorter than τ₀(T) calculated by eqn (2). The JG β-relaxation of D-maltose octa-acetate and carvedilol cannot be resolved due to its proximity to the α-relaxation. Nevertheless, this is irrelevant to this work because we are considering the resolved JG β-relaxations of toluene, etoricoxib, and telmisartan in the change by densification in USG.

Some of the results in Sections (B) and (C) are based on the approximate invariance of τ_β(T,P) to variations of T and P while keeping τ_α(T,P) constant, which follows from relation (7). Only approximate invariance of τ_β(P,T) is predicted because the JG β-relaxation is composed of a distribution of processes involving increasing number of molecules with increasing time,^42,43 and the value of τ_β(P,T) determined from experimental data for different combinations of P and T may not be associated with the same process in the distribution. Fortunately, in all previous experiments involving non-polymeric glass-formers in ref. 15 and 20–28, the JG β-relaxation was resolved in the dielectric spectra as a pronounced loss peak, and the peak frequency f_β(P,T) provides a characteristic time τ_β(P,T) of the distribution. No fit is needed to determine f_β(P,T) and test the approximate invariance. Nearly exact invariance of f_β(P,T) or τ_β(P,T) was found in all cases published in ref. 15 and 20–28 as well as in DGEBA (diglycidyl ether of bisphenol-A, M_w = 380 g mol⁻¹) shown in Fig. 2, and thus overwhelmingly verifying the approximate invariance of τ_β(P,T) for molecular glass-formers.

On the other hand, the dielectric spectra of the polymers, polyisoprene,⁴³ polymethylmethacrylate⁵⁰ (PMMA), and 1,4 polybutadiene,⁵¹ are not as ideal or straightforward as in the case of molecular glass-formers to test invariance of τ_β(P,T). The α-relaxation of PMMA has low dielectric strength and its loss peak is not well resolved for many P and T combinations. Consequently, an assumption of the frequency dispersion of the α-relaxation had to be made,⁵⁰ and used to fit the loss spectra with further assumption that the α-relaxation and the JG β-relaxation represented by a Cole–Cole function are additive. It is remarkable but questionable that practically the same value of τ_α(P,T) was obtained from the fits for different P and T despite the lack of well defined α-loss peaks. Nevertheless, the deviation of the deduced values of τ_β(P,T) from exact invariance is only ± half a decade, while the τ_α(P,T) values change over 7 orders of magnitude. The results therefore are consistent with the approximate invariance of τ_β(P,T) predicted from relation (7).

In the case of 1,4 polybutadiene⁵¹ the data at ambient pressure of 0.1 MPa show a long plateau or a broad shoulder, and f_β(P,T) was deduced from the fit by assuming the Cole–Cole function representing the JG β-relaxation and the data can be represented by the sum of the Cole–Cole function and the α-loss represented by the Fourier transform of the Kohlrausch function with its intensity adjusted in the global fit. This is common practice, but no one can be sure whether the JG β-relaxation is a Cole–Cole function and the assumption of additivity is correct or not, and thus the value of τ_β(P,T) deduced for 0.1 MPa is debatable. The same fits were made to data taken at elevated pressures, where the JG β-relaxation shows up as broad loss peaks. Despite the loss peaks at elevated pressures being very broad, the loss peak frequencies f_β(P,T) are approximately the same. Thus, except for the questionable value of τ_β(P,T) at ambient pressure deduced by fitting data without a β-loss peak, the peak frequencies of data at elevated pressures are approximately the same and consistent with the predicted approximate invariance of τ_β(P,T). Only studies of other polymers having both well resolved α and β loss peaks in the future, as in the molecular glass-formers, can critically test the approximate invariance of τ_β(P,T) for polymers. The present paper deals exclusively with molecular glass-formers, toluene, etoricoxib, and telmisartan. Since all data of molecular glass-formers in ref. 15 and 20–28, as well as those of DGEBA shown in Fig. 2, show consistency with the approximate invariance of τ_β(P,T), the unsettled issue of polymers are not relevant for the present paper.

3. Conclusions

The onset temperature T_on of ultrastable glass (USG) is significantly higher than that of ordinary glass (OG) due to the increase in density and this suggests an increase in the α-relaxation time τ_α by many orders of magnitude. The JG β-relaxation also responds to the higher density by a large reduction in relaxation strength, which would take thousands of years of aging for OG to achieve. However, experimental studies of USG and OG show the increase of the relaxation time τ_β(T) of USG compared to OG is within one decade over the entire common temperature range. Thus, the activation energy of τ_β(T) is practically the same in the two glasses. The insensitivity of these dynamic properties of the JG β-relaxation to the density increase in USG is remarkable and challenging to explain. We have successfully explained these properties by the Coupling Model (CM) after considering that the intermolecular potential in USG remains the same as in OG. High pressure can facilitate the formation of glass with higher density than at ambient pressure, and we demonstrate that the change in properties of τ_β is similar to that in USG compared to OG. This proves that the effects are general and the explanation is given. The explanation also applies to another remarkable finding of approximately the same surface diffusion coefficient D_S in USG, OG, and nanometer thin films of the same glass former.^29–32

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

C. R.-T. and M. P. acknowledge the 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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Footnote

† Electronic supplementary information (ESI) available: Fig. S1–S4. See DOI: 10.1039/c8cp05107k

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