Introduction

Previtreous changes of the structural (primary, alpha) relaxation time (τ), viscosity (η), electric conductivity (σ), heat conductivity (κ), diffusion (d) or chemical reactions rates (k) in systems ranging from low-molecular-weight liquids and polymers to liquid crystals and plastic crystals are the key manifestation of the hypothetical universal dynamics emerging on approaching the glass transition (Tg, Pg)1,2,3,4,5. Similar patterns are observed both for the temperature and pressure paths6,7,8. The temperature path is associated with the Super-Arrhenius (SA) dynamics, and it is governed by changes of the apparent activation energy Ea(T), which strongly increases on approaching the glass transition temperature Tg9,10,11. The non-biased way of determining Ea(T) and its properties are discussed in refs12,13,14 and recalled in Supplementary Info.

This report focuses on the still puzzling case of the (high) pressure-induced glass transition. For compressed glass-formers, general features of the previtreous dynamics are described by the Super-Barus (SB) equation7,8,9,10:

$$\tau (P)={\tau }_{0}^{P}\,{\exp }(\frac{P{V}_{a}(P)}{RT}),\,\,\eta (P)={\eta }_{0}^{P}\,{\exp }(\frac{P{V}_{a}(P)}{RT})$$
(1)

where T = const and P < Pg; Va(P) denotes the apparent activation volume, which changes on compressing. Generally, the name ‘activation volume’ is reserved for the basic Barus15 equation with Va(P) = Va = const in the given domain of pressures.

Prefactors τ0P and η0P in Eq. (1) refer to P = 0, but within the experimental error they can be approximated by atmospheric pressure values, i.e., τ0P = τ(P = 0) ≈ τ(P = 0.1 MPa) for the tested isotherm T. For high-pressure studies the experimental errors are ΔP ≈ ±0.2 MPa (moderate pressures) and ΔP > ±1 MPa (GPa domain). The shift of pressure by 0.1 MPa does not yield detectable changes of dielectric relaxation time6,7,8.

Similar SB dependences describe pressure changes of all physical properties recalled above: pressure dependences of τ(P) and η(P) are parallel (Eq. (1))6,7,8, but for the remaining dynamic properties the translational - orientational decoupling have to be taken into account8,10. For instance, for DC electric conductivity16:

$${\sigma }^{-1}(P)={\sigma }_{0}^{-1}(S\frac{P{V}_{a}(P)}{RT})$$
(2)

where S < 1 is the decoupling exponent associated with the fractional Debye-Stokes-Einstein (f-DSE) dependence σ(P)[τ(P)]S = C = const8,10,16.

The first discussion regarding η(P) or τ(P) behavior in compressed liquids can be associated with the relation \(\eta (P)\propto {\exp }(\alpha P)\) proposed by Barus at the end of the 19th century when studying viscosity of natural oils15. Whalley17,18 and Williams19 applied such description for pressurized polymers and dielectric relaxation time, introducing the activation volume Va, what led to Eq. (1) with Va = const. The Barus (B) or Barus-Williams dependence can be considered as the pressure counterpart of the basic, temperature-related Arrhenius equation which was originally introduced as \(k(T)={k}_{0}{\exp }({E}_{a}/RT)\), where Ea stands for the activation energy and k is the reaction rate coefficient20. Generally, for ultraviscous/ultraslow glass-forming systems one should expect the Super-Barus (SB) behavior (Eqs (1) and (2)), where dynamic properties are governed by changes of the pressure-dependent activation volume: the apparent activation volume.

According to the above discussion one concludes that Va(P) governs the dynamics of ultraviscous/ultraslow systems, and its determination and understanding is the key to the ultimate insight into the glass transition problem6,7,8,9, the behavior of soft matter under pressure7,8,9,21, high-pressure chemistry and biochemistry22, innovative material engineering23, high-pressure biotechnology24, geophysics, and deep Earth science25.

Usually, for the previtreous domain the apparent activation volume Va(P) is calculated from τ(P) or η(P) experimental data via26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55:

$${V}_{a}(P)\to {V}^{\#}(P)=RT[\frac{d\,{ln}\,\tau (P)}{dP}]$$
(3)

under the assumption that Va(P) = V#(P) and for 0.1 MPa < P ≤ Pg.

The analysis exploring the (apparent) activation volume determined via Eq. (3) is the key point of numerous research reports. In the framework of the transition state theory, the activation volume describes the difference between volumes occupied by a molecule in activated and non-activated states22. It is the essential parameter characterizing the sensitivity of the structural relaxation time or other dynamic properties to pressure changes7,8,21. It estimates the local volume required for a given dynamical process (in the case of τ denotes molecular rearrangements)8,26,27,28,29,30,31,32,33,34,35. Hong et al.36,37 indicated that the activation volume correlates with the length scale of dynamical heterogeneities ξ3, which are considered as one of the essential sources of the previtreous ‘universality’ of dynamic previtreous properties. Tests in hydrogen-bonded molecular liquids showed the case-sensitivity of the activation volume, determined via Eq. (3) to subtle features of molecular structures38,39,40. Worth stressing is also the broadly used link between the activation volume at the glass transition, and the fragility: ΔV# = mP × 2,303R(dTg/dP)8,40,41,42, where the fragility \({m}_{P}={[d{lo}{{g}}_{10}\tau (T)/d({T}_{g}/T)]}_{T={T}_{g}}\) is one of key ‘universality’ metrics for the glass transition phenomenon2,3,4,5,9,10,43,44. The analysis via Eq. (3) was also used for showing different activation volumes determined by dielectric and light scattering spectroscopies8,45,46,47,48,49,50,51,52,53. Reasonings based on such analysis can yield important checkpoints for glass transition models8,45,46,47,48,49,50,51,52,53,54,55. The activation volume is also significant for the thermodynamic scaling linking τ(T.P, V) experimental data8,54,55. There are also reports where V#(P) is recalled as the (apparent) activation volume, but the link to the steepness index mT(P) is indicated56,57.

For the validation of Eq. (3) reports by Whalley17,18 and Williams19 are most often cited8,30,31,32,33,34,38,39,40,41,42,48,49,50,51,52,53,54. However, these reports did not consider the Super-Barus dynamics with the pressure depending apparent activation volume but the basic Barus behavior with the constant activation volume. This issue is worth stressing, since the SB Eq. (1) directly yields the differential equation:

$$RT\frac{d\,{ln}\,\tau (P)}{dP}={V}_{a}(P)+P\frac{d{V}_{a}(P)}{dP}$$
(4)

Comparing Eqs (3) and (4) one obtains that generally: Va(P) ≠ V#(P). The second term in Eq. (4) disappears only for two ‘special’ cases: (i) for P = 0, or (ii) for Va(P) = Va = const, i.e., for the basic Barus behavior in the given pressure domain.

Worth recalling is the difference between the free volume (Vf) as the volume not occupied by molecules and the activation volume (Va) as the volume required for the given process, for instance, the molecular rearrangement or reorientation. Then, one can expect Vf  > Va8,9.

Consequently, the question arises for the (proper) estimation of the (apparent) activation volume in the previtreous domain. This report proposes the solution to this problem and discusses the meaning and behavior of both V#(P) and Va(P) for P < Pg. The discussion is supported by the analysis of the τ(P) experimental data for glass-forming low molecular weight liquid diisobutyl phthalate (DIIB, Tg(0.1 MPa) = 196.8K), epoxy resin bisphenol A/epichlorohydrin (EPON 828, Tg(0.1 MPa) = 253.9K) and liquid crystalline isooctyloxycyanobiphenyl (8*OCB, Tg(0.1 MPa) = 220.7K). The latter vitrifies in the isotropic liquid phase, and the possible nematic phase is hidden below the glass transition. In given studies, pressures up to P ≈ 1.2 GPa were reached, i.e. for the domain hardly available in high resolutions tests carried out so far8,10. Experimental details are described in the Methods section.

Results and Discussion

Figure 1 shows the pressure evolution of the structural relaxation time for selected isotherms for three qualitatively different glass formers 8*OCB, DIIP and EPON 828, in the pressure range 0.1 MPa < P < Pg. They served for estimating both V#(P) and Va(P).

Figure 1
figure 1

The isothermal pressure evolution of the structural relaxation time in previtreous domains of tested glass formers recalled in the Figure. Arrows show terminals of the reliable fitting using the simple approximation via Eq. (10a).

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When discussing the physical meaning of V#(P) one can recall the definition of the pressure-related steepness index (the normalized rate of changes of the relaxation time, viscosity, …) in the previtreous domain8, which leads to the relation:

$${m}_{T}(P)=\frac{d\,{lo}{{g}}_{10}\tau (P)}{d\,(P/{P}_{g})}=\frac{{P}_{g}}{{ln}\,10}\frac{d\,{ln}\,\tau (P)}{dP},\,T=const$$
(5)

It terminates at the pressure-related fragility11: \({m}_{T}={m}_{T}({P}_{g})={[d{lo}{{g}}_{10}\tau (P)/d(P/{P}_{g})]}_{P={P}_{g}}\), which is the key metric for glass-forming ultraviscous/ultraslow systems. Then, the pressure-dependent (isothermic) coefficient mT(P) for P < Pg can be called the apparent fragility. Linking Eqs (3) and (5) one obtains the relation showing that V#(P) ∝ mT(P):

$${V}^{\#}(P)=(\frac{RT\,{ln}\,10}{{P}_{g}})\times {m}_{T}(P),\,T=const$$
(6)

Considering further the ratio of fragilities along Tg(P) curve/line: \({m}_{P}(T)/{m}_{T}(P)=[d{lo}{{g}}_{10}\tau (T)/d({T}_{g}/T)]/\)\([d{lo}{{g}}_{10}\tau (P)/d(P/{P}_{g})]=d(P/{P}_{g})/d({T}_{g}/T)\) and linking this with Eq. (6) the following relations are obtained:

$${m}_{P}(T)={m}_{T}(P)\frac{T}{{P}_{g}}{(\frac{d{T}_{g}}{dP})}^{-1}$$
(7a)

and

$${m}_{P}(T)=\frac{{V}^{\#}(P)}{R\,{ln}\,10}{(\frac{d{T}_{g}}{dP})}^{-1}$$
(7b)

Equation (7b), originally derived in ref.40, is broadly used for calculating isobaric fragilities m = mP(Tg) for different isobars, based on the knowledge of the ‘~activation volume V#(P)’ calculated via Eq. (3) and the pressure shift of Tg8,26,27,28,29,30,31,32,33,34,38,39,40,41,42,47,48,49,50,51,52,53,54. Notwithstanding, Eq. (7a) is fundamentally more correct than Eq. (7b), since V#(P) should not be recalled as the apparent activation volume.

Recently, it was shown experimentally that changes of the pressure-related apparent fragility can exhibit a ‘universal’ previtreous behavior5,7:

$${m}_{T}(P)=\frac{A}{{P}^{\ast }-P}$$
(8)

where T = const, the amplitude A = const, and P* is for the extrapolated singular pressure. Regarding pressures: P < Pg and P* > Pg.

Following Eqs (6) and (8) one can conclude that: 1/V#(P) ∝ 1/mT(P) ∝ P* − P. Such behavior is confirmed in the insets in Figs 2–4. One of the basic features of the previtreous domain is the appearance of two dynamical domains, i.e., regions with different SA or SB behavior remote and close to the glass transition, respectively58,59. This is associated with the crossover from the ergodic to the non-ergodic dynamical domain at (TB, PB), where TB ≫ Tg and PB ≪ Pg8,16,58,59. Roland60 showed the pressure-temperature invariance of the dynamic crossover time-scale for a set of glass-forming liquids τ(TB, PB) ~ 10−7±1s. Until recently, the detection of PB was associated with \({\Phi }_{P}(P)={(d{lo}{{g}}_{10}\tau (P)/dP)}^{-1/2}\) vs. P plot8,59, which is parallel to (mT(P))−1/2 vs. P presentation5,7,56. Such analysis assumes a priori the validity of the pressure counterpart of the Vogel-Fulcher-Tammann (VFT) relation for describing τ(P) experimental data5,7,56. Recently, an alternative and a model-free way for detecting the dynamic crossover via 1/mT(P) vs. P analysis was indicated7. This report shows that the pressure evolution of 1/V#(P) follows the pattern noted for 1/mT(P): this is shown in insets in Figs 2 and 3. The lack of PB for 8*OCB in the inset in Fig. 4 results from the limited tested pressure range, between Pg(T) = 0.56 GPa and P = 0.1 MPa. Consequently, in 8*OCB measurements were carried out only in the high-pressure dynamical domain (Pg > P > PB).

Figure 2
figure 2

The pressure evolution of the normalized apparent activation volume Va(P)/RT (Eq. 9) and the reciprocal of V#(P)/RT ∝ mT(P) (Eqs 3 and 8: the inset) in superpressed diisobutylphtalate. In the inset, the manifestation of the dynamical crossover pressure PB is indicated. Results are for T = 317 K isotherm. The thin dashed green line is for the Kornilov et al.62 moderate pressures approximation (Eq. (10b)). The thick blue curve is related to Eq. (13). The vertical arrow shows the glass transition. In the inset, the dynamic crossover pressure is indicated.

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Figure 3
figure 3

The pressure evolution of the normalized apparent activation volume Va(P)/RT (Eq. (9)) and the reciprocal of V#(P)/RT ∝ mT(P) (Eqs (3) and (8): the inset) in superpressed epoxy resin EPON 828. In the inset, the manifestation of the dynamical crossover pressure PB is indicated. Results are for T = 303 K isotherm. The thin dashed green line is for the Kornilov et al.62 moderate pressures approximation (Eq. 10b). The thick blue curve is related to Eq. (13). The vertical arrow shows the glass transition. The dynamic crossover pressure is also indicated.

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Figure 4
figure 4

The pressure evolution of the normalized apparent activation volume Va(P)/RT (Eq. 9) and the reciprocal of V#(P)/RT ∝ mT(P) (Eqs (3) and (8): the inset) in superpressed liquid crystalline 8*OCB (isotropic liquid phase). Results are for T = 298 K isotherm. The thin dashed green line is for the Kornilov et al.62 moderate pressures approximation (Eq. (10b)). The thick blue curve is related to Eq. (13). The vertical arrow indicates the glass transition.

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The above discussion shows the direct link between the magnitude considered so far as the apparent activation volume V#(P) and the apparent fragility mT(P). To determine the ‘real’ apparent activation volume Va(P) the protocol developed for the apparent activation energy Ea(T) can be applied. The latter is given in refs12,13,14 and the Supplementary Info. For the pressure path, this means the numerical solution of the differential Eq. (4) for the given set of τ(P) experimental data. However, for the SB Eq. (1) a more straightforward way of determining Va(P) is possible since the prefactor in SB Eq. (1) is known: τ0P = τ(T)P=0 ≈ τ(T)P=0.1MPa. Consequently, the apparent activation volume can be calculated directly from the SB Eq. (1), via the simple rearrangement:

$${V}_{a}(P)=\frac{RT}{P}\,{ln}(\frac{\tau (P)}{{\tau }_{0}^{P}})\,T=const$$
(9)

Results of such analysis present in main panels of Figs 2–4.

Insets in these Figures show pressure evolutions of 1/V#(P) ∝ 1/mT(P). It is visible that both quantities follow the same pattern, in agreement with Eq. (8) and ref.7. One can also see that V#(P) ~ Va(P) only for P → 0. Some distortion between Va(P → 0) and V#(P → 0) visible in Figs 2–4 can be associated with the distortions sensitive nature of Eq. (3) containing derivatives and used for calculating V#(P). For higher pressures, i.e. P → Pg: V#(P) ≫ Va(P). Interestingly, the pressure evolution of the apparent activation volume Va(P) seems to be poorly, if at all, sensitive to the dynamical crossover phenomenon (see main parts of Figs 2 and 3).

To the best knowledge of the author, similar discussions of the apparent activation volume as introduced via Eq. (9) have appeared only in few reports so far. Beyeler and Lazarus61 considered the diffusion processes during chemical reactions under high compression and introduced a similar apparent activation volume concept. Recently, Kornilov et al.62 experimentally studied reaction rate constants for which the following relation was applied: \(\mathrm{ln}\,k(P)\) = a′P + b′P2, for P < (220–300 MPa). It can be linked to the simple approximation of the SB Eq. (1):

$$\tau (P),\eta (P),k(P)\propto {\exp }\,(P\times \alpha (P))={\exp }(P\frac{a+bP}{RT})$$
(10a)

Consequently, recalling Eq. (9) the activation volume is given via:

$${V}_{a}(P)=a+bP,\,T=const$$
(10b)

Equations (10a) and (10b) describe τ(P) and Va(P) experimental data up 200 MPa for 8*OCB and EPON 828 and even above P ~ 400 MPa for DIIP, as shown by solid curves in Fig. 1 and thin (green) dashed lines in Figs 2–4.

When considering the description of Va(P) in the broad range of pressures one can recall a recently derived relation for the previtreous pressure evolution of τ(P), originating directly from ‘universal’ changes of the apparent fragility described via Eq. (8) 7:

$$\tau (P)={\tau }_{0P}{({P}^{\ast }-P)}^{-\Psi }$$
(11)

where the power exponent \(\Psi =\,{ln}\,10(\varDelta {P}_{g}^{\ast }/{P}_{g}){m}_{T}({P}_{g})\), and the ‘discontinuity’ of the glass transition: ΔPg* = P* − Pg7.

Combining Eqs (9) and (11) one obtains:

$${V}_{a}^{RT}(P)=\frac{{V}_{a}(P)}{RT}=\frac{C}{P}-\frac{\Psi }{P}\,{ln}({P}^{\ast }-P)\approx \frac{C-1}{P}+\frac{{({P}^{\ast })}^{-\psi }}{P}{(1-\frac{P}{{P}^{\ast }})}^{-\Psi }$$
(12)

where \(C={ln}\,{\tau }_{0P}/{ln}\,\tau (T)\), T = const.

However, in Eq. (12) the apparent activation volume Va(P) → ∞ for P → 0, and then the anomalous increase occurs for P → 0.1 MPa. This problem can be avoided when taking into account that for solids, including liquids, the available range of pressures extends from the ‘normal’ positive domain (the isotropic compression) to the negative one, associated with the isotropic stretching63. This concept proved its fundamental significance for the general equation of state for water or critical mixtures and blends63. Angell and Quing experimentally showed the appearance of negative pressures and passing P = 0 without any hallmark, in glass-forming liquids using the centrifugal method64. Consequently, the ‘positive’ (isotropic compressing) and ‘negative’ (isotropic stretching) pressure regions can be considered as the common area, terminating at the absolute stability limit (SL) spinodal, hidden under negative pressures63,64,65. To describe experimental data in both pressure domains, one can consider the transformation P → ΔP = P − PSL57. Introducing the latter to Eq. (11) the modified dependence appears:

$${V}_{a}(P)=\frac{C}{\Delta P}-\frac{\Psi }{\Delta P}\,{ln}({P}^{\ast }-P)\approx \frac{C-1}{\Delta P}+\frac{{({P}^{\ast })}^{-\Psi }}{\Delta P}{(1-\frac{P}{{P}^{\ast }})}^{-\psi }$$
(13)

The above relation describes pressure changes of the apparent activation volume in the whole tested pressure range, up to P ≈ 1.2 GPa, as shown in Figs 2–4. It can be also extended into the negative pressures domain. Fitted parameters are given in Table 1. Values P* were estimated from insets in Figs 2–4 using the condition 1/V#(P*) = 0; this also includes parameters C and Ψ estimated following ref.7.

Table 1 Results of fitting P* via Eq. (13) for experimental data from Figs 2–4.
Full size table

Consequently, the analysis of Va(P) experimental data via Eq. (13) can be reduced solely to the single fitted parameter (PSL). This offers the new route for estimating the absolute stability limit pressure in negative pressures domain, which is considered as one of the most difficult to estimate properties via the experimental determination63,64,65.

Conclusions

Concluding, the activation volume is the key parameter governing the complex previtreous dynamics for the structural relaxation time, viscosity, diffusion, and reaction rates under high pressure, as indicated in the SB Eqs (1) and (2). In numerous research reports, focusing on the glass transition, this property is considered to be given by \({V}^{\#}(P)=RT[d\,\mathrm{ln}\,\tau (P)/dP]\)8,26,27,28,29,30,31,32,33,34,35,38,39,40,41,42,47,48,49,50,51,52,53,54,55, and subsequently used for developing the ‘pressure dimension’ of the glass transition models/theories8,26,27,28,29,30,31,32,33,34,35,38,39,40,41,42,47,48,49,50,51,52,53,54,55: the free volume model66, Adam-Gibbs model67, Cohen-Grest model68, and Avramov-Milchev model69. This report shows that the activation volume Va(P) ≠ V#(P). These magnitudes coincide only for the basic Barus dynamics, i.e. for Eq. (1) with Va = const. The magnitude V#(P) is directly linked to the pressure-related apparent fragility mT(P).

The main result of the given report is the simple and non-biased way of determining the apparent activation volume Va(P) via Eq. (9) and the proposal for the parameterization of Va(P) evolution given by Eq. 13. It is worth mentioning that problems with the estimation and meaning of the activation volume seem to be absent for geophysics/deep Earth science where Murnaghan – O’Connel relation is broadly applied70,71:

$${V}_{a}(P)={V}_{a}^{ref.}{(1+\frac{P}{{K}_{0}/{K^{\prime} }_{0}})}^{1/{K^{\prime} }_{0}}$$
(14)

where K0, \({K^{\prime} }_{0}\) denotes 4/9 of the bulk modulus and its first derivative.

Notable is the similarity of Eq. (14) to approximated Eqs (12) and (13)], as shown above.

Results presented focused on the pressure evolution of the primary relaxation time, but they can also be applied for the viscosity, electric conductivity, diffusion, equilibrium, and reaction rates coefficients, in ultraviscous/ultraslow systems what indicates the broad range of fundamental and practical applications ranging from the glass transition physics and the solid state physics to ‘extreme’ chemistry, geophysics, petrology, innovative material engineering, high pressure preservation of food and biotechnology under pressure.

Methods

In the last decades, the broadband dielectric spectroscopy (BDS) has become the key tool for studying previtreous behavior, including challenging insights from high-pressure studies8,72,73. In this report, BDS is used to determine the pressure evolution of the primary (structural, alpha) relaxation time72,73. Studies were carried out using the Novocontrol impedance analyzer, model 2015. BDS studies were carried out between the atmospheric pressure (P = 0.1 MPa) and the glass transition pressure, estimated via the empirical condition τ(Tg, Pg) = 100 s8,72. The structural relaxation times were determined from the peak frequencies of primary relaxation loss curves ε"(f): τ = 1/2πfpeak8,9,72. Tested samples were placed in the flat-parallel measurement capacitor made from Invar. The gap between plates d = 0.2 mm and their diameter 2r = 16 mm. Samples were entirely isolated from the pressurized medium (Plexol). They were in contact only with Invar, quartz (the spacer between plates) and Teflon. The pressure was transmitted to the sample via the deformation of 50 mm thick Teflon film. The process was supported by the computer–controlled pump, enabling pressure changes and programming with the precision ΔP = ±0.2 MPa, The pressure chamber was surrounded by a special jacket associated with the Julabo high-accuracy thermostat with the external circulation and the volume of the thermostated liquid V = 20L. These enabled temperature changes and control with accuracy ΔT = ±0.02 K. The temperature was monitored using the thermocouple within the pressure chamber and two platinum mini-resistors placed at the bottom and the top of the chamber. The high-pressure system was designed and produced by UnipresEquipment (Poland). Further experimental details are given in refs6,7,74. Notable, that the examined range of pressures was extended up to P ~ 1.2 GPa, the still hardly available range in high-resolution BDS pressure studies8,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54. Experimental results cover time-scales from τ(P = 0.1 MPa) to τ(Pg, Tg) = 100s. The latter is commonly applied as the practical empirical estimation of (Tg, Pg)8,10,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54.