Abstract
In pressurized glassforming systems, the apparent (changeable) activation volume V_{a}(P) is the key property governing the previtreous behavior of the structural relaxation time (τ) or viscosity (η), following the SuperBarus behavior: \({\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{,}}{\boldsymbol{\eta }}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{\propto }}{\bf{\exp }}{\boldsymbol{(}}{{\boldsymbol{V}}}_{{\boldsymbol{a}}}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{/}}{\boldsymbol{R}}{\boldsymbol{T}}{\boldsymbol{)}}\), T = const. It is usually assumed that V_{a}(P) = V^{#}(P), where \({{\boldsymbol{V}}}^{{\boldsymbol{\#}}}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}={\boldsymbol{R}}{\boldsymbol{T}}{\boldsymbol{d}}\,{\boldsymbol{ln}}\,{\boldsymbol{\tau }}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{/}}{\boldsymbol{d}}{\boldsymbol{P}}\) or \({{\boldsymbol{V}}}^{{\boldsymbol{\#}}}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{=}}{\boldsymbol{R}}{\boldsymbol{T}}{\boldsymbol{d}}\,{\boldsymbol{ln}}\,{\boldsymbol{\eta }}{\boldsymbol{(}}{\boldsymbol{P}}{\boldsymbol{)}}{\boldsymbol{/}}{\boldsymbol{d}}{\boldsymbol{P}}\). This report shows that V_{a}(P) ≪ V^{#}(P) for P → P_{g}, where P_{g} denotes the glass pressure, and the magnitude V^{#}(P) is coupled to the pressure steepness index (the apparent fragility). V^{#}(P) and V_{a}(P) coincides only for the basic Barus dynamics, where V_{a}(P) = V_{a} = const in the given pressure domain, or for P → 0. The simple and nonbiased way of determining V_{a}(P) and the relation for its parameterization are proposed. The derived relation resembles Murnaghan  O’Connel equation, applied in deep Earth studies. It also offers a possibility of estimating the pressure and volume at the absolute stability limit. The application of the methodology is shown for diisobutyl phthalate (DIIP, lowmolecularweight liquid), isooctyloxycyanobiphenyl (8*OCB, liquid crystal) and bisphenol A/epichlorohydrin (EPON 828, epoxy resin), respectively.
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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 lowmolecularweight liquids and polymers to liquid crystals and plastic crystals are the key manifestation of the hypothetical universal dynamics emerging on approaching the glass transition (T_{g}, P_{g})^{1,2,3,4,5}. Similar patterns are observed both for the temperature and pressure paths^{6,7,8}. The temperature path is associated with the SuperArrhenius (SA) dynamics, and it is governed by changes of the apparent activation energy E_{a}(T), which strongly increases on approaching the glass transition temperature T_{g}^{9,10,11}. The nonbiased way of determining E_{a}(T) and its properties are discussed in refs^{12,13,14} and recalled in Supplementary Info.
This report focuses on the still puzzling case of the (high) pressureinduced glass transition. For compressed glassformers, general features of the previtreous dynamics are described by the SuperBarus (SB) equation^{7,8,9,10}:
where T = const and P < P_{g}; V_{a}(P) denotes the apparent activation volume, which changes on compressing. Generally, the name ‘activation volume’ is reserved for the basic Barus^{15} equation with V_{a}(P) = V_{a} = const in the given domain of pressures.
Prefactors τ_{0}^{P} and η_{0}^{P} in Eq. (1) refer to P = 0, but within the experimental error they can be approximated by atmospheric pressure values, i.e., τ_{0}^{P} = τ(P = 0) ≈ τ(P = 0.1 MPa) for the tested isotherm T. For highpressure 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 time^{6,7,8}.
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 account^{8,10}. For instance, for DC electric conductivity^{16}:
where S < 1 is the decoupling exponent associated with the fractional DebyeStokesEinstein (fDSE) dependence σ(P)[τ(P)]^{S} = C = const^{8,10,16}.
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 19^{th} century when studying viscosity of natural oils^{15}. Whalley^{17,18} and Williams^{19} applied such description for pressurized polymers and dielectric relaxation time, introducing the activation volume V_{a}, what led to Eq. (1) with V_{a} = const. The Barus (B) or BarusWilliams dependence can be considered as the pressure counterpart of the basic, temperaturerelated Arrhenius equation which was originally introduced as \(k(T)={k}_{0}{\exp }({E}_{a}/RT)\), where E_{a} stands for the activation energy and k is the reaction rate coefficient^{20}. Generally, for ultraviscous/ultraslow glassforming systems one should expect the SuperBarus (SB) behavior (Eqs (1) and (2)), where dynamic properties are governed by changes of the pressuredependent activation volume: the apparent activation volume.
According to the above discussion one concludes that V_{a}(P) governs the dynamics of ultraviscous/ultraslow systems, and its determination and understanding is the key to the ultimate insight into the glass transition problem^{6,7,8,9}, the behavior of soft matter under pressure^{7,8,9,21}, highpressure chemistry and biochemistry^{22}, innovative material engineering^{23}, highpressure biotechnology^{24}, geophysics, and deep Earth science^{25}.
Usually, for the previtreous domain the apparent activation volume V_{a}(P) is calculated from τ(P) or η(P) experimental data via^{26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55}:
under the assumption that V_{a}(P) = V^{#}(P) and for 0.1 MPa < P ≤ P_{g}.
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 nonactivated states^{22}. It is the essential parameter characterizing the sensitivity of the structural relaxation time or other dynamic properties to pressure changes^{7,8,21}. 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 ξ^{3}, which are considered as one of the essential sources of the previtreous ‘universality’ of dynamic previtreous properties. Tests in hydrogenbonded molecular liquids showed the casesensitivity of the activation volume, determined via Eq. (3) to subtle features of molecular structures^{38,39,40}. Worth stressing is also the broadly used link between the activation volume at the glass transition, and the fragility: ΔV^{#} = m_{P} × 2,303R(dT_{g}/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 phenomenon^{2,3,4,5,9,10,43,44}. The analysis via Eq. (3) was also used for showing different activation volumes determined by dielectric and light scattering spectroscopies^{8,45,46,47,48,49,50,51,52,53}. Reasonings based on such analysis can yield important checkpoints for glass transition models^{8,45,46,47,48,49,50,51,52,53,54,55}. The activation volume is also significant for the thermodynamic scaling linking τ(T.P, V) experimental data^{8,54,55}. There are also reports where V^{#}(P) is recalled as the (apparent) activation volume, but the link to the steepness index m_{T}(P) is indicated^{56,57}.
For the validation of Eq. (3) reports by Whalley^{17,18} and Williams^{19} are most often cited^{8,30,31,32,33,34,38,39,40,41,42,48,49,50,51,52,53,54}. However, these reports did not consider the SuperBarus 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:
Comparing Eqs (3) and (4) one obtains that generally: V_{a}(P) ≠ V^{#}(P). The second term in Eq. (4) disappears only for two ‘special’ cases: (i) for P = 0, or (ii) for V_{a}(P) = V_{a} = const, i.e., for the basic Barus behavior in the given pressure domain.
Worth recalling is the difference between the free volume (V_{f}) as the volume not occupied by molecules and the activation volume (V_{a}) as the volume required for the given process, for instance, the molecular rearrangement or reorientation. Then, one can expect V_{f} > V_{a}^{8,9}.
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 V_{a}(P) for P < P_{g}. The discussion is supported by the analysis of the τ(P) experimental data for glassforming low molecular weight liquid diisobutyl phthalate (DIIB, T_{g}(0.1 MPa) = 196.8K), epoxy resin bisphenol A/epichlorohydrin (EPON 828, T_{g}(0.1 MPa) = 253.9K) and liquid crystalline isooctyloxycyanobiphenyl (8*OCB, T_{g}(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 far^{8,10}. 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 < P_{g}. They served for estimating both V^{#}(P) and V_{a}(P).
When discussing the physical meaning of V^{#}(P) one can recall the definition of the pressurerelated steepness index (the normalized rate of changes of the relaxation time, viscosity, …) in the previtreous domain^{8}, which leads to the relation:
It terminates at the pressurerelated fragility^{11}: \({m}_{T}={m}_{T}({P}_{g})={[d{lo}{{g}}_{10}\tau (P)/d(P/{P}_{g})]}_{P={P}_{g}}\), which is the key metric for glassforming ultraviscous/ultraslow systems. Then, the pressuredependent (isothermic) coefficient m_{T}(P) for P < P_{g} can be called the apparent fragility. Linking Eqs (3) and (5) one obtains the relation showing that V^{#}(P) ∝ m_{T}(P):
Considering further the ratio of fragilities along T_{g}(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:
and
Equation (7b), originally derived in ref.^{40}, is broadly used for calculating isobaric fragilities m = m_{P}(T_{g}) for different isobars, based on the knowledge of the ‘~activation volume V^{#}(P)’ calculated via Eq. (3) and the pressure shift of T_{g}^{8,26,27,28,29,30,31,32,33,34,38,39,40,41,42,47,48,49,50,51,52,53,54}. 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 pressurerelated apparent fragility can exhibit a ‘universal’ previtreous behavior^{5,7}:
where T = const, the amplitude A = const, and P^{*} is for the extrapolated singular pressure. Regarding pressures: P < P_{g} and P^{*} > P_{g}.
Following Eqs (6) and (8) one can conclude that: 1/V_{#}(P) ∝ 1/m_{T}(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, respectively^{58,59}. This is associated with the crossover from the ergodic to the nonergodic dynamical domain at (T_{B}, P_{B}), where T_{B} ≫ T_{g} and P_{B} ≪ P_{g}^{8,16,58,59}. Roland^{60} showed the pressuretemperature invariance of the dynamic crossover timescale for a set of glassforming liquids τ(T_{B}, P_{B}) ~ 10^{−7±1}s. Until recently, the detection of P_{B} was associated with \({\Phi }_{P}(P)={(d{lo}{{g}}_{10}\tau (P)/dP)}^{1/2}\) vs. P plot^{8,59}, which is parallel to (m_{T}(P))^{−1/2} vs. P presentation^{5,7,56}. Such analysis assumes a priori the validity of the pressure counterpart of the VogelFulcherTammann (VFT) relation for describing τ(P) experimental data^{5,7,56}. Recently, an alternative and a modelfree way for detecting the dynamic crossover via 1/m_{T}(P) vs. P analysis was indicated^{7}. This report shows that the pressure evolution of 1/V^{#}(P) follows the pattern noted for 1/m_{T}(P): this is shown in insets in Figs 2 and 3. The lack of P_{B} for 8*OCB in the inset in Fig. 4 results from the limited tested pressure range, between P_{g}(T) = 0.56 GPa and P = 0.1 MPa. Consequently, in 8*OCB measurements were carried out only in the highpressure dynamical domain (P_{g} > P > P_{B}).
The above discussion shows the direct link between the magnitude considered so far as the apparent activation volume V^{#}(P) and the apparent fragility m_{T}(P). To determine the ‘real’ apparent activation volume V_{a}(P) the protocol developed for the apparent activation energy E_{a}(T) can be applied. The latter is given in refs^{12,13,14} 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 V_{a}(P) is possible since the prefactor in SB Eq. (1) is known: τ_{0}^{P} = 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:
Results of such analysis present in main panels of Figs 2–4.
Insets in these Figures show pressure evolutions of 1/V^{#}(P) ∝ 1/m_{T}(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) ~ V_{a}(P) only for P → 0. Some distortion between V_{a}(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 → P_{g}: V^{#}(P) ≫ V_{a}(P). Interestingly, the pressure evolution of the apparent activation volume V_{a}(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 Lazarus^{61} 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′P^{2}, for P < (220–300 MPa). It can be linked to the simple approximation of the SB Eq. (1):
Consequently, recalling Eq. (9) the activation volume is given via:
Equations (10a) and (10b) describe τ(P) and V_{a}(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 V_{a}(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}:
where the power exponent \(\Psi =\,{ln}\,10(\varDelta {P}_{g}^{\ast }/{P}_{g}){m}_{T}({P}_{g})\), and the ‘discontinuity’ of the glass transition: ΔP_{g}^{*} = P^{*} − P_{g}^{7}.
Combining Eqs (9) and (11) one obtains:
where \(C={ln}\,{\tau }_{0P}/{ln}\,\tau (T)\), T = const.
However, in Eq. (12) the apparent activation volume V_{a}(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 stretching^{63}. This concept proved its fundamental significance for the general equation of state for water or critical mixtures and blends^{63}. Angell and Quing experimentally showed the appearance of negative pressures and passing P = 0 without any hallmark, in glassforming liquids using the centrifugal method^{64}. 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 pressures^{63,64,65}. To describe experimental data in both pressure domains, one can consider the transformation P → ΔP = P − P_{SL}^{57}. Introducing the latter to Eq. (11) the modified dependence appears:
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}.
Consequently, the analysis of V_{a}(P) experimental data via Eq. (13) can be reduced solely to the single fitted parameter (P_{SL}). 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 determination^{63,64,65}.
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/theories^{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}: the free volume model^{66}, AdamGibbs model^{67}, CohenGrest model^{68}, and AvramovMilchev model^{69}. This report shows that the activation volume V_{a}(P) ≠ V^{#}(P). These magnitudes coincide only for the basic Barus dynamics, i.e. for Eq. (1) with V_{a} = const. The magnitude V^{#}(P) is directly linked to the pressurerelated apparent fragility m_{T}(P).
The main result of the given report is the simple and nonbiased way of determining the apparent activation volume V_{a}(P) via Eq. (9) and the proposal for the parameterization of V_{a}(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 applied^{70,71}:
where K_{0}, \({K^{\prime} }_{0}\) 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 highpressure studies^{8,72,73}. In this report, BDS is used to determine the pressure evolution of the primary (structural, alpha) relaxation time^{72,73}. 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(T_{g}, P_{g}) = 100 s^{8,72}. The structural relaxation times were determined from the peak frequencies of primary relaxation loss curves ε"(f): τ = 1/2πf_{peak}^{8,9,72}. Tested samples were placed in the flatparallel 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 highaccuracy 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 miniresistors placed at the bottom and the top of the chamber. The highpressure system was designed and produced by UnipresEquipment (Poland). Further experimental details are given in refs^{6,7,74}. Notable, that the examined range of pressures was extended up to P ~ 1.2 GPa, the still hardly available range in highresolution BDS pressure studies^{8,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54}. Experimental results cover timescales from τ(P = 0.1 MPa) to τ(P_{g}, T_{g}) = 100s. The latter is commonly applied as the practical empirical estimation of (T_{g}, P_{g})^{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}.
Data Availability
The data supporting the findings of this study are available from the author upon reasonable requests.
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Acknowledgements
This research was carried out due to the support of the National Centre for Science (Poland), project NCN OPUS ref. 2016/21/B/ST3/02203, head by Aleksandra DrozdRzoska.
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DrozdRzoska, A. Activation volume in superpressed glassformers. Sci Rep 9, 13787 (2019). https://doi.org/10.1038/s4159801949848w
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DOI: https://doi.org/10.1038/s4159801949848w
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