Breakdown of the Simple Arrhenius Law in the Normal Liquid State

Letter pubs.acs.org/JPCL

Cite This: J. Phys. Chem. Lett. 2018, 9, 1783−1787

Breakdown of the Simple Arrhenius Law in the Normal Liquid State Erik Thoms, Andrzej Grzybowski, Sebastian Pawlus, and Marian Paluch* Institute of Physics, University of Silesia in Katowice, ul. 75 Pulku Piechoty 1, 41-500 Chorzow, Poland Silesian Center for Education and Interdisciplinary Research, 75 Pulku Piechoty 1A, 41-500 Chorzow, Poland ABSTRACT: It is common practice to discuss the temperature effect on molecular dynamics of glass formers above the melting temperature in terms of the Arrhenius law. Using dielectric spectroscopy measurements of dc conductivity and structural relaxation time on the example of the typical glass former propylene carbonate, we provide experimental evidence that this practice is not justified. Our conclusions are supported by employing thermodynamic density scaling and the occurrence of inflection points in isothermal dynamic data measured at elevated pressure. Additionally, we propose a more suitable approach to describe the dynamics both above and below the inflection point based on a modified MYEGA model.

G

Besides supercooling, the generation of a vitreous state is also possible by limiting the free volume of the molecules via an increase of the pressure p. Close to the glass transition, a pressure equivalent of the VFTH law can be used to describe the p dependence of η and other quantities4

lasses, i.e. solids in a noncrystalline state, and supercooled liquids are substantial materials for a multitude of technical applications. From a physical point of view, however, the glass transition process is still not fully explored, and many aspects and characteristics wait to be explained satisfactorily. In particular, the roles of the free volume on the one hand and thermal energy on the other hand are not yet determined. A common way to transfer a material into the glassy state is supercooling. By decreasing the temperature under the melting point Tm at a rate high enough to avoid crystallization, the viscosity η of the liquid can be increased, until a solid without long-range order forms below the glass temperature Tg. One characteristic property of glass formers in the supercooled state is the non-Arrhenius dependency of η(T) upon cooling. A simple energy barrier activated behavior is given by the Arrhenius equation ⎛E ⎞ η(T ) = η∞ × exp⎜ act ⎟ ⎝ RT ⎠

⎛ Cp ⎞ η(p) = η0 × exp⎜⎜ F ⎟⎟ ⎝ p∞ − p ⎠

η0 = η(p → 0) is the viscosity at very low pressures, p∞ is the divergence pressure, and CF is a pressure equivalent to DVF. At conditions far from the glass transition (i.e., T ≫Tg(p)), however, deviation from this behavior is observed and sometimes described by the pressure counterpart of the Arrhenius law ⎛ pV ⎞ η(p) = η0 × exp⎜ act ⎟ ⎝ RT ⎠

(1)

(2)

Here, TVF < Tg is the Vogel−Fulcher temperature, indicating a divergence point, while the strength parameter DVF quantifies the discrepancy from ideal Arrhenius behavior. In the case of other properties, e.g., the structural relaxation time τα or inverse dc conductivity σ−1 dc , a similar temperature dependence can be observed and be described by analogous models. © XXXX American Chemical Society

(4)

with the constant activation volume Vact.5 While an isobaric change of temperature affects both the available energy and the molecular packing, isothermal modification of pressure only acts on the latter. As a consequence, it is possible to separate the influence of those two factors by a systematic analysis of the pressure and the temperature behavior of a glass former,6 which makes high-pressure studies a key element for understanding the glass transition. Despite its prevalence, the VFTH model met with criticism, in particular, due to the lack of experimental evidence for a divergence of τα at finite temperatures and the resulting breakdown of the model at low temperatures.7,8 On the basis of

where T is the absolute temperature, η∞ the viscosity for very high T, and R the universal gas constant. In contrast, a commonly employed model to describe the super-Arrhenius temperature dependence observed in glass formers is the empirical Vogel−Fulcher−Tammann−Hesse equation (VFTH equation)1−3 ⎛D T ⎞ η(T ) = η∞ × exp⎜ VF VF ⎟ ⎝ T − TVF ⎠

(3)

Received: February 23, 2018 Accepted: March 21, 2018

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DOI: 10.1021/acs.jpclett.8b00583 J. Phys. Chem. Lett. 2018, 9, 1783−1787

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The Journal of Physical Chemistry Letters the Adam−Gibbs model9 and the Phillips−Thorpe constraint theory,10,11 Mauro et al. proposed a three-parameter entropybased model avoiding such divergence,12 which has been tested successfully by other groups.13,14 According to this MYEGA model, the temperature dependence of dynamic properties can be described by ⎡ K ⎛ C ⎞⎤ η(T ) = η∞ × exp⎢ exp⎜ ⎟⎥ ⎝ T ⎠⎦ ⎣T

(5)

with K being related to the activation energy and C to the energy difference between the intact and broken states of network constraints. One of the most interesting findings in recent years in the study of glass-forming materials is thermodynamic density scaling. For several supercooled liquids, it has been reported that a scaling function J(Γ) can describe the thermodynamic dependencies of dynamic properties, e.g., the isothermal and isobaric structural relaxation times τα, close to the glass transition.15−20 Consequently, the measured data for multiple p and T values collapse to one master curve when plotted against Γ. The scaling variable Γ can be expressed by temperature T and specific volume V via

Γ = T −1V −γ

Figure 1. Measured pressure-dependent inverse dc conductivity of PC for constant temperatures of 328 (diamonds) and 350 K (squares). In the inset, the corresponding activation volumes are shown, together with the activation volume derived from the structural relaxation times (circles) given in Figure 2b. The lines are fits according to eq 9.

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where the scaling exponent γ is a material constant independent of thermodynamic conditions and Γ = const. for τα = const.

exponential growth at elevated pressures (positive curvature and convex shape).23 Consequently, the low-pressure range is not well described by a pressure-dependent Arrhenius-like law (eq 4). The 328 K isotherm shows equal behavior, albeit at generally lower conductivity. Similar results have recently been observed in the viscosity η of PC,24 in the conductivity of protic ionic liquids,25 as well as in η or the structural relaxation time τα of other molecular or ionic liquids,26,27 even though the concave progression at low pressures is not necessarily incorporated in the models used in these reports. The presence of the inflection point becomes more obvious when the σdc(p) data is analyzed in terms of the activation −1 volume parameter Vact,σ(p) = RT × (∂ ln(σdc )/∂p)T, as presented in the inset of Figure 1. In this representation, slower and faster than exponential pressure dependences correspond to a decrease or an increase of Vact (p), respectively. Therefore, the inflection point pinfl is being reflected by a minimum of Vact(p). Herbst et al. suggested that the inflection might arise from a nonlinearity of the volume with a change of pressure,28 while Casalini and Bair attributed it to the pressure dependences of the compressibility and of the apparent activation energy at constant volume.24 Bair emphasized the importance of the incorporation of this feature for physically meaningful models23 and proposed a combination of the McEwen model with a pressure equivalent4 of the VF law (eq 2) to give a phenomenological description

(7)

(respectively σdc = const.) is the scaling criterion. These findings sparked high interest as they potentially could bring light to relations between thermodynamics and relaxation dynamics. Conforming to the scaling criterion, Masiewicz et al. generalized the MYEGA model to extend its applicability to T−p-dependent data transformed via V = V(T, p) to the T−V domain22 21

⎡ D ⎛ A ⎞⎤ τα = τ0 × exp⎢ γ exp⎜ γ ⎟⎥ ⎝ TV ⎠⎦ ⎣ TV

(8)

γ

The relaxation time τ0 for high TV , i.e., Γ → 0, and D and A are fitting parameters. In this Letter, we employ the scaling criterion to generate pressure-dependent τα data from ambient pressure dielectric spectroscopy measurements on the example of the canonical glass former propylene carbonate (PC). The generated data is verified via comparison with measured dc conductivity data and is fitted according to a modified MYEGA model. We find an inflection point in the pressure-dependent curves, separating distinctly different p dependency at low- and high-pressure ranges. Using various calculated τα isotherms, we analyzed the temperature and relaxation time dependences of the inflection point. In Figure 1, the inverse dc conductivity σ−1 dc (p) is shown for T = 328 and 350 K in a pressure range up to p ≈ 1380 MPa. For the 350 K isotherm, the conductivity decreases from an ambient pressure value of σdc ≈ 4 × 10−6 S/cm to approximately 3 × 10−8 S/cm over this range. Generally, the decrease of molecular mobility with rising pressure induces an increase in σ−1 dc . It can be seen from the pictured data that the pressure dependence σ−1 dc (p) changes from a slower than exponential growth, i.e., negative curvature and concave shape versus the p-axis in Figure 1 at low pressures, to faster than

q ⎛ Cp ⎞ ⎛ α ⎞ η(p) = η0 × ⎜1 + 0 p⎟ × exp⎜⎜ F ⎟⎟ q ⎠ ⎝ ⎝ p∞ − p ⎠

(9)

Here, η0 is the viscosity at p → 0. The fitting parameters α0 and q originate from the McEwen equation, while CF and the divergence pressure p∞ are taken from the VF-like model. Fits according to eq 9 are presented in Figure 1 and, with identical parameters, in the inset. From the minima in the latter curves, the inflection pressures pinfl(T = 328 K) ≈ 345 MPa and pinfl(T = 350 K) ≈ 440 MPa can be derived. Analyzing the respective 1784

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Reiser et al.,30 to get the best density scaling of the combined structural relaxation and dc conductivity data. It can be seen clearly that the Γ dependence changes from a slow increase at low Γ to a much faster ascent closer to the glass transition. The crossover can be found approximately at 0.009 K−1 gγ cm−3γ and τα ≈ 10−10 s. It is worth noting that the scaling criterion (eq 7) allows one to numerically generate isotherms, isochores, and isobars out of measured data, as shown in detail in the Supporting Information to ref 31. Using this procedure with γ = 4.2 and the Tait EOS from ref 29, τα(p) values were calculated based on the ambient pressure isobar shown in Figure 2a. The isotherm for T = 350 K is presented in Figure 2b exemplary, together with the analogous conductivity data from Figure 1 for comparison. The scales on both axes cover the same number of decades, as above. Both curves show distinct accordance in the p dependence, with negative curvature of the generated curve in the range from τα = 2 × 10−11 s at ambient pressure up to the inflection point at pinfl = 440 MPa and τα = 8.5 × 10−11 s. At higher pressures, up to a value of τα = 3 × 10−9 s at 1400 MPa, the increase in τα(p) becomes faster than exponential. The high conformity of both curves strongly supports the validity of data generation using the density scaling criterion that turned out to be able to predict the inflection point in isothermal conditions, although at first glance the dielectric data collected at ambient pressure have not suggested that such an inflection occurs in isothermal data. Of particular interest is the emergence of an inflection, concurring with the measured conductivity results, in the generated curve, while no such behavior was observed in the ambient pressure source data. The accordance of τα(p) and σ−1 dc (p) becomes even more obvious in a comparison of the respective activation volumes. As can be seen in the inset of Figure 1, both curves are in near perfect agreement over the whole pressure range, implying that the structural relaxation dynamics of PC can be followed by studies of the dc conductivity. No decoupling between τα and σ−1 dc is a promising result because dielectric measurements of fast dynamics under high pressure are difficult experimentally, as opposed to conductivity measurements. Using the Tait EOS from ref 29, eq 8 was adapted for the fitting of isotherms. Separate fits of the τα isotherm at T = 350 K for p < pinfl and p > pinfl can be found in Figure 2b. Below the inflection point, it yields τ0 = 10−11.54 s, D = 382 K(g/cm3)γ, and A = 0 K(g/cm3)γ for a fitted range from ambient pressure to 166 MPa. The Γ value corresponding to 440 MPa and 350 K is 0.0088 K−1 gγ cm−3γ. Using the above parameters, τα(Γ) can be described equally well below that point. This is illustrated by the solid line in Figure 2a. It should be noted that a linear Γ

PC viscosity data from ref 24 in the same way, we receive pinfl(T = 328 K) ≈ 350 MPa, in very good agreement with our results. The phenomenology of eq 9 is, however, insufficient to gain a deeper insight into consequences of the dynamic crossover indicated by the inflection point. To make progress toward a better understanding of this phenomenon, we employ the density scaling idea in this study, as presented in Figure 2a.

Figure 2. Ambient pressure relaxation times from ref 29 versus scaling variable Γ = T−1V−γ (a). In the low-Γ range, conductivity data are added to expand the diagram nearly to the boiling point. The boiling, melting, and glass temperatures Tb, Tm, and Tg (all at ambient pressure) are marked by the arrows. Fits of the low- and high-Γ regime data according to eq 8 are given by the solid and dashed lines, respectively. (b) Isothermal pressure dependence of τα, generated from the density scaling criterion based on the data shown in (a) and pressure−volume−temperature measurements, together with the measured σdc 350 K data from Figure 1. The dashed and solid lines correspond to the fits in (a). (c) Ratio of the isochoric and isobaric activation energy as a function of Γ. The top scale gives the inverse temperature associated with Γ at ambient pressure.

Here, the density scaling of the structural relaxation time τα(Γ) is shown for the ambient pressure data taken from ref 29, with the addition of σdc data from Figure 1 to expand the curve toward higher temperatures. It should be noted that the scales on the τα and σdc axes comprise the same amount of decades but are shifted against each other to make the data comparable. This way, the thermodynamic evolution of dynamic properties can be presented from close to the glass transition to nearly the boiling point (Tg = 156 K and Tb = 515 K at 0.1 MPa). To acquire the volume data, results of pressure−volume−temperature measurements and a Tait-like equation of state (EOS) reported earlier by Pawlus et al.29 have been used. In the past, slightly different values of the scaling exponent have been reported.24,29,30 Indeed, we found γ = 4.2, in accordance with

dependence, as given by τα = τ0 × exp

(

D TV γ

exp

( ATV= 0 )), γ

precludes application of an Arrhenius law (eq 1) to describe the data at high temperatures. This coincides with the observations made on the σdc isotherm at low pressures, showing that eq 4 is likewise improper. As given by the dashed lines in Figure 2a,b, eq 8 can also be used to fit the results above the crossover point (i.e., p > 440 MPa at T = 350 K and generally Γ > 0.0088 K−1 gγ cm−3γ) with τ0 = 10−10.63 s, D = 28.7 K(g/cm3)γ, and A = 186 K(g/cm3)γ, with considering points p > 166 MPa at T = 350 K and generally Γ > 0.0067 K−1 gγ cm−3γ. This implies that a single model is eligible to describe all measured data. 1785

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Another interesting result can be drawn when plotting the structural relaxation times at the inflection point τα,infl against pinfl (Figure 3b). In fact, for all isotherms, log10τα,infl ≈ −10.07 is almost constant, i.e., the inflection can be observed at isochronal conditions. Summing up, we performed isothermal high-pressure dielectric measurements on PC for two isotherms. Additionally, structural relaxation times for T = 350 K in the same pressure range were generated out of existing ambient pressure data employing the density scaling criterion and a Tait-like EOS. A direct comparison of the σdc and τα data interestingly shows remarkable accordance of the pressure-dependent behavior over the whole measured range, suggesting that it is possible to access fast dynamics via conductivity measurements. Because direct measurements of τα are difficult in high-pressure setups in the fast dynamic range, this can offer a new route to expand spectra. A noteworthy observation is the presence of an inflection point, i.e., a transition from slower to faster than exponential pressure dependence in both properties. Similar observations were made before, both in PC and in other materials.24−27 However, the presence of the inflection in the generated τα isotherm is of special significance because no such feature can be observed in the reference isobar. This implies that it is possible to effectively predict high-pressure properties via lowpressure measurements when using the density scaling criterion. Our analysis of the high-T and low-Γ data indicates a clear deviation from an Arrhenius model regularly used19,34−36 to describe the fast dynamics. This can be seen in both the linear Γ dependence in Figure 2a and the slower than exponential p dependence in Figures 1 and 2b in the low-Γ and low-p ranges, as well as the increase of EV,act/Ep,act with T. Instead, we employed a modified MYEGA model (eq 8) to successfully fit the data, using the parameter A = 0 K(g/cm3)γ, which reasonably indicates that the intact and broken constraints are energetically degenerated or the topological constraints are ineffective at τ < τinfl.12 The same model with a positive value of A can be used to appropriately describe the data in the high-p range. Analyzing the temperature dependence of the inflection pressure pinfl, a linear increase with T was observed. Besides, τinfl ≈ 10−10 s is nearly constant for all curves, i.e., the inflection point is found at isochronal conditions. We speculate that this phenomenon can be attributed to the transition from simple to cooperative molecular dynamics in glass formers. The crossover takes place at temperatures higher than the melting point, suggesting that the observed change of dynamics is a precursor of supercooling.

To quantify the relative importance of the influences of temperature and free volume on a dynamic property, the ratio of the isochoric activation energy EV,act = R[∂ ln(τα)/∂T−1]|V and the isobaric equivalent Ep,act can be calculated.32 The ratio can take values between 0 and 1, indicating solely free volume activated and ideal thermally activated behavior, respectively. From Figure 2c, it can be seen that EV,act/Ep,act increases with higher values of 1/TVγ, which implies an increasing relative influence of the temperature over the volume on τα.33 Close to the boiling point, the volume is the more important parameter (EV,act/Ep,act = 0.41), while near the glass transition, a ratio of 0.69 is observed, i.e., the thermal influence on the dynamics is roughly twice as high as the volumetric one. The increase of volumetric influence when approaching the boiling point is contradictory to the purely energy activated behavior as given by eq 1, supporting the employment of a non-Arrhenius model. By using a set of τα isotherms generated as described above, the influence of temperature on the inflection point can be determined (see Figure 3). pinfl(T) is taken at the minimum of

Figure 3. Temperature dependence of pinfl. The line is a linear fit of the data, extrapolated to the inflection temperature at ambient pressure (arrow). For reference, Vact(p) is shown in inset (a) for τα of selected isotherms. Inset (b) presents the relaxation times as a function of pinfl, where the line illustrates log(τα) = 10−10.07 s.

the corresponding Vact(p) curves, shown in inset (a). As elaborated in ref 31, measured base points are needed for the curve generation. Consequently, isotherms between T = 294 and 371 K were taken into account based on the underlying data collected at ambient pressure. In the given range, a linear increase of pinfl with T from 187 to 553 MPa can clearly be observed. Fits of the viscosity data from Casalini and Bair24 according to eq 9 result in pinfl = p(Vact = min) ≈ 235 and 350 MPa for 298 and 328 K, respectively, in excellent agreement with our data. A linear pinfl(T) dependence seems to be common behavior for materials without significant hydrogen bonding.25 A linear fit (line in Figure 3) with a slope of 4.7 MPa/K adequately describes the data. Extrapolation of this fit to ambient pressure results in an estimate of Tinfl(p = 0.1 MPa) = 253 K. Remarkably, Tinfl is nearly 30 K higher than the melting temperature Tm, implicating a change of the molecular dynamics even before supercooling takes place.

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EXPERIMENTAL DETAILS The PC was purchased from Sigma-Aldrich with a purity of >99%. Details on the experimental setup can be found in an earlier work.37

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Erik Thoms: 0000-0002-2215-9974 Notes

The authors declare no competing financial interest. 1786

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ACKNOWLEDGMENTS S.P., M.P., and E.T. acknowledge financial support of Project No. UMO-2015/17/B/ST3/01221 by the National Science Centre, Poland.

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