On the dependence of the properties of glasses on cooling and heating rates II: Prigogine–Defay ratio, fictive temperature and fictive pressure

doi:10.1016/j.jnoncrysol.2010.12.005

Journal of Non-Crystalline Solids

Volume 357, Issue 4, 15 February 2011, Pages 1303-1309

https://doi.org/10.1016/j.jnoncrysol.2010.12.005 Get rights and content

Abstract

In a previous analysis (J. Chem. Phys. 125 (2006) 094505) it was shown by us that – in contrast to earlier believe – a satisfactory theoretical interpretation of the experimentally measured values of the so-called Prigogine–Defay ratio, Π, can be given employing only one structural order parameter. According to this analysis, the value of this ratio has to be, in accordance with experimental findings, larger than one (Π > 1). This analysis is extended here and, in particular, the dependence of the value of Π on cooling and heating rates is studied. Finally, employing the general model-independent definition of internal (fictive) pressure and fictive temperature, developed by us (J. Non-Crystalline Solids 355 (2009) 653), it is shown how these parameters behave in dependence on temperature for different sets of cooling and heating rates. Some further consequences and possible extensions are discussed briefly.

Research Highlights

► The dependence of the Prigogine-Defay ratio on cooling and heating rates is established. ► A general definition of fictive pressure and fictive temperature is given. ► The dependence of fictive pressure and temperature on cooling and heating rates is analyzed.

Introduction

Following the classical concepts developed by Simon [1], [2], vitrification in the cooling of melts is commonly interpreted as the transformation of a thermodynamically (meta)stable equilibrium system into a frozen-in, thermodynamically non-equilibrium system, the glass. Hereby it is assumed in a first approximation that the transformation proceeds at some well-defined sharp temperature, the glass transition temperature, T_g. However, a more detailed experimental and theoretical analysis shows [3], [4], [5], [6] – as stated explicitly already by Tammann [7] – that the transition to a glass via the cooling of a glass-forming liquid proceeds in a more or less broad temperature range, where the characteristic times of change of temperature, τ_T, and relaxation times of the system to the respective equilibrium states, τ, are of similar magnitude. This vitrification interval and the glass transition temperature, T_g, can be defined – according to these considerations – generally via the relation τ_T ≅ τ or, employing the definition of the characteristic time of change of the control parameter $τ_{T} = (T / | \dot{T} |)$ (c.f. [8]), by $(\frac{1}{T} |\frac{d T}{d t}|) τ ≅ constant at T ≅ T_{g},$ where the constant is of the order of unity and depends in its value on the chosen precise definition of T_g.

Eq. (1) implies that the glass transition temperature (and as the result also a broad spectrum of properties of glasses) depend significantly on cooling rates. An extended analysis of the dependence of glass properties for a wide range of cooling and heating rates has been performed in a preceding paper [9]. Here are also the methods are outlined in detail employed in the analysis in the present paper. In this analysis, the structural order-parameter concept as developed by De Donder [10], [11] is utilized. The temperature dependence of a structural order parameter is determined and its effect on the properties of glass-forming melts and glasses is analyzed. Applying the same concepts, it was shown by us earlier in [12] that – in contrast to the previous general believe – the experimental data on the so-called Prigogine–Defay ratio can be given a theoretical interpretation by introducing only one structural order parameter into the description. A detailed discussion of this circle of problems including the analysis of alternative approaches is given also in [13]. This discussion will be continued and extended in the present analysis.

Above statement – that one structural order parameter is sufficient for the theoretical interpretation of the Prigogine–Defay ratio – does not imply necessarily that always only one and not more structural order parameters may be required for an appropriate description of a vitrifying or devitrifying system under investigation (for a detailed discussion of the problem of how many structural order parameters may be required in order to appropriately describe a glass-forming system, see e.g. [2], [13]). In this latter more general case of several structural order parameters, the classical definition of fictive temperature as introduced by Tool [14] (and the similar definition of fictive pressure) becomes ambiguous (see also [15]) as far as for such cases no general definition of these parameters has been available so far. These circumstances are one of the reasons why we searched for and introduced in [13] a new model-independent thermodynamic definition of fictive pressure and temperature. This definition is valid generally independent on the systems considered and in particular valid for any arbitrary numbers of structural order parameters required for their description. The analysis of consequences of this new definition and, in particular, of the dependence of fictive pressure and temperature on temperature for different cooling and heating rates is the second main topic of the present analysis.

The paper is structured as follows: We start the analysis with (i.) a derivation of the Prigogine–Defay ratio including a detailed discussion of details and approximations omitted in [12] and (ii.) the course of fictive pressure and temperature is discussed in dependence on temperature for a large spectrum of cooling and heating rates. A summary of the results and discussion of future possible developments completes the paper.

Section snippets

Behavior of thermodynamic coefficients at glass transition: configurational specific heat

In order to describe glass-forming systems in thermodynamic terms, in addition, to the conventional thermodynamic state parameters one (ξ) or a set of structural order parameters ({ξ_i}) has to be introduced into the description. The values of these order parameters depend on prehistory or (for the assumed here constant cooling and heating rates, q) on the value of q. A typical example for such type of behavior is shown in Fig. 1 (for the details see [9]).

As one of the consequences, the values

Fictive temperature and fictive pressure

For the first time, the concept of a structural order parameter was introduced into glass science by Tool [14] in terms of fictive temperature. Already Davies and Jones [18] noted that instead of fictive temperature one can employ equivalently the concept of fictive pressure as the determining structural order parameter. This concept was further advanced in recent years by Landa et al. [19], [20] who developed an interpretation of the glass transition and accompanying effects employing the

Discussion

In the present and the preceding [9] paper, the behavior of a variety of thermodynamic properties of glass-forming melts and glasses is analyzed in dependence on cooling and heating rates varying its absolute value in a wide range. This renewed interest is partly caused by the fact that such experiments are feasible, now, analyzing glass-forming melts in the range of cooling and heating rates in the range between 10^− 4 K/s up to 10⁵ K/s [23], [24], partly, by the increased interest in the

Conclusions

In terms of the generic approach to relaxation, vitrification and devitrification processes, the thermodynamic properties of glass-forming melts in cooling and heating are analyzed theoretically varying the rates of change of temperature in very wide intervals. A detailed derivation of the theoretical value of the Prigogine–Defay ratio is outlined confirming previously obtained results. In particular it is reconfirmed that the Prigogine–Defay ratio has values larger than one also in the case

Acknowledgments

The present research was supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). The financial support is gratefully acknowledged. One of the authors, J. S., acknowledges the discussions with Drs. Jean-Luc Garden and Herve Guillou, Grenoble, France, at the 11-th Lähnwitzseminar on Calorimetry in June 2010 in Rostock.

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According to equilibrium thermodynamic theory, a liquid of zero configurational entropy would undergo an Ehrenfest type second order transition to another disordered structure at a temperature $T_{2}$ known as the ideal glass transition temperature, but glass formation prevents one from observing it. The heat capacity $C_{p}$ , isothermal compressibility, $κ_{T}$ , and isobaric thermal expansion coefficient, $α_{p}$ , change by an amount $Δ$ at the Ehrenfest temperature, $T_{EH} = T_{2}$ , and also at the glass transition temperature $T_{g}$ . The Ehrenfest ratio $R_{E}$ at $T_{EH} or T_{2}$ , $\frac{Δ C_{p, EH} Δ κ_{T, EH}}{T_{EH} V_{EH} {(Δ α_{p, EH})}^{2}} \equiv 1 .$ In contrast, experimentally determined Prigogine-Defay ratio, $\prod_{P - D}$ at $T_{g}$ , $\frac{Δ C_{p} Δ κ_{T}}{T V {(Δ α_{p})}^{2}} > 1$ , where $V$ is the volume at the respective temperatures. Angell and Sichina [Ann. N. Y. Acad. Sci. 279 (1976) 53-67] discussed how $\prod_{P - D}$ would change to $R_{E}$ if a liquid could be cooled slowly enough to reach $T_{2}$ . After critically reviewing the basic aspects of $R_{E}$ and of experimental and theoretical values of $\prod_{P - D}$ , we use generic variations of $Δ C_{p}$ , $Δ κ_{T}$ , $Δ α_{P}$ and $V$ with the fictive temperature, $T_{f}$ , and the extrapolated values of these quantities for o-terphenyl at $T = T_{2}$ , and calculate $R_{E}$ and $\prod_{P - D}$ . We find that $R_{E} < \prod_{P - D}$ , thus contributing to Angell-Sichina's discussion of the unique structure of the low temperature disordered phase. We then consider how $\prod_{P - D}$ would change if the entropy of a liquid were extrapolated such that it did not violate the third law of thermodynamics and there was no need for the conjecture of an Ehrenfest transition.
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In the commonly used technique of scanning calorimetry, one measures the heat capacity, $C_{p}$ , as a function of temperature $T$ on heating or cooling a material at a fixed rate. In the study of liquid-glass-liquid transition, a calorimetry scan is modelled in terms of a non-exponential parameter, $β$ , and non-linearity parameter, $x$ , which describe the shape of the $C_{p}$ -scan as well as the time- and temperature-dependence of structural dynamics. A liquid also becomes glass on isothermal pressurizing and the glass is expected to become liquid on depressurizing. Here we describe the formalism of a technique in which the volume, $V$ , may be measured as a liquid is isothermally pressurized at a fixed rate, $r$ = ${(d p / d t)}_{T}$ , to form glass, and also as the glass is isothermally depressurized at a fixed rate until it becomes liquid. We name it pressure scanning volumetry (PSV) and simulate the pressurizing and depressurizing scans of ${(d V / d p)}_{T}$ within the framework of non-exponential, non-linear structural relaxation. Hence we show how the features of simulated PSV scans change with change of, (i) the pressurizing and depressurizing rates, (ii) the parameters $β$ and $x,$ and (iii) the volume of activation for characteristic relaxation time, $Δ V^{*}$ . We also describe how the limiting fictive pressure, $p_{f}^{'}$ , can be determined from a PSV scan, and fit the PSV formalism to the available bulk modulus against $p$ plot of atactic poly(propylene) at 297 K [S.P. Andersson, O. Andersson, Int. J. Thermophys. 18 (1997) 845-864] to obtain $β$ , $x$ and $Δ V^{*}$ . In addition to being of academic interest, the formalism would be useful for modelling polymer extrusion processes. The study may also stimulate commercial development of equipment with computer-controlled pressure-change for use in studies of academic and technological significance.
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Thermodynamic analysis of the classical lattice-hole model of liquids
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The requested number of internal variables is classically discussed in relation to the Defay-Prigogine ratio [2,4–9]. Without being exhaustive, it is worth mentioning the single structural order parameter approaches of Tropin et al. [8,10–15] and the one of Garden et al. [9, 16]. In the following, these two approaches will be referred to by the acronyms SGT and GGRW respectively.
A complete thermodynamic analysis of the classical 2-parameter lattice-hole model of liquids is presented. To our knowledge, no such analysis was available before. It is shown that the model depicts a van der Waals like behavior. The calculated phase diagram features a coexistence line between a condensed and a gas phases ending at a critical point. The model is not able to describe a transition between two condensed phases such as melting. Model parametrization to simulate an archetypal fragile glass forming liquid, the ortho-terphenyl, reveals an only qualitative/semi-quantitative agreement with available experimental information. The extreme simplicity of this 2-parameter model restricts its ability to describe a real liquid. Taking into account the existence of a volume difference between a hole and a molecule, inserting a temperature and pressure dependencies of the physical parameters and allowing the energy parameter to depend on the hole concentration are the most promising modeling options.
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Kinetic criteria of vitrification and pressure-induced glass transition: dependence on the rate of change of pressure
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Kinetic criteria are formulated describing vitrification and devitrification of liquids in response to variations of the rate of change of external pressure. General analytical relations are derived allowing one to determine the pressure at the glass transition, p_g, and the width of the glass transition interval, δp_g, in dependence on the rate of change of pressure. In addition, a direct correlation between the pressure at the glass transition and the width of the glass transition interval is established. It is shown that the ratio δp_g/p_g is determined generally by an appropriately defined fragility index being a measure of the steepness of change of relaxation time at the glass transition, p_g, relevant for the selected rate of change of pressure. Similar relations are derived also for dynamic glass transitions in response to oscillations of pressure. All these dependencies are widely determined by the Maxwellian relaxation time of the liquid under consideration. However, they are shown to hold independently of any particular assumptions concerning the particular type of dependence of viscosity and/or relaxation time on pressure and the set of structural order-parameters describing the effect of deviations from (metastable) thermodynamic equilibrium. The results are illustrated here utilizing a recently developed new relation for the pressure dependence of the Newtonian viscosity, respectively, the relation for the pressure dependence of the relaxation time corresponding to it. Similarly to vitrification caused by variations of temperature, the pressure at the glass transition and the width of the glass transition range are shown to be logarithmic functions of the rate of change of pressure. The analytical results are confirmed by numerical computations employing a simple statistical model of glass-forming liquids.

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Journal of Non-Crystalline Solids

Abstract

Research Highlights

Introduction

Section snippets

Behavior of thermodynamic coefficients at glass transition: configurational specific heat

Fictive temperature and fictive pressure

Discussion

Conclusions

Acknowledgments

References (25)

Über den Zustand der unterkühlten Flüssigkeiten und Gläser

Z. Anorg. Allg. Chem.

The vitreous state: thermodynamics, structure, rheology, and crystallization

Thermodynamics and kinetics of the glass transition: a generic geometric approach

J. Chem. Phys.

Generic phenomenology of vitrification and relaxation and the Kohlrausch and Maxwell equations, Proc. XIX-th International Congress on Glass

Phys. Chem. Glasses

Phenomenological theories of glass transition: classical approaches, new solutions and perspectives

J. Non-Cryst. Solids

Aggregatzustände, Leopold Voss Verlag, Leipzig, 1923; Der Glaszustand

Freezing-in and production of entropy in vitrification

J. Chem. Phys.

Thermodynamic theory of affinity

Chemical Thermodynamics

The Prigogine–Defay ratio revisited

J. Chem. Phys.

Cited by (34)

Prigogine-Defay ratio and its change with fictive temperature approaching the ideal glass transition

Pressure scanning volumetry

Understanding the heterogeneous kinetics of Al nanoparticles by simulations method

Thermodynamic analysis of the classical lattice-hole model of liquids

Molecular dynamics factors affecting on the structure, phase transition of Al bulk

Kinetic criteria of vitrification and pressure-induced glass transition: dependence on the rate of change of pressure