Kinetic criteria of glass-formation, pressure dependence of the glass-transition temperature, and the Prigogine–Defay ratio
Introduction
As noted long ago by R. O. Davies and G. O. Jones [1] in connection with their theoretical analysis of the properties of glasses and the glass transition, “The central problem is to explain in molecular terms the way in which the glass differs from the liquid and the nature of the change from the glass to the equilibrium liquid. In view of the complexity of glass-forming substances we cannot hope for a detailed microscopic theory”. Of course, meanwhile significant advances in the statistical–mechanical modeling of glass-forming melts, glasses and the glass transition have been attained, however, a final solution of the problem mentioned by Davies and Jones has not been achieved yet.
In a complementary approach to studying glass transition and the properties of glasses, equilibrium and non-equilibrium thermodynamic methods have been employed widely (for a review c.f. e.g. [2], [3], [4]) to adequately describe these phenomena. As it is always the case in the application of thermodynamic methods, thermodynamics supplies us with the general macroscopic description of the phenomena under consideration. This thermodynamic description has to be supplemented and advanced in its further development by a microscopic interpretation in statistical-mechanical terms.
In order to apply the above mentioned methods to the analysis of vitrification and the specification of the properties of glass, the basic nature of these phenomena has to be clearly understood. Following the classical concepts as developed by Simon [5], Tammann [6] and many others, vitrification in the cooling of glass-forming melts is commonly interpreted as the transformation of a thermodynamically (meta)stable equilibrium system into a frozen-in, thermodynamically non-equilibrium system, the glass (c.f. e.g. also [1], [2], [7], [8]). Hereby it is assumed in a first approximation – suggested by F. Simon – that the transformation takes place at some well-defined discrete temperature, the glass transition temperature, Tg. Simon also noted already that the glass transition proceeds not at a sharp temperature value but over a certain temperature range. He even stated – with reference to Tammann and Kohlhaas [9] and Parks and Huffmann [10] – explicitly that by varying the cooling rate different glasses can be obtained. However, he considered both the width of this glass transition range and the effect of varying cooling rates on glass properties as small and, for this reason, as to be of minor importance [5]. Such reservations are today known to be not adequate, especially if wide ranges of cooling and heating rates are employed as meanwhile available (c.f. also Chapter 1 in [4]). The specification of the dependence of the glass transition temperature and the width of the glass-transition range on the rate of change of external control parameters and its consequences are one of the main topics of the present analysis (c.f. also [2], [11], [12]).
In particular, the glass transition is accompanied by qualitative changes of the response of the system to variations of the external control parameters like temperature and pressure allowing one to experimentally determine the glass transition range and the glass transition temperature. These qualitative changes of the response of the system are reflected by jumps of thermodynamic coefficients like specific heat, thermal expansion coefficient, and compressibility. An appropriate description of the magnitude of the jumps of these thermodynamic coefficients and their particular combination, known as Prigogine–Defay ratio [13], is, for this reason, one major problem in the theoretical analysis of the glass transition [14], [15], [16]. Following Prigogine and Defay [13], these qualitative changes of the response of the systems in glass transition can be understood as a consequence of the freezing-in of the configurational contributions to the thermodynamic coefficients. The existence of a distinct second phase and the knowledge of its properties are, to our point of view (c.f. however [16]), not required for the theoretical interpretation. A discussion of different theoretical approaches in the determination of the Prigogine–Defay ratio is the second of the main topics of the present analysis.
In more detail, the aim of the present paper is the following: As shown in preceding papers, thermodynamic considerations allow one to develop straightforwardly kinetic criteria for the specification of the glass transition temperature, its dependence e.g. on cooling and heating rates and on pressure [11], [12], the upper and lower boundaries and the width of the glass transition range [17]. These criteria are briefly reviewed here and applied to the discussion of a spectrum of problems in the description of vitrification and devitrification not covered in the previous analysis (Section 2). As an example, we consider here as a rule the glass transition resulting from changes of temperature at constant external pressure. However, the methods are also applicable if the glass transition occurs as a consequence of heating or of variations of other external control parameters, for example, of pressure (c.f. e.g. also [8], [12], [17], [18]). In the latter case, at least, for “normal liquids” whose viscosity increases with increasing pressure [19], a pressure increase may lead to a transformation into a glass similarly to cooling a liquid.
Section 3 is devoted to the analysis of theoretical approaches to the determination of the Prigogine–Defay ratio in glass transition. Following the analysis of Prigogine–Defay [13] and Davies and Jones [1], [7] it was widely believed that the experimentally determined values of the Prigogine–Defay (PD) ratio (being as a rule larger than one) can be theoretically explained only if more than one structural order-parameter is employed for the description of the glass-forming system. This point of view was questioned first in [14] demonstrating the possibility to predict theoretically values of the PD-ratio larger than one in glass transition utilizing merely one structural order-parameter. It is shown here that some recent attempts to question our point of view do not affect the basic result of the analysis performed in [14]. A summary of the conclusions completes the paper.
Section snippets
Formulation
As discussed in detail in Section 1, glass transition is a kinetic phenomenon, the transformation of a thermodynamic (metastable) equilibrium system into a frozen-in thermodynamically non-equilibrium system, a glass. Since glasses are non-equilibrium systems, they have to be described thermodynamically – following De Donder [20], Prigogine and Defay [13] – by the introduction of a set of f structural order-parameters, {ξi}, in addition to the conventional thermodynamic state parameters like
Definition
The glass transition, determined theoretically by above discussed kinetic criteria, can be detected experimentally searching for jumps of the thermodynamic coefficients. The theoretical interpretation of the magnitude of these jumps and their combinations – like the Prigogine–Defay (PD) ratio – is by this reason one of the major tasks in the theoretical analysis of the glass transition.
The PD-ratio in glass transition is determined in analogy to Ehrenfest's relations in second-order equilibrium
Conclusions
Particular kinetic criteria of glass transition, derived by different authors, are shown to be (partly approximate) special cases of a general criterion which can be derived in a model-independent way from basic thermodynamic considerations. According to this criterion, the glass transition proceeds when the characteristic time scale of relaxation of the system to thermodynamic equilibrium is of the same order of magnitude as the characteristic time scale of variation of the external
Acknowledgment
The present research was supported by a grant from the Deutsche Forschungsgemeinschaft (DFG: SCHM 937/16-1) and the Heisenberg-Landau program of the German Ministry for Science and Technology (BMBF). The financial support is gratefully acknowledged. J. W. P. S. acknowledges also the stimulating discussions with G. P. Johari.
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2019, Thermochimica ActaCitation Excerpt :They were mainly directed to the description of vitrification and devitrification in cooling and heating of liquids. An overview on existing different approaches of formulation of kinetic criteria of glass-formation was given in [10–12] as the starting point for the development of a general criterion of vitrification and devitrification (see also [2,3,25,31,32]). In the present paper, we will utilize this approach in order to derive relations for the glass transition pressure for vitrification and devitrification in dependence on the rate of change of external pressure.