Prigogine-Defay Ratio Research Articles

Thermodynamics of the glassy state with multiple order parameters

Experimental data, both old and more recent, indicate the need for more than a single order parameter for adequate description of glassy materials. The thermodynamics of the glassy state with multiple order parameters, previously discussed by others, has been examined and developed further systematically. The result shows that the Prigogine-Defay ratio can be defined at every point on the (T,p) plane near the glass transition line and is generally greater than unity. One has to define a number of glass transition lines (Tg versus p), depending on which of the order parameters are assumed frozen-in at the glass transition. Even for a definition of the glass transition line operationally in terms of the relaxation time, one has to specify not only the time scale of the experiment but also the particular property observed and the mode by which the system is disturbed from equilibrium. A one-dimensional lattice gas model is utilized to obtain an expression for the Gibbs free energy as a function of T, p, and two order parameters, and this is then used to illustrate and test out the concepts and the thermodynamic relations derived here.

Comments on a paper entitled ’’Prigogine–Defay ratio for systems with more than one order-parameter’’

Comments on a paper entitled ’’Prigogine–Defay ratio for systems with more than one order-parameter’’

Prigogine–Defay ratio for systems with more than one order parameter

The Prigogine–Defay ratio, Π≡ΔCp Δκ/TV (Δα)2, has been derived for systems whose states need to be specified in terms of a number of order parameters z by considering the conditions of tangency of the equilibrium free energy surface and the free energy surface for constant values of the order parameters. It is shown that Π?1. The equality applies for the case of a single order parameter or for the case of more than one order parameter which satisfies the constraint (∂V/∂zi)/ (∂S/∂zi) = (∂V/∂zk)/(∂S/∂zk). The inequality applies for the case of more than one order parameter which does not satisfy this constraint. The relevance of the Prigogine–Defay ratio to the kinetics of the structural relaxation in glass is discussed.

Degenerate Excited State in the Structure of B2O3

A degenerate excited state is essential to the description of structural rearrangements which account for the relaxational heat capacity, thermal expansion, and compressibility of vitreous B2O3. These relaxational responses cannot be simultaneously described by a two-state model, or by any model involving a single ordering parameter, as shown by the magnitude of the Prigogine—Defay ratio (π≠1). The apparent two-state behavior of the temperature derivatives, Δα and Δcp, and failure of a two-state model to predict a sufficiently large compressibility (pressure derivative), suggest an analysis of structural rearrangements in vitreous B2O3 in terms of a single ground state and a twofold degenerate excited state which is characterized by different volumes.

Two remarks concerning Dr. Finck's reply

The glass transition is theoretically described in terms of a generic approach employing de Donder’s structural order parameter method, appropriate expressions for the relaxation behavior of glass-forming systems and an order parameter related to the free volume of the system. Employing this approach the behavior of a variety of thermodynamic quantities describing glass-forming systems in vitrification and devitrification processes is interpreted theoretically. Particular attention is devoted here to the estimation of the value of the Prigogine–Defay ratio Π and the analysis of the behavior of the entropy in vitrification. In contrast to previous findings it is shown that treating vitrification correctly as proceeding in some interval of temperature and/or pressure the use of only one structural order parameter is sufficient to explain the generally obtained experimentally results Π > 1. In addition, an analysis of some alternative approaches of introduction of different order parameters is performed and relationships between different structural order parameters are analyzed. It is shown how the concept of an internal (fictive) pressure can be introduced straightforwardly in terms of the generic approach described. Finally, within the framework of the generic approach utilized here, a general model-independent definition of internal (fictive) pressure and fictive temperature is given for the case that an arbitrary number of structural order parameters is required for the description and some further consequences are discussed briefly.