Prigogine-Defay criterion

Yet another important property of fractals which distinguishes them from traditional Euclidean objects is that at least three dimensions have to be determined, namely, d, the dimension of the enveloping Euclidean space, df, the fractal (Hausdorff) dimension, and d the spectral (fraction) dimension, which characterises the object connectivity. [For Euclidean spaces, d = d = d this allows Euclidean objects to be regarded as a specific ( degenerate ) case of fractal objects. Below we shall repeatedly encounter this statement] [27]. This means that two fractal dimensions, d( and d are needed to describe the structure of a fractal object (for example, a polymer) even when the d value is fixed. This situation corresponds to the statement of non-equilibrium thermodynamics according to which at least two parameters of order are required to describe thermodynamically nonequilibrium solids (polymers), for which the Prigogine-Defay criterion is not met [28, 29]. 

The important argument in favor of fractal approach application is the usage of two order parameter values, which are necessary for correct description of polymer mediums structure and properties features. As it is known, solid phase polymers are thermodynamically nonequilibrium mediums, for which Prigogine-Defay criterion is not fulfilled, and therefore, two order parameters are required, as a minimum, for their structure description. In its turn, one order parameter is required for Euclidean object characterization (its Euclidean dimension d). In general case three parameters (dimensions) are necessary for fractal object correct description dimension of Euclidean space d, fractal (Hausdorff) object dimension d and its spectral (fraction)... 

The results stated above suppose that simbateness of the values of Oy and as a function of any parameter (temperature, crosslinking density and so on) can be an individual case only. Equation 6.13 demonstrates that the indicated simbateness realisation condition is the criterion = const, or, as follows from Equation 6.14, (p = const. In other words, invariance in polymer structure is the condition of the realisation of the change in the simbateness of Oy and . Let us be reminded that amorphous polymers are thermodynamically non-equilibrium solids, for the description of which two parameters of order, as a minimum, are required according to the principle of Prigogine-Defay. It is obvious that Equation 6.13 satisfies this principle, whereas the linear correlation Oy( ) does not [36]. 

In a recent paper, S. R. Brinkley, J. Chem. Phys., 14, 563, 686 (1946) has proposed another criterion for determining the number of independent components c —c-r starting from the stoichiometric coefficients of the elements in each compound. For a criticism of this method c/. I. Prigogine and R. Defay, J. Chem. Phys., 15, 614 (1947). For a discussion of the relation between Brinkley s criterion and that of Jouguet cf. A. Penecloux, C. R. 228, 1729 (1949). 

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