Phase Heterogeneity and Criticality

As discussed in Section 10.3, equilibrium phase homogeneity is associated with existence of a null eigenvector t] of the full (c + 2)-dimensional metric matrix 

Let us now consider the case of a heterogeneous system of distinct phases a, (3. Each phase is associated with its own GD vector 

According to the positivity property (9.27c) of a Euclidean scalar property, a vector nr ) is considered zero if and only if it is of zero length, namely, 

Alternatively, for a column vector iq such as (11.141b), the equivalent condition is 

any vector satisfying M r = 0 ( null eigenvector of M) is a zero vector of Ms, even if some or all of its elements r/, are nonvanishing. Indeed, the trivial case of all r/j = 0 in (11.141b) is never considered (no system ). We may therefore write iq 0 to distinguish such a vector of zero length from a vector with all zero elements. 

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The new non-equilibrium thermodynamic theory of heterogeneous polymer systems [37] is aimed at giving a basis for an integrated description for the dynamics of dispersion and blending processes, structure formation, phase transition and critical phenomena. Our new concept is derived from these more general non-equilibrium thermodynamics and has been worked out on the basis of experiments mainly with conductive systems, plus some orienting and critical examples with non-con-ductive systems [72d]. The principal ideas of the new general non-equilibrium thermodynamical theory of multiphase polymer systems can be outlined as follows. 

The new nonequilibrium thermodynamic theory of heterogeneous polymer systems [7] is aimed at providing a basis for an integrated description of the dynamics of dispersion and blending processes, structure formation, phase transition, and critical phenomena. 

The EMMS model was first proposed for the hydrodynamics of concurrent-up particle-fluid two-phase flow. Though it is based on a rather simplified physical picture of the complex system (Li, 1987 Li and Kwauk, 1994), it harnesses the most intrinsic complexity in the system, the meso-scale heterogeneity, and this is why it allows better predictions to the critical phenomena in the system which is obscured in other seemingly more comprehensive models. 

In heterogeneous catalysis the catalyst is present as a phase distinct from the reaction mixture. The most important case is the catalytic action of certain solid surfaces on gas-phase and solution-phase reactions. A critical step in the production of sulfuric acid relies on a solid oxide of vanadium (V2O5) as catalyst. Many other solid catalysts are used in industrial processes. One of the best studied is the addition of hydrogen to ethylene to form ethane ... 

In a gas solid system, some adsorbate species are generally distributed between a solid surface and a gas phase. This is due to the fact that below critical temperature, all gases tend to be adsorbed on a solid as a result of the van der Waals interactions with the solid surface. This type of adsorption is called physical adsorption (physisorption). In this case, the important factors affecting adsorption include the magnitude and nature of adsorbent-adsorbate and adsorbate-adsorbate interactions. The degree of surface heterogeneity and the translational and internal degrees of freedom which the adsorbed molecules possess can also be important. 

We therefore believe that the Elovich equation may be used as a basis for a quantitative interpretation of rates of adsorption and desorption both from the single-gas phase, and from binary mixtures, and that it is a useful expression, like that for a Freundlich isotherm in equilibrium adsorption studies, as a means of describing the heterogeneous nature of many rate processes. We have not attempted to describe, in detail, the extensive experimental data that are available in the literature since this has been thoroughly and critically assessed up to 1960 by Low (5) who has written an excellent and comprehensive review in which he provides references to the original papers. 

Furthermore, the physical state of the second component at the time of matrix nucleation is of importance. It may be presumed that the mode of nucleation of a polymer in the presence of solidified domains of the second polymeric phase is heterogeneous, and therefore the nucleation rate should be higher than in the pure homopolymer. The effect of blending on the nucleation behavior is more subtie and complex in the presence of a molten second component. Factors such as miscibility, relative melt-viscosity, and inherent crystallizability all influence the formation of critical size nuclei [Nadkami and Jog, 1991]. 

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