Random mixing theory

The random mixing theory and mixing processes have been extensively explored (2). Randomization requires equally sized and weighted particles, with little or no surface effects, showing no cohesion or interparticle interaction, to achieve the best results it cannot be applied to all practical mixing situations, especially where cohesive or interacting particles are mixed. 

Making use of the random mixing theory, the canonical ensemble partition function becomes... 

Even though the random-mixing theory played an important role in the development of mixture theories, its predictions are not reliable due to its unrealistic physical basis (random assignment of molecules). However, the one-fluid concept, which states that the properties of a mixture can somehow be equated to those of a hypothetical pure-fluid whose properties can be evaluated from corresponding-states, has persisted and forms the basis for what are currently the most accurate corresponding-states models for mixtures. 

On the other hand, upon closer examination even the copper-gold solid solutions evince serious discrepancies with the quasichemical theory. There is a composition range where the entropy of solution is larger than that for random mixing (see Fig. 1) where... 

By using the liquid lattice approach to treat the random mixing of a disoriented polymer and a solvent, the so-called Flory-Huggins theory is often used to correlate the penetrant activity and the composition of the solution ... 

Taking into account the modes in which the water can be sorbed in the resin, different models should be considered to describe the overall process. First, the ordinary dissolution of a substance in the polymer may be described by the Flory-Huggins theory which treats the random mixing of an unoriented polymer and a solvent by using the liquid lattice approach. If as is the penetrant external activity, vp the polymer volume fraction and the solvent-polymer interaction parameter, the relationship relating these variables in the case of polymer of infinite molecular weight is as follows ... 

According to Hill (8), due to the small correction for excess entropy Ase, "the approximation of random mixing can appropriately be introduced, for molecules of like size, in lattice or cell solution theories that are otherwise fairly sophisticated."... 

In systems with specific interactions random mixing cannot be assumed. Hence, the thermodynamic theories traditionally used to interpret ternary system properties, such as the Flory - Huggins formalism or the equation of state theory of FI ory, are expected not to apply to such systems. 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

The second major difference between the Panayiotou-Vera and the Sanchez-Lacombe theories is that Sanchez and Lacombe assumed that a random mixing combinatorial was sufficient to describe the fluid. Panayiotou and Vera developed equations for both pure components and mixtures that correct for nonrandom mixing arising from the interaction energies between molecules. The Panayiotou-Vera equation of state in reduced variables is... 

The heat of mixing could be temperature dependent. Though favourable at low temperatures it may become less so at higher temperatures. This could arise from a specific interaction which tends to dissociate at higher temperatures. There is experimental evidence that this is an important factor but as yet no theory describes it. Certain theories of non-random mixing do however have some of the features of such an effect. . 

It is also appropriate to return briefly to the work by Poser and Sanchez of which the theory just discussed was a variant. The liquid consists of r-mers and vacancies, randomly mixed, with next neighbour interactions. This energy, together with the lattice volume and r define the fluid. For the Interface Cahn-... 

In the model employed in this paragraph, the interaction curves differ only in the depth of the minimum, while the equilibrium distances or effective molecular radii are the same for both components. The extension of the theory to molecules of different size, where nd s different, has been made recently.ff Solutions in which one kind of molecule can be regarded as an r-mer of the other have also been considered. The effect of non-random mixing has been shown to be relatively unimportant, especially for non-polar molecules. l ... 

Transport properties of ionomer blends, characterized by a given type of spheroids and the aspect ratio, e/a, can now be analyzed by the effective medium theory discussed in the previous section. In this theory, the two phases are assumed randomly mixed and the probability of finding each phase is equal to its volume fraction f.. The effective conductivity, o, of the composite for either Na+ of OH ions is given by (15) ... 

Theories more recent than the simple Langmuir (random-mixing) adsorption statistics and usually regarded as more realistic can handle the entropic terms in Equations 10 and 12. Two of these, which have been used for adsorbed ions at the mercury/electrolyte interface, are based on Flory-Huggins (12,13,25, 26) and scaled-particle statistics (31,... 

The spinodal, binodal and critical point Equations derived on the basis of this theory will be discussed later. When the theory has been tested it has been found to describe the properties of polymer blends much better than the classical lattice theories 17 1B). It is more successful in interpreting the excess properties of mixtures with dispersion or weak attraction forces. In the case of mixtures with a strong specific interaction it suffers from the results of the random mixing assumption. The excess volumes observed by Shih and Flory, 9, for C6H6-PDMS mixtures are considerably different from those predicted by the theory and this cannot be resolved by reasonable alterations of any adjustable parameter. Hamada et al.20), however, have shown that the theory of Flory and his co-workers can be largely improved by using the number of external degrees of freedom for the mixture as ... 

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