Non-random mixing

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. . 

There has been an attempt by Renuncio and Prausnitz to introduce a partition function by modifying the energy of interaction to take into account the non-random mixing of two components. They introduced two site fractions between segment i and j as 0, and 0. where... 

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 ... 

The non-random mixing term containing (ArG°) can cause an S-shaped character of the In Yrix versus xg+xy-plot. For instance, when xg+ +xx, the In yay curve will have a positive contribution from this term up to xg+ = 0.333 and negative at Xg+ > 0.333. An example for this behavior is the LiF-KCl system. 

In binary solutions - in the case under consideration, water sorbed in a polymer - non-random mixing is also described as clustering. For this case, the cluster size is rarely, if ever, sufficient to produce visual opacity, and the evidence for the phenomenon is found in peculiarities of the sorption isotherm. 

The enhancement number Is a measure, then, of the extent to which the sorption of water Is Increased by the abnormalities of the process which result from non-random mixing. Another index Is available from the calculation of the cluster size, based on the Zlimn-Lundberg concepts. 

The theories based on the equation of state are more versatile. The model developed by Simha and many of his collaborators is most useful. By contrast with the H-F theory it leads to two binary interaction parameters, one energetic the other volumetric, that are constant in the full range of independent variables. Furthermore, it has been found that the numerical values of these two parameters can be approximated by the geometric and algebraic averages, respectively. The non-random mixing can easily be incorporated into the theory. 

There are many papers in the literature that applied the Prigogine-Flory-Patterson theory to polymer solutions as well as to low-molecular mixtures. Various modifications and improvements were suggested by many authors. Sugamiya introduced polar terms by adding dipole-dipole interactions. Brandani discussed effects of non-random mixing on the calculation of solvent activities. Kammer et al. added a parameter reflecting differences in segment size. Shiomi et assumed non-additivity of the number of external... 

The density and composition dependent correlation effects present in the R-MPY theory arise from non-random mixing intermolecular packing and... 

Huggins has developed a more genoal model which considers non-randomness of mixing but as yet this appears to have not been applied to polymers. It could no doubt be useful in analysing the wealth of data collected by Starkweather on non-random mixing (clustering) in polymer solutions. 

All models need some binary interaction parameters that have to be adjusted to some thermodynamic equihbriirm properties since these parameters are a priori not known (we will not discuss results from Monte Carlo simulations here). Binary parameters obtained from data of dilute polymer solutions as second virial coefficients are often different from those obtained from concentrated solutions. Distinguishing between intramolecular and intermolecular segment-segment interactions is not as important in concentrated solutions as it is in dilute solutions. Attempts to introduce local-composition and non-random-mix-ing approaches have been made for all the theories given above with more or less success. At least, they introduce additional parameters. More parameters may cause a higher flexibility of the model equations but leads often to physically senseless parameters that cause troubles when extrapolations may be necessary. Group-contribution concepts for binary interaction parameters in equation of state models can help to correlate parameter sets and also data of solutions within homologous series. 

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