Random mixing approximation

The calculation of Zconf makes use of the random mixing approximation for the fully disordered state. Several authors [23-27] have reported improved estimates of Zconf that take into account the effect of local ordering at low temperatures however, the resulting improvement in the prediction of the... 

It is now well known (4) that the so-called random mixing approximation, which is based on the Lennard-Jones potential, greatly exaggerates the effect of size differences of the molecules on Since the mixtures are random in all the models considered in this chapter, it is more appropriate to call that in which u(r) varies with (cr/r)6 the Lennard-Jones (LJ) approximation. 

An alternative derivation of Eq. (16), based on the amphoteric reactions of the surface hydroxide groups, given as Eq. (1), has been proposed by Levine and Smith [16]. Using the random mixing approximation for the distribution of the three types of surface sites [i.e., neutral, positive, and negative sites, as shown in Eq. (1)], these authors were able to derive an expression for the dependence of cm on the oxide surface. The derivation has also been outlined by Smith [2], and the modified Nemst equation can be written as... 

In the spirit of the local random mixing approximation, we neglect fluctuations and determine the counterion charge density through a functional minimization of the free energy, subject to the constraint of charge conservation. We thus minimize the grand potential, G ... 

It should be noted that the probability of a vacant coordination site surrounding 7th monomer is pij = 1 —(77 + i)/N, which is on the basis of assumpticm two, i.e. the so-called random-mixing approximation. On this point, FlOTy and Huggins made different treatments. Let s consider the 7th monomer that has been put into a previously vacant site, the (7 + l)th monomer has to be put into a vacant coordination site surrounding the previously vacant site. Therefore, should be the fraction of two consecutively connected vacant sites in the total pairs of two neighboring sites containing one vacant site. The total vacant sites are N—rj—i, and their total coordination number is q N—rj—i), each with the vacant probability 1 (ry + i) N, so the total number of two consecutively connected vacant sites is... 

Apparently, the random-mixing approximation in the classical lattice statistical theory is not applicable to the dilute solutions of polymers. In 1950, Flory and Krigbaum treated the suspended polymer coils in dilute solutions as rigid spheres with an effective excluded volume u (Flory and Krigbaum 1950). The combinatorial entropy depends upon the total number of ways Q to put N2 spheres in a big volume V (supposed in the unit of u). The number of ways to put the first sphere is V, to put the second sphere is V—u, to put the third V—2m, and so on so forth, then the total number of ways... 

Buta et al. (2001) tested the Monte Carlo approach for the lattice cluster theory to derive the thermodynamic properties of binary polymer blends. They considered the two polymers to have the same polymerization indices, i.e., M = 40, 50, or 100. The results confirm that this lattice cluster theory had a higher accuracy compared to the Flory-Huggins theory and the Guggenheim s random mixing approximation. However, some predictions for the specific heat were found to be inaccurate because of the low order cutoff of the high temperature perturbative expansion. 

Simplistic models of the type sketched above are able to reproduce, qualitatively, some of the main experimental results. (If one goes further than the random mixing approximation to study local correlations, then these include the characteristic peaked S(q) seen for microemulsions. ) Certainly these models can help us decide what are the important interactions to include in a more complete approach also they... 

In 8 we consider bri y critical solution phenomena. Here we use the relations derived in Ch. IX and X for the excess free energy, including the random mixing approximation. Iherefore the e q)ressions we give should lead to a critical temperature which is too high (cf. Ch. Ill, 3). 

We finally wish to mention that as we have used the random mixing approximation, the critical temperatures estimated by (12.8.1), (12.8.2) and (12.8.3) should be too high (probably by 20 till 30 %). 

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