Binary mixtures on a lattice

This expression accounts for the configurational entropy of an ideal binary mixture with identical molecular sizes, but not for that of a polymer solution, since polymer chains are large and flexible. For that case, more contributions arise from the chain conformational entropy, first considered by Meyer [19] and then derived by Huggins [20] and Flory [21]. In analogy with a nonreversing random walk on a lattice, the conformational contribution of polymer chains to the partition function is given by... 

We now extend the previous discussion of pure confined lattice fluids to binary (A-B) mixtures on a simple cubic lattice oi J f = nz sites, whose lattice constant is again i. We deviate from our previous notation (i.e., M = nx%r ) because we concentrate on chemically homogeneous substrates where n = n riy located in a plane at some fixed distance from the substrate, which are energetically equivalent. Moreover, our subsequent development will benefit notationally by replacing henceforth n by just z. 

Similarly to the description of real phase behavior of mixtures of low-molar-mass components, mixture models based on activity coefficients can be formulated. Whereas in the case of low-molar-mass components the models describe the deviations from an ideal mixture, the models for polymer solutions account for the deviations from an ideal-athermic mixture. As a starting point for the development of a model, all segments are placed on a lattice (Figure 10.3). Polymer chains will be arranged on lattice sites of equal size, where the number of occupied lattice sites depends on the segment number r. For a quasi-binary polymer solution, all other places are occupied with solvent segments. 

We consider a binary mixture containing Na molecules A and Nb molecules 5 on a lattice of N — Na + b) sites with a coordination number z. Each molecule has z closest neighbours and there are in all I zN pairs of closest neighbours. Such pairs are of three kinds namely AA, AB, BB. We shall denote their respective numbers by Naa, Nab, NbB Slimming all neighbours of the Na molecules, we have... 

Fig. 4.29 Binary mixture of linear polymers represented by paths on a lattice...

Phase Equilibrium (PE) Binary mixtures of a polymer in a single solvent phase-separate at various temperatures, Tsep, depending on the volmne fi-action (/12 of the polymer. The maximmn of the 7 sep=/(< 2) fiuiction is called the critical solution temperature Test-The experiment is repeated for a series of dilute solutions of polymers of the same constitution and configmation but of different molar mass. The relation between the eritieal solution temperature and the molar mass of the polymer is based on the Flory-Huggins lattice theory which predicts that... 

The Kumar equation of state (Kumar, 1986 Kumar et al., 1987) is a modification of the Panayiotou-Vera model that was developed to simplify the calculations for multicomponent mixtures. Since the Panayiotou-Vera equation is based on the lattice model with the quasichemical approach for the nonrandomness of the molecules in the mixture, the quasichemical expressions must be solved. For a binary system the quasichemical expressions reduce to one quadratic expression with one unknown, but the number of coupled... 

However, one-dimensional confined fluids with purely repulsive interactions can be expected to be only of limited usefulness, especially if one is interested in phase transitions that cannot occur in any one-dimensional system. In treating confined fluids in such a broader context, a key theoretical tool is the one usually referred to as mean-field theory. This powerful theory, by which the key problem of statistical thermodynamics, namely the computation of a partition function, becomes tractable, is introduced in Chapter 4 where we focus primarily on lattice models of confined pure fluids and their binary mixtures. In this chapter the emphasis is on features rendering confined fluids unique among other fluidic systems. One example in this context is the solid-like response of a confined fluid to an applied shear strain despite the absence of any solid-like structure of the fluid phase. 

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