Interaction Change Upon Mixing

Af/njix depends on the interaction through x- A system with the same x has the same AU j. For instance, a mixture with epp = ej and Sps = e s = 0 is thermodynamically equivalent to a mixture with epp = ess = 0 and eps = -ej2. 

Contact Energy Probability Before Mixing Probability After Mixing 

A solution with = 0 is called an athermal solution. There is no difference between the P-S contact energy and the average energy for the P-P and S-S contacts. In the athermal solution, = 0 regardless of / . We can regard that 

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The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

M = 10 g mol , with 1 mol of cyclohexane at 34 °C Note that 34 °C is the -temperature for a polystyrene solution in cyclohexane (Flory interaction parameter x = 1 /2). The molar volume of polystyrene is Vps = 9.5 X 10 cm mol , the molar volume of cyclohexane is Vcyc= 108 cm mol . Assume no volume change upon mixing and assume that the volume of one solvent molecule is the lattice site volume Vq. 

It is to note that this simple equation was deduced assuming a random mixing process, a volume change upon mixing that vanishes, and an interaction parameter xi,2 independent of composition, among others [58, 59]. In spite of these simplification criteria, and others of subtle nature, the equation gives a qualitative insight into the nature of polymer solutions. 

Thermodynamic descriptions of polymer systems are usually based on a rigid lattice model. The relevant expression for the free energy change upon mixing, aG, was published in 1941 indepentently by Staverman and Van Santen, Huggins and Flory. In the usual notation (Flory [16]) the symbol x is used to express the interaction function. We prefer to use g((p) as interaction parameter, where x Ifg does... 

Following our prior nomenclature, dispersion would dictate a negative interaction energy change (upon mixing), which corresponds to a positive interfacial tension difference (yas Yps)- For an apolar (y 0) alkyl surfactant (e.g., dode-cane to nonadecane, y 26 mJ/m ) used to organically modify a typical silicate (e.g., montmorillonite, with y 66 mJ/m, y + 0.7 mJ/m, and yf 36 mJ/m ), miscibility would be achieved with any polymer for which... 

An important excess property is the excess Gibbs energy GE. Many models have been developed to describe and predict GE from the properties of the molecules in the mixture and their mutual interactions. GE models often refer to the condensed state, the solid and liquid phases. In case significant changes in the volume take place upon mixing, or separation, the Helmholtz energy A, defined as... 

The ideal solution assumes equal strength of self- and cross-interactions between components. When this is not the case, the solution deviates from ideal behavior. Deviations are simple to detect upon mixing, nonideal solutions exhibit volume changes (expansion or contraction) and exhibit heat effects that can be measured. Such deviations are quantified via the excess properties. An important new property that we encounter in this chapter is the activity coefficient. It is related to the excess Gibbs free energy and is central to the calculation of the phase diagram. 

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