Mixing, athermal

We concluded the last section with the observation that a polymer solution is expected to be nonideal on the grounds of entropy considerations alone. A nonzero value for AH would exacerbate the situation even further. We therefore begin our discussion of this problem by assuming a polymer-solvent system which shows athermal mixing. In the next section we shall extend the theory to include systems for which AH 9 0. The theory we shall examine in the next few sections was developed independently by Flory and Huggins and is known as the Flory-Huggins theory. 

Athermal mixing is expected in the case of 61 - 62. Since polymers generally decompose before evaporating, the definition 6 = (AUy/V°) is not useful for polymers. There are noncalorimetric methods for identifying athermal solutions, however, so the 6 value of a polymer is equated to that of the solvent for such a system to estimate the CED for the polymer. The fact that a range of 6 values is shown for the polymers in Table 8.2 indicates the margin of uncertainty associated with this approach. 

Linking this molecular model to observed bulk fluid PVT-composition behavior requires a calculation of the number of possible configurations (microstmctures) of a mixture. There is no exact method available to solve this combinatorial problem (28). ASOG assumes the athermal (no heat of mixing) FIory-Huggins equation for this purpose (118,170,171). UNIQUAC claims to have a formula that avoids this assumption, although some aspects of athermal mixing are still present in the model. 

Fig. 8 Theoretical liquid-liquid demixing curve (solid line) and the bulk melting temperature (dashed line) of a flexible-polymer blend with one component crystallizable and with athermal mixing. The chain lengths are uniform and are 128 units, the linear size of the cubic box is 64, and the occupation density is 0.9375 [86]...

Hence, the viscosity of the solution of high-molecular weight polymers substantially decreases on addition of the oligomer salt and the viscometrically determined average degree of association, N, should allow a reliable determination of the equilibrium constant, K, of the athermal mixing process. 

Problem 3.1 Based on the lattice model for low-molecular-weight mixtures, determine the effect of non-athermal mixing on Raoult s law. 

This expression for polymer solutions is seen to be very similar to the cor responding expression for ideal mixing, i.e., Eq. (3.29). The only difference is that for polymer solutions, the mole fractions in Eq. (3.29) are replaced by the corresponding volume fractions. It may be noted that Eq. (3.48) is a more general expression for athermal mixing and reduces to Eq. (3.29) when <7 = 1. 

The last term in Equation 8 is the athermal mixing of components 1 and 2 which differ in molecular size. In our case (2) (Figure 2) di > a2 thus, the contribution of this term to Alii is positive. However, if a reorientation of 1 occurs so that a < a2> this term may become negative. 

As suggested above, the character of atomic environments can be changed by athermal mixing in which the LL environment fraction can increase with plastic shear rate yP in proportion to the availability of the SL environments, i.e., 1 — as... 

Real solutions may deviate from ideality, particularly since the condition of athermal mixing is not usually encoimtered. The model can be modified to... 

Examining this phase diagram in more detail we consider athermal mixing, i.e. where xi is equal to zero. In this case there is no net interaction between the... 

Ideal solutions are mixtures of molecules (i) that are identical in size and (ii) for which the energies of like (i.e. 1-1 or 2-2) and unlike (i.e. 1-2) molecular interactions are equal. The latter condition leads to athermal mixing (i.e. A// = 0), which also means that there are no changes in the rotational, vibrational and translational entropies of the components upon mixing. Thus AS depends only upon the combinatorial (or configurational) entropy change, which is positive because the number of... 

Real solutions may deviate from ideality, particularly since the condition of athermal mixing is not usually encountered. The model can be modified to account for intermolecular interactions, which, within the lattice model, are of the first-neighbour-type. For a binary system, three kinds of interactions need to be considered [1,1], [2, 2], and [1, 2] contacts and so the process of dissolution is given by... 

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