Athermal entropy of mixing

Changes in the flexibility of polymer coils owing to concentration - variation may effect the entropy. Huggins [21] introduced two corrections for the athermal entropy of mixing, that take into account the influence a second polymer has on the stiffness of the other polymer... 

Chain Flexibility. Up to here we only have discussed flexible chains. Changes in the flexibility of polymer coils due to concentration variations may effect the orientation energy of the system. Huggins [21] introduced two corrections for the athermal entropy of mixing, that take into account the influence a second polymer has on the stiffness of the other polymer chain and vice versa. From this model one can conclude that a stiff chain will be enhanced in its orientational possibilities when surrounded by an increasing amount of a flexible polymer while the reverse will occur when a polymer is embedded in stiffer polymer surroundings. 

Since the 0 s are fractions, the logarithms in Eq. (8.38) are less than unity and AGj is negative for all concentrations. In the case of athermal mixtures entropy considerations alone are sufficient to account for polymer-solvent miscibility at all concentrations. Exactly the same is true for ideal solutions. As a matter of fact, it is possible to regard the expressions for AS and AGj for ideal solutions as special cases of Eqs. (8.37) and (8.38) for the situation where n happens to equal unity. The following example compares values for ASj for ideal and Flory-Huggins solutions to examine quantitatively the effect of variations in n on the entropy of mixing. 

Combine your information to make a single, rough plot of AS i /R versus At what weight fraction is the difference in the entropy of mixing between an ideal and athermal polymer solution the greatest for this system ... 

In another approximation athermal mixtures are discussed with a zero heat of mixing just as ideal solutions, but with a different entropy of mixing, owing to a difference in size of the two kinds of molecules. The most simple expression is that derived by Flory ... 

Finally we may observe that we have defined perfect solutions through equation (20.1) for the chemical potentials, and from this we have established the properties discussed in this paragraph. Conversely, for a solution to be perfect, all these properties must be satisfied simultaneously. Thus it is not sufficient that the mixture can be made without heat effect, and without change in volume. The entropy of mixing must also have the form (20.17). Indeed later on we shall discuss solutions (athermal solutions) for which the deviations from ideality arise entirely from the entropy term. 

Equation 8.22 is the expression for the combinatorial entropy of mixing of an athermal polymer solution, and comparison with Equation 8.7 shows that they are similar in form except for the fact that now the volume fraction is foimd to be the most convenient way of expressing the entropy change rather than the mole fraction used for small molecules. This change arises from the differences in size between the components, which would normally mean mole fractions close to unity for the solvent, especially when dilute solutions are being studied. 

Among compounds of low molecular weight and similar size athermal mixtures or solutions usually show an entropy of mixing amounting to —J In Ni] they behave like ideal solutions. This means, apparently, that in the absence of forces between the molecules of the two species, there are also no additional entropy effects. 

If, however, one component is large compared with the other, and especially if chain-like molecules are involved, athermal solutions can show anomalous entropies of mixing in the sense that the change of entropy is always more negative than —R In Ni, regardless of whether A i is less or more than zero. ... 

Because for most systems the entropy of mixing is small, attractive interactions between both components are needed to obtain a homogeneous mixed state. In the opposite case miscible polymer blends for which k 0 (no or weak interactions) are called athermal blends. 

The systematic study of athermal solutions was initiated by Fowler and Eushbrooke [40] who proposed the lattice model as the only one capable of providing the quantitative and explicit formulas for thermodynamic quantities, proving at the same time that, in contrast to previous assumptions, the athermal mixtures are not ideal. The entropy of mixing of these systems is not ideal. 

Now, if the influence of molecular shape upon entropy of mixing is heglected, one can calculate the deviation from ideality of athermal solutions due to differences in molecular size. 

It should be mentioned that Eq. (4.3) is only one of several possible thermodynamic routes to the entropy of mixing in the athermal blend. Another possible route is through the charging formula of Chandler used earlier in Eq. (3.9b) for the one-component polymer melt. 

The favorable effect on polyolefin miscibility of statistical segment length asymmetry due to the entropy contributions required for conformational adjustments has also been emphasized by Bates et al. [87]. In a series of papers. Bates and Fredrickson [88] attributed the miscibility of athermal or nearly athermal polymer mixtures mainly to these conformational asymmetries which contribute substantially to a nonlocal conformational excess entropy of mixing. The effect is exemplified for the amorphous polyethylene/poly-(ethylethylene) blend. Due to the fact that unperturbed PE and PEE molecules cannot be randomly interchanged, a positive excess free energy of mixing caused by nonlocal excess entropy contribution is anticipated by the authors. The effect of asymmetry on polymer miscibility is also supported by computer simulations, which suggest additional contributions due to entropy density differences of the pure polymeric phases [89]. 

In the same way as we defined a regular solution as one which has the same entropy of mixing as a perfect solution, we shall define an athermic solution as one which has the same enthalpy of mixing as a perfect solution - i.e. zero. Of course, its excess molar enthalpy is also null. 

An athermal solution is also characterized by an enthalpy of mixing equal to zero but its entropy of mixing is higher than the conformational entropy. 

It should be emphasized that this expression of the entropy of mixing is applicable only to athermic systems or to mixtures exhibiting only weak interactions between molecules—that is, solutions with low enthalpy of mixing. Deviations from ideality could arise in particular in the following situations, which will be... 

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