Solution, athermal general

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. 

In the general case of a solute B in a plastic matrix P the parameter AP is a function of temperature and produces a more or less significant deviation from the activation energy, EA - 86.923 kJ mol-1 in reference Eq. (6-20). Consequently we can write Ap = Ap -Xp/T, with the athermal, dimensionless number AP and the parameter Xp with the dimension of a temperature, respectively. Both values, AP and Xp can be obtained from two diffusion measurements at different temperatures, using a reference solute B in matrix P (see Chapter 15). 

This assumption is based on the fact that the polymer-solvent interaction parameter [see Eq. (8)] of the tributyrin-cellulose tributyrate system, as evaluated from melting-point depressions, is nearly zero at about 100° C [Mandelkern and Flory (160)]. It does not follow, however, that the system is athermal, for the parameter generally involves an entropy contribution. Furthermore, the heat and entropy parts of this parameter vary with the concentration in a complicated way, especially in polar systems [see, for example, Takenaka (243) Zimm (22) Kurata (154)]. Thus it is extremely hazardous to predict dilute solution properties from concentrated solution properties such as the melting-point depression, at least on a highly quantitative level as in the present problem. 

Another problem of the combinatorial term of the model is that it predicts complete miscibility for athermal polymer solutions. This is not in agreement with the general observation of the LCST for even nonpolar (athermal) polymer solutions, as we mentioned previously. However, even the more recent combinatorial expressions, which are discussed in Section 16.4, suffer from this deficiency. 

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. 

In the phenomenological characterization of small deviations from SI solutions, the concepts of regular and athermal solutions were introduced. Normally, the theoretical treatment of these two cases was discussed within the lattice theories of solutions. Here, we discuss only the very general conditions for these two deviations to occur. First, when Pt ab does not depend on temperature, we can differentiate (6.19) with respect to T to obtain... 

In practical cases, we have two components which are quite similar (in the usual sense) but not similar (in the sense of the definition of the previous section) and therefore we expect (4.118) to be a valid approximation. In fact, first-order theories of regular and athermal solutions are special cases of (4.118), although their phenomenological characterization is more general [for more details, see Guggenheim (1952)]. 

Although they are partially ideal systems, the athermal solutions are completely different in character from regular solutions while regular solutions consist of molecules of similar shape and size, the constituents of athermal solutions are polymers whose molecular weights are much larger than those of the solvents, which are in general common organic molecules. 

This approach to athermal solutions may be generalized for i components. If is the number of molecules of i species, each taking ri sites, < is given by equation (3.52). 

The term general solution was introduced by Flory to characterize polymer solutions whose enthalpy of mixing is not zero. The model of general solutions borrows the formula of excess enthalpy from regular systems and the excess entropy from athermal solutions. Thus, a treatment of non-ideal polymer solutions arises which is simpler than the conventional methods applied to real systems this allows the deduction, on the basis of the known relationships, of the expressions of functions of deviation from ideality. Thus, for the activity coefficients of components in a binary system the following relations were established ... 

Athermal solutions, for example, polyolefins with alkanes offer a way of testing FV terms and numerous such investigations have been presented. EV models perform generally better than those that do not contain volume-dependent terms. ° Better FV terms than that of... 

Using the general expression of the excess entropy [A.2.33], for the excess molar entropy of an athermic solution, we obtain ... 

Hre butterfly patterns were observed at various tempera-mres and concentrations and for various solvents such as DOP, tydohexane, and diethyl malonate, which are theta solvents for PS at 35°C, as wdl as dibutyl phthalate and tricresyl phosphate, which are good solvents for PS. The patterns were also observed for semidilute solutions of polyethylene with paraflin as the solvent (athermal solution) and for sheared colloidal srrspensions. They are thus quite general for sheared dynamically asymmetric systems. The butterfly pattern was also formd for PS/PVME mixtures by Mani et and Gerard et and for other polymer mixtures. ... 

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