Solution, athermal models

The simplest type of solutions which exhibit non-randomness are those in which the non-randomness is attributable solely to geometric factors, i.e. it does not come from non-ideal energetic effects, which are assumed equal to zero. This is the model of an athermal solution, for which... 

The Qnsager model is only able to describe the phase behaviour of low concentrated solutions. A second drawback is that it is athermal, and thus is not able to describe temperature dependent phase transitions, and accordingly cannot describe the occurrence of a clearing temperature. 

Beginning with the present sub section we shall give a number of model elaborations. We will start with the Langmuir case, l.e. the simple situation that ideality prevails, all molecules are monatomic, have the same sizes and mix athermally in the solution. The adsorption Is localized. 

Tnteractions are not important. The dynamics on these intermediate scales (for r < t< Te) are described by the Rouse model with stress relaxation modulus similar to the Rouse result for unentangled solutions [Eq. (8.90) with the long time limit the Rouse time of an entanglement strand Tg]. At Te, the stress relaxation modulus has decayed to the plateau modulus Gg[kT per entanglement strand, Eq. [(9.37), see Fig. 9.9)]. The ratio of osmotic pressure and plateau modulus at any concentration in semidilute solution -in athermal solvents is proportional to the number of Kuhn monomers in ... 

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. 

Some typical results are shown in Figure 16.2 (LLE for PS/acetone), Table 16.4 (infinite dilution activity coefficient for PBMA solutions in a variety of solvents). Tables 16.5 and 16.6 (activity coefficients of low- and heavy-molecular-weight alkanes in asymmetric athermal-alkane solutions). Table 16.7 (VLB for ternary polymer-solvent solutions). Figure 16.6 (VLB calculations for systems containing the commercial epoxy resin Araldit), and Table 16.8 (comparison of LLE results from various thermodynamic models). 

The activities of alkane solvents in either alkane or athermal polymer (PE, PIB) solutions are very satisfactorily predicted (much better than the Entropic-FV formula) by some of the modified equations cited above (Chain-EV, p-EV, R-UNIEAC). However, these models cannot be extended to multicomponent systems. " This is apparently a serious limitation for these models. 

In the FH formula, the characteristic volume V can be the molar volume or a specific hard-core volume of the molecule such as the van der Waals volume Vw. The last of the above equations is the Entropic-FV formula, discussed earlier (Equation 16.51). As shown, in the Entropic-FV model the van der Waals volume is used, as a measure of the molecular volume with increments by Bondi. The Entropic-FV formula was shown to work satisfactorily for athermal polymer solutions and offers a substantial improvement over the FH term. The van der Waals repulsive term and the Entropic-FV form are functionally identical. 

Hildebrand and Rotariu [14] have considered differences in heat content, entro])v and activity and classified solutions as ideal, regular, athermal, associated and solvated. Despite much fundamental work the theory of binary liquid mixtures is still e.ssentiaUy unsatisfactory as can be seen from the. systematic treatment of binar> mi.Ktures by Mauser-Kortiim [15]. The thermodynamics of mixtures is presented most instructively in the books of Mannchen [16] and Schuberth [17]. Bittrich et al. [17a] give an account of model calculations concerning thermophysical properties of juire and mixed fluids. 

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. 

This simple model accommodates the existence of rod-like structures needed for liquid crystal formation, without recourse to anisotropy of individual molecules. The secretions are biphasic (mixed isotropic/liquid crystalline) over a narrow concentration range implying that they have the thermodynamic characteristics typical of an athermal liquid crystalline solution (Figure 12.10). 

At the end of this chapter the calculated and exjKjrimental (chromatographic) data are compared this is preceded by a presentation of some important models which have been put forward for regular solutions and athermal solutions, together with the theories of contact points and segments, perturbation and Plory s equation of state. It should be mentioned that only theories verified by chromatographic data are considered. 

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. 

Chang [41] started up from lattice models and applied the Bethe method to give a statistical solution for some mixtures of monomers and dimers Miller [42, 43] applied the same procedure for mixtures of monomers and trimers. Later Huggins [44—47] extended the method to various types of athermal mixtures. The lattice model proposed assumes that monomeric molecules may take one single lattice site while those consisting of r monomeric units and referred to as r-mers may take r sites. 

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 ... 

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

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... 

Entropic-FV (Equation 3.13) have been proposed ° and they may serve as the basis for future developments resulting in even better activity coefficient models for polymer solutions. A rigorous test for newly developed FV expressions is provided by athermal alkane systems, especially the activity coefficients of heavy alkanes in short-chain ones. 

Gaussian thread limit for N—For many physical problems (e.g., polymer solutions and melts, liquid-vapor equilibria, and thermal polymer blends and block copolymers), the Gaussian thread model has been shown to be reliable in the sense that it is qualitatively consistent with many aspects of the behavior predicted by numerical PRISM for more realistic semiflexible, nonzero thickness chain models. However, there are classes of physical problems where this is not the case. The athermal stiffness blend in certain regions of parameter space is one case, both in... 

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