Athermic solutions

Evaluate ASj for ideal solutions and for athermal solutions of polymers having n values of 50, 100, and 500 by solving Eqs. (8.28) and (8.38) at regular intervals of mole fraction. Compare these calculated quantities by preparing a suitable plot of the results. 

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. 

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... 

Considering an athermal solution (/ mixing = 0 and G xing 3.8.3) composed of moles of AM and % moles of BM, where M is the common anionic group, we obtain... 

Figure 10.2D shows normalized distributions of elements A and B for various values of r. The more the forbidden zone (in terms of r sites) is large, the more precocious are deviations from ideality. Although the approximation to athermal solutions is not strictly valid in the case of trace elements in crystals, solid/hquid... 

We further assume that the blocks in the aggregates as well as in the continuous phase would have the lowest possible free energy if their coil dimensions would equal those of the corresponding homopolymers in athermic solutions. Conformation restrictions (8, 15, 16, 72), which are caused by a block end being connected with a second block that does not mix with the first one, are thus neglected. (At least one end of each block is situated in the interface.) For blocks of polystyrene and polybutadiene the following equations are valid (19, 24, 33, 56, 65) (with the end-to-end distance h = hi2) ... 

Thus, in the athermal limit the only difference between the equilibrium free energies of the solutions of separate rods and long chains of rods is due to the translational entropy term. Consequently, we can immediately conclude (analogously to Sect. 2) that the liquid-crystalline transition for the athermal solution of semiflexible chains takes place at 1/p. 

Fig. 2. The partition function Z for athermal solutions of hard rods having an axial ratio x = 100 shown as a function of the disorder index y for the volume fractions indicated. Calculations were carried out according to Eq. (C-1) of the Appendix with x = 0. The logarithm of the partition function relative to the state of perfect alignment (y = 1) is plotted on the ordinate...

The above calculations assume that the gross chain conformations are those of a random walk, which is the case in the melt. However, for an isolated polymer molecule in a dilute solution, the average conformation is affected by excluded-volume interactions between one part of the chain and another. Because the chain must avoid self-intersection, the conformation of the chain will be that of a self-avoiding walk, rather than a random walk, if the solution is athermal—that is, if all interactions are negligible except excluded volume. Self-avoiding walks lead, on average, to more expanded coil dimensions, since expanded configurations are less likely than contracted ones to lead to self-intersection of the chain. Thus, in an athermal solution, the mean-square end-to-end dimension of a polymer molecule scales as... 

Activity Coefficient at Infinite Dilution. A procedure similar to that employed by Wilson will be used here to obtain an expression for the excess Gibbs energy. Wilson started from the Flory and Huggins expression" 2 for the excess free energy of athermal solutions, but expressed the volume fractions in terms of local molar fractions. We selected Wilson s approach from a number of approaches, because it provided a better description of phase equilibria and because the interactions that count the most are the local one, but started from the more... 

The comparison of the experimental solubilities [4,5] of Ar, CH4, C2H6 and CsHg in the binary aqueous mixtures of PPG-400, PEG-200 and PEG-400 with the calculated ones is presented in Figs. 1-3 and Table 2. They show that Eq. (4) coupled with the Flory-Huggins equation, in which the interaction parameter x is used as an adjustable parameter, is very accurate. The Krichevsky equation (1) does not provide accurate predictions. While less accurate than Eq. (4), the simple Eq. (2) provides very satisfactory results without involving any adjustable parameters. It should be noted that Eq. (4) coupled with the Flory-Huggins equation with X (athermal solutions) does not involve any adjustable parameters and provides results comparable to those of Eq. (2). 

Eq. (4) combined with the Flory-Huggins equation with adjustable parameter x (the value of the parameter x is given in parenthesis). ° Eq. (4) combined with the Flory-Huggins equation with parameter = 0 (athermal solution). 

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. 

Conversely (24.40) means that A = 0 as is obvious from (24.4). The activity coefficients of athermal solutions are independent of temperature. 

REGULAR SOLUTIONS AND ATHERMAL SOLUTIONS 1. Regular Solutions. 

Statistical methods lead, in the case of athermal solutions, or where the heat of mixing is sufficiently small, to the following approximate expression for the excess entropy. ... 

Starting from (25.37) we can calculate the activity coefficients, since (24.12) gives for athermal solutions... 

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