Activity coefficient, segment-molar

Finally, segment-molar activity coefficients Ta and fa, are introduced to describe all deviations from the Flory-Huggins mixture (with x = 0). Within the continuous treatment the segment-mole fraction Vb, will be replaced by [29]... 

Further treatment depends on how the distribution function WB(rB, Y ) influences the segment-molar activity coefficients f and fg and the segment-molar excess Gibbs free energy G . In the most general case, G may be a functional of WjfrB, Yb) itself, and the unknown scalar ipg and the unknown distribution function WB(rB, Yg) are coupled in complicated ways, because QB(rB, Yg) in Eq. (39) contains depending on the unknown distribution function Wg(rg, Yg). 

No special reference is made here to the methods which calculate the segment-molar activity coefficients for the general case treated in this chapter, because the general expressions are given by Eqs. (18) and (19). All special details depend on the G -model to be applied and the details in which this model is influenced by moments of the distribution function. The principal way of taking different moments into account is given by Eqs. (54)-(61), which may analogously be extended to the case of more moments. 

With Eqs. (62), (64) and (81), the segment-molar activity coefficients are given by... 

The segment-molar activity coefficients now read (generalizing Eqs. (82) and (83))... 

The segment-molar activity coefficients have to be expressed by two terms. 

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. 

Often r is approximated by the ratio of the molar volumes of pure liquid polymer and pure solvent. The segment fractions of solvent and polymer are then equal to the volume fractions of solvent and polymer, and p. Equation (27) is derived using many assumptions and approximations (for a discussion, see Ref. 8), but on the basis of this rather simple expression many features of the phase behavior of polymer solutions can be explained. The expressions for the mole fraction based activity coefficients of solvent and polymer are Eqs. (30) and (31), respectively. 

The influence of the chemical composition in (5) can be derived using a simplified versicai of Barker s lattice theory [55], The most important consequence of (5) is the fact that the segment-molar excess Gibbs free energy of mixing and, hence, the activity coefficients depend only oti the average value (yw) of the distribution function, but not on the distribution function itself. In continuous thermod5mamics, the phase equilibrium conditions read ... 

The activity coefficients in (9) can be derived using standard thermodynamics in combination with (5). The replacement of the segment-molar chemical potential of the copolymer species in (7) according (4) leads to ... 

Solution activity data obtained by osmometry on dilute solutions showed that the second virial coefficient is dependent on molar mass, contradicting the Flory-Huggins theory. These problems arise from the mean-field assumption used to place the segments in the lattice. In dilute solutions, the polymer molecules are well separated and the concentration of segments is highly non-uniform. Several scaling laws were therefore developed for dilute (c < c is the polymer concentration in the solution, c is the threshold concentration for molecular overlap) and semi-dilute (c > c ) solutions. In a good solvent the threshold concentration is related to molar mass as follows ... 

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