Helmholtz energy of mixing

Sum of the effects of all intermolecular interactions between polymer and solvent molecules in solution that are reflected in the values of the Gibbs and Helmholtz energies of mixing. 

Note The excluded volume of a segment depends on the Gibbs and Helmholtz energies of mixing of solvent and polymer, i.e., on the thermodynamic quality of the solvent, and is not a measure of the geometrical volume of that segment. 

According to Gibbs-Helmholtz relationship, the Helmholtz energy of mixing of system can be obtained by integrating the internal energy... 

The second term on the right-hand side of Equation (1) is the residual Helmholtz energy of mixing for an Ising mixture (AmixFrIsing). Its expression will be discussed below. 

Residual Helmholtz energy of mixing for multicomponent Ising mixture... 

For a binary Ising lattice, we introduced a nonrandom factor that was observed from simulation to have a linear relation with composition. The characteristic parameter of the linear relation was found by combining a series expansion and the infinite dilution properties. On this basis, an accurate expression for the Helmholtz energy of mixing... 

Equation (12) can be used to calculate the residual Helmholtz energy of mixing (AmixFrIsing) of monomer mixture of polymer system in step 2 of Figure 1. However, the mole fraction x, should be replaced by volume fraction ,. We have... 

Finally, by substituting Equations (7), (13), and (14) into Equation (7), we obtain the Helmholtz energy of mixing for multicomponent polymer systems... 

To extend a close-packed lattice model Equation (15) to a lattice fluid model, we adopt a two-step process as shown in Figure 15 to establish an EOS (Hu et al., 1992). In the first step, pure chain molecules at close-packed lattice are mixed to form a close-packed mixture. In the second step, the close-packed mixture is mixed with N0 holes to form an expanded realistic system with volume V at temperature T and pressure p. According the two-step process, the Helmholtz energy of mixing can be expressed as... 

Figure 4.3.1. The excess Gibbs and Helmholtz energies of mixing for the methanol and benzene binary system at 373 K calculated u ith the Wong-Sandler fWS) mixing rule and the PRSV equation of state at 1 and 1000 bar.

We can now calculate the Helmholtz energy of mixing and the Gibbs energy of mixing, which are equal because the mixing volume is null, by ... 

---