Although the goal of this project was the development of an equation of state for real substances, the Perturbed-Chain SAFT model was first derived for chain molecules interacting with a square-well potential. This preliminary step was taken, since in the case of a square-well fluid, it is easy to compare modeling results to "exact" computer simulation results. The accuracy of the model theory can so be tested.

### Comparison to molecular simulation data of pure square-well chains

### Comparison to molecular simulation of data mixtures

Note: Results for the square-well chain fluid were obtained applying an analytical expression for ghc(m;x,r) (Chiew, 1991; Tang and Lu, 1996) in the following relation:

The other equations remain as summarized in the equation-pages, except, that di=si for a square-well potential.

Figure 1 and 2 show a comparison between molecular simulation data of square-well chains and the equation of state theory at reduced temperatures T*=T/(e/k)=1.5, 2 and 3 . For both chain-lengths, the theory is in close agreement to the simulation data.

Fig. 1: Compressibility factor of pure square-well dimers (m=2) at three reduced temperatures of T* = 1.5, 2, and 3.

The solid lines are from theory. Symbols are Monte Carlo simulation results of Tavares et al., 1995.

Fig. 2: Compressibility factor of pure 8-mer (m=8) square-well chains at three reduced temperatures of T* = 1.5, 2, and 3.

The solid lines are from theory. Symbols are Monte Carlo simulation from the Literature

A comparison of the theory to computer simulation results of a dimer-dimer mixture at T*=2 with s22/s11=1 and e22/e11=1.5 is given in Fig. 3 for three compositions (x1=0.2, 0.5 and 0.8). The influence of composition on the compressibility factor is well described by the theory. The two components in this example (Fig. 3) differ in their potential strength. Figure 4 shows a dimer-dimer mixture of equal sized molecules at T*=2 with the ratios of the potential depth ranging from e22/e11=0.5 to 1.5 . This diagram indicates that – as for spherical molecules – energetic effects onto mixture behavior of chains are well captured by the one-fluid mixing rule.

The influence of segment size on the compressibility factor of a dimer-dimer mixture is illustrated in Fig. 5 for two reduced temperatures of T*=2 and T*=3.

The other influence of molecules geometry – the effect of chain length onto mixture behavior – is illustrated for a monomer-dimer system. Figure 6 compares the proposed equation of state to simulation results of such a system at T*=2 with sd/sm=1 and ed/em=1. Figure 6 proves the one-fluid theory in close agreement to the simulation data for the given mixture. In Fig. 6 mixtures with parameter ratios of (sd/sm=0.5, ed/em=1) and (sd/sm=1, ed/em=0.5) are also displayed.

Fig. 3: Compressibility factor of mixtures of square-well dimers at reduced temperature T* = 2 at three compositions of x1=0.8, x1=0.5 and x1=0.2, where e22/e11=1.5 and s22/s11=1. Comparison of Monte Carlo simulation from Gulati and Hall, 1997 (symbols) and predictions of the perturbation theory (solid curve).

Fig. 4: Compressibility factor of mixtures of square-well dimers at reduced temperature T* = 2, where s22/s11=1 and the composition of all mixtures is x1=0.5. Three mixtures of different well-depth ratios ranging from e22/e11=0.5 to e22/e11=1.5 are shown. The symbols and lines are defined as in Fig. 6

Fig. 5: Compressibility factor of mixtures of square-well dimers with size ratio s22/s11=2 at two reduced temperatures of T* = 2 and T*=3, where e22/e11=1 and the composition of all mixtures is x1=0.5. The symbols and lines are defined as in Fig. 6.

Fig. 6: Compressibility factor of square-well dimer - monomer mixtures at T* = 2, where the composition of all mixtures is xd=0.33. Three mixtures of different well-depth ratios ranging from ed/em=0.5 to ed/em=1 and size ratios ranging from sd/sm=0.5 to sd/sm=1 are shown. Comparison of Monte Carlo simulation from Gulati and Hall, 1997 (symbols) and predictions of the perturbation theory (solid curve).

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