Knowledge of phase behavior (thermodynamics) of polymer solutions is important for the design of many processes and products, including many specific applications in colloid and surface chemistry.
Knowledge of phase behavior (thermodynamics) of polymer solutions is important for the design of many processes and products, including many specific applications in colloid and surface chemistry.
Among the many applications, we can mention the following:
The aforementioned list shows some of the many applications where polymer thermodynamics plays a key role. Polymer solutions and blends are complex systems: frequent existence of liquid–liquid equilibria (LLE) (UCST, LCST, closed loop, etc.), the significant effect of temperature and polymer molecular weight including polydispersity in phase equilibria, freevolume (FV) effects, and other factors may cause difficulties. For this reason, many different models have been developed for polymer systems and often the situation may seem rather confusing to the practicing engineer.
The choice of a suitable model will depend on the actual problem and depends, specifically, on
This chapter will present key tools in polymer thermodynamics at three different levels from the “simpler” but also “more easytouse” methods up to the “more advanced” but also “more complex” and potentially more accurate approaches:
A summary of some rules of thumb for predicting polymer–solvent miscibility, with focus on the screening of solvents for polymers, is presented here. These rules are based on wellknown concepts of thermodynamics (activity coefficients, solubility parameters) and some specific ones for polymers (the FH parameter). It can be roughly stated that a chemical (1) will be a good solvent for a specific polymer (2), or in other words, the two compounds will be miscible if one (or more) of the following “rules of thumb” are valid:
They are widely used. The starting point (in their derivation and understanding) is the equation for the Gibbs free energy of mixing:
A negative value implies that a solvent–polymer system forms a homogeneous solution, that is, the two components are miscible. Since the contribution of the entropic term (−TΔS) is always negative, it is the heat of mixing term that determines the sign of the Gibbs energy. The heat of mixing can be estimated from various theories, for example, the Hildebrand regular solution theory for nonpolar systems, which is based on the concept of the solubility parameter. For a binary solvent(1)–polymer (2) system, according to the regular solution theory,
where φ _{i} is the socalled volume fraction of component i. This is defined via the mole fractions x_{i} and the molar volumes V_{i} as (for binary systems):
According to Equation 3.4, the heat of mixing is always positive. For some systems with specific interactions (hydrogen bonding), the heat of mixing can be negative and Equation 3.4 does not hold. Thus, the regular solution theory is strictly valid for nonpolar–slightly polar systems, without any specific interactions.
According to Equations 3.3 and 3.4, if solvent and polymer have the same solubility parameters, the heat of mixing is zero and they are therefore miscible at all proportions. The lower the solubility parameter difference, the larger the tendency for miscibility. Many empirical rules of thumb have been proposed based on this observation. Seymour ^{24} suggests that if the difference of solubility parameters is below 1.8 (cal/cm ^{3} )^{1/2}, then polymer and solvent are miscible (Equation 3.1).
Similar rules can be applied for mixed solvent–polymer systems, which are very important in many practical applications, for example, in the paints and coatings industry and for the separation of biomolecules using aqueous twophase systems.
The solubility parameter of a mixed solvent is given by the equation:
Barton ^{25} ^{,} ^{26} provides empirical methods based on solubility parameters for ternary solvent systems.
Charles Hansen introduced the concept of 3D solubility parameters, which offers an extension of the regular solution theory to polar and hydrogen bonding systems. Hansen observed that when the solubility parameter increments of the solvents and polymers are plotted in 3D plots, then the “good” solvents lie approximately within a sphere of radius R (with the polymer being in the center). This can be mathematically expressed as shown in Equation 3.2. The quantity under the square root is the distance between the solvent and the polymer. Hansen found empirically that a universal value 4 should be added as a factor in the dispersion term to approximately attain the shape of a sphere. This universal factor has been confirmed by many experiments. Hansen ^{27} in his book provides a review of the method together with extensive tables of parameters.
The Hansen method is very valuable. It has found widespread use particularly in the paints and coatings industry, where the choice of solvents to meet economical, ecological, and safety constraints is of critical importance. It can explain some cases in which polymer and solvent solubility parameters are almost perfectly matched and yet the polymer will not dissolve. The Hansen method can also predict cases where two nonsolvents can be mixed to form a solvent. Still, the method is approximate; it lacks the generality of a full thermodynamic model for assessing miscibility and requires some experimental measurements. The determination of R is typically based on visual observation of solubility (or not) of 0.5 g polymer in 5 cm ^{3} solvent at room temperature. Given the concentration and the temperature dependence of phase boundaries, such determination may seem a bit arbitrary. Still the method works out pretty well in practice, probably because the liquid–liquid boundaries for most polymer–solvent systems are fairly “flat.”
Since in several practical cases concerning polymer–solvent systems, the “solvent” is only present in very small (trace) amounts, the socalled infinite dilution activity coefficients are important. On a molar and weight basis, they are defined as follows:
The latter part of the equation is valid for a binary solvent–polymer solution and γ^{∞} _{1} is the infinite dilution activity coefficient of the solvent.
The weightbased infinite dilution activity coefficient, Ω^{∞} _{1}, which can be determined experimentally from chromatography, is a very useful quantity for determining good solvents. As mentioned in the previous section, low values (typically below 6) indicate good solvents, while high values (typically above 10) indicate poor solvents according to rules of thumb discussed by several investigators. ^{28} ^{,} ^{29} The derivation of this rule of thumb is based on the FH model.
This method for solvent selection is particularly useful, because it avoids the need for direct liquid–liquid measurements and it makes use of the existing databases of solvent infinite dilution activity coefficients, which is quite large (e.g., the DECHEMA and DIPPR databases ^{30} ^{,} ^{31} ).
Moreover, in the absence of experimental data, existing thermodynamic models (such as the FH, the EntropicFV, and the UNIFACFV discussed later) can be used to predict the infinite dilution activity coefficient. Since, in the typical case today, existing models perform much better for VLE and activity coefficient calculations than directly for LLE calculations, this method is quite valuable and successful, as shown by sample results in Table 3.1.
This rule of thumb makes use of either experimental or predicted, by a model, infinite dilution activity coefficients. However, the results depend not only on the accuracy of the model, but also on the rule of thumb, which in turns depends on the assumptions of the FH approach. A thermodynamically more correct method is to employ the activity–concentration diagram. The maximum indicates phase split, while a monotonic increase of activity with concentration indicates a single liquid phase (homogeneous solutions).
The FH model for the activity coefficient, proposed in the early 1940s by Flory and Huggins, ^{32} ^{,} ^{33} is a famous Gibbs free energy expression for polymer solutions. For binary solvent–polymer solutions and assuming that the parameter of the model, the socalled FH interaction parameter χ_{12}, is constant, the activity coefficient is given by the equation:
where
Appendix 3.A presents the general expression for the FH model suitable for multicomponent systems.
Using standard thermodynamics and Equation 3.8, it can be shown that for high molecular weight polymer–solvent systems, the polymer critical concentration is close to zero and the interaction parameter has a value equal to 0.5. Thus, a good solvent (polymer soluble in the solvent at all proportions) is obtained if χ_{12} ≤ 0.5, while values greater than 0.5 indicate poor miscibility. Since the FH model is only an approximate representation of the physical picture and particularly the FH parameter is often not a constant at all, this empirical rule is certainly subject to some uncertainty. Nevertheless, it has found widespread use and its conclusions are often in good agreement with experiment.
Solvent 
S/NS 
EFV 
UFV 
GCFL 

Hexane 
NS 
7.1 
7.0 
10.7 
nOctane 
NS 
6.7 
6.3 
10.4 
nDecane 
NS 
6.5 
6.0 
10.7 
nDodecane 
NS 
6.6 
6.0 
11.3 
nHexadecane 
NS 
6.8 
6.1 
13.2 
Toluene 
S 
3.2 
4.4 
4.7 
Xylene 
S 
2.3 
3.6 
5.7 
Methylene dichloride 
S 
3.3 
2.5 
3.0 
Chloroform 
S 
1.9 
2.1 
1.7 
Carbon tetrachloride 
S 
2.2 
2.2 
2.9 
Ethylene dichloride 
S 
3.5 
3.0 
— 
Trichloroethylene 
S 
2.5 
2.9 
33.9 
Chlorobenzene 
S 
2.5 
3.0 
3.0 
oDichlorobenzene 
S 
1.3 
2.5 
2.7 
Acetone 
S 
10.9 
14.2 
11.1 
MEK 
S 
8.4 
10.5 
8.2 
MIBK 
S 
6.3 
7.7 
5.7 
Acetophenone 
S 
8.1 
9.3 
8.6 
Ethyl acetate 
S 
6.7 
6.7 
60.3 
Butyl acetate 
S 
5.3 
5.1 
31.4 
Diethyl ether 
S 
5.2 
5.8 
11.6 
THF 
S 
3.8 
4.0 
— 
1,4Dioxane 
S 
4.1 
4.4 
159.4 
Methanol 
NS 
43.7 
57.7 
35.7 
Ethanol 
NS 
29.2 
31.3 
17.3 
1Butanol 
NS 
18.1 
17.1 
8.1 
Cyclohexanol 
NS 
24.3 
20.1 
3.0 
Ethylene glycol 
NS 
277.8 
— 
15,947.0 
Propylene glycol 
NS 
212.6 
— 
1,879.2 
1,3Butanediol 
NS 
158.5 
— 
525.5 
Glycerol 
NS 
294.6 
— 
2,282.4 
Isopropanol 
NS 
23.4 
21.6 
10.6 
Isobutanol 
S 
19.0 
17.9 
7.9 
Diethylene glycol 
NS 
240.1 
— 
2,470.4 
Dipropylene glycol 
NS 
127.9 
945.7 
827.8 
Nitromethane 
NS 
16.7 
17.2 
— 
1Nitropropane 
S 
4.7 
5.2 
— 
N,Ndimethylformamide 
S 
3.8 
— 
— 
Source: Modified from Lindvig, Th. et al., AIChE J., 47(11), 2573, 2001.
Notes: EFV, EntropicFV; UFV, UNIFACFV; GCFL, GCFlory; S, solvent; NS, nonsolvent (experimental observation).
There are several, still rather obscure issues about the FH model, which we summarize here together with some recent developments:
The FH model provides a first approximation for polymer solutions. Both the combinatorial and the energetic terms require substantial improvement. Many authors have replaced the random vanLaar energetic term by a nonrandom localcomposition term such as those of the UNIQUAC, NRTL, and UNIFAC models. The combinatorial term should be modified to account for the FV differences between solvents and polymers.
The improvement of the energetic term of FH equation is important. Localcomposition terms like those appearing in NRTL, UNIQUAC, and UNIFAC provide a flexibility, which cannot be accounted for by the singleparameter van Laar term of FH. However, the highly pronounced FV effects should always be accounted for in polymer solutions.
The concept of FV is rather loose, but still very important. Elbro ^{28} demonstrated, using a simple definition for the FV (Equation 3.12), that the FV percentages of solvents and polymers are different. In the typical case, the FV percentage of solvents is greater (40%–50%) than that of polymers (30%–40%). There are two notable exceptions to this rule; water and PDMS: water has lower FV than other solvents and closer to that of most of the polymers, while PDMS has quite a higher FV percentage, closer to that of most solvents. LCST is, as expected, related to the FV differences between polymers and solvents. As shown by Elbro, the larger the FV differences, the lower is the LCST value (the larger the area of immiscibility). For this reason, PDMS solutions have a LCST, which are located at very high temperatures.
Many mathematical expressions have been proposed for the FV. One of the simplest and successful equations is
originally proposed by Bondi ^{39} and later adopted by Elbro et al. ^{35} and Kontogeorgis et al. ^{40} in the socalled EntropicFV model (described later). According to this equation, FV is just the “empty” volume available to the molecule when the molecules’ own (hardcore or closedpacked V*) volume is subtracted.
The FV is not the only concept, which is loosely defined in this discussion. The hardcore volume is also a quantity difficult to define and various approximations are available. Elbro et al.^{35 } suggested using V* = V_{w} , that is, equal to the van der Waals volume (V_{w} ), which is obtained from the group increments of Bondi and is tabulated for almost all existing groups in the UNIFAC tables. Other investigators ^{41} interpreted somewhat differently the physical meaning of the hardcore volume in the development of improved FV expressions for polymer solutions, which employ Equation 3.12 as basis, but with V* values higher than V_{w} (about 1.2–1.3V_{w} ).
The original UNIFAC model does not account for the FV differences between solvents and polymers and, as a consequence of that, it highly underestimates the solvent activities in polymer solutions. One of the most successful and earliest such models for polymers is the UNIFACFV by Oishi and Prausnitz. ^{42} The UNIFACFV model was developed for solvent activities in polymers, but it cannot be successfully applied to LLE.
A similar to UNIFACFV but somewhat simpler approach, which can be readily extended to multicomponent systems and LLE, is the socalled EntropicFV model proposed by Elbro et al. ^{35} and Kontogeorgis et al. ^{40} :
As can been seen from Equation 3.13, the FV definition given by Equation 3.12 is employed. The combinatorial term of Equation 3.13 is very similar to that of FH. However, instead of volume or segment fractions, FV fractions are used. In this way, both combinatorial and FV effects are combined into a single expression. The combinatorial–FV expression of the EntropicFV model is derived from statistical mechanics, using a suitable form of the generalized van der Waals partition function.
The residual term of EntropicFV is taken by the socalled new or linear UNIFAC model, which uses a lineardependent parameter table ^{43} :
This parameter table has been developed using the combinatorial term of the original UNIFAC model. As with UNIFACFV, no parameter reestimation has been performed. The same group parameters are used in the “linear UNIFAC” and in the EntropicFV models.
A common feature for both UNIFACFV and EntropicFV is that they require values for the volumes of solvents and polymers (at the different temperatures where application is required). This can be a problem in those cases where the densities are not available experimentally and have to be estimated using a predictive groupcontribution or other method, for example, GCVOL ^{44} ^{,} ^{45} or van Krevelen methods. These two estimation methods perform quite well and often similarly even for low molecular weight compounds or oligomers such as plasticizers.
Both UNIFACFV and EntropicFV, especially the former, are rather sensitive to the density values used for the calculations of solvent activities.
As already mentioned, the EntropicFV model has been derived from the van der Waals partition function. The similarity of the model with the van der Waals equation of state P = RT/(V − b) − a/V ^{2} becomes apparent if the van der Waals equation of state is written (when the classical van der Waals one fluid mixing and classical combining rules are used) as an activity coefficient model:
where φ _{i} is the volume fraction as defined in Equation 3.5. The first term in Equation 3.15 is the same as in EntropicFV with V_{w} = b, while the latter term is a regular solution theory or van Laartype term.
Table 3.2 presents an overview of the results with EntropicFV model for different applications together with the corresponding references. Selected results are shown in Tables 3.3 through 3.9 and Figures 3.1 and 3.2.
The most important general conclusions can be summarized as following:
Application  References 

VLE binary solutions  [40,41] 
VLE complex polymers–solvents  [48] 
VLE ternary polymer–solvents  [37] 
Paints  [61] 
Dendrimers  
VLE copolymers  [47] 
VLE athermals systems  [28,35,50,62] 
SLE hydrocarbons  [51,52] 
SLE polymer–solvents  [49] 
Comparison with other models  [40,46,53,60] 
LLE polymer–solvents  [58] 
LLE ternary polymer–mixed solvents  [8] 
Polymer blends  [59] 
EFV + UNIQUAC 
System  Experiment  EntropicFV  UNIFACFV 

PBMA/nC_{10}  NS  6.5 (–)  6.1 (–) 
PBMA/xylene  S  2.3 (S)  3.6 (S) 
PBMA/CHCl_{3}  S  1.9 (S)  9.1 (NS) 
PBMA/acetone  S  0.2 (NS)  14.1 (NS) 
PBMA/ethyl acetate  S  6.7 (–)  6.7 (–) 
PBMA/ethanol  NS  29.2 (NS)  31.3 (NS) 
PMMA/acetone  S  10.0 (NS)  16.5 (NS) 
PMMA/ethyl acetate  S  6.6 (–)  8.4 (NS) 
PMMA/butanol  NS  26.8 (NS)  14.4 (NS) 
PEMA/MEK  S  8.1 (NS)  11.7 (NS) 
PEMA/diethyl ether  S  5.8 (S)  7.6 (–) 
PEMA/nitropropane  NS  4.5 (S)  1.4 (S) 
PVAc/hexane  NS  38.7 (NS)  38.6 (NS) 
PVAc/methanol  S  18.9 (NS)  19.4 (NS) 
PVAc/ethanol  NS  15.2 (NS)  38.9 (NS) 
PVAc/nitromethane  S  3.9 (S)  3.8 (S) 
PVAc/THF  S  8.4 (NS)  5.6 (S) 
Notes: S, good solvent; NS, nonsolvent; –, no answer according to the rule of thumb.
PIP Systems  Exper. Value  EntropicFV  UNIFACFV 

+Acetonitrile  68.6  47.7 (31%)  52.3 (24%) 
+Acetic acid  37.9  33.5 (12%)  17.7 (53%) 
+Cyclohexanone  7.32  5.4 (27%)  4.6 (38%) 
+Acetone  17.3  15.9 (8%)  13.4 (23%) 
+MEK  11.4  12.1 (6%)  10.1 (12%) 
+Benzene  4.37  4.5 (2.5%)  4.4 (0%) 
+1,2Dichloroethane  4.25  5.5 (29%)  6.5 (54%) 
+CCl_{4}  1.77  2.1 (20%)  1.8 (0%) 
+1,4Dioxane  6.08  6.3 (4%)  5.9 (2%) 
+Tetrahydrofurane  4.38  4.9 (14%)  3.9 (10%) 
+Ethylacetate  7.47  7.3 (2%)  6.6 (11%) 
+nHexane  6.36  5.1 (20%)  4.6 (27%) 
+Chloroform  2.13  3.00 (41%)  2.6 (20%) 
Experimental values and calculations are at 328.2 K.
Model  % AAD (Systems in Database)  % AAD Araldite 488  % AAD Eponol55 

FH/Hansen, Volume (Equation 3.10)  22  31  28 
EntropicFV  35  34  30 
UNIFACFV  39  119  62 
Source: Adapted from Lindvig, Th. et al., Fluid Phase Equilib., 203, 247, 2002. The second column presents the systems used for optimization of the universal parameter (solutions containing acrylates and acetates). The last two columns show predictions for two epoxy resins.
% AAD Infinite Dilution Conditions γ^{∞} _{1} (Ω^{∞} _{1} for Polymers)  EntropicFV  UNIFACFV  FloryFV 

Short nalkanes/long alkanes  8  15  20 
Short branched, cyclic alkanes/long alkanes  10  17  20 
Alkanes/polyethylene  9  23  19 
Alkanes/polyisobutylene  16  12  38 
Organic solvent/PDMS, PS, PVAc  20  29  26 
Overall  13  19  25 
Source: Kouskoumvekaki, I. et al., Fluid Phase Equilib., 202(2), 325, 2002. With permission.
EntropicFV  UNIFACFV  FloryFV  

% AAD infinite dilution conditions, γ^{∞} _{2}  
Symmetric long alkanes/short alkanes  36  47  10 
Medium asymmetric long alkanes/short alkanes  34  48  12 
Asymmetric long alkanes/short alkanes  44  54  37 
Overall  38  50  20 
% AAD finite concentrations, γ_{2}  
Symmetric long alkanes/short alkanes  14  17  6 
Medium asymmetric long/short alkanes  23  31  11 
Asymmetric long/short alkanes  40  55  16 
Overall  26  34  11 
Source: Kouskoumvekaki, I. et al., Fluid Phase Equilib., 202(2), 325, 2002. With permission.
Sys. No.  Variable  SAFT  EFV/UQ  FH  PaVe  EFV/UN  UFV  GCFL  GCLF  FHHa 

1  P  11  6  6  6  2  1  21  8  2 
y  4  3  3  3  3  3  3  3  3  
2  P  4  2  14  8  2  2  —  2  11 
y  5  2  5  1  2  2  —  4  5  
3  P  —  —  —  —  3  3  12  5  18 
y  —  —  —  —  3  3  2  4  4  
4  P  —  —  —  —  4  4  13  9  5 
y  —  —  —  —  18  18  19  19  15  
5  P  14  16  11  4  17  19  93  5  52 
y  17  17  16  11  17  13  2  18  14 
Sources: Katayama, T. et al., Kagaku Kogaku, 35, 1012, 1971; Matsumara, K. and Katayama, T., Kagaku Kogaku, 38, 388, 1974; Tanbonliong, J.O. and Prausnitz, J.M., Polymer, 38, 5775, 1997.
Based on the results shown by Lindvig et al. ^{37}
Notes: 1, PS–toluene–ethylbenzene at 303 K; 2, PS–toluene–cyclohexane at 303 K; 3, PVAc–acetone–ethyl acetate at 303 K; 4, PVAc–acetone–methanol at 303 K; 5, PS–chloroform–carbon tetrachloride at 323.15 K.
Polymer System  Molecular Weight  EntropicFV  N0065w UNIFAC  GCFlory 

PS/acetone  4,800  84  21  75 
PS/acetone  10,300  98  8  42 
PS/cyclohexane  20,400  38  62  — 
PS/cyclohexane  37,000  26  59  — 
PS/cyclohexane  43,600  24  63  — 
PS/cyclohexane  89,000  11  60  — 
PS/cyclohexane  100,000  15  62  — 
PS/cyclopentane  97,200  27  105  — 
PS/cyclopentane  200,000  12  103  — 
HDPE/nbutyl acetate  13,600  10  82  72 
HDPE/nbutyl acetate  20,000  22  97  70 
HDPE/nbutyl acetate  61,100  29  107  71 
PMMA/1chloro butane  34,760  53  —  — 
PBMA/npentane  11,600  Hourglass  —  — 
PBMA/noctane  11,600  155  —  — 
Notes: PS, polystyrene; HDPE, highdensity polyethylene; PMMA, poly(methyl methacrylate); PBMA, poly(butyl methacrylate).
Figure 3.1 Experimental and predicted LLE diagram for the system polystyrene/acetone at three polymer molecular weights (4,800, 10,300, 19,800). The points are the experimental data and the lines are the predictions with the EntropicFV model. (From Kontogeorgis, G.M. et al., Ind. Eng. Chem. Res., 34, 1823, 1995. With permission.)
Figure 3.2 Ternary LLE for PS(300,000)/methyl cyclohexane/acetone, T = 298.15 K. (Reprinted from Pappa, G.D. et al., Ind. Eng. Chem. Res., 36, 5461, 1997. With permission.)
Some additional observations for specific cases are hereafter presented:
Both UNIFACFV and EntropicFV are group contribution models. This renders the models truly predictive, but at the same time with little flexibility if the performance of the models for specific cases is not satisfactory. Two interesting alternative approaches are discussed here, which still maintain the FV terms but use different residual terms.
The first approach is to employ the UNIQUAC expression for the residual term. This EntropicFV/ UNIQUAC model has been originally suggested by Elbro et al. ^{35} and has shown to give very good results for polymer solutions if the parameters are obtained from VLE data between the solvent and the low molecular weight monomer (or the polymer’s repeating unit).
The EntropicFV/UNIQUAC model has been recently further developed and extended independently by two research groups.^{55–57} Both VLE and LLE equilibria are considered, but the emphasis is given to LLE. Very satisfactory results are obtained as can be seen for two typical systems in Figures 3.3 and 3.4. It has been demonstrated that the EntropicFV/UNIQUAC approach can correlate both UCST/LCST and closedloop behavior and even show the pressure dependency of critical solution temperatures (UCST and LCST).
Figure 3.3 Correlation of liquid–liquid equilibria for the PVAL/water system with the EntropicFV/ UNIQUAC model. (•) Exp. data (M n = 140,000 g/mol); (—) correlation. (From Bogdanic, G. and Vidal, J., Fluid Phase Equilib., 173, 241, 2000. With permission.)
Figure 3.4 Correlation and prediction of liquid–liquid equilibria for the PBD/1octane system with the EntropicFV/UNIQUAC model. (▪) Exp. data (Mv = 65,000 g/mol), (—) correlation; (▴) Exp. data (Mv = 135,000 g/mol), ( ) prediction; and (•) Exp. data (Mw = 44,500 g/mol), (   ) prediction. (From Bogdanic, G. and Vidal, J., Fluid Phase Equilib., 173, 241, 2000. With permission.)
The second approach proposed by Thorlaksen et al. ^{63} is based on a combination of the EntropicFV term with Hildebrand’s regular solution theory and developed a model for estimating gas solubilities in elastomers. The socalled Hildebrand–EntropicFV model is given by the equation:
where
Finally, the gas solubility in the polymer is estimated from the equation:
Calculations showed that the hypothetical gas “liquid” volumes are largely independent to the polymer used, and moreover, for many gases (H_{2}O, O_{2}, N_{2}, CO_{2} and C_{2}H_{2}), these are related to the critical volume of the gas by the equation:
Very satisfactory results are obtained as shown in Table 3.10.
A final note for these “classical” activity coefficient models is that, despite the advent of advanced SAFT and other equations of state discussed next (Section 3.4), they are still quite popular and widely used in practical applications. They are also well cited in literature. For example, the historical articles by Flory and Huggins (Refs. [32,33]) are cited 998 (13.5) and 1034 (14) and the citations of the articles by Elbro et al. ^{35} : 164 (6.6), Lindvig et al. ^{36} : 44 (3.4), Kontogeorgis et al. ^{40} : 121 (5.5), and Oishi and Prausnitz ^{42} : 353 (9.5). The citations are per May 2014 and the numbers in parenthesis are citations per year.
“Statistical mechanics is that branch of physics which studies macroscopic systems from a microscopic or molecular point of view. The goal of statistical mechanics is the understanding and prediction of macroscopic phenomena and the calculation of macroscopic properties from the properties of the individual molecules making up the system.”
This is the opening paragraph of “Statistical Mechanics” written by McQuarrie, ^{64} already in 1976. Attempts to achieve this goal of statistical mechanics have been around for a long time. For example, Wertheim ^{65} was the first to derive an equation of state for hardsphere systems. Carnahan and Starling ^{66} made an empirical modification to Wertheim’s solution based on molecular simulation data to arrive at what is by now the famous Carnahan–Starling equation of state. As well as being an early attempt to arrive at an engineering model using results from “hard” science (or what we might cynically call “impractical” science), this work also showed the usefulness of using results from molecular simulation. In a sense, molecular simulation fulfils the goal of statistical mechanics, in that it predicts (some) macroscopic properties using only molecular properties as input. However, molecular simulation is system specific, time consuming, and ultimately can only be as successful as the molecular model it is based upon. Thus, while it is certainly a step forward to be able to predict the properties of a system of hard spheres, the hard sphere as a model is itself an incomplete description of a real molecule. Nevertheless, increasing computer power and ever more detailed knowledge of molecular properties, extending even to the quantum level, means that molecular simulation will continue to be an important tool in engineering thermodynamics. See Economou ^{67} for a review of current industrial applications of molecular simulation.
Polymer 
Gas 
Michaels/Bixler 
Tseng/Lloyd 
Hildebrand/Scott 
Hildebrand EntropicFV1 
Hildebrand EntropicFV2 

PIP 
N_{2} 
14.7 
73 
3.9 
−7.9 
−4.6 
O_{2} 
−16.1 
−4 
14 
10.8 
11.8 

Ar 
−32.5 
−23 
— 
29.4 
−22.2 

CO_{2} 
−3.2 
13 
4.5 
8.7 
4.6 

PIB 
N_{2} 
−2.5 
— 
6.8 
3.1 
5.0 
O_{2} 
−6.1 
— 
−1.7 
−8.3 
1.7 

Ar 
— 
— 
— 
— 
— 

CO_{2} 
32.8 
— 
−1.9 
41.1 
35.2 

PBD 
N_{2} 
22.3 
— 
8.1 
8.1 
12.6 
O_{2} 
14.9 
— 
−6 
8.7 
10.8 

Ar 
12.1 
— 
— 
111.1 
24.0 

CO_{2} 
−9.7 
— 
−4.6 
0.4 
−4.0 

PDMB 
N_{2} 
— 
— 
−23 
−7.5 
−3.1 
O_{2} 
— 
— 
−32 
−16.8 
−15.9 

Ar 
— 
— 
— 
— 
— 

CO_{2} 
— 
— 
−24 
2.3 
−2.2 

PCP 
N_{2} 
58.1 
— 
49 
−7.0 
−4.2 
O_{2} 
43.7 
— 
60 
−1.4 
−1.4 

Ar 
— 
— 
— 
— 
— 

CO_{2} 
8.8 
— 
27 
−13.3 
−17.1 

AAD 
19.8 
28 
18 
16.8 
10.6 
Errors associated with models for predicting gas solubilities in polymers.
Hildebrand EntropicFV1: The liquid volume of the gas is determined from its relationship with the critical volume, Equation 3.26. Hildebrand EntropicFV2: The average hypothetical liquid volume of a gas is used.
However, equations of state, too, will be an essential component of chemical engineering theory and practice for the foreseeable future, and as ever, the balance will need to be struck between rigorous theory and engineering applicability. One equation of state, which seems to have done an admirable job of bridging the gap between molecular theory and engineering application, is statistical associating fluid theory (SAFT) and it is with this equation of state and its spinoffs that the remainder of this discussion is concerned.
A series of four seminal papers once again written by Wertheim^{68–71} appeared in 1984 and 1986. These papers laid the foundation for the associating theory (or thermodynamic perturbation theory [TPP]), which was to become the key feature in the novel molecularbased equation of state known as SAFT. The theories presented in Wertheim’s papers are highly complex and almost intractable—“essentially incomprehensible,” as one author has put it. ^{1} However, in the period 1988–1990 at Cornell University, Chapman and coworkers^{72–75} performed the monumental task of transforming the abstruse theory of Wertheim into workable equations, and finally into an engineering equation of state. SAFT is by no means the only molecularbased equation of equation of state out there—PHSC is another ^{76} but it differs from the vast majority of other similar equations of state in one important respect—it is extensively used. In their review of SAFT published in 2001, Müller and Gubbins ^{77} estimate that 200 articles dealing with SAFT had appeared. Since that review appeared, a further 70 or so articles have appeared dealing directly with SAFT or one of its variants. Significantly, SAFT is also now available in industrial process simulators such as ASPEN, PRO/II, and ChemCad as well as in the SPECS thermodynamics package of IVCSEP at the Technical University of Denmark.
One of the drawbacks arising from the extensive use of SAFT is that many versions have appeared, with the result that the literature is complex and can be confusing. Here, we will try to outline the theoretical development in some detail of the original SAFT model (this too is not unambiguous, since “original SAFT” is often used to describe the version due to Huang and Radosz, ^{78} ^{,} ^{79} which is slightly different from that of Chapman), as well as some of its modified versions. We will then summarize some of the more interesting results obtained using SAFT and its variants.
The foundation for what was to become SAFT was laid in two papers ^{72} ^{,} ^{75} which appeared in the journal Molecular Physics in 1988. The first of these papers developed the theory required for associating fluids, while the second focused on chain formation. However, the first paper to contain an equation of state that can realistically be called SAFT appeared in the August, 1990, issue of Industrial and Engineering Chemistry Research. ^{74} It is interesting that the Huang and Radosz paper appeared in the November issue of the same journal. ^{78}
To understand exactly what occurs in SAFT, we refer to Figure 3.5, taken from Fu and Sandler. ^{80} Initially, a pure fluid is assumed to consist of equalsized hard spheres (b). Next a dispersive potential is added to account for attraction between the spheres (c). Typical potentials are the squarewell or Lennard–Jones potential. Next each sphere is given two (or more) “sticky” spots, which enables the formation of chains (d). Finally, specific interaction sites are introduced at certain positions in the chain, which enable the chains to associate through some attractive interaction (hydrogen bonding) (e). This interactive energy is often taken to be a squarewell potential. The final single molecule is shown in Figure 3.5a. Each of these steps contributes to the Helmholtz energy. The residual Helmholtz energy is given by
where
Figure 3.5 Procedure to form a molecule in the SAFT model. (a) The proposed molecule. (b) Initially the fluid is a hardsphere fluid. (c) Attractive forces are added. (d) Chain sites are added and chain molecules appear. (e) Association sites are added and molecules form association complexes through association sites. (From Fu, Y.H. and Sandler, S.I., Ind. Eng. Chem. Res., 34, 1897, 1995. With permission.)
It is this separation of the Helmholtz energy into additive components that is partly responsible for the fact that SAFT has so many variants—each of the contributions can be considered (and modified) separately, or new terms may be added (such as polar, electrostatic, or other contributions). The individual terms may also be used outside of the context of SAFT. Thus, the term that accounts for association has been combined (with minor modification) with the SRK equation to create CPA, an equation of state, which has had remarkable success in a number of applications. ^{81}
It is worth noting that both the chain formation and the association term derive from Wertheim’s TPT. However, chains (i.e., covalent bonds) are formed in the limit of complete association. It is these two terms that make up the innovative development in SAFT—the first because now we have available a rational method for considering polymer molecules (linear chains with very many bonded segments) and the second because we can now consider associating molecules in a way that more closely resembles the actual physical picture. The calculation of useful thermodynamic properties from a^{assoc} was initially a complex procedure, although Michelsen and Hendriks ^{82} simplified the computations considerably by recasting the equations in a simpler, although mathematically equivalent form.
We now consider each of the terms individually. We follow the original notation of Chapman et al. ^{74} Each pure component is characterized by a chain length m, a sizeparameter σ, and an energy parameter, ɛ. If the molecule is selfassociating, there are two further parameters, which characterize the volume (κ^{ AiBi }) and energy (ɛ^{ AiBi }) of association. The association term is given for mixtures by
where
We have
where
Here d_{ij} is a temperaturedependent sizeparameter related to σ _{ij} by
where
and
This temperature dependence is incorporated to account for the fact that real molecules are not hard spheres, but rather there is some degree of interpenetration between molecules, particularly at high temperatures. Thus, the “effective” hardsphere diameter of a segment is smaller at higher temperatures. The radial distribution function in Equation 3.25 is given by the mixture version of the Carnahan–Starling equation of state for hardsphere mixtures:
where
For the chain term in Equation 3.22, we have
Finally, for the segment term in Equation 3.22, we have
where the 0 subscript indicates a nonassociated segment. The segment energy consists of a hardsphere reference and a dispersion contribution:
The Carnahan–Starling equation ^{66} is used for both pure components and mixtures to give
where for mixtures η = ζ_{3} as defined by Equation 3.30. The dispersion term is given by
where
The reduced quantities are given by T_{R} = kT/ɛ and ρ _{R} = (6/2^{0.5}π)η.
Most of the results of this initial paper are comparisons with simulation data for chains with various parameters, although purecomponent parameters for six hydrocarbons and two associating fluids were fitted. No results for mixtures of real fluids are presented.
Probably the main contribution of the Huang and Radosz ^{78} version of SAFT was the regression of purecomponent parameters for over 100 different fluids. There are also some notational differences. Thus, instead of a size parameter σ, they use a volume parameter v ^{00}, which is related through the equation
Here τ = 0.74048 is the highest possible packing fraction for a system of pure hard spheres. They also use the notation u ^{0} instead of ɛ for the energy parameter, although these terms are completely equivalent.
The rather complex temperature dependence of the hardsphere diameter given by Equations 3.26 through 3.28 was simplified by Huang and Radosz, following Chen and Kreglewski ^{83} to
The dispersion term is also different from that of Chapman et al. ^{74} and is given by
where
Another important contribution of Huang and Radosz is the presentation of detailed tables discussing bonding schemes for different associating fluids. These schemes are presented as Tables 3.11 and 3.12 and have been widely adopted in the literature of SAFT and other equations of state for associating fluids.
In their paper on mixture properties, ^{79} Huang and Radosz also use the full mixture version of the Carnahan–Starling equation for the hardsphere mixtures reference system:
One of the reasons that the Huang and Radosz version of SAFT has been adopted (and is widely referred to as “original” SAFT) is that they undertook an extensive purecomponent parameterization for over 100 pure fluids. This meant that their equation of state could be used immediately for real fluids of industrial interest without any intermediate steps. Both Huang and Radosz were employed by Exxon during the development of SAFT. The fact that the model had the backing of a major oil company may also help explain its rapid adoption and use as an engineering tool.
Besides it is interesting to note that the first paper to appear describing PCSAFT also had purecomponent parameters for about 100 species. ^{84} This fact, coupled with the success of the model, has certainly been partly responsible for the rapid adoption of PCSAFT in the few years it has been in existence, both in industry and in academia.
Type 
Δ Approximations 
X ^{A} Approximations 
X ^{A} 

1 
Δ^{AA} ≠ 0 
$\frac{1+(1+4\rho \mathrm{\Delta}{)}^{1/2}}{2\rho \mathrm{\Delta}}$ 

2A 
Δ^{AA} = Δ^{AB} = Δ^{BB} ≠ 0 
X ^{A} = X ^{B} 
$\frac{1+(1+8\rho \mathrm{\Delta}{)}^{1/2}}{4\rho \mathrm{\Delta}}$ 
2B 
Δ^{AA} = Δ^{BB} = 0 Δ^{AB} ≠ 0 
X ^{A} = X ^{B} 
$\frac{1+(1+4\rho \mathrm{\Delta}{)}^{1/2}}{2\rho \mathrm{\Delta}}$ 
3A 
Δ^{AA} = Δ^{AB} = Δ^{BB} = Δ^{AC} = Δ^{BC} = Δ^{CC} ≠ 0 
X ^{A} = X ^{B} = X ^{C} 
$\frac{1+(1+12\rho \mathrm{\Delta}{)}^{1/2}}{6\rho \mathrm{\Delta}}$ 
3B 
Δ^{AA} = Δ^{AB} = Δ^{BB} = Δ^{CC} = 0 Δ^{AC} = Δ^{BC} ≠ 0 
X ^{A} = X ^{B} 
$\frac{(1\rho \mathrm{\Delta})+(1+\rho \mathrm{\Delta}{)}^{2}+4\rho \mathrm{\Delta}{)}^{1/2}}{4\rho \mathrm{\Delta}}$ 
4A 
Δ^{AA} = Δ^{AB} = Δ^{BB} = Δ^{AC} = Δ^{BC} = Δ^{CC} = Δ^{AD} = Δ^{BD} = Δ^{CD} = Δ^{DD} ≠ 0 
X ^{A} = X ^{B} = X ^{C} = X ^{D} 
$\frac{1+(1+16\rho \mathrm{\Delta}{)}^{1/2}}{8\rho \mathrm{\Delta}}$ 
4B 
Δ^{AA} = Δ^{AB} = Δ^{BB} = Δ^{AC} = Δ^{BC} = Δ^{CC} = Δ^{DD} = 0 Δ^{AD} = Δ^{BD} = Δ^{CD} ≠ 0 
X ^{A} = X ^{B} = X ^{C} X ^{D} = 3X ^{A} − 2 
$\frac{(12\rho \mathrm{\Delta})+\left(\right(1+2\rho \mathrm{\Delta}{)}^{2}+4\rho \mathrm{\Delta}{)}^{1/2}}{6\rho \mathrm{\Delta}}$ 
4C 
Δ^{AA} = Δ^{AB} = Δ^{BB} = Δ^{CC} = Δ^{CD} = Δ^{DD} = 0 Δ^{AC} = Δ^{AD} = Δ^{BC} = Δ^{BD} ≠ 0 
X ^{A} = X ^{B} = X ^{C} = X ^{D} 
$\frac{1+(1+8\rho \mathrm{\Delta}{)}^{1/2}}{4\rho \mathrm{\Delta}}$ 
Source: Huang, S.H. and Radosz, M., Ind. Eng. Chem. Res., 29, 2284, 1990. With permission.
Species 
Formula 
Rigorous Type 
Assigned Type 

Acid 

1 
1 
Alkanol 

3B 
2B 
Water 

4C 
3B 
Amines  
Tertiary 

1 
Nonselfassociating 
Secondary 

2B 
2B 
Primary 

3B 
3B 
Ammonia 

4B 
3B 
Source: Huang, S.H. and Radosz, M., Ind. Eng. Chem. Res., 29, 2284, 1990. With permission.
The key idea in the work of Fu and Sandler is the simplification of the dispersion term. All other terms from Huang and Radosz are retained. Since the dispersion term given by Huang and Radosz, Equation 3.40, contains 24 constants, it seems reasonable to attempt to simplify this term. For mixtures, the dispersion Helmholtz free energy is
where
The remaining terms have the same meaning as in Huang and Radosz. ^{78} Generally, simplified SAFT performs as well as Huang and Radosz SAFT, although it requires refitting all the purecomponent parameters. Fu and Sandler provide parameters for 10 nonassociating and 8 associating fluids. Table 3.13 is reproduced from Fu and Sandler. It is interesting because it presents different types of crossassociation in some detail. This scheme is completely general and applicable to any equation of state incorporating association. Combining rules for crossassociation also need to be introduced, however, and are far from selfevident.
The main change in the SAFT version of Kraska and Gubbins ^{85} ^{,} ^{86} is that they use Lennard–Jones (LJ) spheres for the reference term, rather than hard spheres. The remaining terms are unchanged, except that the radial distribution function used in the calculation of the chain and association contributions in Equations 3.25 and 3.31 is the radial distribution function for LJ spheres rather than hard spheres. Thus, an equation of state for LJ spheres is required. The equation used is that of Kolafa and Nezbeda. ^{87} The Helmholtz energy for the reference (LJ) system is (for a pure fluid)
where
Mixture 
Component 1 
Component 2 
Association Type 

Acid–acid 


∈^{A1A1 } ≠ 0 ∈^{A2A2 } ≠ 0 ∈^{A1A2 } ≠ 0 
Alcohol–alcohol 


∈^{A1B1 } ≠ 0,∈^{A2B2 } ≠ 0 ∈^{A1B2 } = ∈^{A2B1 } ≠ 0 ∈^{A1A1 } =∈^{A2A2 } =∈^{A1A2 } =∈^{A2A1 } = 0 ∈^{B1B1 } =∈^{B2B2 } =∈^{B1B2 } =∈^{B2B1 } = 0 
Acid–alcohol 


∈^{A1B1 } ≠ 0,∈^{A2B2 ≠ 0} ∈^{A1A2 } =∈^{A1B2 } ≠ 0 ∈^{A2A2 } =∈^{B2B2 } = 0 
Water–acid 


∈^{A1C1 } =∈^{B1C1 } ≠ 0,∈^{A2A2 } ≠ 0 ∈^{A1A2 } =∈^{B1A2 } =∈^{11A2 } ≠ 0 ∈^{A1A1 } =∈^{B1B1 } =∈^{C1C1 } ∈^{A1B1 } = 0 
Water–alcohol 


∈^{A1C1 } =∈^{B1C1 } ≠ 0,∈^{A2A2 } ≠ 0 ∈^{A1B2 } =∈^{B1B2 }∈^{C1A2 } ≠ 0 ∈^{A1A1 } =∈^{B1B1 } =∈^{C1C1 } =∈^{A1B1 } = 0 ∈^{A2A2 } =∈^{B2B2 } = 0 ∈^{A1A2 } =∈^{B1A2 } =∈^{C1B2 } = 0 
Source: Fu, Y.H. and Sandler, S.I., Ind. Eng. Chem. Res., 34, 1897, 1995. With permission.
Apart from using a different reference system, the notation in Kraska and Gubbins does not follow the customary SAFT notation, nor are the purecomponent parameters defined in the same way (e.g., the energy parameter with units of 1/K is defined inversely as a temperature parameter with units K) and care should be taken in using it. They also incorporate a term to account for dipole– dipole interactions.
SAFTVR is the version of SAFT developed by George Jackson and coworkers first at the University of Sheffield and currently at Imperial College (GilVilegas et al. ^{88} ; McCabe et al. ^{89} ). SAFTVR is identical to the Huang and Radosz version except in the dispersion contribution. This term incorporated attraction in the form of a squarewell potential. Thus, in addition to a segment being characterized by a size and an energy parameter, the squarewell width (λ) is also included as a purecomponent parameter. Thus, changing the parameter λ changes the range of attraction of the segment (hence the name VR for “variable range”). It is the introduction of this extra term that gives SAFTVR greater flexibility, since we now have an extra pure component to “play with.” Although it is generally desirable to describe purecomponent liquid densities and vapor pressure with the minimum number of parameters, the extra variablerange parameter may be necessary for the description of certain anomalous behaviors in systems containing water. The Helmholtz energy for the dispersion energy is given by
where
The subscript s refers to segment rather than molecule properties. We have
and g^{HS} is the radial distribution function for hard spheres as before except that the arguments are different:
The constants c_{i} in Equation 3.53 are given by
The secondorder term in Equation 3.50 is given by
where
A recent version of SAFT that has appeared is that due to Gross and Sadowski ^{84} developed at the Technical University of Berlin. Once again, most of the terms in PCSAFT are the same as those in the Huang and Radosz version. The term that is different is the dispersion term. However, it is not simply a different way of expressing the dispersion attraction between segments, but rather it tries to account for dispersion attraction between whole chains. Referring to Figure 3.5 should make this clear. Instead of adding the dispersion to hard spheres and then forming chains, we first form hardsphere chains and then add a chain dispersion term, so the route in Figure 3.5 would be (b)–(d)–(c)–(e). To do this, we require interchain rather than intersegment radial distribution functions. These are given by O’Lenick et al. ^{90} The Helmholtz energy for the dispersion term is given as the sum of a first and secondorder term:
where
where
The radial distribution function g^{hc} is now an interchain function rather than a segment function as before. This is a key point in PCSAFT. The term involving compressibilities is given by
We still need to solve the integrals in Equations 3.60 and 3.61. Setting
we can substitute the Lennard–Jones potential and the radial distribution function of O’Lenick et al. ^{90} This was done for the series of nalkanes and the integrals were fit as a power series:
with
Equations 3.67 and 3.68 require a total of 42 constants, which are adjusted to fit experimental purecomponent data of nalkanes. This direct fitting to experimental data to some extent accounts for errors in the reference equation of state, the perturbing potential, and the radial distribution function, which appear in the integrals of Equations 3.63 and 3.64. The dispersion potential given by Equations 3.60 and 3.61 is readily extended to mixtures using the van der Waals onefluid theory.
Since this first PCSAFT paper appeared, the authors rapidly published a series of further papers applying PCSAFT to polymers, ^{91} ^{,} ^{92} associating fluids ^{93} and copolymers. ^{94}
Two simplifications to PCSAFT have been proposed, which simplify phase equilibrium calculations substantially for mixtures. For pure components, this simplified PCSAFT becomes original PCSAFT, so the simplifications may be considered as a particular set of mixing rules. The advantage of this is that existing purecomponent parameters can be used in simplified PCSAFT—no refitting is required. The targets of the simplifications are Equations 3.29 and 3.41. In other words, this simplified PCSAFT targets the hardsphere reference equation of state. The remaining terms are the same, except as mentioned previously, the simpler radial distribution function will affect both the chain and association terms, since the radial distribution function appears in both of these terms.
By setting η ≡ ζ_{3}, Equations 3.29 and 3.40 reduce to
and
respectively. In the initial paper, ^{95} use of Equation 3.69 only is called modification 1, while use of both Equations 3.69 and 3.70 is called modification 2. This and subsequent work has shown that there is no loss of accuracy using the most simplified version of PCSAFT, so modification 2 is used throughout and is called “simplified PCSAFT.” Simplified PCSAFT has since been applied in our group to several polymer systems including: polymer VLE, ^{96} polymer–solvent binary LLE, ^{97} ternary polymer–solvent and blend systems, ^{98} and highpressure gas solubility in polymers. ^{99} ^{,} ^{100}
One of the interesting points about SAFT in general is the ability to extrapolate the properties of higher molecular weight substances from knowledge of similar shorter chain compounds. This is most evident for the nalkane series. Purecomponent parameters for polyethylene can in principle be predicted by extrapolating from the properties of the nalkanes. In practice, this is problematic, since for very long chains, effects such as entanglement of the individual polymer chains start to influence the behavior. Figure 3.6 shows PCSAFT parameters for the alkanes up to eicosane (C20). The parameters m, mσ ^{3} , and mɛ/k are all linear with molecular weight. Tables 3.14 and 3.15 show computing times for simplified PCSAFT compared with original PCSAFT, as well as with other equations of state.
Figure 3.6 The groups m, mσ 3 , and mɛ/k vs. molecular weight for linear alkanes up to eicosane. Points are PCSAFT parameters, lines are linear fits to these points, excluding methane. (From von Solms, N. et al., Ind. Eng. Chem. Res., 42, 1098, 2003. With permission.)
Calculation 
Computing Time (ms) 

Phaseenvelope calculation, 36component mixture, full PCSAFT 
48 
Phaseenvelope calculation, 36component mixture, PCSAFT modification 2 
32 
Source: von Solms, N. et al., Ind. Eng. Chem. Res., 42, 1098, 2003. With permission.
Computations were performed on a 2.0 GHz Pentium IV machine with DVF compiler.
Computing Time (μs) 


Mixture Model 
Fugacity Coefficients Only 
All Derivatives (T, P, Composition Residual Heat Capacity) 
SRK, 6 components 
1.9 
3.4 
SRK, 6 components 
3.1 
5.6 
CPA, 15 components, 0 sites 
15 
20.1 
CPA, 15 components, 2 sites 
18.5 
24.8 
CPA, 15 components, 4 sites 
24.4 
32.2 
CPA, 15 components, 6 sites 
32.3 
40.9 
CPA, 15 components, 8 sites 
40.7 
50.9 
Modification 2, 15 components, 0 sites 
15.6 
23.1 
Modification 2, 15 components, 6 sites 
39.1 
54.4 
Source: von Solms, N. et al., Ind. Eng. Chem. Res., 42, 1098, 2003. With permission.
Number of sites refers to number of association sites on a molecule when employing an equation of state with association.
The additive contributions within SAFT mean that the equation of state is quite versatile—contributions can be added to account for effects not included in the discussion earlier. Thus, papers have appeared accounting for polar, ^{101} quadrupolar, ^{102} and dipolar ^{103} molecules, as well as polarizable dipoles. ^{104}
Versions of SAFT have also appeared, which include an electrostatic contribution, the intention being to develop a version of SAFT suitable for modeling electrolytes. ^{105} ^{,} ^{106}
Care should be taken when adding very many contributions, since each extra contribution will almost always require one or more purecomponent parameters. The actual physical picture should always be borne in mind when considering which contributions to include.
Another recent development is the application of group contribution methods to SAFT (see, e.g., Le Thi et al. ^{107} and references therein). Rather than the addition of extra terms, which then require more purecomponent parameters, the use of group contribution methods is an attempt to generalize the parameters in SAFT (extending even to binary interaction parameters). The hope is that in this way SAFT becomes a more predictive tool, relying less on fitting of parameters to experimental data.
The remainder of this discussion looks at some polymer applications of SAFT. Figure 3.7 shows the results of using parameters extrapolated based on Figure 3.6. The system is methane–tetratetracontane (C44), where the C44 parameters are obtained by extrapolation of the lines in Figure 3.6. While this is not a polymer system, the method of obtaining parameters by extrapolation is applicable to polymer systems. A more sophisticated method for finding polymer parameters based on extrapolation of the monomer properties and polymer density data has been published recently. ^{108}
Figures 3.8 and 3.9 show a comparison of the various modifications and original PCSAFT for VLE in the systems polystyrene–propyl acetate (Figure 3.8) and polypropylene–diisopropyl ketone (Figure 3.9). In general, PCSAFT and simplified PCSAFT performed similarly, as can be seen from Table 3.16, which gives the errors in prediction for a large number of polymer–solvent VLE systems. Figure 3.10 is a pressure–weight fraction diagram (VLE) for the polymer poly(vinyl acetate) in the associating solvent 2methyl1propanol. A small value of the binary interaction parameter correlates the data well.
Figure 3.11 is an illustration of a novel method developed by von Solms et al. ^{97} for finding LLE in polymer systems, known as the method of alternating tangents. This figure shows the Gibbs energy of mixing for two binary systems as a function of the mole fraction of component 1. The method will be illustrated with reference to the system methanol(1)–cyclohexane(2), since this curve clearly shows the existence of two phases. The composition of methanol in each phase is found by locating a single line, which is a tangent to the curve in two places (the common tangent). In Figure 3.11, these compositions are given by x ^{ eq 1} _{1} and x ^{ eq 2} _{1}. In fact the curve for the system PS(1)–acetone(2) also shows the existence of two phases, although this is not visible. The first step in the procedure is to determine whether a spinodal point exists (this is a necessary condition for phase separation). In the figure, the two spinodal points are given by the compositions x ^{ sp 1} _{1} and x ^{ sp 2} _{1}. The spinodal condition is given by ((∂ ^{2} g^{mix} /RT)/∂x ^{2} ) = 0, that is, an inflection point on the curve. Once a spinodal point has been found (using a Newton–Raphson method), the next step is to find the point of tangent of a line originating at the spinodal point. In the figure, this is the line connecting x ^{ sp 1} _{1} and x _{1}. This point is just to the left of x ^{ eq 2} _{1} (i.e., we are not yet quite at the equilibrium concentration after one step). Once the first tangent has been found, the point of tangent opposite is then found in a similar way. This process is repeated until the change in the composition at the tangent point is within a specified tolerance. At this point, the equilibrium values have been calculated.
Figure 3.7 Vapor pressure curve for the system methane(1)–tetratetracontane(2) at T = 423.2 K. The dashed line is PCSAFT, the solid line is PCSAFT with a binary interaction parameter kij = 0.04. The points are experimental data. (From von Solms, N. et al., Ind. Eng. Chem. Res., 42, 1098, 2003. With permission.)
Figure 3.8 Vapor pressure curve for the system propyl acetate(1)–polystyrene(2) at T = 343.15 K. The dashed line is PCSAFT (the lowest line on the plot), the solid line is modification 1, and the dotted line is modification 2. Points are experimental data. All lines are pure predictions (kij = 0). (From von Solms, N. et al., Ind. Eng. Chem. Res., 42, 1098, 2003. With permission.)
Figure 3.9 Pressure–weight fraction plot of polypropylene(1)–diisopropyl ketone(2) at T = 318 K. Polypropylene molecular weight = 20,000. Comparison of experimental data with the predictions of original (solid line) and the simplified version (dotted line) of PCSAFT. In both curves, the interaction parameter kij = 0. (From Fluid Phase Equilib., 215, Kouskoumvekaki, I.A., von Solms, N., Michelsen, M.L., and Kontogeorgis, G.M., Application of a simplified perturbed chain SAFT equation of state to complex polymer systems, 71–78, Copyright 2004, with permission from Elsevier.)
PCSAFT 


% AAD 
Simplified Version 
Original Version 
Cyclic hydrocarbons  
PS–cyclohexane 
13 
16 
PS–benzene 
28 
13 
PS–ethyl benzene 
3 
6 
PS–mxylene 
25 
16 
PS–toluene 
18 
7 
PVAc–benzene 
6 
10 
Chlorinated hydrocarbons  
PS–carbon tetrachloride 
18 
12 
PS–chloroform 
30 
11 
PP–dichloromethane 
59 
74 
PP–carbon tetrachloride 
55 
47 
Esters  
PS–propyl acetate 
5 
21 
PS–butyl acetate 
3 
25 
PVAc–methyl acetate 
3 
2 
PVAc–propyl acetate 
19 
18 
Ketones  
PS–acetone 
6 
26 
PS–diethyl ketone 
7 
28 
PS–methyl ethyl ketone 
14 
12 
PVAc–acetone 
4 
7 
PP–diethyl ketone 
16 
27 
PP–diisopropyl ketone 
4 
11 
PVAc–methyl ethyl ketone 
7 
6 
Amines  
PVAc–propylamine 
4 
3 
PVAc–isopropyl amine 
17 
16 
Alcohols  
PVAc–1propanol 
56 
54 
PVAc–2propanol 
84 
73 
PVAc–1butanol 
59 
59 
PVAc–2butanol 
39 
36 
PVAc–2methyl1propanol 
29 
29 
Overall 
23 
24 
Source: Kouskoumvekaki, I.A. et al., Fluid Phase Equilib., 215, 71, 2004. With permission.
Average percentage deviation between experimental and predicted equilibrium pressure curves.
Figure 3.10 Pressure–weight fraction plot of poly(vinyl acetate)(1)–2methyl1propanol(2) at T = 313 K. Comparison of experimental data with prediction (kij = 0) and correlation (kij = −0.012) results of simplified PCSAFT. Poly(vinyl acetate) molecular weight = 167,000. (From Fluid Phase Equilib., 215, Kouskoumvekaki, I.A., von Solms, N., Michelsen, M.L., and Kontogeorgis, G.M., Application of a simplified perturbed chain SAFT equation of state to complex polymer systems, 71–78, Copyright 2004, with permission from Elsevier.)
Figure 3.11 Illustration of the method of alternating tangents. The solid line is the system methanol(1)–cyclohexane(2). The dotted line is the system PS(1)–acetone. The two spinodal points are indicated by x sp1 1 and x sp2 1. The equilibrium (binodal) points are indicated by x eq1 1 and x eq2 1. Starting from a spinodal point, the equilibrium values can be calculated by solving for only one point at a time. (From Fluid Phase Equilib., 222–223, von Solms, N., Kouskoumvekaki, I.A., Lindvig, T., Michelsen, M.L., and Kontogeorgis, G.M., A novel approach to liquidliquid equilibrium in polymer systems with application to simplified PCSAFT, 87–93, Copyright 2004, with permission from Elsevier.)
Figures 3.12 through 3.15 are examples of binary LLE for polymer systems. Figure 3.12 shows results for the system polystyrene–methylcyclohexane for different molecular weights of polystyrene. The experimental data are from the classic work of Dobashi et al. ^{109} The lines are simplified PCSAFT correlations with k_{ij} = 0.0065 for polystyrene molecular weights 10,200, 46,400 and 719,000 in order of increasing critical solution temperature. The data are reasonably well correlated over a very large range of molecular weight with a single value of the binary interaction parameter, k_{ij} . The binary interaction parameter was adjusted to give the correct upper critical solution temperature. However, the correct critical solution concentration is not obtained, although the experimental trends are correctly predicted by the model: The critical solution temperature increases, and the polymer weight fraction at the critical solution temperature decreases with increasing molecular weight.
Figure 3.13 shows results for the system polyisobutylene–diisobutyl ketone at different polymer molecular weights. The experimental data are from Shultz and Flory. ^{110} The lines are simplified PCSAFT correlations. A single binary interaction parameter (k_{ij} = 0.0053) was used for all three systems, although it seems that there is a weak dependence of molecular weight on k_{ij} . Incorporating a functional dependence of k_{ij} on molecular weight (e.g., a linear fit) would improve the correlation. It should also be noted that these three systems represent a very large range of molecular weights.
Figure 3.14 shows the results for a single molecular weight of HDPE in five different nalkanol solvents from npentanol up to nnonanol. The results are well correlated using simplified PCSAFT with a small value of the binary interaction parameter k_{ij} . A k_{ij} value of around 0.003 gives a good correlation for all the systems, except HDPE–npentanol. In the figure, a small value (k_{ij} = 0.0006) was used to correlate the data, although the data is also well predicted by simplified PCSAFT (k_{ij} = 0), giving an error in the upper critical solution temperature of 3 K in the case of HDPE–npentanol.
Figure 3.15 shows the results for the system HDPE–butyl acetate. This system displays both UCST and LCST behaviors. A single binary interaction parameter (k_{ij} = 0.0156) was used to correlate the data for both molecular weights shown. The binary interaction parameter was adjusted to give a good correlation for the UCST curve at the higher molecular weight (64,000). As mentioned earlier, the LCST curve is rather insensitive to k_{ij} . Nevertheless, the LCST curve is reasonably well correlated using this value. The prediction (k_{ij} = 0) is almost as good for the LCST curve, although the UCST will then be substantially underpredicted.
Figure 3.12 Liquid–liquid equilibrium in the system polystyrene–methyl cyclohexane for different molecular weights of polystyrene. The experimental data are from Dobashi et al. 109 The lines are simplified PCSAFT correlations with kij = 0.0065 for polystyrene molecular weights 10,200, 46,400 and 719,000 in order of increasing critical solution temperature. (From Fluid Phase Equilib., 222–223, von Solms, N., Kouskoumvekaki, I.A., Lindvig, T., Michelsen, M.L., and Kontogeorgis, G.M., A novel approach to liquidliquid equilibrium in polymer systems with application to simplified PCSAFT, 87–93, Copyright 2004, with permission from Elsevier.)
Figure 3.13 Liquid–liquid equilibrium in the system polyisobutylene–diisobutyl ketone. PCSAFT parameters for diisobutyl ketone were obtained by fitting to experimental liquid density and vapor pressure data in the temperature range 260–600 K. This data was taken from the DIPPR database. The parameters were: m = 4.6179, ɛ/k = 243.72 K, and σ = 3.7032 Å. Average percent deviations were 1.03% for vapor pressure and 0.64% for liquid density. Lines are simplified PCSAFT correlations with kij = 0.0053, the same at all three molecular weights. (From Fluid Phase Equilib., 222–223, von Solms, N., Kouskoumvekaki, I.A., Lindvig, T., Michelsen, M.L., and Kontogeorgis, G.M., A novel approach to liquid–liquid equilibrium in polymer systems with application to simplified PCSAFT, 87–93, Copyright 2004, with permission from Elsevier.)
Figure 3.14 Liquid–liquid equilibrium for HDPE with nalkanols. Lines are simplified PCSAFT correlations for each of the five solvents (pentanol highest, nonanol lowest). Polymer molecular weight is 20,000. Binary interaction parameters are as follows: pentanol: 0.0006; hexanol: 0.003; heptanol: 0.0025; octanol: 0.0033; and nonanol: 0.0029. (From Fluid Phase Equilib., 222–223, von Solms, N., Kouskoumvekaki, I.A., Lindvig, T., Michelsen, M.L., and Kontogeorgis, G.M., A novel approach to liquid–liquid equilibrium in polymer systems with application to simplified PCSAFT, 87–93, Copyright 2004, with permission from Elsevier.)
Figure 3.15 Liquid–liquid equilibrium in the system HDPE–butyl acetate. The system displays both upper and lower critical solution temperature behaviors. The experimental data for molecular weights 13,600 and 64,000. Lines are simplified PCSAFT correlations with kij = 0.0156 for both molecular weights. (From Fluid Phase Equilib., 222–223, von Solms, N., Kouskoumvekaki, I.A., Lindvig, T., Michelsen, M.L., and Kontogeorgis, G.M., A novel approach to liquid–liquid equilibrium in polymer systems with application to simplified PCSAFT, 87–93, Copyright 2004, with permission from Elsevier.)
Figure 3.16 shows a ternary phase diagram for the system polystyrene–acetone– methylcyclohexane. The binary interaction parameters were obtained by fitting to the individual binary systems. The ternary coexistence curves are predictions. The algorithm for finding ternary LLE in systems containing polymers is an extension of the binary algorithm discussed earlier and was developed by Lindvig et al. ^{98}
Figure 3.16 Ternary phase diagram for the system polystyrene–acetone–methylcyclohexane. The binary interaction parameters were obtained by fitting to the individual binary systems. The ternary coexistence curves are predictions.
Figure 3.17 Highpressure equilibrium for mixtures of poly(ethylenecomethyl acrylate) (EMA) and propylene for different repeatunit compositions of the EMA. Comparison of experimental cloudpoint measurements to calculation results of the PCSAFT equation of state. (EMA [0% MA] is equal to LDPE: open diamonds and dashed line.) (From Gross, J. et al., Ind. Eng. Chem. Res., 42, 1266, 2003. With permission.)
Figure 3.17 from Gross et al. ^{94} shows high pressure equilibrium for mixtures of poly(ethylenecomethyl acrylate) (EMA) and propylene for different repeat unit compositions of EMA. As repeat units of methyl acrylate (MA) are added to the polyethylene chain, the demixing pressure at first declines, but then increases as the composition of MA increases. This effect is correctly predicted by PCSAFT.
Finally, Figure 3.18 shows a comparison of SAFTVR and simplified PCSAFT in a recent study ^{100} where the two models were compared in their ability to model multicomponent phase equilibrium in systems typical of real polyethylene reactors. The system shown here is polyethylene/nitrogen/1butene. The results of the simplified PCSAFT and SAFTVR calculations for this ternary are consistent. From the figure, one can see that as butene in the vapor is replaced by nitrogen, the calculated absorption of butene decreases (not surprisingly—there is less of it to absorb). However, it is also clear that as nitrogen in the vapor is replaced by butene, absorption of nitrogen increases, even though there is less nitrogen to absorb. This suggests that there may be some enhancement/ inhibition of absorption effect.
In the time since the previous edition of this work (2008), around 150 articles have appeared applying SAFT to various polymer systems. Certain newer themes become apparent here:
The use of density functional theory combined with equations of state to examine surface and structural properties is a relatively new phenomenon, resulting in equations such as iSAFT. ^{111} ^{,} ^{112} For example, Jain et al. ^{111} used SAFT and density functional theory to probe the structure of tethered polymer chains.
Perhaps the most frequently occurring, relatively new idea is that of groupcontribution SAFT, where there has been a great deal of activity. ^{107} ^{,} ^{113} ^{–} ^{122} Part of the motivation for a groupcontribution method is the need for an equation that can be readily extended to complex molecules, where purecomponent properties may be unavailable, and for which SAFT types of equation are well suited. Examples of complex systems where SAFT has been applied are biopolymers, ^{123} hyperbranched polymers, ^{124} and asphaltenes. ^{125}
Figure 3.18 Gas absorptions in amorphous PE calculated with SAFTVR and simplifiedPCSAFT for a range of vapor compositions of the ternary mixture of (but1ene + nitrogen + the reference PE [MW = 12,000 g mol−1]) at T = 80°C: (a) 100 vapor mol% butene (binary mixture); (b) 75% but1ene, 25% nitrogen; (c) 50% but1ene, 50% nitrogen (with a vertical scale chosen to highlight butene absorption); (d) 50% but1ene, 50% nitrogen (with a vertical scale chosen to highlight nitrogen absorption); (e) 20% butene, 80% nitrogen; and (f) 100% nitrogen (binary mixture). In each case, solid curves represent SAFTVR calculations and dashed curves represent simplifiedPCSAFT calculations. (From Haslam, A.J. et al., Fluid Phase Equilib., 243, 74, 2006. With permission).
Another encouraging feature is the use of SAFTtype models in industrial applications. In addition to the asphaltene example noted earlier, ^{125} work has appeared relevant to refrigeration ^{126} ^{,} ^{127} and catalytic polymerization processes. ^{128}
Finally, it can be mentioned that since the review article by Müller and Gubbins, ^{77} several monographs reviewing various aspects of SAFT have appeared.^{129–131}
Attempting to summarize in a few words the current status in polymer thermodynamics, we could state
Some of the future challenges in the area of polymer thermodynamics will involve
The FH model was originally developed as a model for the entropy of mixing for mixtures containing molecules of different size, but it was soon modified also to account for energetic interactions. The model can be formulated in terms of the excess Gibbs energy as follows (Lindvig et al. ^{37} ):
Using basic thermodynamics, the following expression for the activity coefficient is obtained:
where the combinatorial term is given by
and the residual term is
The aforementioned formulation of the FH model is slightly different from the conventionally used formulation using the FH interaction parameter (χ_{12}), although there is an interrelationship based on the simple equation shown above.
For a binary mixture, the multicomponent equation reduces to the traditional FH residual term:
BR 
butadiene rubber 
CPA 
cubic plus association 
CST 
critical solution temperature 
EAC 
ethyl acetate 
EoS 
equation of state 
EFV 
entropicfree volume 
EMA 
poly(ethylenecomethyl acrylate) 
FH 
Flory–Huggins (model/equation/interaction parameter) 
FV 
freevolume 
GC 
group contribution (method/principle) GC Fl(ory) group contribution Flory equation of state 
GCVOL 
group contribution volume (method for estimating the density) 
HDPE 
high density polyethylene 
LCST 
lower critical solution temperature LJSAFT Lennard–Jones SAFT 
LLE 
liquid–liquid equilibria 
MA 
methyl acrylate 
MCSL 
Mansoori–Carnahan–Starling–Leland 
PBMA 
polybutyl methacrylate PCSAFT perturbed chainSAFT 
PDMS 
polydimethylsiloxane 
PE 
polyethylene 
PEMA 
polyethylmethacrylate 
PEO 
polyethylene oxide 
PHSC 
perturbed hardsphere chain 
PIB 
polyisobutylene 
PS 
polystyrene 
PVAC 
polyvinyl acetate 
PVC 
polyvinyl chloride 
SAFT 
statistical associating fluid theory SAFTVR SAFTvariable range 
SLE 
solid–liquid equilibria 
SLLE 
solid–liquid–liquid equilibria 
SRK 
Soave–Redlich–Kwong 
TPT 
thermodynamic perturbation theory 
UCST 
upper critical solution temperature 
UNIFAC 
UNIQUAC functional activity coefficient (a method for estimating activity coefficients) 
UFV 
UNIFACFV 
vdW1f 
van der Waals one fluid (mixing rules) 
VLE 
vapor–liquid equilibria 
VOC 
volatile organic content 