3.2.1  Rules of Thumb Based on Solubility Parameters

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 so-called 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 two-phase 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.”

3.2.2  Rule of Thumb Based on the Infinite Dilution Activity Coefficient

Since in several practical cases concerning polymer–solvent systems, the “solvent” is only present in very small (trace) amounts, the so-called 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 weight-based 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 Entropic-FV, and the UNIFAC-FV 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).

3.2.3  Rule of Thumb Based on the Flory–Huggins Model

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 so-called FH interaction parameter χ₁₂, is constant, the activity coefficient is given by the equation:

where

• φ _i can be volume or segment fractions
• r is the ratio of the polymer volume to the solvent volume V ₂/V ₁ (approximately equal to the degree of polymerization)

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 χ₁₂ ≤ 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.

There are several, still rather obscure issues about the FH model, which we summarize here together with some recent developments:

1. The FH parameter is typically not a constant and should be estimated from experimental data. Usually, it varies with both temperature and concentration, which renders the FH model useful only for describing experimental data. It cannot be used for predicting phase equilibria for systems for which no data is available. Moreover, when fitted to the critical solution temperature, the FH model cannot yield a good representation of the whole shape of the miscibility curve with a single parameter.
2. Accurate representation of miscibility curves is possible with the FH model using suitable (rather complex) equations for the temperature and the concentration dependence of the FH parameter. ³⁴
3. In some cases, a reasonable value of the FH parameter can be estimated using solubility parameters via the equation:
   3.9 $${\chi }_{12}={\chi }_s+{\chi }_h=0.35+\frac{V_1}{RT}({\delta }_1-{\delta }_2{)}^2$$

   
   Equation 3.9, without the empirical 0.35 factor, is derived from the regular solution theory. The constant 0.35 is added for correcting for the deficiencies of the FH combinatorial term. These deficiencies become evident when comparing experimental data for athermal polymer and other asymmetric solutions to the results obtained with the FH model. A systematic underestimation of the data is observed, as discussed extensively in the literature, ^{28 , 35} which is often attributed to the inability of the FH model in accounting for the FV differences between polymers and solvents or between compounds differing significantly in size such as n-alkanes with very different chain lengths. The term, which contains the “0.35 factor,” corrects in an empirical way for these FV effects. However, and although satisfactory results are obtained in some cases, we cannot generally recommend Equation 3.9 for estimating the FH parameter. Moreover, for many nonpolar systems with compounds having similar solubility parameters, the empirical factor 0.35 should be dropped.
4. Lindvig et al. ³⁶ proposed an extension of the FH equation using the Hansen solubility parameters for estimating activity coefficients of complex polymer solutions.
   3.10 $$\begin{array}{c}\text{ln}{\gamma }_1=\text{ln}\frac{{\phi }_1}{x_1}+1-\frac{{\phi }_1}{x_1}+{\chi }_{12}{\phi }_2^2\hfill \\ {\begin{array}{c}\chi \end{array}}_{12}=0.6\frac{V_1}{RT}[({\delta }_{d1}-{\delta }_{d2}{)}^2+0.25({\delta }_{p1}-{\delta }_{p2}{)}^2+0.25({\delta }_{h1}-{\delta }_{h2}{)}^2]\end{array}$$

   
   In order to achieve that, Lindvig et al. ³⁶ have, as shown in Equation 3.10, employed a universal correction parameter, which has been estimated from a large number of polymer–solvent VLE data. Very good results are obtained, especially when the volume-based combinatorial term of FH is employed, also for ternary polymer–solvent systems. ³⁷
5. Based on the FH model, several techniques have been proposed for interpreting and for correlating experimental data for polymer systems, for example, the so-called Schultz–Flory (SF) plot. Schultz and Flory ³⁸ have developed, starting from the FH model, the following expression, which relates the critical solution temperature (CST), with the theta temperature and the polymer molecular weight:
   3.11 $$\frac{1}{\text{CST}}=\frac{1}{\mathrm{\Theta }}[1+\frac{1}{\psi }(\frac{1}{\sqrt{r}}+\frac{1}{2r})]$$

   
   where
   • ψ = ((1/2)−χ _s ) is the entropic parameter of the FH model (Equation 3.9)
   • r is the ratio of molar volumes of the polymer to the solvent
   This parameter is evidently dependent on the polymer’s molecular weight. The SF plot can be used for correlating data of critical solution temperatures for the same polymer–solvent system, but at different polymer molecular weights. This can be done, as anticipated from Equation 3.11, because the plot of 1/CST against the quantity in parentheses in Equation 3.11 is linear. The SF plot can also be used for predicting CST for the same system but at different molecular weights than those used for correlation as well as for calculating the theta temperature and the entropic part of the FH parameter. It can be used for correlating CST/molecular weight data for both the UCST and LCST areas. Apparently different coefficients are needed.

3.3.1  The Free-Volume Concept

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 van-Laar energetic term by a nonrandom local-composition 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. Local-composition terms like those appearing in NRTL, UNIQUAC, and UNIFAC provide a flexibility, which cannot be accounted for by the single-parameter 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 so-called Entropic-FV model (described later). According to this equation, FV is just the “empty” volume available to the molecule when the molecules’ own (hard-core or closed-packed V*) volume is subtracted.

The FV is not the only concept, which is loosely defined in this discussion. The hard-core 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 hard-core 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 ).

3.3.2  Entropic-FV Model

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 UNIFAC-FV by Oishi and Prausnitz. ⁴² The UNIFAC-FV model was developed for solvent activities in polymers, but it cannot be successfully applied to LLE.

A similar to UNIFAC-FV but somewhat simpler approach, which can be readily extended to multicomponent systems and LLE, is the so-called Entropic-FV 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 Entropic-FV model is derived from statistical mechanics, using a suitable form of the generalized van der Waals partition function.

The residual term of Entropic-FV is taken by the so-called new or linear UNIFAC model, which uses a linear-dependent parameter table ⁴³ :

This parameter table has been developed using the combinatorial term of the original UNIFAC model. As with UNIFAC-FV, no parameter re-estimation has been performed. The same group parameters are used in the “linear UNIFAC” and in the Entropic-FV models.

A common feature for both UNIFAC-FV and Entropic-FV 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 group-contribution 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 UNIFAC-FV and Entropic-FV, especially the former, are rather sensitive to the density values used for the calculations of solvent activities.

As already mentioned, the Entropic-FV 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 Entropic-FV with V_w = b, while the latter term is a regular solution theory or van Laar-type term.

3.3.3  Results and Discussion

Table 3.2 presents an overview of the results with Entropic-FV 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:

1. Satisfactory predictions are obtained for solvent activities, even at infinite dilution, and for nonpolar, as well as for complex polar and hydrogen bonding systems including solutions of interest to paints and coatings. Rather satisfactory predictions are also achieved when mixed solvents and copolymers are present.

   Table 3.2   Applications of the Entropic-FV Model

   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

   Table 3.3   Prediction of the Solubility for Characteristic Polymer–Solvent Systems Using the Ω∞ 1 Rule of Thumb and Two FV Models for Solvent Selection

   System	Experiment	Entropic-FV	UNIFAC-FV
   PBMA/nC10	NS	6.5 (–)	6.1 (–)
   PBMA/xylene	S	2.3 (S)	3.6 (S)
   PBMA/CHCl3	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.

   Table 3.4   Prediction of Infinite Dilution Activity Coefficients for Polyisoprene (PIP) Systems with Two Predictive Group Contribution Models

   PIP Systems	Exper. Value	Entropic-FV	UNIFAC-FV
   +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%)
   +1,2-Dichloroethane	4.25	5.5 (29%)	6.5 (54%)
   +CCl4	1.77	2.1 (20%)	1.8 (0%)
   +1,4-Dioxane	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%)
   +n-Hexane	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.

   Table 3.5   Average Absolute Deviations between Experimental and Calculated Activity Coefficients of Paint-Related Polymer Solutions Using the FH/ Hansen Method and Two FV Models

   Model	% AAD (Systems in Database)	% AAD Araldite 488	% AAD Eponol-55
   FH/Hansen, Volume (Equation 3.10)	22	31	28
   Entropic-FV	35	34	30
   UNIFAC-FV	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.

   Table 3.6   Mean Percentage Deviations between Experimental and Calculated Activity Coefficients of Solvents in Various Nearly Athermal Solutions

   % AAD Infinite Dilution Conditions γ∞ 1 (Ω∞ 1 for Polymers)	Entropic-FV	UNIFAC-FV	Flory-FV
   Short n-alkanes/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.

   Table 3.7   Mean Percentage Deviations between Experimental and Calculated Activity Coefficients of Heavy Alkanes Solutes in Alkane Solvents

   	Entropic-FV	UNIFAC-FV	Flory-FV
   % 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, γ2
   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.

   Table 3.8   Average Absolute Logarithmic Percentage Deviations between Experimental and Predicted Equilibrium Pressures and Average Absolute Deviation (×100) between Calculated and Experimental Vapor Phase Compositions (Mole Fractions) for Various Ternary Polymer–Mixed Solvent Systems

   Sys. No.	Variable	SAFT	EFV/UQ	FH	Pa-Ve	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.

   Table 3.9   Prediction of Liquid–Liquid Equilibria for Polymer–Solvent Systems with Various Thermodynamic Models

   Polymer System	Molecular Weight	Entropic-FV	N0065w UNIFAC	GC-Flory
   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/n-butyl acetate	13,600	10	82	72
   HDPE/n-butyl acetate	20,000	22	97	70
   HDPE/n-butyl acetate	61,100	29	107	71
   PMMA/1-chloro butane	34,760	53	—	—
   PBMA/n-pentane	11,600	Hourglass	—	—
   PBMA/n-octane	11,600	155	—	—

   Notes: PS, polystyrene; HDPE, high-density 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 Entropic-FV 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.)

2. Qualitatively correct and occasionally also quantitatively satisfactory representation of LLE for binary and ternary polymer–solvent systems is achieved, especially for the UCST-type phase behavior. A single investigation for SLLE also shows good results.
3. Less satisfactory results are obtained for polymer blends, where FV effects are not as dominant as for polymer solutions.

Some additional observations for specific cases are hereafter presented:

1. Athermal solutions, for example, polyolefins with alkanes offer a way of testing FV terms and numerous such investigations have been presented. FV models perform generally better than those that do not contain volume-dependent terms. ⁶⁰ Better FV terms than that of Entropic-FV (Equation 3.13) have been proposed ^{41 , 51 , 52 , 57 , 60} and they may serve as the basis for future developments resulting in even better activity coefficient models for polymer solutions. A rigorous test for newly developed FV expressions is provided by athermal alkane systems, especially the activity coefficients of heavy alkanes in short-chain ones.
2. A difference of 10°–30° should be expected in UCST predictions with Entropic-FV, see Table 3.9.
3. An alternative to Entropic-FV successful approach is the FH/Hansen model presented previously (Equation 3.10). In this case, best results are obtained when the original FH combinatorial rather than a FV term is used together with HSP. It seems that HSP incorporate some FV effects.

3.3.4  Correlative Versions of the Entropic-FV model

Both UNIFAC-FV and Entropic-FV 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.

3.4.1  Introduction

“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 hard-sphere 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.

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 spin-offs that the remainder of this discussion is concerned.

3.4.2  SAFT Equation of State—History and Introduction

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 molecular-based 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 molecular-based 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 IVC-SEP 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.

3.4.3  Original SAFT 74

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 equal-sized hard spheres (b). Next a dispersive potential is added to account for attraction between the spheres (c). Typical potentials are the square-well 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 square-well 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

• a ^seg is the Helmholtz energy of the segment, including both hard-sphere reference and dispersion terms
• a ^chain is the contribution from chain formation
• a ^assoc is the contribution from association

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 size-parameter σ, and an energy parameter, ɛ. If the molecule is self-associating, there are two further parameters, which characterize the volume (κ ^A_iB_i ) and energy (ɛ ^A_iB_i ) of association. The association term is given for mixtures by

where

• X^A_i is the mole fraction of molecules i not bonded at site A
• M_i is the number of association sites on molecule i

We have

where

• N_AV is the Avogadro’s number
• ρ _j is the molar density of j
• Δ ^A_iB_j is the association strength given by
  3.25 $${\mathrm{\Delta }}^{A_i{B}_j}=d_{ij}^3{g}_{ij}(d_{ij}{)}^{seg}{\kappa }^{A_i{B}_j}[\exp \left(\frac{{\epsilon }^{A_i{B}_j}}{kT}\right)-1]$$

  

Here d_ij is a temperature-dependent size-parameter 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” hard-sphere 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 hard-sphere 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 hard-sphere reference and a dispersion contribution:

The Carnahan–Starling equation ⁶⁶ is used for both pure components and mixtures to give

where for mixtures η = ζ₃ 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 = (6/2^0.5π)η.

Most of the results of this initial paper are comparisons with simulation data for chains with various parameters, although pure-component parameters for six hydrocarbons and two associating fluids were fitted. No results for mixtures of real fluids are presented.

3.4.4  “Original” SAFT 78

Probably the main contribution of the Huang and Radosz ⁷⁸ version of SAFT was the regression of pure-component 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 hard-sphere 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

• D_ij are universal constants
• u is the temperature-dependent energy parameter given by u = u ⁰(1 + e/kT) and e/k is a constant set to −10, with a few exceptions

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 hard-sphere 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 pure-component 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 PC-SAFT also had pure-component 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 PC-SAFT in the few years it has been in existence, both in industry and in academia.

3.4.5  Simplified SAFT 80

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

• “average” chain length m ∑ _i x _i m _i
• Z_M = 36 is the maximum coordination number
• v_s = 1/ρm is the total molar volume of a segment
  3.43 $$⟨v^{*}Y⟩=\frac{N_{AV}{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_j{x}_i{x}_j{m}_j(d_{ij}^3/\sqrt{2})[\exp (u_{ij}/kT)-1]}{{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_j{x}_i{x}_j{m}_i{m}_j}$$

  

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 pure-component 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 cross-association in some detail. This scheme is completely general and applicable to any equation of state incorporating association. Combining rules for cross-association also need to be introduced, however, and are far from self-evident.

3.4.6  LJ-SAFT 85 , 86

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

Apart from using a different reference system, the notation in Kraska and Gubbins does not follow the customary SAFT notation, nor are the pure-component 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.

3.4.7  SAFT-VR 88

SAFT-VR is the version of SAFT developed by George Jackson and coworkers first at the University of Sheffield and currently at Imperial College (Gil-Vilegas et al. ⁸⁸ ; McCabe et al. ⁸⁹ ). SAFT-VR is identical to the Huang and Radosz version except in the dispersion contribution. This term incorporated attraction in the form of a square-well potential. Thus, in addition to a segment being characterized by a size and an energy parameter, the square-well width (λ) is also included as a pure-component 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 SAFT-VR greater flexibility, since we now have an extra pure component to “play with.” Although it is generally desirable to describe pure-component liquid densities and vapor pressure with the minimum number of parameters, the extra variable-range 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 second-order term in Equation 3.50 is given by

where

3.4.8  PC-SAFT 84

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 PC-SAFT 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 hard-sphere 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 second-order term:

where

where

• x = r/σ
• $\tilde{u}\left(x\right)=u\left(x\right)/\epsilon$
  is the reduced intermolecular potential

The radial distribution function g^hc is now an interchain function rather than a segment function as before. This is a key point in PC-SAFT. 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 n-alkanes 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 pure-component data of n-alkanes. 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 one-fluid theory.

Since this first PC-SAFT paper appeared, the authors rapidly published a series of further papers applying PC-SAFT to polymers, ^{91 , 92} associating fluids ⁹³ and copolymers. ⁹⁴