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A Predictive Group-Contribution Simplified PC-SAFT Equation of State: Application to Polymer Systems Amra Tihic, Georgios M. Kontogeorgis,* Nicolas von Solms, and Michael L. Michelsen Centre for Phase Equilibria and Separation Processes (IVC-SEP), Department of Chemical Engineering, Technical UniVersity of Denmark, DK-2800 Lyngby, Denmark
Leonidas Constantinou Frederick Research Center, P.O. Box 24729, Nicosia, Cyprus
A group-contribution (GC) method is coupled with the molecular-based perturbed-chain statistical associating fluid theory (PC-SAFT) equation of state (EoS) to predict its characteristic pure compound parameters. The estimation of group contributions for the parameters is based on a parameter database of 400 low-molecularweight compounds estimated by fitting experimental vapor pressures and liquid densities. The method has been successfully used for estimating the PC-SAFT parameters for common polymers. Specifically, using the new polymer parameters as calculated from the proposed GC scheme, the simplified PC-SAFT yields rather good predictions of polymer densities and gives promising modeling results of various binary polymer mixtures exhibiting both vaporsliquid and liquidsliquid phase equilibria. In summary, the data required for calculating polymer phase equilibria with the proposed method are the molecular structure of the polymer of interest in terms of functional groups and a single binary interaction parameter for accurate mixture calculations. 1. Introduction The thermodynamics and phase behavior of polymer systems play a key role in the integration of polymer production, processes, and material development.1 The more advanced the properties and performances of the final polymer product are, the more complex its phase behavior may be. Accurate prediction of the phase equilibria of polymer mixtures requires thermodynamic models that are able to deal with large size differences between molecules, to deal with polydisperse polymers, and to account for strong deviations from ideal behavior over wide ranges of operating conditions. For the past several decades, investigations of polymer systems with equations of state show that such models are efficient tools but their application is often constrained by the availability of the required input parameters. Advances in statistical mechanics have allowed developments of different semiempirical equations of state based on molecular principles with a strong theoretical foundation. Among the different approaches available to date, those based on the Wertheim perturbation theory2–5 resulted in a family of equations of state (EoS) known as statistical association fluid theory (SAFT). Since the original SAFT model,6–8 several modifications have appeared with one of the most recent versions being perturbed-chain SAFT (PC-SAFT).9 Two reviews10,11 provide more detailed discussions of recent developments and applications of the various types of SAFT. In order to reduce the computational time and programming efforts without essentially affecting the performance of the model, we have modified the original PC-SAFT9 and developed the simplified PC-SAFT12 EoS that has already been applied to vaporsliquid (VLE) and liquidsliquid equilibria (LLE) of various polymer systems.13–15 Both those versions of the PC-SAFT model, the original and simplified, are able to predict the effects of molecular weight, copolymerization, and hydrogen bonding on the thermodynamic * To whom correspondence should be addressed. Tel.: +45 45 25 28 59. Fax: + 45 45 88 22 58. E-mail:
[email protected].
properties and phase behavior of complex fluids including solvents, monomers, and polymer solutions and blends. Complete description of these systems requires three physically meaningful temperature-independent parameters: the segment number (m), the hard-core segment diameter (σ), and the segmentssegment interaction energy parameter (/k) for each pure nonassociating fluid. They are typically fitted to vapor pressure and liquid density data. Associating fluids require two additional compound parameters, the association energy (AB) and the association volume (κAB). Since high-molecular-weight compounds, such as polymers, do not have a detectable vapor pressure and commonly undergo thermal degradation before exhibiting a critical point, determination of EoS parameters is often based solely on experimental density data.16 Unfortunately, derivation of polymer parameters based only on density usually results in poor prediction of phase equilibria with the SAFT EoS. Recent publications demonstrate that this shortcoming remains even in the newest versions of the SAFT EoS, such as PC-SAFT and simplified PC-SAFT.17–21 Hence, there is a need for a predictive calculation method for polymer EoS parameters. As PC-SAFT models molecules as chains of united atom segments with their own parameters, the model is adaptable for developing a group-contribution scheme. Group-contribution (GC) methods are the most widely used techniques for estimating and predicting thermophysical properties of pure compounds and mixtures.22 Use of the GC principle in models for phase equilibrium calculations can greatly enhance their predictive capabilities. The GC methods can be divided into two classes. Methods in the first class are those that estimate the property of a compound as a summation of the contributions of simple first-order groups (FOG) that may occur in the molecular structure, such as CH2 and OH. This approach is necessary where there is no theoretical basis for the groups’ identification and the groups are not able to capture proximity effects or isomer differences. The second class includes methods that attempt to capture fine structural differences by additionally
10.1021/ie0710768 CCC: $40.75 2008 American Chemical Society Published on Web 12/28/2007
Ind. Eng. Chem. Res., Vol. 47, No. 15, 2008 5093 12
Table 1. Modifications of the Simplified PC-SAFT
EoS Compared to the Original PC-SAFT9 EoS
9
simplified PC-SAFT12
original PC-SAFT
gijhs )
( )
( )
2
didj 2 3ζ2 didj 2 2ζ2 1 + + 2 1 - ζ3 di + dj (1 - ζ ) di + dj (1 - ζ )3 3 3
[
( )
ghs(η) )
]
1-η⁄2 (1 - η)3
4η - 3η a˜hs ) (1 - η)2
ζ23 ζ23 1 3ζ1ζ2 a˜hs ) + + - ζ0 ln(1 - ζ3) ζ0 1 - ζ3 ζ (1 - ζ )2 ζ32 3 3
2
Table 2. First-Order Group (FOG) Contributions for the m, mσ3, and mE/k Parameters contributions first-order groups (FOG) “sCH3” “sCH2” “sCHC
