The thermodynamics of polymeric systems play an important role in the polymer industry and are often a key factor in polymer production, processing and material development, especially for the design of advanced polymeric materials. Many polymeric products are produced with a solvent or diluent (or a mixture of them) and often with other low molecular weight compounds (plasticizers, among others). A problem which often arises is how to remove the low molecular weight constituent(s) from the final product (polymer). The solution to this problem involves, among other tasks, solving the vapor-liquid equilibrium (VLE) and/or the vapor-liquid-liquid equilibrium (VLLE) problem. Other applications of polymer thermodynamics directly involve the polymerization processes. For example, several processes such as the production of PET (polyethylene terephthalate) are carried out in two-phase (vapor-liquid) reactors. Phase equilibrium compostions of the reacting components will determine their phase concentrations and thus the outcome of the polymerization reaction. Another example is the case of LDPE (Low Density PolyEthylene) made in autoclave reactors where it may be desirable to perform the polymerization reaction nearby but outside the two-liquid phase region, but close to it, which makes accurate liquid-liquid equilibrium (LLE) information at high pressure essential. During PE (polyethylene) or PP (polypropylene) industrial processing, for example, deposition of the polymer on the reactor surface, heat exchangers and flash drums frequently occurs and this can cause clogging in pipelines. Modeling solid-liquid equilibrium (SLE) is a useful basis from which to gain a better understanding of these industrial polymer problems and thus to avoid their occurrence.
Analogous to the modeling of conventional phase equilibrium, there are two basic approaches available to describe phase equilibrium of polymer-solvent mixtures: activity coefficient models and equations of state (EOS). There are several drawbacks to the activity coefficient approach, for example: it is hard to define standard states, especially for supercritical components; the parameters of the activity coefficient models are very temperature dependent, and critical phenomena are not predicted because different models are used for the vapor and liquid phases. Furthermore, other thermodynamic properties such as densities, enthalpies, entropies, among others, cannot usually be obtained from the same model because the excess Gibbs free energy is rarely known as a function of temperature and pressure.
EOS are powerful tools for investigating thermodynamic properties and phase behavior of pure fluids and their mixtures. There are many well-tested EOS available for fluid mixtures of conventional substances. For mixtures of polymers with solvents, on the other hand, problems arise due to the different characteristics of the components. To address these many polymer-specific EOS have been proposed, which focus on the polymer component(s) of the mixture. Efforts to represent conventional systems with these EOS have not always been very successful; indeed some of these models perform less successfully than traditional cubic EOS in this regard. This may be a handicap when these models are used for the VLE of the polymer-solvent mixtures. In such cases, little or no polymer is present in the vapor phase and the solvent compressibility plays an important role in the phase behavior. Consequently, there is a strong incentive to extend the conventional EOS developed for small molecules to polymers.
There are two basic issues in extending cubic EOS to apply to polymers and their use. The first issue is the description of the pure component EOS parameters for polymers. To obtain these parameters, various techniques have been suggested. The second issue in extending cubic EOS to apply to polymers is the selection of mixing rules (MR) for the EOS parameters. The classical mixing rules of van der Waals (vdW) have already been tested for polymer solvent mixtures, however, it has been observed that, in order to fit the experimental data, some unrealistic values are necessary for the binary interaction parameters (BIP).
The use of equations of state in phase equilibrium modeling instead of activity coefficient models is mainly a result of the recent development of a class of mixing rules that enable the use of liquid activity coefficient models in the EOS formalism. The implication of this change is far-reaching as an EOS offers a unified approach in thermodynamic property modeling. With this approach, the applicability of simple cubic EOS has been extended to complex systems, such as polymeric systems, if coupled with the appropriate activity coefficient model. Therefore, there is much interest in mixture EOS models capable of describing higher degrees of nonideality than that possible with the van der Waals one-fluid model and its modifications.
Future development of EOS for polymer mixtures is unclear and some contradictory statements can be found in the literature. Some authors indicate that cubic equations can be extended to correlate and predict VLE in polymer mixtures accurately. On the other hand, others state that, considering the complexity of this type of mixture, simplicity is not a necessary requirement for an EOS, as the calculation of parameters for the mixture components is more important. There is agreement, however, on the fact that future development of EOS for polymer mixtures must emphasize the study of mixing rules and that EOS input parameters should be related to the commonly measured properties of the polymers.
The most apparent progress toward EOS with the ability to describe phase behavior with polymers has been made by applying statistical mechanics. Some early models derived from statistical thermodynamics assumed molecules to be arranged in a lattice, whereas many of the more recent theories picture molecules to be moving freely in continuous space. In lattice models, the molecules are assumed to have one or more segments, and the partition function of the system can be obtained by counting the possible configuration when these segments are arranged in hypothetical cells which are like the lattices in solid materials. Then the thermodynamic quantities can be calculated from the partition function on the basis of statistical mechanics.
A huge amount of work has been done on the understanding of phase behavior in polymeric mixtures, either from an experimental or theoretical point of view. As well as supplying important data, experiments enable the evaluation of EOS models for the correlation and/or prediction of phase behavior. A model, on the other hand, takes much less experimental effort and can guide the researcher/analyst in the right direction.
A detailed review of the different lines of developing equations of state for the calculation of fluid phase equilibria is given by . Recently  presented and discussed in depth both classical and novel thermodynamic models, which have been developed and can potentially be used for industrial applications. A review of the use of some equations of state (EOS) for LDPE process simulation can be found in Orbey
Although there have been some analyses on equations of state that can describe the phase equilibria involving polymers, additional assessments are necessary. In general, the available works concern a specific approach, not taking into account others. In addition, these reviews and surveys focus on detailed model theory or theoretical possibilities of model variations, with a few quotes from practical applications. This chapter therefore presents an overview of the progress on EOS models for polymer systems considering the following approaches:
Cubic EOS (mixing rules incorporationg excess Gibbs free energy models)
Lattice models [Sanchez-Lacombe (SL) equation of state]
Perturbation theory (SAFT equation: the original version and its variants)
In EOS applications only works dealing with phase equilibrium are discussed, other types of applications, such as solvent absorption and/or polymer swelling, are not addressed. The timeline diagram in Figure 1 shows some of the key developments and outstanding papers related to the development of equations of state for polymer systems, which are discussed in this chapter. The following notation is used:
Cubic Equations of State [Huron and Vidal (HV) Mixing Rule (MR); Sako-Wu-Prausnitz (SWP) Equation; Wong and Sandler (WS) MR; vdW Applied to Polymer Solutions (vdW-P) by Kontogeorgis, Harismiadis, Fredeslund and Tassios; Linear Combination of Vidal and Michelsen (LCVM) Mixing Rules by Boukouvalas, Spiliots, Coutsikos, Tzouvaras and Tassios; Zhong and Masuoka (ZM) MR]
Lattice Models [Flory-Huggins (FH); Sanchez and Lacombe (SL) equation; key Modifications and Applications (MA) of SL by KIeintjens and KoningmeId, Panayiotou and Vera, Kiran and Xiong and Zhuang, Koak and Heidemann, Gauter and Heidemann, Krenz and Heidemann and de Loos]
Perturbation theory [Wertheim Thermodynamic Perturbation Theory (TPT); Statistical Associating Fluid Theory (SAFT) by Chapman, Gubbins, Jackson, Radosz; Chen and Kreglewski  SAFT (CK-SAFT) by Huang and Radosz; Perturbed-Chain SAFT (PC-SAFT) by Gross and Sadowski; Simplified PC-SAFT (sPC-SAFT) by von Solms, Michelsen and Kontogeorgis]
2. Cubic equations of state and mixing rules
The first group of models to describe the phase behavior (by calculating the equilibrium constant) corresponds to the van der Waals equations of state, known as cubic equations, in either the original version or variants thereof. They are extremely simple and efficient for experimental data correlation. In this group, modifications of the Redlich-Kwong equation stand out, especially the Soave-Redlich-Kwong (SRK)  and the Peng-Robinson (PR) , which can calculate, often successfully, the vapor-liquid equilibrium for normal fluid and mixtures. However, application of the cubic equations of state for polymeric blends is not immediately obvious as this application does not follow standard procedures. The conventional method for calculating the pure parameters in cubic equations of state requires components' critical properties and vapor pressure, which do not exist for polymers. Therefore, two basic issues should be addressed when extending cubic equations of state for polymers and their mixtures. The first, presented in the following paragraphs is the description of the parameters of pure components, and the second is the choice of the mixing rule, which will be discussed later in sections 2.1-2.3.
There are four conditions to be satisfied when selecting the pure component parameters of a cubic equation of state. First, a polymer is non-volatile and therefore should not exhibit any vapor pressure. If there are oligomers in the mixture though, low vapor pressures might be considered. Therefore, critical properties may be assigned for the oligomers, treating them as conventional components. The second condition is that the equation of state should predict densities of molten polymers. The third condition requires that the parameters reflect the polymers' basic characteristics such as the degree of polymerization. This is important because experimental data demonstrate that these polymer characteristics directly affect the vapor-liquid equilibrium in polymer-solvent mixtures. The fourth point, somewhat connected to the third, requires easily accessible and physical meaning characteristics as input parameters for calculating the parameters of the equation of state. As stated before, the SRK equation of state is expressed by:
and the PR equation is given by:
The first attempt to apply a cubic equation to polymers was made by Sako
The SWP equation was used with relative success by Tork
The performance of cubic equations of state is directly related to the efficiency of mixing rules to represent the phase equilibria at high pressures. Basically, the mixing rules can be divided into two classes: van der Waals-type and those that incorporate excess Gibbs energy (
2.1. van der Waals mixing rules
In order to extend the application of PR and SRK cubic equations of state for polymer-solvent systems, the conventional mixing rules employed are those from van der Waals (vdW) , which are expressed as:
It should be stressed that these rules are limited to non-polar fluids and therefore are unable to represent the highly non-ideal behavior of polar or associative fluids.
An empirical approach to overcome the shortcomings of the vdW mixing rule has been to simply add new parameters and composition dependence to the combination rule for parameter
2.2. Mixing rules for excess free energy (GE) models
Like conventional phase equilibrium modeling, there are two basic modeling tools for dealing with polymer-solvent mixtures: excess Gibbs free energy (
Over the last two decades several methods combining activity coefficient models with equations of state have emerged. These methods are useful for correlating/predicting the phase equilibria of conventional mixtures, and are promising for mixtures containing polymers. Moreover, they allow us to investigate an activity coefficient model in two ways: first as a conventional model (i.e. in the approach
Huron and Vidal , pioneers in this field, incorporated an excess Gibbs free energy model into a mixing rule. Their method is based on three assumptions: (i) the excess Gibbs free energy, calculated from an equation of state at infinite pressure equals the excess Gibbs free energy calculated from an activity coefficient model for the liquid phase (ii) the co-volume parameter
where and for SRK equation. Huron and Vidal  showed that their mixing rule gives good results for non-ideal mixtures. Soave  showed that the Huron and Vidal rule represents an improvement over the classical quadratic mixing rules and can accurately correlate the vapor-liquid equilibrium for highly nonideal systems. The Huron and Vidal mixing rule has also been applied to several polar and asymmetric systems -.
The Huron and Vidal mixing rule has some undesirable characteristics, such as: it does not reproduce the quadratic dependence of the second virial coefficient (QDSVC) with the composition at low pressure; it has no predictive value because the parameters of the activity coefficient model, estimated at low pressure, have to be re-estimated at high pressure; furthermore, these parameters are temperature dependent. Various proposals - have tried to cope with these constraints, however, they fail to succeed as discussed below.
Mollerup  suggested an alternative method to that of Huron and Vidal, assuming that the excess volume is zero at low pressure and that the excess Gibbs free energies calculated from an equation of state and from an activity coefficient model can be matched in this condition. Therefore, the activity coefficient parameters do not need be re-estimated if pressure and temperature conditions correspond to those at which they are fitted. However, since this theory cannot be applied to supercritical fluids, as well as the difficulty of computing roots (of the equation of state) for the liquid phase at zero pressure, its application is restricted.
Heidemann and Kokal , in accordance with Mollerup , also take the reference state at null pressure, attempting, however, to overcome the problem of calculating the root for the liquid phase at zero pressure. The major contribution of this method is to propose an extrapolation procedure from the system pressure, enabling calculation at temperatures near and above the critical point. Important to mention is that this method requires the solution of a transcendental equation when calculating the mixing rule. Comparative studies demonstrate a better performance of the Heidemann and Kokal rule when compared to the Huron and Vidal rule .
A method very similar to Heidemann and Kokal’s  was proposed by Michelsen . The main difference between them lies in the extrapolation method used for supercritical components. The mixing rule in the Michelsen approach also requires the solution of a transcendental equation. The SRK-Wilson model (using SRK equation of state and Wilson activity coefficient model), obtained using this method, was tested to obtain the phase envelope, including the critical points, and to calculate the phase diagrams at high pressure, without re-estimating the parameters of the Wilson model. Good results were achieved.
Michelsen  modified his own method, considering an explicit mixing rule, i.e. avoiding the solution of the transcendental equation. The only drawback of this modification lies in the impossibility of ensuring the accurate reproduction of the
Dahl and Michelsen  found out that replacing the linear approach (MHV1) by a quadratic approximation considerably improves the reproductibility of the
Attempting to straighten out the theoretical inconsistency of the aforementioned mixing rules, Wong and Sandler  proposed a new method in which the rules fulfill the QDSVC with the composition at low pressure condition. The basic idea was to consider the excess Helmholtz free energy as much less dependent on pressure than the excess Gibbs free energy. In this way, the excess Helmholtz free energy at high pressure might be equal to the excess Gibbs free energy at low pressure. Therefore, this mixing rule is given by:
where and are given by Vidal and Michelsen rules, respectively. The contributions related to
Zhong and Masuoka , based on experimental data, evaluated the MHV1 mixing rule with SRK equation of state and the original UNIFAC model for
A new mixing rule for cubic equations of state, particularly suitable for highly symmetric systems, was proposed by Zhong and Masuoka . It was validated by two cubic equations of state: a modification of Peng-Robinson equation proposed by Stryjek and Vera (PRSV)  and the SRK equation. As there is no critical point for polymers, parameters
In recent years, many studies have focused on improving hybrid models, i.e. equations of state which embody
2.3. Modeling polymeric systems with equations of state embodying Gibbs free energy (GE) models
Orbey and Sandler  applied the PRSV cubic EOS, along with the mixing rules proposed by Wong and Sandler , to correlate vapor-liquid equilibrium data for some polymer solutions. For pure solvents, they used the conventional method to determine the parameters of the equation of state from the critical properties and the acentric factor. For polymers, however, in order to determine these parameters, they chose an arbitrary value for the vapor pressure, 10-7 MPa, and used experimental data of molten polymer densities. As expected, the parameters
An equation of state based on ASOG (Analytical Solution Of Groups), called PRASOG (Peng-Robinson-ASOG), was developed by Tochigi  to predict the vapor-liquid equilibrium of non-polymeric and polymeric solutions. It makes use of the zero-pressure
An evaluation of vapor-liquid equilibrium in polymer-solvent systems with cubic equation of state was performed by Louli and Tassios . In this study the parameters
Using the PRSV cubic equation of state, Haghtalab and Espanani  studied the vapor-liquid equilibrium in polymer binary solutions with different molecular weights and temperatures. The parameters of the cubic equation of state were calculated using the Wong-Sandler mixing rule  incorporating the FH-NRTL-NRF (Flory-Huggins Non-Random Two Liquid Non-Random-Factor) excess Gibbs free energy model. The total vapor pressure of the polymer solutions was correlated using two adjustable energy parameters as functions of temperature with six constants for the entire temperature range. The modeling results showed very good agreement with the experimental data of several binary polymer solutions.
The SRK and the Sanchez and Lacombe (SL) equations of state were applied by Costa
3. Lattice models
In the second group of models for calculating the equilibrium constant, it is assumed that the molecules have one or more segments, and that the partition function of the system can be obtained by counting the number of possible configurations when these segments are arranged in hypothetical cells that resemble the crystal lattice of a solid. The thermodynamic functions can be calculated using the formalism of statistical mechanics. These crystal lattices can be considered compressible or incompressible. Incompressible lattices are generally used to model liquids at low pressures, a condition in which the concept of activity coefficient is used. The most widely used activity coefficient models are based on this formalism, e.g. , -. For compressible lattices, equations of state based on lattice models result. An example of such models is the lattice fluid theory , .
The lattice model, originally developed to describe the liquid phase, considers the liquid in a quasi-crystalline state, in which the molecules do not translate fully chaotically as in a gas, but each one tends to stay in a small region, a more or less fixed position in space, around which it vibrates back and forth. The quasi-crystalline picture of the liquid state supposes that the molecules are regularly arranged in space as in a lattice, and therefore models for liquid and liquid mixtures are called lattice models. Molecular considerations suggest that deviations from ideal behavior in liquid solutions are mainly due to the following effects: first, the attraction forces between unlike molecules are quantitatively different from those between alike molecules, giving rise to a nonzero enthalpy of mixing; second, if the molecules differ significantly in size or shape, the molecular arrangement in the mixture can be appreciably different from that for pure liquids, resulting in a non-ideal entropy of mixing; finally, in binary mixtures, if the attraction forces in one among the three possible interaction pairs are much stronger (or much weaker) than the other two pairs, there will be some preferred orientation of the molecules in the mixture what, in extreme cases, can lead to instability or incomplete miscibility .
The most simple lattice model considers a mixture of two liquids whose molecules are small, symmetrically spherical and similar in size (the ratio of their sizes is close to one). This model assumes that the molecules of each pure liquid are regularly arranged and equidistant from each other in the lattice. The molecular movement is limited to vibrations around equilibrium positions and is not affected by the mixing process. This model also assumes that for a fixed temperature the lattice spacing in both pure liquids and in the mixture are the same, regardless of composition (excess volume is null). The first step is to obtain an expression for the potential energy of a pure liquid or a mixture, assuming that the potential energy is pair-to-pair additive for every pair of molecules and that only the nearest neighbors are considered in this sum. This means that the potential energy of a large number of molecules in the lattice is given by the sum of the potential energy of all pairs of molecules situated immediately next to each other. Therefore, considering the excess volume and the excess entropy as null, the excess Gibbs free energy for the two-suffix Margules model can be obtained from the total potential energy in the lattice .
This lattice model is particularly useful for describing polymeric solutions in liquid solvents. Flory and Huggins  independently developed a theory for polymeric solutions which have formed the foundation of most subsequent developments in the last fifty years. In the Flory-Huggins  model the system polymer-solvent is modeled as a lattice structure, where each site is occupied by a molecule of solvent or a polymer segment. The combinatorial contributions to the thermodynamic mixing functions are calculated from the number of possible arrangements of the polymer molecules and solvent in the lattice. These combinatorial contributions correspond to the entropy of mixing. The combinatorial contributions of Flory-Huggins  model implicitly state that the mixing volume and the enthalpy of mixing are zero. The number of possible molecular arrangments leads to the well-known Flory-Huggins expression for the entropy of mixing . The Flory-Huggins theory and its variations have been successful in correlating and/or predicting the UCST behaviour and loop phase behavior. Variations of this theory include making the interaction parameter of the enthalpy of mixing dependent on composition and/or temperature. In this context, the works of Cheluget
3.1. Sanchez and Lacombe (SL) equation
The lattice fluid theory for liquid and gaseous mixtures developed by Sanchez and Lacombe ,  is formally similar to the Flory-Huggins theory. However, the essential and important difference is that the Sanchez and Lacombe theory introduces holes to account for variations in compressibility and density, i.e. the mixture density may vary by increasing the fraction of holes in the lattice. The Sanchez and Lacombe equation uses a random mixing expression for the attractive energy term. Random mixture means that the composition everywhere in the solution equals the total composition, i.e. there are no effects of local composition. The energy of the lattice depends only on nearest neighbors interactions. For a pure component the only non-zero interaction energy corresponds to mer-mer pair interaction. The interaction energies of types mer-hole and hole-hole are zero. The Sanchez and Lacombe equation assumes a random mixture of holes and mers. Therefore, the number of mer-mer nearest neighbors is proportional to the probability of finding two neighboring mers in the system. The Sanchez and Lacombe EOS  is given by:
where the segment fraction of component
The cross parameters are:
Thus, the Sanchez and Lacombe equation obtains the PVT properties of pure component assuming that it is broken into parts or mers which are placed on a lattice and can interact with intermolecular potential. In order to calculate the density of the system correctly, an appropriate number of holes are also placed at specific sites in the lattice. In principle, this equation of state is appropriate to describe the thermodynamic properties of fluids in a wide range of conditions, from normal liquid or gaseous state to supercritical fluid at high temperatures and pressures. A real fluid is characterized by three molecular parameters or by three equation of state parameters, which must be known if the equation of state is to be used. In fact these parameters can be determined through any configurational thermodynamic property obtained experimentally. Vapor pressure data though, are particularly useful for solvents because they are readily available for a wide variety of fluids. For polymers, these characteristic parameters can be estimated by experimental data of the liquid density over a wide range of pressures and temperatures, using for example a numerical procedure based on non-linear least squares. When few PVT data are available, the parameters can be estimated from experimental values of density, thermal expansion coefficient and compressibility factor at ambient temperature and pressure.
Gauter and Heidemann  proposed a procedure to obtain parameters of the pure solvent from the critical temperature, critical pressure and acentric factor, as usually done with cubic equations of state. The polymer parameters were determined through PVT data regression.
Gauter and Heidemann  suggested that the polymer's parameters can be adjusted to simultaneously reproduce cloud-point data of polymer-solvent equilibrium and PVT data. They managed to obtain parameters for the Sanchez and Lacombe equation for polyethylene that could be applied for different samples, regardless of molecular weight and molecular weight distribution. The degree of branching and/or the presence of comonomers may also influence the parameters of the polymer.
3.2. Modeling polymeric systems using the Sanchez and Lacombe equation
Although there are few references in the literature for vapor-liquid equilibrium (e.g. , ) and one using industrial plant data , a large number of SL EOS evaluations have been reported in the literature regarding liquid-liquid equilibrium.
Xiong and Kiran  modeled ternary systems of [polyethylene-(n-pentane)-(carbon dioxide)] using the Sanchez and Lacombe equation. Phase diagrams were generated for pressures up to 300 MPa and temperatures up to 460 K. The results show that the system can exhibit two or three phases depending on the pressure. At a given temperature, the three phase region disappears with increasing pressure. Depending on the pressure, the calculations also predict the displacements observed experimentally from LSCT to UCST, which are illustrated in ternary diagrams as displacements of the phase boundaries with temperature. It was shown that for polymer samples with high molecular weight, ternary calculations can be simplified by assuming that the polymer-poor phase is essentially free of polymer. Xiong and Kiran - investigated polyethylene binary systems with n-butane, n-pentane and CO2.
Koak and Heidemann  studied the phase behavior of polymer-solvent systems under conditions close to the vapor pressure curve of the solvent where the vapor-liquid-liquid equilibrium can occur. Experimental data of High Density Polyethylene (HDPE) in n-hexane were modeled using the following equations: Sanchez and Lacombe, Kleintjens and Koningsfeld  and the Perturbed Hard-Sphere-Chain (PHSC) . The phenomena of interest include the LCST behavior and the liquid solvent, vapor solvent and polymer three-phase equilibrium. All the three models examined provided a reasonable representation of the cloud-point for the system HDPE and n-hexane along the three-phase line in the conditions investigated. Phoenix and Heidemann  used the SL EOS to develop an algorithm to determine the cloud and shadow point curves of polydisperse polymer/solvent systems using continuous thermodynamics to represent the polymer.
The applicability of equations of state for the modeling and simulation of phase equilibria in polymer production processes is investigated by Orbey
In order to verify if a single set of parameters can be used to obtain useful correlations for different polyethylene resins with different solvents, Gauter and Heidemann  used the Sanchez and Lacombe equation to model the cloud point isotherms for two systems of ethylene and polyethylene and a system of polyethylene in n-hexane. The three polyethylene samples examined differ considerably in average molecular weight and polydispersity. The polymer parameters were obtained by adjusting volumetric data of pure polyethylene, using an additional volume displacement coefficient. The results showed that the cloud point behavior of the polymer-solvent equilibrium for a variety of polymers and solvents can be correlated with the same set of polymer parameters. The required interaction parameters are relatively small in magnitude. Unfortunately, the calculated results are extremely sensitive to these numbers, even to the third decimal place. In addition, the temperature dependence, although slight, is essential to obtain a reasonable data fit.
The effect of using different molar mass distributions to represent the same polymer on the HDPE-ethylene cloud points, was examined by Krenz
Cloud points for three hydrogenated PolyButaDiene (hPBD)-(n-hexane) mixtures were calculated using the SL equation by Schnell
Correlation and prediction of miscibility involving binary blends of a variety of homopolymers [polypropylene, polybutadiene, polyisoprene, poly(methyl methacrylate), polystyrene, among others] were investigated by Voutsas
One or more polyethylene samples with varying molecular configurations can be mixed to produce a blend with different physical characteristics. Krenz and Heidemann  used the MSL (Modified Sanchez Lacombe) equation  to calculate the cloud points of a blend of two polydisperse LLDPE resins in a hydrocarbon solvent. The MSL equation is a lattice equation that can be used to calculate polydisperse polymer solutions. The considered polyethylene resins were hPBD type. The cloud points were compared with experimental data available for the systems (hPBD-1)-(hPBD-2)-(n-hexane) and (hPBD-3)-(hPBD-4)-(n-pentane). The four hPBD samples have different molecular weight distributions, although the other properties of the mixture are unknown (degree, type and frequency of branching in the polyethylene molecule). The temperature dependent BIP for LLDPE-hydrocarbon were previously fitted to binary mixture cloud points.
4. Perturbation theory
Equations of state based on molecular structures not only provide a useful thermodynamic basis for deriving chemical potentials or fugacities (necessary for phase equilibrium simulation) but they can also help separate and quantify the effects of molecular structure and interactions on global properties and phase behavior. Examples of these effects are the molecular size and shape (e.g. chain length and chain branch), energy of association (e.g. hydrogen bonding), average field energy (e.g. dispersion and induction). Ideally, a single equation of state should incorporate all these effects .
Much progress has been made in the development of molecular theories of associative solutions and those containing macromolecules. The essence of this progress is the use of statistical mechanics methods, such as perturbation theory, to correlate the molecular properties with the macroscopic properties of the system under study. In perturbation models, a simple system is initially used as reference, which should characterize the essential aspects of the system and it is usually obtained using a theory with well-defined assumptions. The difference between the actual and ideal system (i.e. the reference system) is then computed using some correction terms, called perturbation terms, which are often based on semi-empirical models. The complexity and magnitude of these perturbations depend on the degree of accuracy with which the reference term, representing the ideal system, can be specified.
With this method, Beret and Prausnitz  used the results of Carnahan and Starling  for hard spheres, which can be characterized by square well potential to describe the reference state, and proposed the so-called Perturbed Hard-Chain Theory (PHCT). A further refinement, the Perturbed Asinotropic Chain Theory (PACT), was made by Vilmalchand and Donohue  and Vilmalchand et al. . The PACT equation of state takes into account the effects of different molecular sizes, shape and intermolecular forces, including anisotropic dipole and quadrupole forces. The calculations from Vimalchand et al.  show that the explicit inclusion of multipolar forces can predict the properties of highly non-ideal mixtures with reasonable accuracy, without the use of binary interaction parameters. However, for pure fluids, the prediction behavior of the PACT equation of state is similar to other comparable equations of state. Kim et al.  developed a Simplified version of the PHCT equation (SPHCT), replacing the attractive term of the PHCT equation with a simpler theoretical expression. This simpler equation has been used in a large number of applications, including mixtures of molecules which greatly differ in size. Ikonomou and Donohue  derived the Associated PACT equation (APACT). This equation takes into account isotropic repulsive and attractive interactions, anisotropic interactions due to the dipole and quadrupole moments of molecules and hydrogen bonding, and it can predict the thermodynamic properties of associative pure components as well as associative multicomponent mixtures.
4.1. SAFT equation
More recently, a new model in the family of perturbation models was developed by Chapman
The essence of the SAFT equation is to use a reference system which incorporates the chain length (molecular size and shape) and the molecular association, rather than the reference fluid with hard (rigid) spheres, which is much simpler. It is expected that the effects due to other types of intermolecular forces (dispersion, induction, among others) are weaker, and therefore, considered through a perturbation term. Thus, it is expected that this theory is able to describe most real fluids, including polymers and polar fluids. In the SAFT model, the molecules are interpreted as a mixture of spherical segments of equal size, interacting according to a square-well potential. In addition, two types of bonds between these spheres can occur: covalent bonds to form chains and association bonds for specific interactions , .
When developing the equation of state, it is assumed that the molecules are formed from segments of rigid spheres, according to the diagram in Figure 2. Initially, the fluid is composed only of rigid spheres of equal size, and only the effect of rigid spheres are considered. The reference fluid consists of rigid spheres forming chains (tetramers) via covalent bonds. Hydrogen bonds between terminal sites of different chains result in oligomer chains. The last step takes into account weak dispersion forces.
The equation is derived in terms of the residual Helmholtz free energy:
Formation of rigid spheres (a
For a pure fluid, the formation of one mol of each chain, consisting of m segments, requires m moles of hard sphere. For the hard-sphere term a
4.2. PC-SAFT equation
Gross and Sadowski - developed a modification to the SAFT equation referred to as PC-SAFT (Perturbed-Chain SAFT). In the structure of the PC-SAFT equation, molecules are assumed to be chains of spherical segments, freely linked and exhibiting attraction forces among them. The repulsive interactions are described by an expression of rigid chain (hard sphere + chain) developed by Chapman et al. , which is the same as used in the SAFT equation of state. The attraction interactions are in turn divided into dispersion interactions and a contribution due to association. Figure 2 illustrates the formation of a molecule according to the PC-SAFT theory. Earlier versions of SAFT assume that the dispersive interactions of molecule chains are the same as those of spherical molecules. Further investigation, however, demonstrates that equations of state may be improved when the dependence on chain length is considered in the dispersive interactions. A new version, which explicitly takes into account this dependence, has been developed, leading to the PC-SAFT equation. The dispersion term was obtained by extending Barker and Henderson's  theory for chain molecules. This theory considers that a chain segment is connected to neighboring segments. It also considers the effect of the nearest neighbor segments in segment interactions.
When the systems under investigation do not contain associative fluids, the term that takes into account such interactions can be ignored.
In the literature, pure component parameters for various substances can be found, either small molecules or macromolecules. The model is already available in commercial software.
4.3. Other modifications of the SAFT equation
Although the PC-SAFT equation provides excellent results when simulating polymeric systems, a brief survey of other modifications involving the original form of the SAFT equation is given in this section. Four comprehensive reviews of the development and application of the various types of SAFT have appeared recently , , , .
Instead of using the hard sphere fluid as reference, Blas and Veja  used the Lennard-Jones fluid, leading to the soft-SAFT equation. The chain and association terms remained similar to those in the original SAFT formulation. The soft-SAFT equation was successfully applied to pure n-alkanes, 1-alkenes, 1-alcohols and binary and ternary mixtures of n-alkanes including the critical region. In the case of mixtures, two binary parameters should be used even for mixtures of n-alkanes.
Another version of the SAFT equation is the SAFT-VR (SAFT Variable Range) equation -. The differences between SAFT-VR and PC-SAFT arise from the specific treatment of the attractive interactions between segments and the choice of the reference fluid. The SAFT-VR takes as reference the hard-sphere fluid, while PC-SAFT takes the hard-sphere-chain fluid. The SAFT-VR equation describes a fluid of associating molecules with the chain segments interacting through attractive forces of variable range (VR). In SAFT-VR a reference system with interacting monomers is used to build the molecule.
The Simplified PC-SAFT (sPC-SAFT) equation , ,  is not in fact a new equation of state, rather it is a simplified version of the original PC-SAFT regarding mixing rules. Therefore, the parameters of the pure components of the original and simplified PC-SAFT are the same. The sPC-SAFT equation assumes that all segments in the mixture have the same average diameter which provides a volume fraction of the mixture which is identical to the actual mixture. This simplified version is simpler to implement and improves computational performance compared to the original PC-SAFT with negligible differences in accuracy.
4.4. Polymeric systems modeling using the SAFT equation and its modifications
This section presents a brief review of studies that used the SAFT equations of state and its modifications for the modeling of polymeric systems. A crucial aspect for the success of modeling with SAFT and PC-SAFT equations is the correct selection of the parameters of the pure components. In general, the estimation of these parameters may not be easy for macromolecular compounds . Moreover, the fit to experimental data through the estimation of binary interaction parameter k
From the work of Chapman et al. - and Huang and Radosz  several applications of the SAFT model can be found in the literature. Table 1 presents a summary of some applications of the SAFT model for systems consisting of homopolymers and copolymers. A more detailed review of the application of SAFT model to polymeric systems can be found in  and .
Chen et al. - studied different phase transitions, from liquid to liquid-vapor (L to LV) and liquid to liquid-liquid (L to LL) in binary, ternary and quaternary systems containing the solvents ethylene, propylene, 1 -butene, 1-hexene, n-hexane and methylcyclopentane, and Poly(Ethylene-co-Propylene) (PEP). SAFT modeling was used for PEP of varying molecular weights at low and moderate pressures [(0-500) bar]. The BIP set was defined as an exponential function of the polymer molecular weight and adjusted by three parameters.
Xiong and Kiran  compared the performance of SAFT with SL to model cloud point curves in polyethylene systems with n-butane and n-pentane. The pure component parameters were taken from literature  and the BIP were assumed to be equal to zero. For all temperature-pressure-composition ranges, the SAFT model was superior to SL.
The approach used by Han et al.  was to measure the cloud point and the coexistence pressures in propylene and ethylene solutions of alternating PEP of well-controlled polydispersity directly, from monodisperse to broadly polydisperse. These experimental data were modeled using the SAFT equation. More specifically they fited the cloud point pressure for monodisperse PEP and used the model for predicting the cloud point and coexistence pressure of bimodal polydisperse PEP, without any refitting.
Pan and Radosz  used the SAFT equation for copolymers to describe the fluid-liquid and solid-liquid transitions in solutions of polyethylene and poly(ethylene-co-olefin-1) in propane as well as the fluid-liquid transition in solutions of polystyrene in n-hexane. The parameters of the pure solutes were estimated based solely on the molecular weight and on the structure. Copolymer SAFT EOS has been also used to model SLE in systems containing polyethylene, m-xylene and amyl acetate .
|Poly(ethylene-glycol) (PEG)||Propane, nitrogen, CO2|||
|PolyEthylene (PE)||Ethylene||, , , , |
|Propane (also SLE)||, |
|n-butane, n-pentane||, |
|Hexane, heptane, octane|||
|SLE in amyl acetate and m-xylene|||
|PolyPropylene (PP)||1-butene, n-butane, propane, propylene|||
|PolyStyrene (PS)||Cyclohexane, metylcyclohexane, ethylbenzene, chlorobenzene, CO2|||
|Cyclohexane, CO2||, |
|Polyisobutylene (PIB)||Ethane, ethylene, propane, propylene, dimethyl ether||-|
|PolyCarbonate (PC)||n-alkane (C8-C12), alcohol (C3-C10), benzene, toluene, o, m, p-xylene, ethylbenzene|||
|Poly(ethylene-co-vinyl-acetate) (EVA)||vinyl acetate, ethylene, alkanes||, , |
|Poly(vinyl-acetate) (PVA)||Benzene, Vinyl acetate|||
|Poly(ethylene-co-olefin)||Ethylene, propilene, propane (also SLE), 1-butene, 1-hexene, n-alkane (C6-C8)||, , |
|Poly-methyl methacrylate (PMMA)||CO2–methyl methacrylate|||
|Poly(Ethylene-co-Propylene) (PEP)||Ethylene, ethane, propylene, 1-butene, and 1-hexene, methylcyclopentane||, , , , |
Dariva  applied the SAFT equation of state to model PolyPropylene (PP)-solvent systems at low and moderate pressures. Two non-metallocene polypropylenes (molecular weights: 476745 and 244625 g/mol; and polydispersities: 4.4 and 5.0) and a metallocene polypropylene (molecular weight: 197150 g/mol; and polydispersity: 2.9) were used. As solvents, propylene, n-propane, 1-butene and n-butane were used. The author modeled transitions L to LV, L to LL and LL to LLV using the SAFT model with and without fitting the BIP. A large amount of experimental data for these systems can be found in this work.
Horst et al.  studied the influence of supercritical gases in the phase behavior of the systems polystyrene-cyclohexane-gas and polyethylene-cyclohexane-gas, modeling the experimental data with the SAFT equation of state. As supercritical gases, the authors used ethane, propane and nitrogen. The experimental data were collected at moderate pressures, and the binary interaction parameters used for the adjustment were defined as quadratic functions of temperature. Good results were obtained in the modeling, although a larger model mismatch in regions of higher polymer concentration can be obseved.
Jog et al.  used the SAFT equation to model the liquid-liquid equilibrium of LLDPE with hexane, octane and heptanes . The effects of temperature, pressure, polymer concentration and molecular weight on the phase separation were successfully evaluated. The effect of polydispersity on cloud point was also considered. Although the SAFT predictions are sensitive to the binary interaction parameters, a constant value for the binary parameters was considered to model the cloud point in varying conditions (temperature, pressure and polymer concentration) and varying solvents. The SAFT equation showed a good predictive capacity for this system.
Besides the works already mentioned, the SAFT equation of state has also been applied in more recent works as a “reference” model, its performance being compared to PC-SAFT model, as will be shown below. After the work of Gross and Sadowski -, some applications of the PC-SAFT model may be found in literature. Table 2 presents a summary of some of the applications of the PC-SAFT model for systems consisting of homopolymers and copolymers. A more detailed review of the application of PC-SAFT model to polymeric systems can be found in .
Tumakaka et al.  used the PC-SAFT equation to model cloud point curves for polymeric systems consisting of polyolefins, using ethane, ethene, propane, propylene, n-butane, 1-butene and CO2 as solvent. Here, the good results obtained for modeling the systems LDPE-solvents and HDPE-ethylene at high pressures should be highlighted. As well as modeling systems consisting of polyolefins, polyethylene copolymers and PVA (polyvinyl acetate vinyl) systems were also modeled. Good results were obtained when representating a system consisting of polypropylene with moderate polydispersity (MW/MN = 2.2), assuming that the polypropylene was monodisperse. The monodisperse assumption was also considered for the LDPE-solvent system, whereas for the HDPE-ethylene system the polyethylene was modeled using pseudocomponents.
|Polyethylene (PE)||Ethylene, ethane, propylene, propane, butane, 1-butene, hexane||, , , , , , |
|SLE in n-alkanes and in m-xylene|||
|SLE with a variety of solvents|||
|Polypropylene (PP)||Propane, n-pentane, CO2||, , |
|SLE with a variety of solvents|||
|Polystyrene (PS)||Cyclohexane, CO2, metylcyclohexane, ethylbenzene, chlorobenzene||, |
|Ethylbenzene, butyl acetate|||
|Polyamide (PA)||Caprolactam, water|||
|Poly(methyl acrylate) (PMA)||2-octanone||, |
|Poly(ethylene-co-methacrylic acid) (EMA or PE-co-MA)||Ethylene, propylene, butane, 1-butene||, |
|Poly(ethylene-co-acrylic acid) (EAA or PE-co-AA)||Ethylene|||
|Poly(ethylene)-co-olefine||Ethylene, propylene, propane, 1-butene, 1-hexene, n-alkane (C6-C8)||, , |
|Poly(vinyl-acetate) (PVA)||Methyl ethyl ketone, propyl acetate, 1-propylamine, 2-propylamine, 2-methyl-1-propanol, 2-propanol|||
|Polyolefins and PVA||Ethane, ethene, propane, propylene, n-butane, 1-butene, CO2|||
|PE, PP, PB, PIB, PS||Ethylene, n-butane, 1-butene, n-pentane, cyclohexane|||
|Biopolymers: poly(d,l-lactide) (PLA), poly(butylene succinate) (PBS) and poly(butylenes succinate-co-adipate) (PBSA)||Cloro difluorometane, CO2, dimetil ether, difluiorometane, trifluorometane, tetrafluorometane|||
|Polycarbonate (PC)||CO2, cyclohexene oxide|||
|CCl4, CH2Cl2, methyl acetate, methyl ethyl ketone, 1-propanol, 4-heptanone, chlorobutane, octane, cyclohexane, benzene, toluene, ethybenzene, xilene, acetone, diethylketone|||
|PDMS, PE, PS, PBMA, PIB, PB, poly(alpha-methylstyrene) (P-MS), PMMA, PVA||Benzene, toluene, alkane (C5-C8), cyclohexane, methylcyclohexane, 1-propanol, 2-propanol, 1-butanol, 2-butanol, 2-methyl-1-propanol, methyl acetate, prothyl acetate, methyl ethyl ether, acetone, propylamine, isopropylamine|||
Gross and Sadowski  used the fractionation of LDPE in three pseudocomponents to represent the cloud point curves for the ethylene-LDPE system (MW/MN = 8.56). The pure component parameters for ethylene and polyethylene and one binary interaction parameter k
Gross and Sadowski  proposed changes to the PC-SAFT equation, adding two more parameters concerning the association term. Simulations of liquid-liquid and vapor-liquid equilibrium of systems consisting of simple molecules were compared with the SAFT model. Slightly better results were observed for the PC-SAFT equation. The pure component parameters were obtained from the simultaneous regression of vapor pressure and liquid phase density data. Thus, a total of five parameters were optimized for each component i: segment diameter (σ
Cheluget et al.  applied the PC-SAFT equation of state to model a flash separation system of an industrial LLDPE plant. The system under study consisted of ethylene, 1-butene, cyclohexane and polymer at 267 °C and 33 bar. The PC-SAFT parameters used in this study were estimated from binary system liquid-liquid and vapor-liquid equilibrium data obtained from literature. The authors did not re-estimate the binary interaction parameters to fit the model to the industrial data, to compare the predicted (using interaction parameters from literature) and industrial data. The lack of parameter re-estimation is the most likely cause of the significant deviations between experimental and predicted values.
Gross et al.  extended the PC-SAFT model - for copolymers. The authors modeled phase equilibrium for ethylene copolymer systems with random alternating chains in a wide range of compositions (including homopolymer) and with molecular weights ranging between 709 and 242000 g/mol. The studied polymers were composed of repeating apolar [poly([ethylene oxide]-co-propylene) and poly([ethylene oxide]-co-[butene-1])] and polar [poly([ethylene oxide]-co-[vinyl acetate]) and poly(ethylene-co-[methyl acrylate])] units. Additionally, the authors reported binary interaction parameters of phase equilibrium for systems consisting of homopolymers, whose repeating units are present in the copolymers, and varying solvents, and some of these interaction parameters also considered the composition of the copolymer. The BIP were estimated from equilibrium data of binary polymer-solvent systems.
Kouskoumvekaki et al.  implemented a simplified version of the PC-SAFT equation of state, developed by von Solms et al.  with little repercussion for polymeric systems made up of a variety of solvents, including polar, apolar and associative compounds. Pure component parameters were estimated from vapor pressure and liquid phase density data. The simplified model showed similar results to those obtained by the original PC-SAFT equation, thus presenting some advantages due to its simplicity and lower computational cost.
In the work of Tumakaka and Sadowski , the PC-SAFT equation was applied to pure polar compounds as well as to the vapor-liquid and liquid-liquid equilibrium of binary mixtures containing polar compounds, with low molecular weight, and polar copolymers. As the original PC-SAFT is unable to describe polar systems, the authors used an extended version of the equation for polar systems. The dipolar interactions, which contribute significantly to the total intermolecular forces, are explicitly explained in molecular theory . Due to the inclusion of a term of polar interactions in the molecular theory, it was also necessary to include a pure component parameter in the term. When dealing with mixture modeling, the authors defined the binary interaction parameter either as an independent term or as a function of the comonomer molar fraction.
The sPC-SAFT equation (Simplified PC-SAFT equation) was applied by Kouskoumvekaki et al.  to the vapor-liquid equilibrium of binary and ternary systems of polyamide-6 with several solvents (water, caprolactam, ethyl benzene and toluene). Binary interaction parameters between polyamide-6, caprolactam and water were estimated using experimental data of the binary mixtures. The estimated parameters were used to predict and correlate the ternary mixture of polyamide-6, caprolactam and water. When optimizing the pure parameters of polyamide-6, the corresponding values of caprolactam were considered as initial estimates, and just the segment diameter needed to be adjusted using experimental data of liquid volume. The results showed that the sPC-SAFT equation is a versatile tool for modeling multi-component systems containing polyamide.
Arce and Aznar  modeled the systems PP-(n-pentane) and PP-(n-pentano)-CO2 using the PC-SAFT equation of state. In this work resins of low molecular weight (MW = 50400 and 95400) at moderate pressure (below 350 bar) were considered. The PC-SAFT, Sanchez-Lacombe and Peng-Robinson models were used to predict the cloud point pressures from experimental data on each system. Although all the models were able to describe the system, the PC-SAFT equation showed superior performance. For all the models, the authors used the temperature dependent BIP.
Spyriouni and Economou  evaluated the performance of SAFT and PC-SAFT equations of state to describe the phase behavior of mixtures containing polydisperse polymers and copolymers at high pressure. Although there are several studies showing the application of both equations in modeling the phase behavior of polymer systems, the major contribution of this work was to compare the performance of both models for a wide variety of homopolymers and copolymers. The authors concluded that both models show a similar performance in modeling the equilibrium, however, from the data presented, the PC-SAFT model shows superior results for most systems.
Pedrosa et al.  presented phase equilibrium calculations for polyethylene solutions with varying solvents using two different versions of the SAFT equation: PC-SAFT and the soft-SAFT. The soft-SAFT equation uses the spherical fluid of Lennard-Jones as a reference, including attractive and repulsive interactions, while the reference term in the PC-SAFT equation is the rigid sphere chain. The studies carried out by Pedrosa et al. , which also dealt with vapor-liquid equilibrium, showed that results with soft-SAFT equation are slightly more accurate than those obtained with PC-SAFT in some cases.
Buchelli et al.  investigated the performance of the PC-SAFT equation of state for modeling the HPS and LPS units downstream from a low-density polyethylene tubular reactor. Plant data were used to validate the equilibrium stage model prediction for the two gas-liquid flash separators, however, the pure component parameters and BIP of this model were obtained exclusively from experimental data published in the literature. The authors achieved good agreement between the model and LPS plant data, although the predicted solubility was not in agreement with plant-measured values for the HPS.
Guerrieri  investigated the behavior of polymeric systems in two industrial polyethylene plants, a LDPE plant and a HDPE/LLDPE plant, using the PC-SAFT equation. The liquid-liquid equilibrium at high pressure, observed in the reactor, and the vapor-liquid equilibrium, observed in the low-pressure separator, were investigated in the LDPE plant. For this study, 8 commercial resins were considered. In the HDPE/LLDPE plant, the vapor-liquid equilibrium in the intermediate pressure separator was investigated. Here, 25 commercial resins were investigated. The experimental data were taken from measurements and mass/energy balances available in both plants, and the modeling of binary and multicomponent systems consisting of ethylene, ethane, propylene, propane, 1-butene, cyclohexane, 1-octene and polymer was carried out.
Tihic et al. - developed a group contribution method to be used in PC-SAFT equation to predict their pure parameters. If pure polymer parameters in SAFT-type equations are obtained only from density data, poor predictions of phase equilibrium may result. Therefore, the group contribution method for parameter estimation was developed through the adjustment of vapor pressure and density data based on a database of 400 components of low molecular weight. The data required to calculate the phase equilibrium for polymers using this contribution method are the polymer molecular structure in terms of functional groups and a single interaction parameter for accurate mixture calculations.
Understanding the phase behavior of polymer solutions is of great theoretical and practical importance. Some work has also been done on the development of algorithms for real-time prediction of SLE in solution polymerization of polyethylene based on PC-SAFT EOS and to study the effects of monomer and polymer polydispersity in solution polymerization processes . Costa et al.  also modeled the SLE in polyolefins (polyethylene and polypropylene) solutions using PC-SAFT EOS for a variety of different polymer-solvent systems at atmospheric and high pressure with very good results. Pressure versus temperature (P-T) isopleths can be used to determine the number of phases present at a given T, P, and overall mixture composition. The PC-SAFT EOS was applied by Costa et al.  to simulate the curves that describe the borderlines between several distinct regions depicted in P-T isopleths for polyethylene solutions. A new strategy was used and the simulation results show good agreement with experimental cloud point isopleths data from the literature. In order to track the operational performance of industrial reaction systems safely, a strategy to calculate the distance between a given operational point (specified through a given pressure and a given temperature) and the corresponding point in the interface, for a fixed molecular weight and a fixed polymer fraction weight, has been developed which could also be extended for real-time prediction applications.
Kleiner et al.  extended the association term of the PC-SAFT EOS to account for the polydispersity of the copolymer samples. This EOS was used to model cloud-point curves of the systems poly(ethene-co-acrylic acid)-ethene and poly(ethene-co-methacrylic acid)-ethene. Both copolymer composition and molecular weight distribution were varied. To account for polydispersity the concept of pseudocomponents has been applied and they were generated such as to match the molecular weight distribution. An algorithm has been developed for calculating phase equilibria of polydisperse associating copolymer-solvent systems. The PC-SAFT approach turns out to be capable of adequately modeling and even predicting the phase behavior of the polydisperse polymeric systems by using two pseudocomponents for each copolymer, but no additional adjustable parameters.
Phase-dependent BIP were computed by Costa et al.  with PC-SAFT EOS by correlating the flash results for both high and low pressure separators (HPS and LPS) for the industrially significant mixture ethylene-ethane-propane-propylene-LDPE. HPS and LPS data were correlated for five of eight LDPE resins. A pressure, composition and molar mass dependent binary interaction parameter model was proposed for both the vapour and liquid phase. The resulting model was able to provide a good representation of the experimental data. The polydispersity and branching of the LDPE resins, as well as the temperature, were lumped into the BIP. Clearly, the proposed phase-dependent BIP model provides a good representation of the phase behaviour in two industrial separators for very complex polydisperse mixtures.
This chapter has presented a review of equation of state models for polymers, demonstrating their increasing evolution in performance for describing phase equilibria in polymer systems. The development of EOS for polymers remains a very active area of research and it is difficult to recommend a specific EOS .
In general the equations of state using G
Regardless of the model used, the calculation of the parameters of the pure components for polymers is a major challenge. When using an equation of state, the need for other applications besides the description of phase equilibria and PVT data, such as the calculation of Joule-Thompson coefficient and general energy balances should be borne in mind. Despite its importance, little research has been done in this respect with polymer systems. Note that the simpler the equation and its mixing rule, the easier it will be to obtain these properties and other important thermodynamic properties.
Although the PC-SAFT equation seems to show certain superiority in performance when compared to other models, no agreement was observed in the reviewed literature on which concept and/or thermodynamic model structure is better. Thus the choice of the model remains dependent on the system and its conditions. Although many studies address polymeric systems with a high concentration of polymers in a wide range of molecular weights, very few studies can be found that model oligomers (low molecular weight polymers), despite being a relevant problem as in many cases the quality of its measurement/prediction is critical for the proper functioning of important process equipment.Acknowledgement
The authors would like to thank CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FAPESB (Fundação de Amaro à Pesquisa do Estado da Bahia) for financial support.