Perturbed-hard-spheres-chain theory

FEN Feng, W., Wen, H., Xu, Z., and Wang, W., Comparison of perturbed hard-sphere-chain theory with statistical associating fluid theory for square-well fluids, Ind. Eng. Chem. Res., 39, 2559, 2000. 

Subsequently, various perturbation theories were developed which are also based on Eq. (3) but differ in the use of specific expressions for the different types of perturbations. Examples are the Perturbed Hard-Sphere-Chain Theory (PHSC) [64], as well as the models proposed by Chang and Sandler [65], Gil-ViUegas et al. [66], and Hino and Prausnitz [67]. 

The decisive advantage of the original Elory-Huggins theory [1] lies in its simplicity and in its ability to reproduce some central features of polymer-containing mixtures qualitatively, in spite of several unrealistic assumptions. The main drawbacks are in the incapacity of this approach to model reality in a quantitative manner and in the lack of theoretical explanations for some well-established experimental observations. Numerous attempts have therefore been made to extend and to modify the Elory-Huggins theory. Some of the more widely used approaches are the different varieties of the lattice fluid and hole theories [2], the mean field lattice gas model [3], the Sanchez-Lacombe theory [4], the cell theory [5], different perturbation theories [6], the statistical-associating-fluid-theory [7] (SAET), the perturbed-hard-sphere chain theory [8], the UNIEAC model [9], and the UNIQUAC [10] model. More comprehensive reviews of the past achievements in this area and of the applicability of the different approaches are presented in the literature [11, 12]. 

PHSC Perturbed Hard-Sphere-Chain Theory... 

Contemporary Approaches. Numerous advanced theories have been formulated in the last decades to reproduce or even predict experimental findings for polymer containing mixtures. Most of them are particularly suitable for the description of some phenomena and special kinds of systems, but all have in common that they have lost the straightforwardness characterizing the Flory-Huggins theory. The following, incomplete collocation states some of the wider used approaches These are the different forms of the lattice fluid and hole theories (38), the mean field lattice gas model (39), the Sanchez-Lacombe theory(40), the cell theory (41), various perturbation theories (42), the statistical-associating-fluid-theory (43) (SAFT), the perturbed-hard-sphere chain theory (44), the... 

For the calculations, different EoS have been used the lattice fluid (LF) model developed by Sanchez and Lacombet , as well as two recently developed equations of state - the statistical-associating-fluid theory (SAFT)f l and the perturbed-hard-spheres-chain (PHSC) theoryt ° . Such models have been considered due to their solid physical background and to their ability to represent the equilibrium properties of pure substances and fluid mixfures. As will be shown, fhey are also able to describe, if not to predict completely, the solubility isotherms of gases and vapors in polymeric phases, by using their original equilibrium version for rubbery mixtures, and their respective extensions to non-equilibrium phases (NELF, NE-SAFT, NE-PHSC) for glassy polymers. 

The perturbed-hard-sphere-chain (PHSC) equation of state is a hard-sphere-chain theory that is somewhat different to SAFT. It is based on a hard-sphere chain reference system and a van der Waals-type perturbation term using a temperature-dependent attractive parameter a(T) and a temperature-dependent co-volume parameter b(T). Song et ap-... 

The above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

Additional examples of equation of state models include the lattice gas model (Kleintjens et al, [33,34], Simha-Somcynsky hole theory [35], Patterson [36], the cell-hole theory (Jain and Simha [37-39], the perturbed hard-sphere-chain equation of state [40,41] and the modified cell model (Dee and Walsh) [42]. A comparison of various models showed similar predictions of the phase behavior of polymer blends for the Patterson equation of state, the Dee and Walsh modified cell model and the Sanchez-Lacombe equation of state, but differences with the Simha-Somcynsky theory [43]. The measurement and tabulation of PVT data for polymers can be found in [44]. 

To address the hmitations of ancestral polymer solution theories, recent work has studied specific molecular models - the tangent hard-sphere chain model of a polymer molecule - in high detail, and has developed a generalized Rory theory (Dickman and Hall (1986) Yethiraj and Hall, 1991). The justification for this simplification is the van der Waals model of solution thermodynamics, see Section 4.1, p. 61 attractive interactions that stabilize the liquid at low pressure are considered to have weak structural effects, and are included finally at the level of first-order perturbation theory. The packing problems remaining are attacked on the basis of a hard-core model reference system. 

Jog, P.K. and Chapman, W.G., Application of Wertheim s thermodynamic perturbation theory to dipolar hard sphere chains, Mol. Phys., 97(3), 307-319, 1999. 

In the perturbed hard-chain theory (PHCT) of Beret and Prausnitz [14] and Donohue and Prausnitz [15], the reference fluid is modeled as chains of tangential hard spheres. Since the fluid is stiU composed of hard spheres, albeit bonded, the CS eos is applied with modification to account for the bonding. The hard-sphere-chain equation is... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

The best-known model of this kind is the Statistical Associated Fluid Theory (SAFT) model [58-61]. Here, a non-spherical molecule (solvent or polymer) is assumed to be a chain of identical spherical segments. Starting from a reference system of m hard spheres (A ), this model considers three perturbation contributions, which are assumed to effect independently attractive interactions of the (non-bonded) segments (A ), hard-sphere chain formation (A ), and association (A ° ) ... 

Donahue and Prausnitz [23] developed the perturbed-hard chain theory (PHCT) based on perturbed hard-sphere theory for small molecules and Prigogine s theory for chain molecules. In order to account for attractive and repulsive forces among molecules, empirical parameters such as the c parameter were introduced by subsequent investigators. More accurate expressions for the repulsive and attractive forces were... 

The statistical-associated fluid theory (SAFT) of Chapman et al. [25, 26] is based on the perturbation theory of Wertheim [27]. The model molecule is a chain of hard spheres that is perturbed with a dispersion attractive potential and association potential. The residual Helmholtz energy of the fluid is given by the sum of the Helmholtz energies of the initially free hard spheres bonding the hard spheres to form a chain the dispersion attractive potential and the association potential,... 

The recently proposed Perturbed-Chain SAFT (PC-SAFT) model [75, 76] adopts the opposite idea here, a perturbation theory of second order is applied to the reference system of hard chains instead of hard spheres to develop a dispersion term for chain-Hke molecules ... 

To describe the measured cloud-points, the SAFT equation of state (eos) has been used. The SAFT eos [6] is based on the perturbation theory (see Chapter 3), and, in spite of the rather complex derivation of the model equations, the basic idea and the application of the model is less complex. The SAFT eos can be written as a sum of Helmholtz energies. The first contribution is the Helmholtz energy of an ideal gas, followed by a correction for a mixture of hard spheres, a correction for chain formation, and a correction for the dispersion and association forces ... 

EDMD and thermodynamic perturbation theory. Donev et developed a novd stochastic event-driven molecular dynamics (SEDMD) algorithm for simulating polymer chains in a solvent. This hybrid algorithm combines EDMD with the direa simulation Monte Carlo (DSMC) method. The chain beads are hard spheres tethered by square-wells and interact with the surrounding solvent with hard-core potentials. EDMD is used for the simulation of the polymer and solvent, but the solvent-solvent interaction is determined stochastically using DSMC. 

In all the equations that include the double power-series expansion (or variations thereof), such as the family of perturbed-hard-chain equations, the parameters are related to molecular rather than critical properties, and the mixing and combining rules are quadratic in composition for the attractive term and based on hard-sphere theory for the repulsive term. 

In contrast to the other popular versions of the SAFT equation, the PC-SAFT " " approach considers a hard-chain, rather than a hard-sphere, as the reference system for the application of standard high-temperature perturbation theory to obtain the dispersion contribution. Thus the Helmholtz... 

Segura el al. combines Tarazona s WDA DFT for hard-spheres with Wertheim s thermodynamic perturbation theory and has been used in a number of studies of associating fluids in pores and with functionalized walls in the limit of complete association a DFT for polymeric fluids is obtained in this method. Based on these works, Chapman and co-workers have presented the interfacial-SAFT (iSAFT) equation, which is a DFT for polyatomic fluids formulated by considering the polyatomic system as a mixture of associating atomic fluids in the limit of complete association this approach allows the study of the microstructure of chain fluids. Interfacial phenomena in complex mixtures with structured phases, including lipids near surfaces, model lipid bilayers, copolymer thin films and di-block copolymers, have all been studied with the iSAFT approach. 

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