Polymer fluids associative fluid model

The contribution "Application of Meso-Scale Field-based Models to Predict Stability of Particle Dispersions in Polymer Melts" by Prasanna Jog, Valeriy Ginzburg, Rakesh Srivastava, Jeffrey Weinhold, Shekhar Jain, and Walter Chapman examines and compares Self Consistent Field Theory and interfacial Statistical Associating Fluid Theory for use in predicting the thermodynamic phase behavior of dispersions in polymer melts. Such dispersions are of quite some technological importance in the... 

Equations of state offer a number of advantages over activity coefficient models for example, they can be applied to both low and high pressures, for properties other than phase equilibria, and the density is not required as an input parameter. However, often they are more difficult to develop for complex fluids and mixtures than are activity coefficient models. Very many equations of state have been proposed for polymers Section 16.7 discusses the reason. Recent reviews have been presented. " " We will not attempt to cover all the various approaches, but essentially discuss in detail only two of them, which seem promising for polymer solutions and blends the cubic equations of state and the SAFT (Statistical Associating Fluid Theory) method. 

Very many noncubic equations of state have been proposed for polymers. Most of them are rather complicated and a few of them, such as several of the models discussed previously, are based on the GC concept, like the GC-Flory and the GCLF equation of state. Several of these equations of state are reviewed elsewhere, and we will restrict here our presentation to one equation of state that is very promising for polymer systems and has already found widespread acceptance. This is the Statistical Associating Fluid Theory (SAFT). 

Orbey, H., Bokis, C.P., and Chen, C.-C., Equation of state modeling of phase equilibrium in the low-density polyethylene process the Sanchez-Lacombe, Statistical Associating Fluid Theory, and the polymer-SRK equation of state, Ind. Eng. Chem. Res., 37, 4481, 1998. 

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 polymer solutions warrant use of a special class of lattice models such as Florry-Huggins. For correlation purposes Sanchez-Lacombe " method is sufficient but one may also use Statistical Association Fluid Theoiy (SAFT) models to obtain a better representation. 

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 ° ) ... 

The model assumes that the reaction mixture consists of a polymer phase swollen with ethylene and CO2, and an ethylene-C02 phase. The assumption that no polymer dissolves in the ethylene-C02 phase within the experimental conditions has previously been confirmed experimentally for a similar polymer [3]. The swollen polymer phase, i.e. the polymer-ethylene-C02 system, is modeled using the Statistical Associating Fluid Theory (SAFT) eos [6, 7], see Section 8.2.2. The supercritical phase, i.e. the ethylene-C02 system, is either modeled with the Lee-Kessler-Plocker (LKP) eos [30] or with the Peng-Robinson (PR) eos [31], because the use of the SAFT eos for the simulation of both phases results in physically inconsistent behavior [9]. The temperatures used in this work are just above the critical tern-... 

The phase behavior of polymer/SCF mixtures can be described using versions of the lattice fluid (LF) model such as that developed by Sanchez and Lacombe [17]. The LF equation of state is relatively simple, and has been successfully used to describe either polymers dissolved in SCFs, or SCFs dissolved in polymers [18,19], including phenomena such as retrograde vitrification. The statistical associating fluid theory (SAFT) [20] can also describe the phase behavior of polymers dissolved in SCFs. The SAFT model, while somewhat more cumbersome to implement than the LF model, is especially well-suited for polymers with varying backbone architecture, such as branched polymers or copolymers. Both the Sanchez-Lacombe and SAFT models have been incorporated into commercially available modeling software [21]. 

A number of equation of state theories have been used to model phase behavior of polymers in supercritical fluids. For example the lattice-fluid theory of Sanchez and Lacombe[4U 42] includes holes on the lattice in order to model compressibility. The lattice-fluid theory has been applied to model phase behavior of both homopolymers and copolymers in supercritical fluids[32, 38, 43, 44]. The statistical associating fluid theory (SAFT)[43,45-48] and corresponding state models[49] have also been employed to model compressible polymer-solvent mixtures. Figure 1 gives the pressure-concentration phase diagram for poly(dimethyI siloxane) in CO2 modeled with the lattice-fluid equation of state[50]. 

The first and earliest class of new approaches to polymer fluids is the associating fluid model pioneered by Wertheim. In the associating fluid model one considers a liquid of particles interacting with a potential given by... 

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]. 

Simulation of polymerization and copolymerization kinetits requires additional information about the phase behavior of the monomer-polymer system. Cloud-point smdies oncopolymer-ethene as well as copolymer-ethene-comonomer systems have been carried out over wide p and T ranges under visual inspection of the reaction mbcmre. Part of the measured cloud-point curves have been modeled by permrbed chain -statistical associating fluid theory (PC-SAFT) 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... 

Molecularly motivated empiricisms, such as the solubility parameter concept, have been valuable in dealing with mixtures of weakly interacting small molecules where surface forces are small. However, they are completely inadequate for mixtures that involve macromolecules, associating entities like surfactants, and rod-like or plate-like species that can form ordered phases. New theories and models are needed to describe and understand these systems. This is an active research area where advances could lead to better understanding of the dynamics of polymers and colloids in solution, the rheological and mechanical properties of these solutions, and, more generally, the fluid mechaiucs of non-Newtonian liquids. 

One of the common problems associated with underwater pelletizers is the tendency of the die holes to freeze off. This results in nonuniform polymer melt flow, increased pressure drop, and irregular extrudate shape. A detailed engineering analysis of pelletizers is performed which accounts for the complex interaction between the fluid mechanics and heat transfer processes in a single die hole. The pelletizer model is solved numerically to obtain velocity, temperature, and pressure profiles. Effect of operating conditions, and polymer rheology on die performance is evaluated and discussed. 

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