Perturbed hard sphere model

The modeling of the (temperature dependent) densities is described briefly in Sect. 6.1.2, using equations of state (EoS) derived from various modifications of the statistical associated fluid theory (SAFT), the COSMO-RS model, the Sanchez-Lascomb lattice fluid model (SL), or the perturbed hard sphere model (PHS). Each... 

The perturbed hard sphere fluid model used in this study is an extension of the the work of Chandler and coworkers (23 J2). In this model the chemical potential of the solute molecule and the resulting Raman frequency shift are separated into two terms. 

The temperature independence of the CH frequency shifts is also reflected in the nearly constant attractive force parameters (see Table I). In fact, the frequency shifts predicted using the average attractive force parameter, Ca = 0.973, reproduce the experimental results to within 3% throughout the experimental density and temperature range. It thus appears that the attractive force parameter may reasonably be treated as a temperature and density independent constant. This behavior is reminiscent of that found for attractive force parameters derived from high pressure liquid equation of state studies using a perturbed hard sphere fluid model (37). 

Therefore, the hard sphere model cannot be applied to a very concentrated dispersion without introducing a perturbation due to the van der Waals attraction. 

Thus, if we take the packing fraction of the glass transition in the hard-sphere model as a reference, a straightforward prediction of the glass transition volume in real systems based on a complete neglect of temperature effects can be regarded as a perturbation treatment of zeroth order. This... 

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. 

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 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. 

The transport phenomena discussed in Chapters 2 and 3 are the simplest examples of kinetic phenomena, if not the most familiar. For the cases considered, a single macroscopic property (number, charge, momentum, or energy density) was displaced from its equilibrium value. This perturbation caused a flux in the opposite direction, proportional to the displacement. The proportionality constant is the transport coefficient. For simple gaseous systems the relations between displacement and flux and the transport coefficient were determined using a hard-sphere model to describe molecular interaction. 

Fig. 42 illustrates the potential difference between the hard sphere model and a more realistic energy profile. As mentioned at the beginning of the x theory derivation, if the quantum number is held constant and a small perturbation is made in one part of the potential energy well, then the area averaged potential energy wiU be observed. This principle can be applied to make a correction... 

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]. 

A disadvantage of the MSA is that it applies only to charged hard-sphere models. Models with more realistic or elaborate potentials are not easily treated by this approximation except by using it as the leading term in a perturbation theory. 

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]. 

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... 

A perturbed hard sphere equation of state (PHS-EoS) was used by Hosseini et al. [50, 51] to model the volumetric properties of RTILs. This EoS has the form ... 

Hosseini SM, Moghadasi J, Papari MM, Nobandegani EF (2011) Modeling the volumetric properties of mixtures involving ionic liquids using perturbed hard-sphere equation of state. J Mol Liq 160 67-71... 

It is now well-established that for atomic fluids, far from the critical point, the atomic organisation is dictated by the repulsive forces while the longer range attractive forces serve to maintain the high density [34]. The investigation of systems of hard spheres can therefore be used as simple models for atomic systems they also serve as a basis for a thermodynamic perturbation analysis to introduce the attractive forces in a van der Waals-like approach [35]. In consequence it is to be expected that the anisotropic repulsive forces would be responsible for the structure of liquid crystal phases and numerous simulation studies of hard objects have been undertaken to explore this possibility [36]. 

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