Modeling phase behavior

Vapor pressure is a key property in modeling phase behavior because it represents the vapor-liquid equilibria (VLE) of the pure fluid. Accurate vapor pressure characterization is essential to VLE correlation. As the most common operation in chemical processing is... 

Modeling Phase Behavior with the Peng-Robinson Equation of State... 

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]. 

These fascinating bicontinuous or sponge phases have attracted considerable theoretical interest. Percolation theory [112] is an important component of such models as it can be used to describe conductivity and other physical properties of microemulsions. Topological analysis [113] and geometric models [114] are useful, as are thermodynamic analyses [115-118] balancing curvature elasticity and entropy. Similar elastic modulus considerations enter into models of the properties and stability of droplet phases [119-121] and phase behavior of microemulsions in general [97, 122]. 

An essential component of cell membranes are the lipids, lecithins, or phosphatidylcholines (PC). The typical ir-a behavior shown in Fig. XV-6 is similar to that for the simple fatty-acid monolayers (see Fig. IV-16) and has been modeled theoretically [36]. Branched hydrocarbons tails tend to expand the mono-layer [38], but generally the phase behavior is described by a fluid-gel transition at the plateau [39] and a semicrystalline phase at low a. As illustrated in Fig. XV-7, the areas of the dense phase may initially be highly branched, but they anneal to a circular shape on recompression [40]. The theoretical evaluation of these shape transitions is discussed in Section IV-4F. 

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

Solids can be crystalline, molecular crystals, or amorphous. Molecular crystals are ordered solids with individual molecules still identihable in the crystal. There is some disparity in chemical research. This is because experimental molecular geometries most often come from the X-ray dilfraction of crystalline compounds, whereas the most well-developed computational techniques are for modeling gas-phase compounds. Meanwhile, the information many chemists are most worried about is the solution-phase behavior of a compound. 

One proposed approach (75) to modeling the phase behavior for hydrogen bonding pairs uses the following expression for the free energy of mixing (eq. 7). 

Although modeling of supercritical phase behavior can sometimes be done using relatively simple thermodynamics, this is not the norm. Especially in the region of the critical point, extreme nonideahties occur and high compressibilities must be addressed. Several review papers and books discuss modeling of systems comprised of supercritical fluids and soHd orHquid solutes (rl,i4—r7,r9,i49,r50). 

One of the simplest cases of phase behavior modeling is that of soHd—fluid equilibria for crystalline soHds, in which the solubility of the fluid in the sohd phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component are equal for all phases at equilibrium under constant temperature and pressure (51). The soHd-phase fugacity,, can be represented by the following expression at temperature T ... 

A number of theoretical models have been proposed to describe the phase behavior of polymer—supercritical fluid systems, eg, the SAET and LEHB equations of state, and mean-field lattice gas models (67—69). Many examples of polymer—supercritical fluid systems are discussed ia the Hterature (1,3). 

To illustrate the complexity of the phase behavior in a more compact way it is instructive to employ a mean-field lattice-gas model. The relative simplicity of the grand potential... 

In some of these models (see Sec. Ill) the surfactants are still treated as flexible chains [24]. This allows one to study the role of the chain length and chain conformations. For example, the chain degrees of freedom are responsible for the internal phase transitions in monolayers and bilayers, in particular the hquid/gel transition. The chain length and chain architecture determine the efficiency of an amphiphile and thus influence the phase behavior. Moreover, they affect the shapes and size distributions of micelles. Chain models are usually fairly universal, in the sense that they can be used to study many different phenomena. 

Stadler et al. [150,151] have performed Monte Carlo simulations of this model at constant pressure and calculated the phase behavior for various different head sizes. It turns out to be amazingly rich. The phase diagram for chain length N = 1 and heads of size 1.2cr (cr being the diameter of the tail beads) is shown in Fig. 8. A disordered expanded phase is found as well as... 

Whereas chain models still allow for a relatively unified treatment of various aspects of amphiphilic systems, such as their bulk phase behavior and the properties of monolayers and bilayers, this is no longer true for the even more idealized models at the next level of coarse graining. These usually have to be adapted very specifically to the problem one wishes to study. 

Here the functions g(0) and /(0) are defined in a suitable way to produce the desired phase behavior (see Chapter 14). The amphiphile concentration does not appear expHcitly in this model, but it influences the form of g(0)— in particular, its sign. Other models work with two order parameters, one for the difference between oil and water density and one for the amphiphile density. In addition, a vector order-parameter field sometimes accounts for the orientional degrees of freedom of the amphiphiles [1]. 

Thermodynamics gives limited information on each of the three coefficients which appear on the right-hand side of Eq. (1). The first term can be related to the partial molar enthalpy and the second to the partial molar volume the third term cannot be expressed in terms of any fundamental thermodynamic property, but it can be conveniently related to the excess Gibbs energy which, in turn, can be described by a solution model. For a complete description of phase behavior we must say something about each of these three coefficients for each component, in every phase. In high-pressure work, it is important to give particular attention to the second coefficient, which tells us how phase behavior is affected by pressure. 

The dilated van Laar model is readily generalized to the multicomponent case, as discussed in detail elsewhere (C3, C4). The important technical advantage of the generalization is that it permits good estimates to be made of multicomponent phase behavior using only experimental data obtained for binary systems. For example, Fig. 14 presents a comparison of calculated and observed -factors for the methane-propane-n-pentane system at conditions close to the critical.7... 

While the dilated van Laar model gives a reliable representation of constant-pressure activity coefficients for nonpolar systems, the good agreement between calculated and experimental high-pressure phase behavior shown in Fig. 14 is primarily a result of good representation of the partial molar volumes, as discussed in Section IV. The essential part of any thermodynamic description of high-pressure vapor-liquid equilibria must depend,... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

Next, the VLE was calculated using these parameters and the results together with the experimental data are shown in Figure 14.4. The erroneous phase behavior has been suppressed. However, the deviations between the experimental data and the EoS-based calculated phase behavior are excessively large. In this case, the overall fit is judged to be unacceptable and one should proceed and search for more suitable mixing rules. Schwartzentruber et al. (1987) also modeled this system and encountered the same problem. 

Blanco et al. have also correlated the results with the van Laar, Wilson, NRTL and UNIQUAC activity coefficient models and found all of them able to describe the observed phase behavior. The value of the parameter ai2 in the NRTL model was set equal to 0.3. The estimated parameters were reported in Table 10 of the above reference. Using the data of Table 15.7 estimate the binary parameters in the Wislon, NRTL and UNIQUAC models. The objective function to be minimized is given by Equation 15.11. 

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