Sanchez-Lacombe model

KI2 Kiran, E., Xiong, Y., and Zhunag, W., Modeling polyethylene solutions in near and supercritical fluids using the Sanchez-Lacombe model,/. Supercrit Fluids, 6, 193, 1993. 

X11 Xiong, Y. and Kiran, E., Prediction of Mgh-pressnre phase behavionr in polyethylene/n-pentane/carbon dioxide ternary system with the Sanchez-Lacombe models Polymer, 35, 4408, 1994. 

For the thermodynamic description of low-molecular-weight species, i.e. monomer, solvent, and initiator, the Sanchez-Lacombe model [46, 47] was used for the first two species, while a simple partition coefficient was assumed for the initiator. Since all equations related to these species are reported in detail in [41, 42, 48], here we summarize only the main equations, which are the mass balances of solvent, initiator and monomer... 

Figure 9. Pressure-composition (left) and temperature-composition (right) phase diagrams for polyethylene + n-pentane systems for three different molecular weight polyethylene samples (M =16,400 108,000 and 420,000). The P-x diagram corresponds to the 460 K cut and the P-x diagram corresponds to the 10 MPa cut from Figure 8. Solid curves are generated by the Sanchez-Lacombe model. [Ref. 32].

Kiran, E., Xiong, Y. and Zhuang, W. (1993) Modeling Polyethylene Solutions in Near and Supercritical Fluids Using the Sanchez-Lacombe Model, The Journal of Supercritical Fluids 6, 193-203. 

Fig. 2.14. Corresponding states behavior of various liquid-polymer PVT data according to the Sanchez-Lacombe model. Symbols represent experimental data curves are calculated from Eq. (71). Reproduced A/ith permission from Ref 62.

Fig. 2.16. Isothermal cloudpoint curves of the HOPE + ethylene system. M = 43 kg mol = 118 kg moh, Ml = 231 kg mol . Symbols experimental data curves modeling results (a) SAFT model (b) Sanchez-Lacombe model. Reproduced with permission from Ref 77.

The Sanchez-Lacombe model [48-50] is a lattice fluid model in which each component is divided into parts that are placed in a lattice. The different parts are allowed to interact with a mean-field intermolecular potential. By introducing an appropriate number of vacant sites (holes) in the lattice, the correct solution density can be obtained. SAFT [51-53] is based on the perturbation theory. The principle of perturbation theory is that first a model is derived for some idealized fluid with accurately known properties, called the reference fluid . Subsequently, the properties of this model are related to those of a real dense fluid. By expanding this reference fluid into power series over a specified parameter, the power terms can be regarded as corrections or "perturbations for the reference fluid as compared to reality. Obviously, the more the reference model approaches reality, the smaller the corrections are. Therefore, the key issue for applying perturbation theory is deriving the most suitable reference fluid. 

The model appears to describe accurately sorption isotherms when the equation of state parameters of both polymer and penetrant are determined. Like the Flory-Huggins modef, the Sanchez-Lacombe model assumes that the different components mix randomly in a lattice. Unlike the Flory-Huggins model, the Sanchez-Lacombe model permits some lattice sites to be empty, which allows holes or free volmne in the fluid. The addition of free volume to the lattice permits volume changes upon mixing components. The amount of absorbed penetrant in the polymer is determined by equating the chemical potential of the penetrant and the chemical potential of the penetrant in the mixture and by satisfying the equation of state of the pure penetrant phase and of the polymer-penetrant mixture. At fixed temperature and pressure, these conditions are met by equations 5-7. 

Optimizing solvents and solvent mixtures can be done empirically or through modeling. An example of the latter involves a single Sanchez-Lacombe lattice fluid equation of state, used to model both phases for a polymer-supercritical fluid-cosolvent system. This method works well over a wide pressure range both volumetric and phase equilibrium properties for a cross-linked poly(dimethyl siloxane) phase in contact with CO2 modified by a number of cosolvents (West et al., 1998). 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Costas et al. (1981) and Costas and Sanctuary (1981) reformulated the Sanchez-Lacombe equation of state so that the parameter r is not a regression parameter, but is actually the number of segments in the polymer molecule. In the original Sanchez-Lacombe treatment, r was regressed for several n-alkanes, and it was found that the r did not correspond to the carbon number of the alkanes. In addition, the Sanchez-Lacombe equation of state assumes an infinite coordination number. Costas et al. (1981) replaced the segment length r as an adjustable parameter with z. This modification involves the same number of adjustable parameters, but allows r to be physically significant. Thus, the model is more physically realistic, but there have been no definitive tests to determine whether this improves the correlative results from the model. 

Xiong, Y, and Kiran, E., 1995. Comparison of Sanchez-Lacombe and SAFT model in predicting solubility of polyethylene in high-pressure fluids. J. Appl. Polym. Set, 55 1805-1818. 

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. 

Kiszka, M. B., M. A. Meilchen, and M. A. McHugh. 1988. Modeling high-pressure gas-polymer mixtures using the Sanchez-Lacombe equation of state. J. Appl. Polym. Sci. 36 583. 

Flory-Orwoll-Vrij, 1964] (FOV), [Sanchez-Lacombe, 1976-8] (S-L), and [Simha-Somcynsky, 1969] (S-S). Large deviations (< 0.01 mL/g) were observed for S-G, the two following relationships were useful only at low P and over small P-ranges, whereas S-S consistently provided the best representation of data over extended ranges of T and P, with deviations < 0.003 mL/g, comparable to the experimental uncertainties. The FOV model can be expressed as ... 

NAG Nagy, I., Loos, Th.W.de, Krenz, R.A., and Heidemami, R.A., High pressure phase equilibria in the systems linear low density polyethylene -1- n-hexane and linear low density polyethylene + n-hexane + ethylene Experimental results and modelling with the Sanchez-Lacombe equation of state, J. Supercrit Fluids, 37,115,2006. 

Sanchez-Lacombe [1976,1978] The Sanchez and Lacombe [1976,1978] equation of state (S-L) is based on the Ising fluid model. The authors followed the Guggenheim [ 1966] approach, placing A -mers and No holes in an A -lattice. Hard-core volumes of the s-mer, as well as its flexibility, were assumed independent of Tand P. Furthermore, only the nearest neighbors of nonbonded mers contributed to the lattice energy ... 

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. 

As mentioned above, the model was adapted to continuous reactors (CSTR) in order to further vahdate its prediction ability. In particular, the effect of changing the inlet monomer concentration on the MWD was investigated. Simulations were carried out using the identical set of model parameter values already considered for the batch simulations above. However, because of the higher operating temperature in the CSTR reactions (see Table 6.5) with respect to the batch experiments, the binary interaction parameters used in the frame of the Sanchez-Lacombe equation of state have to be updated. This has been done... 

---