Physical Properties Calculation
The reduced density is the ratio between the density r(T,P) and the extrapolated density at 0 Kelvin, r* (also called hard-core density):
For the purposes of our calculations, the variations of density with temperature are defined as:
where a is the classical volumetric coefficient of thermal expansion. Where no experimental PVT data was available, either as tables or as an empirical relation such as the Tait equation,1,2 a was estimated by group contribution methods. The method for the estimation of a is based in the work of D.Boudoirs, L. Constantiou and C. Panayiotou.3 In this method, the Lattice Fluid theory (LF) characteristic parameters, T0, P0 and r0 are estimated using group contributions and used to calculate the density in the desired temperature range. The coefficient of thermal expansion is the exponential regression of the densities calculation. The hard-core density,r*, is assumed to be the intercept of the regression with zero Kelvin temperature.
For the polymer construction section of the applet a simple empirical approach for the estimation of the coefficient of thermal expansion based on the polymer glass transition, Tg is used:4
The coefficient obtained by this equation is not exact, but is an ok estimate if no other information is available.
The hard-core volume (n) is the extrapolation of the repeat unit volume at 0 Kelvin. It can be related to hard-core density, r* by:
where Munit is the molecular weight of the repeating unit (expressed in g/mol). The parameter used in the applet is a molar hard-core volume, which is obtained by multiplying the molecular value by Avogadroís number.
For the polymer construction section of the applet, the hard-core density is obtained from the hard-core parameter, ro, calculated using the Boudoris group contribution method. 3
The Hildebrand solubility parameter or cohesive energy density is defined as:
This parameter is a measure of the interacting behavior of the chemical species. In the Hildebrand Solubility Parameter Theory, chemical species with similar solubility parameters are miscible. For our phase behavior prediction, the solubility parameter for each component is estimated using the group contribution method described by Van Krevelen5. From this method, a solubility parameter at 298 K is obtained. The repeat unit volume, V repeat unit, is also obtained using Van Krevelenís group contribution method.
The molecular weights of the polymer, Mpolymer, components are incorporated into the CRS model in a similar fashion as in the Flory-Huggins model. The parameter used for the calculation is the number of repeat units, N, which can be easily obtained if the molecular weight for the repeat unit is known: