Phase equilibrium computations, polymer solution

The phase relations for quasi-binary solutions outlined in Section 1 are general and exact under the basic assumptions made. However, the computational work with them becomes exponentially difficult as the number of components increases. In fact, it is virtually impossible to solve the phase equilibrium equations for solutions of actual synthetic polymers, which contain an almost infinite number of components. We thus need a novel approach to analyze phase equilibrium data on such systems. The discipline called continuous thermodynamics has emerged to meet this requirement. It deals with mixtures of molecules whose physical properties such boiling point, molecular weight, and so forth vary continuously, and is the correct method for treating solutions of a truly polydisperse polymer (see Section 1.1 of this chapter for its definition). 

The computational problem of polymer phase equilibrium is to provide an adequate representation of the chemical potentials of each component in solution as a function of temperature, pressure, and composition. 

An algebraic equation relating the fundamental state variables of a fluid P, V and T is known as an equation of state, abbreviated here by EOS. The simplest EOS is the ideal gas law PV=RT. The models based on equations of state are widespread in simulation because allow a comprehensive computation of both thermodynamic properties and phase equilibrium with a minimum of data. EOS models are applied not only to hydrocarbon mixtures, as traditionally, but also to mixtures containing species of the most various chemical structures, including water and polar components, or even to solutions of polymers. The most important equations of state are presented briefly below, but they will be examined in more detail in other sections. 

Let s compute the vapor pressure of a solvent over a polymer solution by using the Flory-Huggins theory. Let component A represent the pohnner and B represent the small molecule. To describe the equilibrium, follow the strategy of Equation (16.2). Use Equation (31.20), and set the chemical potential of B in the vapor phase equal to the chemical potential of B in the polymer solution. The vapor pressure of B over a polymer solution is... 

Still, the use of the hydrodynamic volume, a size based on dynamical properties, in the interpretation of SEC data is not fully understood. This is because SEC is typically run under low flow rate conditions where hydrodynamic factor should have little effect on the separation. In fact, both theory and computer simulations assume a thermodynamic separation principle the separation process is determined by the equilibrium distribution (partitioning) of solute macromolecules between two phases — a dilute bulk solution phase located at the interstitial space and confined solution phases within the pores of column packing material. Based on this theory, it has been shown that the relevant size parameter to the partitioning of polymers in pores is the mean span dimension (mean maximal projection onto a line). Although this issue has not been fully resolved, it is likely that the mean span dimension and the hydrodynamic volume are strongly correlated. 

In the DDC process, the reduction in temperature of the system depends not only on the properties but also on the concentration of the polymer/solvent mixtures, because the polymer melts supercool primarily by the latent heat of the evaporating solvent phase. The values of temperature reduction have been computed using Eq.(l) for the LDPE solutions (Figure 1). It was assumed that during decompression, the process of thermal equilibrium happens so fast throughout the system that the temperature in the melt volume is uniform. The AHf m of 35.7 cal/g for the highly crystalline DDC foams and the Cp m of 0.6122 cal/K g at 110°C were taken (16),... 

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