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Simulation of Baker-Williams Fractionation By Continuous Thermodynamics
Abstract
Based on continuous thermodynamics and its application to the theory of successive polymer fractionation procedures, a theory of column fractionation is developed. In continuous thermodynamics the polydispersity of polymers is accounted for by the direct use of the continuous distribution function in the thermodynamic equations. In this way equations which are favorable for computer simulations are obtained. As an example, Baker-Williams fractionation is chosen for presenting theory and computer simulation. The generalization to other column fractionations based on solubility differences is easily possible.
... With the help of continuous thermodynamics, a theory to model stepwise fractionation of homopolymers was developed [45][46][47]. This theoretical framework could be extended to fractionation in columns [48][49][50]. The application of the developed theory was able to contribute to the improvement of the fraction technique [49]. ...
... Hence, the column fractionation is considered as a combination of many local LLEs and treated in an analogous way as successive fractionation procedures. Rätzsch et al. [50] developed a model in order to simulate the fractionation of homopolymers according to the molar mass in a BW column by a number of local equilibria, similar to the model suggested by Smith [76] and by Mac Lean and White [77]. ...
... Starting the fractionation, the total polymer is assumed to be precipitated at stage m ¼ m P ¼ 0 or to be distributed evenly among the m P + 1 stages from m ¼ 0 to m ¼ m P . The temperature gradient is expressed by [50]: ...
Random copolymers show polydispersity both with respect to molecular weight and with respect to chemical composition, where
the physical and chemical properties depend on both polydispersities. For special applications, the two-dimensional distribution
function must adjusted to the application purpose. The adjustment can be achieved by polymer fractionation. From the thermodynamic
point of view, the distribution function can be adjusted by the successive establishment of liquid–liquid equilibria (LLE)
for suitable solutions of the polymer to be fractionated. The fractionation column is divided into theoretical stages. Assuming
an LLE on each theoretical stage, the polymer fractionation can be modeled using phase equilibrium thermodynamics. As examples,
simulations of stepwise fractionation in one direction, cross-fractionation in two directions, and two different column fractionations
(Baker–Williams fractionation and continuous polymer fractionation) have been investigated. The simulation delivers the distribution
according the molecular weight and chemical composition in every obtained fraction, depending on the operative properties,
and is able to optimize the fractionation effectively.
KeywordsContinuous thermodynamics-Fractionation in column-Theory of copolymer fractionation
The enormous diversity of polymers with respect to molecular weight, molecular architecture, and – in the case of copolymers – also the content and arrangement of dissimilar monomers requires well-targeted methods for the separation of the different species that are contained in a given sample. This chapter presents the most abundant fractionation procedures, which are either based on thermodynamic driving forces (like in the case of liquid–liquid phase separation) or on kinetic effects (as with field-flow fractionation).
The theoretical part of this article demonstrates how the original Flory– Huggins theory can be extended to describe the thermodynamic behavior of polymer-containing mixtures quantitatively. This progress is achieved by accounting for two features of macromolecules that the original approach ignores: the effects of chain connectivity in the case of dilute solutions, and the ability of polymer coils to change their spatial extension in response to alterations in their molecular environment. In the general case, this approach leads to composition dependent interaction parameters, which can for most binary systems be described by means of two physically meaningful parameters; systems involving strongly interacting components, for instance via hydrogen bonds, may require up to four parameters. The general applicability of these equations is illustrated in a comprehensive section dedicated to the modeling of experimental findings. This part encompasses all types of phase equilibria, deals with binary systems (polymer solutions and polymer blends), and includes ternary mixtures; it covers linear and branched homopolymers as well as random and block copolymers. Particular emphasis is placed on the modeling of hitherto incomprehensible experimental observations reported in the literature.
For the first time, a quantitative theoretical analysis (liquid/liquid phase equilibria treated by means of the continuous thermodynamics) of the operating characteristics of continuous polymer fractionation (CPF) was performed. The results of these calculations were compared with data published for CPF of polyethylene. It turned out that the efficiency of the conventional CPF corresponds to approximately two theoretical plates only. For this reason, several improvements, suggested by theoretical considerations, were realized experimentally, for which purpose the system dichloromethane/diethylene glycol/bisphenol-A polycarbonate was chosen. The pulsating sieve-bottom column was replaced by a nonpulsating column filled with glass beads. In this manner, the number of theoretical plates could be raised considerably. A further improvement of the fractionation efficiency results from the reflux of part of the polymer contained in the sol phase. In praxi, this situation was realized by putting a condensor on top of the column and introducing the feed somewhere near its upper third. After predictive calculations and orienting experiments, 125 g of a polycarbonate with Mw = 29 kg/mol and a nonuniformity U = 1.3 were fractionated in four consecutive CPF runs (where the gels were directly used as feed for the next step) into five fractions of approximately equal weight. Except for the lowest-molecular-weight fraction, one obtains nonuniformities on the order of 0.1.
Continuous thermodynamics was developed in recent years and applied successfully to the liquid-liquid equilibrium of poly disperse polymer solutions. Continuous thermodynamics is based on the direct use of continuous distribution functions in the thermodynamic equations. There is no need for a reduction to pseudocomponents. This paper presents the application of continuous thermodynamics to successive polymer fractionation procedures based on solubility differences. In this way, simple equations for the distribution function of the different polymer fractions are obtained. Furthermore, the other equations describing the fractionation possess a lucid structure favorable for computer simulations of the fractionation procedures.