Continuous thermodynamics for polydisperse polymer solutions

Abstract

Continuous thermodynamics is a framework which combines continuum modeling for the compositions of complex and multicomponent mixtures with molecular thermodynamic models and efficient numerical methods. In this work, a generalized molecular-thermodynamic model for polydisperse polymer solutions is developed; it is formally similar to the classical Flory-Huggins theory but with a polymer-size dependent and polymer-concentration dependent Flory parameter. Most existing lattice models and equation-of-state models such as the Guggenheim, Orifino-Flory, Koningsveld-Kleintjens, Sanchez-Lacombe and Revised Freed models can be cast into this generalized model but with different polymer-size and polymer-concentration dependence for the Flory parameter. A generalized continuous-thermodynamics framework based on this generalized model is also presented; expressions for chemical potentials, spinodals and critical points are derived using both the discrete multicomponent method and the continuous functional procedure. Internally consistent results are obtained. Criteria for multiple critical points are also derived. Computer programs are established for polydisperse-polymer solutions with either a standard or an arbitrary distribution for the polymer's molecular weight; in the latter case, the derivative method is applied, based on a previously developed spline fit. To illustrate the framework developed here, calculated liquid-liquid-equilibrium phase diagrams are shown, including UCST, LCST and hour-glass-shaped cloud-point curves, shadow curves, spinodals, critical points and their dependence on molecular parameters, on pressure and on molecular-weight distribution properties. used in those equations derived from Δ mix G. A binary energy parameter and a binary size parameter are used to fit experimental critical coordinates. Cloud-point curves, shadow curves, spinodals and critical points have been calculated under various conditions which cover different types of liquid-liquid-equilibrium phase equilibria including UCST, LCST and hour-glass shaped phase diagrams. If we have experimental critical points, spinodals, cloud-point curves and shadow curves, we can obtain the binary energy parameter and the binary size parameter as well as their temperature dependence if the experimental data cover a significant temperature range. Considering the difficulty of obtaining monodisperse polymer samples, we can now use a polydisperse polymer sample with a known molecular-weight distribution for experimental work (e.g. cloud points) and then use the parameters obtained experimentally to calculate phase-equilibrium properties for the same system but with a different molecular-weight distribution. Two different procedures, i.e., the discrete multicomponent method and the continuous functional method, have been used for deriving equations. Consistent results have been obtained by both methods. The discrete method is rigorous because polymer components are discrete. A continuous distribution function is only used to calculate various moments in those expressions. The functional method has the merit of mathematical integrity such that the continuous distribution function is built inside the whole framework. While the composition is, in fact, discrete, the continuous distribution provides a good approximation. The present work is only for solutions containing one solvent and one polydisperse polymer. Work in progress extends the methods discussed here to systems containing mixed solvents or mixed polymers and polymer blends with polydisperse polymers.