Polymer phase equilibrium thermodynamics

In order to begin this presentation in a logical manner, we review in the next few paragraphs some of the general features of polymer solution phase equilibrium thermodynamics. Figure 1 shows perhaps the simplest liquid/liquid phase equilibrium situation which can occur in a solvent(l)/polymer(2) phase equilibrium. In Figure 1, we have assumed for simplicity that the polymer involved is monodisperse. We will discuss later the consequences of polymer polydispersity. 

The design engineer dealing with polymer solutions must determine if a multicomponent mixture will separate into two or more phases and what the equilibrium compositions of these phases will be. Prausnitz et al. (1986) provides an excellent introduction to the field of phase equilibrium thermodynamics. 

Thermodynamic Model for Phase Equilibrium between Polymer Solution and 6/W MlcroemulslonsT figures 6 and / show that when phase separation first occurs, most of the water is in the microemulsion. With an increase in salinity, however, much of the water shifts to the polymer solution. Thus, a concentrated polymer solution becomes dilute on increasing salinity. The objective of this model is to determine the partitioning of water between the microemulsion and the polymer-containing excess brine solution which are in equilibrium. For the sake of simplicity, it is assumed that there is no polymer in the microemulsion phase, and also no microemulsion drops in the polymer solution. The model is illustrated in Figure 12. The model also assumes that the value of the interaction parameter (x) or the volume of the polymer does not change with salinity. 

This brief survey of applied phase-equilibrium thermodynamics can do no mote than summarize the main ideas that constitute ihe present state of ibe art. Attentina is restricted to relatively simple mixtures as encountered in the patrolenm. natural gas. and petmchemical industries unfortunately, limited space does not allow discussion of other important systems such as polymer mixtures, electrolyte solutions, metallic alloys, molien salts, refractories (such as ceramics), or aqueous solutions of biologically impurlant solutes, However, it is not only lack of space that is responsible for these omissions because, at present, thermodynamic knowledge is severely limited for lhese more complex systems,... 

Initial works on the phase equilibrium of polymer solutions were concerned with nonpolar solutions using carefully prepared quasi-monodisperse polymer fractions [78]. The theory and practice was later extended to molecularly heterogeneous polymers [84], multicomponent solutions (ternary mixtures) such as polymer/solvent mixture [16, 85] and polymer mixture/solvent [86], and polymer blends [79, 80], among others [87]. Improvements on predicting thermodynamic properties were particularly proposed for polymer solutions of industrial importance, including those having polar and hydrogen-bonded components [16]. 

The notoriously poor polymer crystals described in Chap. 5 and their typical microphase and nanophase separations in polymer systems have forced a rethit ing of the application of thermodynamics of phases. Equilibrium thermodynamics remains important for the description of the limiting (but for polymers often not attainable) equilibrium states. Thermal analysis, with its methods described in Chap. 4, is quite often neglected in physical chemistry, but unites thermodynamics with irreversible thermodynamics and kinetics as introduced in Chap. 2, and used as an important tool in description of polymeric materials in Chaps. 6 and 7. 

This portion of the chapter can be summarized by noting that there is a substantial body of evidence demonstrating that formal phase-equilibrium thermodynamics can be successfully applied to the fusion of homopolymers, copolymers, and polymer-diluent mixtures. This conclusion has many far-reaching consequences. It has also been found that the same principles of phase equilibrium can be applied to the analysis of the influence of hydrostratic pressure and various types of deformation on the process of fusion [11], However, equilibrium conditions are rarely obtained in crystalline polymer systems. Usually, one is dealing with a metastable state, in which the crystallization is not complete and the crystallite sizes are restricted. Consequently, the actual molecular stmcture and related morphology that is involved determines properties. Information that leads to an understanding of the structure in the crystalline state comes from studying the kinetics and mechanism of crystallization. This is the subject matter of the next section. 

There is a long history of observations of phase equilibrium in polymer solutions. Phase separation in polymer solutions was first considered in detail from a statistical thermodynamic view by Flory [27,40,41). In his original papers on the statistical thermodynamics of these systems, the conditions for equilibrium between two separated phases are the classical conditions of when the partial molar Gibbs free energies are the same for each phase. The partial molar free energies are... 

In this chapter we shall consider some thermodynamic properties of solutions in which a polymer is the solute and some low molecular weight species is the solvent. Our special interest is in the application of solution thermodynamics to problems of phase equilibrium. 

The flow behavior of the polymer blends is quite complex, influenced by the equilibrium thermodynamic, dynamics of phase separation, morphology, and flow geometry [2]. The flow properties of a two phase blend of incompatible polymers are determined by the properties of the component, that is the continuous phase while adding a low-viscosity component to a high-viscosity component melt. As long as the latter forms a continuous phase, the viscosity of the blend remains high. As soon as the phase inversion [2] occurs, the viscosity of the blend falls sharply, even with a relatively low content of low-viscosity component. Therefore, the S-shaped concentration dependence of the viscosity of blend of incompatible polymers is an indication of phase inversion. The temperature dependence of the viscosity of blends is determined by the viscous flow of the dispersion medium, which is affected by the presence of a second component. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

From the outset, Flory (6) and Huggins (4,5 ) recognized that their expressions for polymer solution thermodynamics had certain shortcomings (2). Among these were the fact that the Flory-Huggins expressions do not predict the existence of the LCST (see Figure 2) and that in practice the x parameter must be composition dependent in order to fit phase equilibrium data for many polymer solutions 3,8). 

Our model predicts destabilization of colloidal dispersions at low polymer concentration and restabilisation in (very) concentrated polymer solutions. This restabilisation is not a kinetic effect, but is governed by equilibrium thermodynamics, the dispersed phase being the situation of lowest free energy at high polymer concentration. Restabilisation is a consequence of the fact that the depletion thickness is, in concentrated polymer solutions, (much) lower than the radius of gyration, leading to a weaker attraction. 

As there exists a phase equilibrium both phases must have reached in the internal thermodynamic equilibrium with respect to the arrangement and distribution of the molecules the measuring time. Therefore, no time effects or path dependencies of the thermodynamic properties in the liquid crystalline phase should be expected. To check this point for the l.c. polymer, a cut through the measured V(P) curves at 2000 bar has been made (Fig. 6) and the volume values are inserted at different temperatures in Fig. 7, which represents the measured isobaric volume-temperature curve at 2000 bar 38). It can be seen from Fig. 7 that all specific volumes obtained by the cut through the isotherms in Fig. 6 he on the directly measured isobar. No path dependence can be detected in the l.c. phase. From these observations we can conclude that the volume as well as other properties of the polymers depend only on temperature and pressure. The liquid crystalline phase of the polymer is a homogeneous phase, which is in its internal thermodynamic equilibrium within the normal measuring time. 

With monomeric molecules, the aggregation number of micelles is determined by equilibrium thermodynamics. In polymeric molecules, however, topological constraints are imposed on the system. If the degree of polymerization exceeds the aggregation number of the monomeric micelle, unsaturated sites of the polymeric molecules become available (directed to the aqueous phase) and inter-molecular interactions (agglomeration) occur. In the case of polymer with Mw= 6.23x105, typical surfactant behavior was found. 

Chapter 2 is an in depth discussion of the various theories important to phase equilibria in general and polymer thermodynamics specifically. First a review of phase equilibria is provided followed by more specific discussions of the thermodynamic models that are important to polymer solution thermodynamics. The chapter concludes with an analysis of the behavior of liquid-liquid systems and how their phase equilibrium can be correlated. 

Sluckin adopted a quasi-equilibrium thermodynamic approach to understanding the effect of a strain rate field on the isotropic-anisotropic transition in polymer solutions. He derived a Clausius-Clapeyron-like equation which connects the shift in the critical polymer mole fraction C, and Cj, which are concentrations of isotropic and nematic phases, respectively, to the applied strain rate. 

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