Polydisperse thermodynamics

DE2 DeSousa, H.C. and Rebelo, L.P.N., A continuous polydisperse thermodynamic algorithm for a modified Flory-Huggins model The (polystyrene + nitroethane) example, J. Polym. Sci. PartB Polym. Phys., 38, 632, 2000. 

M. Jaric, G. F. Toothill. Thermodynamic polydispersity and the Flory exponent. Phys Rev Lett 55 2891-2894, 1985. 

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. 

In order to understand polymer solution behaviour, the samples have to be characterised with respect to their molecular configuration, their molar mass and polydispersity, the polymer concentration and the shear rate. Classical techniques of polymer characterisation (light scattering, viscometry, ultracentrifugation, etc.) yield information on the solution structure and conformation of single macromolecules, as well as on the thermodynamic interactions with the solvent. In technical concentrations the behaviour of the dissolved polymer is more complicated because additional intramolecular and intermolecular interactions between polymer segments appear. 

Now, when these two species are added to each other, in a given relative concentration, a new species appears with a much narrower size distribution. This is shown in Figure 10.18, where the P-index (a measure of the polydispersity) is plotted against the molar fraction of DDAB. The P-index drops from the initial value of 0.20 (a very broad distribution) to 0.04, a very narrow distribution (stable for months), at a relative percent of 0.4 DDAB to 0.6 oleate (Thomas and Luisi, 2004). Between DDAB molar fractions of 0.41 and 0.60, flocculation occurs, which indicates a thermodynamic instability, in agreement with other cationic systems (Kaler etal., 1989 Marques etal., 1998 Kondo etal, 1995). 

For a polydisperse polymer, analysis of sedimentation equilibrium data becomes complex, because the molecular weight distribution significantly affects the solute distribution. In 1970, Scholte [62] made a thermodynamic analysis of sedimentation equilibrium for polydisperse flexible polymer solutions on the basis of Flory and Huggins chemical potential equations. From a similar thermodynamic analysis for stiff polymer solutions with Eqs. (27) for IT and (28) for the polymer chemical potential, we can show that the right-hand side of Eq. (29) for the isotropic solution of a polydisperse polymer is given, in a good approximation, by Eq. (30) if M is replaced by Mw [41],... 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

The polydispersity of polymers results in competing adsoiption of the thermodynamically favored larger molecules for surface sites filled initially by smaller molecules. Different segments of a block copolymer may exhibit quite different adsoiption characteristics, complicating the rearrangement process farther. This is an effect of considerable interest in protein adsoiption, and is referred to as the rearrangement of a protein layer to maximize hydrophobic interaction of "oily" patches with low energy surfaces such as medical implant polymers. 

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