The thermodynamics of polymer solutions

In a poor solvent, the polymer segments prefer to remain attached to other segments of the polymer molecule thus while separating from other polymer molecules in the solid, the molecule coils upon itself (Fig. 35.3b). These different conformations have enormous influence on the viscosity, f or example. The viscosity of a solution of long uncoiled chains is very much larger than that of a solution containing the coiled molecules. 

The equation for the Gibbs energy of mixing of any solution is given by 

Since the Gibbs-Duhem equation requires E the sum is zero and we have 

For a long time it was thought that if there was no heat of mixing a mixture would behave ideally. However, even if the heat of mixing is zero, if there are large differences between the molar volumes of the two constituents the mixture will not be ideal. 

By considering the number of arrangements of polymer and solvent molecules on a lattice, we can calculate the entropy of the mixture and from that the Gibbs energy (if we assume some value for the heat of mixing). A simplified two-dimensional model of a polymer molecule arranged on alatticeis shown in Fig. 35.5. We assume that a solvent molecule 

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Before concluding this section, there is one additional thermodynamic factor to be mentioned which also has the effect of lowering. Since we shall not describe the thermodynamics of polymer solutions until Chap. 8, a quantitative treatment is inappropriate at this point. However, some relationships familiar from the behavior of low molecular weight compounds may be borrowed for qualitative discussion. The specific effect we consider is that of chain ends. The position we take is that they are foreign species from the viewpoint of crystallization. 

In Chap. 8 we discuss the thermodynamics of polymer solutions, specifically with respect to phase separation and osmotic pressure. We shall devote considerable attention to statistical models to describe both the entropy and the enthalpy of mixtures. Of particular interest is the idea that the thermodynamic... 

A number of chapters have been overhauled so thoroughly that they bear only minor resemblance to their counterparts in the first edition. The thermodynamics of polymer solutions is introduced in connection with osmometry and the drainage and spatial extension of polymer coils is discussed in connection with viscosity. The treatment of contact angle is expanded so that it is presented on a more equal footing with surface tension in the presentation of liquid surfaces. Steric stabilization as a protective mechanism against flocculation is discussed along with the classical DLVO theory. 

In this chapter, subsequent to an introduction to devolatilization equipment, we review the thermodynamics of polymer solution equilibrium, which determines the maximum amount of volatiles that can be separated under a given set of processing conditions the phenomena associated with diffusion and diffusivity of small molecules in polymeric melts, which affects the rate of mass transfer the phenomena and mechanisms involving devolatilization and their modeling and the detailed and complex morphologies within the growing bubbles created during devolatilization of melts. 

The preceding section illustrates the variety of phenomena that may be observed in polymer-colloid-solvent mixtures. Polymer dissolved in a colloidal suspension is in some ways similar to ionic solutes responsible for electrostatic effects. Interactions between colloidal particles and polymer generate nonuniform distributions of polymer throughout the solution. Particle-particle interactions alter the equilibrium polymer distribution, producing a force in which sign and magnitude depend on the nature of the particle-polymer interaction. The major difference between polymeric and ionic solutions lies in the internal degrees of freedom of the polymer. Thus, a complete treatment of particle-polymer interactions requires detailed consideration of the thermodynamics of polymer solutions. 

Figure 10.18 is a schematic representation of depletion stabilization in which the polymer is prevented from the zone of close approach between two particles. As a result of this low polymer concentration between the particles due to size exclusion, there is a lower osmotic pressure, which results in (1) an attractive force for greater than theta solvents and (2) a repulsive force for less than theta solvents. Theta solvents will be discussed in the section on the thermodynamics of polymer solutions, but first a discussion of pol3naaer properties. 

The book is divided into four parts. After an introduction in Chapter 1, where the necessary concepts from a first course on polymers are summarized, the conformations of single polymer chains are treated in Part 1. Part 2 deals with the thermodynamics of polymer solutions and melts, including the conformations of chains in those states. Part 3 applies the concepts of Part 2 to the formation and properties of polymer networks. 

Despite the drawbacks, Flory-Huggins lattice theory was a major step forward towards understanding the thermodynamics of polymer solutions and is the basis of many other theories. Since it was first proposed, other workers have elaborated on it to improve on the assumptions involved. Some of these more refined theories will be given brief consideration, but in general are beyond the scope of the book. 

To characterize the thermodynamic behavior of the components in a solution, it is necessary to use the concept of partial molar or partial specific functions. The partial molar quantities most commonly encountered in the thermodynamics of polymer solutions are partial molar volume Vi and partial molar Gibbs free energy Gi. The latter quantity is of special significance since it is identical to the quantity called chemical potential, pi, defined by... 

Special books on the thermodynamics of polymers or, at least, on the thermodynamics of polymer solutions, at least, have not been widely scattered around the scientific world. In contrast, there is a great number of large reviews or original papers on this subject in the international or Russian scientific journals. Many of them are of a general character but the principle itself of writing problematic or review papers prevents a relatively complete consideration of any branch of science on the wholr. 

There are ample books in the literature considering in detail the thermodynamics of polymer solutions, i.e. the state of the P-I-LMWL system above the 9 point (for systems with an upper critical solution temperature) (see the bibliography). With the exception of riory (J953) and Tompa (1956), the other authors cither did not deal with phase separation or mentioned it only in its relation to fractionation. A need has, therefore, arisen to look into the questions of liquid-liquid phase separation (including multiphase separation) as c.arefully as possible, the more so that many applications of these problems can be introduced into the technology of polymer materials. 

The method is also useful in the study of the thermodynamics of polymer solutions at finite concentrations of the injected sample and in three component systems in which the vapours of the volatile substance interact with the blends of polymers or polymer-plasticizer blends. 

In this section the basic principles of membrane formation by phase inversion will be described in greater detail. All phase inversion processes are based on the same thermodynamic principles, since the starting point in all cases is a thermodynamically stable solution which is subjected to demixing. Special attention will be paid to the immersion precipitation process with the basic charaaeristic that at least three components are used a polymer, a solvent and a nonsolvent where the solvent and nonsolvent must be miscible with each other. In fact, most of the commercial phase inversion membranes are prepared from multi-component mixtures, but in order to understand the basic principles only three component systems will be considered. An introduction to the thermodynamics of. polymer solutions is first given, a qualitatively useful approach for describing polymer solubility or polymer-penetrant interaction is the solubility parameter theory. A more quantitative description is provided by the Flory-Huggins theory. Other more sophisticated theories have been developed but they will not be considered here. 

We have given this discussion at the outset and in some detail, because the question when and which model is a central question for any MC simulation of polymers Actually a large variety of models for macromolecules exists, differing in the details of how the coarse-graining is done. Actually, simple lattice models like the SAW have been central to the formulation of the first theories for the thermodynamics of polymer solutions and polymer blends, such as the Flory-Hu ns the-ory2-14,15,39 extensions, although excluded volume in... 

In this chapter it is not possible to give an in-depth treatment of the thermodynamics of polymer solutions. For further reading, see Refs. 1-6. 

For a much fuller discussion on the thermodynamics of polymer solutions the reader is referred to the texts by Flory (1953) and Richards (1980). 

A detailed presentation of the thermodynamics of polymer solutions is given in Chapter 4 here only the final Flory-Huggins equation (eq. (3.45)) is presented. The decrease in free energy comes from the mixing enthalpy and entropy and the molar free energy of mixing (AGj J becomes ... 

A number of different compounds are used as polymer additives to tailor the properties of materials for their end-use. As discussed in detail in Chapter 18, these additives include dyes, flllers, plasticizers, and a variety of other compounds. Because each of these additives gets between polymer chains (at least to some extent), they reduce the number of polymer-polymer interactions and cause an increase in chain mobility (when compared to the pure polymer at the same temperature). This causes a drop in Tg (which is exactly what a plasticizer is designed to do. Figure 6.7). This decrease in Tg depends on the thermodynamic compatibility of the additive with the polymer. For some fillers, this may be quite low, causing little change in Tg, but for chemical additives that interact with the polymer, Tg can be reduced significantly. The thermodynamics of polymer solutions and polymer... 

By definition, a solution contains more than one component. A solution can Ije gaseous, liquid, or solid. The term macromolecular (or polymer) solution will be used to indicate a mixture of a polymer with a small-molecule solvent and polymer blend when solvent and solute are both polymers. In this chapter the thermodynamics of polymer solutions and of solid polymer blends will thus be discussed separately. 

If the Flory theory is indisputably a reference for the thermodynamics of polymer solutions, it suffers from a lack of accuracy in its description of dilute polymer solutions as previously mentioned. Well suited to the case of concentrated solutions, this theory depicts the behavior of dilute solutions and describes the forces due to excluded volume as the result of a perturbation to random walk statistics for example, it does not account for the significant variations experienced by the density of segments in dilute media. Indeed, the replacement of the radial variation of this function (which describes the density of interaction in the medium) by an average value is not satisfactory. 

Polymers are thus materials with peculiar physical properties which are controlled by their methods of synthesis and their internal structure. The first chapters (I to III) introduce the notions of configuration and conformation of polymers, their dimensionality, and how their multiple interactions contribute to their overall cohesion. The three next chapters are concerned with physical chemistry, namely the thermodynamics of polymer solutions (IV), the structures typical of polymer assemblies (V), and the experimental methods used to characterize the size, the shape and the structures of polymers (VI). Four chapters (VII to IX) then follow that elaborate on the methods of synthesis and modification of polymers, and the engineering of complex architectures (X). Chapters XI to XIII subsequently describe the thermal transitions and relaxations of polymers, their mechanical properties and their rheology. These thirteen chapters are rounded off by monographs (chapters XIV to XVI) of natural polymers and of some common monodimensional and tridimensional polymers. 

In his fundamental work, based on that of Meyer, Flory extended the thermodynamics of polymer solutions to cover swollen, crosslinked polymers. The expressions developed include the number of network chains per unit volume or, in terms of the density... 

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