Thermodynamics of polymer solution

Ideal solutions are mixtures of molecules (i) that are identical in size and (ii) for which the energies of like (i.e. 1-1 or 2-2) and unlike (i.e. 1-2) molecular interactions are equal. The latter condition leads to athermal mixing (i.e. A// = 0), which also means that there are no changes in the rotational, vibrational and translational entropies of the components upon mixing. Thus AS depends only upon the combinatorial (or configurational) entropy change, which is positive because the number of 

From statistical mechanics, the fundamental relation between the entropy S of an assembly of molecules and the total number ft of distinguishable degenerate (i.e. of equal energy) arrangements of the molecules is given by Boltzmann s equation 

1 Schematic representation of a liquid lattice, (a) mixture of molecules of equal size, (b) mixture of solvent molecules with a polymer molecule showing the connectivity of polymer segments. 

For ideal mixing of molecules of solvent with JV2 molecules of solute in a lattice with (Ni + JV2) cells, the total number of distinguishable spatial arrangements of the molecules is equal to the number of permutations of (A i 4- N2) objects which fall into two classes containing Ni identical objects of type 1 and N2 identical objects of type 2 respectively 

It is more usual to write thermodynamic equations in terms of numbers of moles, n, and mole fractions, X, which here are defined by = Ni/Na, 2 = 2/N/j, 2f = /( + 2) 3nd X2 = n2/(ni+n2), where is the Avogadro constant. Thus Equation (3.8) becomes 

For a binary polymer-solvent solution, the simplest expression for the Gibbs free energy of mixing is based on the Flory-Huggins Theory (Flory, 1942 Huggins, 1942), 

For a mixture that is applicable to the FRRPP process, one can write the multi-component Flory-Huggins equation as 

For a ternary mixture that usually applies to the FRRPP reaction fluid, the Gibbs 

The above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

Modified Flory-Huggins equations used different techniques to account for the concentration dependence of x or without considering the true cause of the deficiency of the theory thus, their accuracy to represent experimental data must 

The interaction of long-chain molecnles with Uquids is of considerable interest from both a practical and theoretical viewpoint. For Unear and branched polymers, liquids that will dissolve the polymer completely to form a homogeneous solution can usually be found, whereas cross-linked networks will only swell when in contact with compatible liqnids. In this chapter, we shall deal with linear or branched polymers and treat the swelUng of networks in Chapter 14. 

When an amorphons polymer is mixed with a suitable solvent, it disperses in the solvent and behaves as thongh it too is a Uquid. In a good solvent, classed as one that is highly compatible with the polymer, the liquid-polymer interactions expand the polymer coil from its nnpertnrbed dimensions in proportion to the extent of these interactions. In a poor solvent, the intraactions are fewer, and coil expansion or pertnrbation is restricted. 

Only a short description of the basic theory is given here, so that experimental results can be interpreted qualitatively. A more detailed mathematical treatment can be found elsewhere (Flory, 1953). The basic theory for interpreting the thermodynamic properties of polymer solutions is that due to Flory and Huggins (Flory, 1953). According to this theory the chemical potential is given by  

The value of the interaction parameter, /, is important in the present discussion. When 0 x i th polymer-solvent interaction is strong, so the polymer coil expands to attain maximum solvent-polymer contact and the solution has a relatively high viscosity. In this situation the solvent is said to be good . When x = i the polymer is said to be in its 9 state and the polymer has exactly the molecular dimensions that theory would predict if there were no polymer-solvent interaction (i.e. a = 1 see discussion in Sections 3.3.2 and 3.3.3). When x h polymer-polymer contacts are thermodynamically preferred to polymer-solvent ones, so the polymer coil contracts, producing a low-viscosity solution in this situation the solvent quality is said to be poor . Eventually x will become so high that precipitation of the polymer will occur. 

The Flory-Huggins theory is only intended to work with non-ionic polymers dissolved in non-polar solvents, so that its use to interpret the results obtained from polyelectrolytes in water must be approached with care. For instance, when changing the calcium concentration the solvent quality will be altered, as well as changing the charge on the chain as a result of ion binding and altering the thickness of the electrical double layer. 

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

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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... 

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