Flory-Huggins Interaction Parameter Discussion

Section 2 reviewed the relevant contributions to the understanding of polymer swelling and permeation reported by earlier investigators. It also discusses the important parameters that have bearing on these phenomena, namely the Hildebrand Solubility Parameters, 8, the Flory-Huggins Interaction Parameters,... 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

Predicting blend miscibility correctly is, however, often considerably more challenging than one might guess by looking at the equations given above or even their somewhat more refined versions. Flory-Huggins interaction parameters, their more elaborate versions, and alternative methods such as the equation-of-state theories [9] discussed in Section 3.E, all provide correct predictions in many cases, but unfortunately provide incorrect predictions in many other cases. 

Just a few years ago, the limitations of solubility parameter calculations and measurements discussed above were serious impediments to modeling the phasic and interfacial behaviors of polymeric systems. The coming of age of atomistic simulation methods over the last few years has improved this situation dramatically. As discussed in Section 5.A.3, whenever accuracy is important in calculating the phasic or the interfacial behavior of a system, it is nowadays strongly preferable to use atomistic simulations employing modem force fields of the highest available quality instead of solubility parameters in order to estimate the Flory-Huggins interaction parameters (%) between the system components as input for further calculations. 

The Hildebrandt solubility parameter can be related to the Flory-Huggins interaction parameter by use of regular solution theory. The Hildebrandt solubility parameter can be calculated from one of the EOS theories discussed in Chapter 2 for polymeric systems. It was shown that... 

The above discussion on the two components of should lead to a better understanding of physical adsorption. Theoretically, polymer adsorption(so) can be treated by the Scheutjens-Fleer (SF)(si) mean-field theory, the Monte Carlo (MC) method,( 2) or the scaling approach. (83) In Figure 10, two profiles are given for the cases of adsorption (x = 1) and depletion (x = 0) using the SF theory, where x is the Flory-Huggins interaction parameter(84) between a polymer and a solvent with respect to pure components. The polymer coil expands if X < 0.5 and contracts if x These two cases are referred to as good and poor solvents, respectively. From the volume fraction profile c )(z), we can calculate other adsorption parameters, such as F, the adsorbed amount ... 

Here Vp is the volume fraction of polymer (related to the conversion), X is the number average degree of polymerisation of the polymer, x is the Flory-Huggins interaction parameter between the monomer and the polymer, R is the gas constant and T the temperature. Um is the molar volume of the monomer, y is the particle-water interfacial tension and To is the radius of the unswollen micelles, vesicles and/or latex particles. [M]a is the concentration of monomer in the aqueous phase and [M]a,sat the saturation concentration of monomer in aqueous phase. Figure 3.3 shows the contributions of the different terms of Equation 3.10 to the Vanzo equation. For a more detailed discussion see also Section 4.2 and Figure 4.5. 

However, in practice, to fit experimental phase diagrams, one has to make x a more complex (often, non-monotonic) function of T. Sometimes, researchers use a concentration-dependent Flory-Huggins interaction parameter - while this is clearly contrary to the spirit of the original Flory-Huggins theory, it can be useful to successfully fit experimental diagrams (see, e.g.. Reference [38]). For more detailed discussion see, for example. References [38-40]. 

The role of the solvent in polymer adsorption has been the subject of much discussion. For example, theories have made predictions about the effect of the polymer/solvent interaction (i.e. Flory Huggins x parameter) on adsorption. For many systems, x parameters had already been tabulated so that a number of adsorption studies focused attention on this parameter. In spite of much effort, available data are ambiguous, sometimes verifying and sometimes contradicting the trends predicted by theory. 

Here we rather focus on effects of external surfaces (e.g. hard walls) on polymer blends. In general, one expects that the forces between the wall and monomers of type A will differ from those between the wall and monomers of type B, as it generally occurs at the surfaces of small-molecule mixtures as well [365]. For polymer mixtures that are partially compatible, the interactions in the bulk (as described by the Flory-Huggins x-parameter) must be relatively small, however, since the entropy of mixing is down by a factor of N (for simplicity, the following discussion is restricted to a symmetric mixture, Na = Nb = N). However, there is no reason that the difference of wall-A and wall-B forces is similarly small [125]. Thus one may expect rather pronounced surface enrichment effects in polymer mixtures [125], Indeed some experimental evidence for this prediction has been found [37, 38, 126, 127]. 

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