Flory-Huggins mean-field theory, polymer

Flory-Huggins mean-field theory. A similar mean-field theory successfully describes thermodynamics of polymer blends and, with some modifications, diblock copolymers and their blends with homopolymers. 

We shall in section 4.2 deal with regular solutions of small-molecule substances. The construction of phase diagrams from the derived equations is demonstrated. The Flory—Huggins mean-field theory derived for mixtures of polymers and small-molecule solvents is dealt with in section 4.3. It turns out that the simple Flory—Huggins theory is inadequate in many cases. The scaling laws for dilute and semi-dilute solutions are briefly presented. The inadequacy of the Flory-Huggins approach has led to the development of the equation-of-state theories which is the fourth topic (section 4.6) Polymer-polymer mixtures are particularly complex and they are dealt with in section 4.7. 

Fig. 2 A graphical visualization of the Flory-Huggins mean field theory. The grating positions with white circles illustrate the solvent and the gray circles illustrate the PE. It s assumed that Each field has the same size, no overlap of fields or chains, all field positions are occupied, all polymer-polymer interactions are the same (all chain parts are the same)...

While the Flory-Huggins mean-field theory [13, 14, 15] of Sect. 2.1 describes the generic, qualitative behavior of incompressible polymer h-solvent mixtures, it invokes three important simplifications that restrict its application ... 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

The previous discussion, whose microscopic origins are in a lattice gas theory of mixtures, is applicable to mixtures of compact, molecules with comparable dimensions. In order to treat mixtures of long, chain, flexible polymers in small molecule solvents, the mean-field theory described above must be modified to take into account that the polymers and the solvent molecules are not of the same size and that the polymers are flexible macromolecules. The resulting Flory-Huggins free energy per monomer, ffn, is written... 

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 ... 

It is perhaps advisable to reiterate here that all of the thermodynamic analysis of SANS data is based on a mean-field theory, essentially that of Flory and Huggins, and this is known to be inadequate. It may be that, if a proper account were taken of the different expansibilities of the polymer mixture components, as in equation-of-state or lattice-fluid theories of polymer thermodynamics, then the dependencies on microstructure and a g-dependent % might disappear. 

Experimentally, there are many miscible polymer pairs in which at least one of the components is a random copolymer but specific interactions are not present [40]. This phenomenon is attributed to the so-called intramolecular repulsion effect. Within the familiar Flory-Huggins description, and in the case of a random copolymer A Bi blended with a homopolymer C, three interaction parameters, Xab,Xac and Xbc are required to describe the enthalpy of mixing. Using a mean field theory, the mixture can be described in terms of one parameter XeS given by ... 

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