Flory-Huggins treatment

The summative-fractioruition method was extended to apply to narrow-distribution polymers with polydispersity (Mw/ Mn) less than 1.12. A fractionation parameter H, previously defined and calculated for theoretical molecular weight distributions for normal polymers, was computed for narrow-distribution polymers. The calculations were made both with and without correction for fractionation errors, using the Flory-Huggins treatment. The method was applied to a well-characterized anionic polystyrene with Mw = 97,000, for which the polydispersity was estimated by this technique to be 1.02 (in the range 1.014-1.027, 95% confidence limits). 

Accepting the conventional Flory-Huggins treatment, one deduces then the free energy of polymerization, A G1(S, of a liquid monomer into solid monomer to be... 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Volume fraction of species a in Flory-Huggins treatment of polymer mixtures. 

Interaction parameter in Flory-Huggins treatment of polymer mixtures after normalization on a per monomer basis this becomes Xay also susceptibilities in discussion of density functional theories. Applied external field acting on species a in conformation X". Applied external field as a function of position acting on species a. Contribution of attractive interactions to the second virial coefficient for species pair ay also van der Waals coefficient. 

At temperatures above Tg, the magnitude of Vg° is a measure of the solubility of the probe in the stationary phase. From the Flory-Huggins treatment of solution thermodynamics, one can obtain the x parameter, which is a measure of the residual free energy of interaction between the probe and the polymer QZ,. 18). 

Tlhis study was made to reconcile the behavior of low molecular weight hydrocarbon resins and the behavior of their plasticizers with the solubility parameter and with the Flory-Huggins treatment of phase separation from polymer solutions. These resins are widely used industrially for coatings, floorings, adhesives, rubber compounds, and many other applications. Since they are usually hard and brittle, they are used with rubber, drying oils, plastics, or with plasticizers. 

The behavior is predicted in the Flory-Huggins treatment of swelling, which leads to a relation between the degree of swelling Q for a particular solvent and the shear modulus G of the unswollen rubber... 

The Flory-Huggins treatment can be extended to the calculation of an interaction parameter (X,) used in many theories dealing with polymer miscibility. The concept of a theta (or Flory) condition is a major consequence. As mentioned in Chapter 2 in the discussion on chain dimensions, the use of the theta condition removes the complication of long chain interaction in assessing chain conformations. 

The expression for the entropy of mixing as derived from the Flory-Huggins treatment is applied in deriving an equation relating the free energy of mixing to the interaction parameter X. The latter can be... 

Equation 3.108 states that the difference in chemical potential between a standard state and the crystalline phase equals the difference in chemical potential between the same standard state and the diluted amorphous polymer unit. An equation similar to equation 3.80 derived from the Flory-Huggins treatment can be applied to the case of an amorphous phase containing diluent... 

A completely different approach to modeling the behavior of systems of the above type was developed by Lindman and co-workers (64). It draws on the Flory-Huggins treatment of polymer solutions (see Chapters 1-3) and utilizes careful determinations of the phase diagrams of these systems—approaches that have recently been refined, analyzed, and extended by Ranganathan and Kwak (63). 

See Chapter 31 for the Flory-Huggins treatment for particles of different sizes, such as polymers in simple solvents.)... 

The decomposition of A mix in these two contributions points to the two main aspects of the mixing process but this alone would not be of much value. What is needed, for practical uses, are explicit expressions for ASt and A ioc, so that the sum of the two contributions can be calculated. The Flory-Huggins treatment is based on approximate equations for both parts. We formulate them first and then discuss their origins and the implications. The equations have the following forms ... 

Although a modelling of a liquid polymer mixture on a lattice may first look rather artificial, it makes sense because it retains the important aspects of both the entropic and enthalpic part of A mix- In recent years, lattice models have gained a renewed importance as a concept which is suitable for computer simulations. Numerical investigations make it possible to check and assess the validity range of the Flory-Huggins treatment. In fact, limitations exist and, as analytical calculations are difficult, simulations are very helpful and important. We shall present one example in a later section. 

Af (PS) = 2 10, M(PVME) — 4.7 10. For molecular weights in this range the contribution of the translational entropy becomes very small indeed and mixing properties are mostly controlled by x- The curved appearance of the binodal, which contrasts with the result of the model calculation in Fig. 3.18 where we obtained for polymers with medium or high molecular weights a nearly horizontal line, is indicative of a pronounced compositional dependence of X- This represents a case where the Flory-Huggins treatment does not provide a comprehensive description. Interactions in this mixture are of a complex nature and apparently change with the sample composition, so that it becomes impossible to represent them by just one constant. 

Hence, we obtain an explicit expression for the collective response coefficient in terms of the known single chain response coefficients and a . Next, we turn to a non-athermal mixture. The difference in the interaction between like and unlike chains may be approximately accounted for in the spirit of the Flory-Huggins treatment, by introduction of the x-parameter. This is achieved by changing Eq. (A.92) into... 

Finally, in the Flory-Huggins treatment the strength of the interactions is frequently expressed in terms of the interaction parameter, %, defined by Flory as the ratio between the interaction energy and the thermal energy ... 

The Flory-Huggins Treatment of Polymer Mixtures 111 and add up to unity,... 

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