Flory-Huggins treatment, of polymer

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. 

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). 

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

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. 

A general treatment of the dependence of M,. on the nature of solvent and the volume fraction of polymer was outlined by Ivin and Leonard184. Adopting the Flory-Huggins theory of polymer solutions to ternary systems, they derived the relation... 

The thermodynamic properties of concentrated polymer solutions were studied by Floryi and independently by Huggins. The Flory-Huggins theory of polymer solutions still forms the basis for much discussion of these solutions in industry and even in academic research. Understanding this model is important for making coimections to much of the literature. Flory also substantially improved this model to include compressible fluids. The Flory-OrwoU theory of polymer solutions is still transparent and easily applicable, predicting both upper and lower critical solution temperatures. More-empirically adequate theories of concentrated solutions do not lend themselves to simple lecture presentation and often require detailed computer calculations to obtain any results. Concentrated solutions also introduce the phenomenon of viscoelasticity. An extensive treatment of the full distribution of relaxation times necessary to imderstand the dynamic properties of polymers in concentrated solution is presented. 

Dealing with the osmotic pressure we first consider the case of an electrically neutral network in a good solvent. We refer here to the Flory-Huggins treatment of A/B-polymer mixtures (Sect. 4.1) and use it for a network(A)-solvent(B) system, by just setting... 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

In addition to the solubility parameter model to treat SEC adsorption effects, an approach based on Flory-Huggins interaction parameters has also been proposed (24-27). For an excellent review of both mechanisms, see reference 28.- A general treatment of polymer adsorption onto chromatographic packings can be found in Belenkii and Vilenchik s recent book (29). 

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... 

Paralleling this experimental work has been considerable activity in the theoretical treatment of polymer solutions. The original work of Flory and Huggins is often used for the calculation... 

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. 

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... 

There are some smdies (23,24) which link the magnitude of the Huggins and Sehulz-Blasehka coefficients to the coefficient of polymer- solvent expansion,, of the Fox and Flory treatment of polymer coil properties (25). 

The classical treatment of polymer solution thermodynamics due to Flory and Huggins [4] is based on a lattice model which assumes a uniform polymer segment concentration throughout the entire system. The free energy of mixing of a polymer solution is given by... 

The thermodynamic treatment of polymer/solvent systems is based on the Flory-Huggins parameter, %, which measures polymer power dissolution and takes into account the specific interaction that occurs among the polymer segments and the solvent molecules (Schuld and Wolf 1999, 2001). Increasing polymer concentration causes preponderantly polymer/polymer interactions, and starting with a concentration value characteristic for each polymer/solvent system, the Flory-Huggins parameter presents values higher than 0.5 (Schneider et al. 2004). 

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

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. 

Empirical Description of the Binary Interaction Parameter. As noted earlier, the most important contribution to for blends of high polymers is due to enthalpic effects. Contrary to the assumptions of the original Flory-Huggins treatment, there now is experimental evidence that indicates that x 2 dependent on concentration and has a temperature dependence other than the inverse relation of Eq. (15) [33]. 

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