Huggins expression

Considering an (incompressible) polymer mixture with volume fraction (j)A = (j) of A-monomers and volume fraction (j)B = 1 - (j) of B monomers, the mean-field expression for the excess-free energy of mixing is given by the well-known Flory-Huggins expression " ... 

Equivalent expressions for equations (3) and (6) exist for the polymer (7). The Flory-Huggins expressions can also be extended to multicomponent systems (2) ... 

From the outset, Flory (6) and Huggins (4,5 ) recognized that their expressions for polymer solution thermodynamics had certain shortcomings (2). Among these were the fact that the Flory-Huggins expressions do not predict the existence of the LCST (see Figure 2) and that in practice the x parameter must be composition dependent in order to fit phase equilibrium data for many polymer solutions 3,8). 

AG/6V. =0 (spinodal) using the Flory-Huggins expression (14,7) for a 50 50 AB polymer blend. Here M is the number average molecular weight for the two component system, which allows for the possibility that the two monodisperse homopolymers in the blend may not be of identical molecular weight. 

The solvent chemical potential is taken to be that given by the Flory-Huggins expression (18) ... 

The Flory-Huggins expression for the enthalpy of mixing is given by Equation (3.71). In the present notation it becomes... 

In spite of this, Rijke assumes that the Flory-Huggins expression for AGdil is sufficiently adequate and then finds a molecular expansion factor, a from ... 

Afit = 0. Employing (Eq. (IV-10) in conjunction with the Flory-Huggins expression for AGdU (Eq. III-15) we find ... 

Recently Leonard and Ivin (28) have pointed out that the application of equation (1) is valid only if the mixture of monomer and polymer behaves ideally over the range of compositions covered by the experiment. They reexamined some of the early data, making allowance for non-ideal mixing by use of the Flory-Huggins expression. They derived a equation for the free energy of polymerization in terms of the volume fractions of the polymer and monomer, tp2 and polymer-monomer interaction parameter ... 

The form derived by Scheutjens and Fleer (1979) depends on the Flory-Huggins expression for the free energy of a solution [Eq. (22)]. The Helfand (1975b) SCF [Eq. (39)] can incorporate any equation-of-state indeed, supplementing Eq. (22) by a term accounting for adsorption energy and substituting into Eq. (39) produces... 

Note, however, that if the molecules were of equal size (i.e., m = 1, so that we were ouly mixing one blue molecule with 75 reds ), then the mole fraction would equal the volume fraction. The Flory-Huggins expression for the entropy of mixing is clearly far more general and is the one we will use. 

It makes things easier to see if we consider the free energy on a per mole of lattice sites basis by dividing the Flory-Huggins expression for the free energy by the number of moles of lattice sites, VI V. Recall that the... 

Consider a polymeric species with degree of polymerization i in solution. The homogeneous solution can be caused to separate into two phases by decreasing the affinity of the solvent for the polymer by lowering the temperature or adding some poorer solvent, for example. If this is done carefully, a small quantity of polymer-rich phase will separate and will be in equilibrium with a larger volume of a solvent-rich phase. Tlie chemical potential of the i-mer will be the same in both phases at equilibrium, and the relevant Flory-Huggins expression is... 

In the concentrated and marginal regimes of fig 5,3 a mean-field description, which neglects any spatial fluctuations, is appropriate. In these regimes, the solution is homogeneous and there is no chain-length dependence. Neither does the persistence p of the chains play a role since the Floiy-Huggins expressions do not contain the chain flexibility. This is so because the flexibility is assumed to be the same in the solution and in the reference state, so that p cancels in the entropy difference between the two states. 

Flory-Huggins expression. Substituting in Eq.(8) one obtains for the semiequilibrium state discussed above... 

Activity Coefficient at Infinite Dilution. A procedure similar to that employed by Wilson will be used here to obtain an expression for the excess Gibbs energy. Wilson started from the Flory and Huggins expression" 2 for the excess free energy of athermal solutions, but expressed the volume fractions in terms of local molar fractions. We selected Wilson s approach from a number of approaches, because it provided a better description of phase equilibria and because the interactions that count the most are the local one, but started from the more... 

Since we will focus below primarily on adhesion at interfaces between two immiscible polymers, it is appropriate to describe briefly what is known about such interfaces. The Gibbs free energy of mixing (per segment) of any two homopolymers A and B is given approximately by the Flory-Huggins expression ... 

When using IQC for the evaluation of the polymer-polymer interaction coefficient X 23, the free energy of mixing is routinely expressed by an extension of the Flory-Huggins expression (11) to a three component system (12) ... 

The critical point Equation is derived from the spinodal Equation using the critical condition given earlier. The differential of the spinodal Equation with respect to 2 is zero at the critical point. This calculation is perfectly feasible but so far no one has made any attempt to use these Equations to predict the critical point. In the past it has usually been approximated using the simple Flory-Huggins expression for the critical point. 

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