Flory-Huggins parameter critical value

To analyze the stability of the ordered microphases, the simplest incompressible random-phase approximation [132] can be employed. Using this approach, the critical value of the Flory-Huggins parameter, x > and the corresponding spinodal temperature, T = l/x > can be determined by the condition that the scattering intensity S(q) reaches its maximum value at a nonzero wave vector q. Within the RPA the scattering intensity is given by [132,142]... 

The excess free energy per solvent molecule of polymer solutions is characterized by a semi-empirical Flory-Huggins parameter, X) which is a function of temperature for a given polymer-solvent pair. To estimate the compatibility parameter experimentally, it is necessary to define the x value for each polymer-solvent pair and compare it to its critical value calculated by the equation... 

Using the simplest incompressible random phase approximation, the critical value of the Flory-Huggins parameter, / , and the corresponding transition temperature, T are... 

The factor fo is positive for %Xc (where Xc is a critical value of the Flori-Huggins parameter), fi reverses its sign when crosses binodal, where it has zero value, and the factor f2 is alw s positive. It matches requirement of the fiee energy minimization with one minimum in a solution area and two minimums in a two-phase area. The gradient part of the free energy ... 

The polymer solubility can be estimated using solubility parameters (11) and the value of the critical oligomer molecular weight can be estimated from the Flory-Huggins theory of polymer solutions (12), but the optimum diluent is still usually chosen empirically. 

According to Flory-Huggins theory, in the limit of x the critical x parameter is 0.5.(H) Below this value the polymer and solvent will be miscible in all proportions. Above this value, the solvent will not dissolve the polymer, but will act only as a swelling solvent. Thus, the pure solvent may not dissolve the polymer even though it is not crosslinked. If x is not , the critical value of x is larger, but the same qualitative arguments regarding mutual solubility of the solvent and polymer hold. Thus, the application of Equation 1 does not require that the pure solvent be able to completely dissolve the polymer, only that the solvent dissolve into the polymer by an amount that can be measured. 

The tendency of differing blocks to microseparate from each other is quantified by Flory s chi parameter /, introduced in Chapter 2. An increasing, positive value of x implies an increasing tendency for the two chemically dissimilar species to segregate from each other. As discussed in Section 2.3.1.2, for a blend of two different homopolymers (A and B) of equal degree of polymerization Na — Ag at a 50/50 composition, the Flory-Huggins theory predicts that phase separation should occur at a critical value of Xc = For block... 

In the early 1940s, Flory and Huggins proposed, separately, a lattice model to describe polymer solutions and introduced the interaction parameter This parameter increases as solvent power decreases hence, a thermodynamically good solvent is characterized by a low interaction parameter. In practice, most polymer-solvent combinations result in x-values ranging from 0.2 to 0.6. Moreover, the theory predicts that a polymer will dissolve in a solvent only if the interaction parameter is less than a critical value Xc. which, at a given temperature, depends on the degree of polymerization (x) of the dissolved polymer ° ... 

Fig. 24a. Phase diagram of the asymmetric polymer mixture (A. = 2.0, NA = NB = N = 32, 4, = 0.5) in the plane of variables reduced temperature and relative concentration 4a/(4a + 4b) of component A. The dashed lines are the histogram extrapolations for three simulated system sizes, the full line denotes the binodal, and the circle denotes the critical point. From Deutsch and Binder [93]. b Phase diagram of asymmetric polymer mixtures for NA = NB = N = 32, 4 = 0.5 in the (T, Ap) plane. Three choices of the asymmetry parameter A are shown as indicated. The first order transitions are shown as a full line, the critical points as circles. Temperature is normalized such that in the Flory-Huggins-approximation the critical temperature would occur for the same abscissa value. From Deutsch [266]...

Block and graft copolymers based on two or more incompatible polymer segments phase separate and self-assemble into spatially periodic structure when the product ixN) of the Flory-Huggins interaction parameter and the total number of statistical segments in the copolymer exceeds a critical value [25,262,263]. Since the incompatible... 

The critical value for the Flory-Huggins interaction parameter is obtained from the combination of Equations (6-94) and (6-95) ... 

The Flory-Huggins interaction parameter can, however, exceed its critical value above a certain temperature for solutions in poor solvents. In this case, separation into two liquid phases also occurs with crystalline... 

The Flory-Huggins interaction parameter can, however, exceed its critical value above a certain temperature for solutions in poor solvents. In this case, separation into two liquid phases also occurs with crystalline polymers, as is shown, for example, by poly(ethylene) in nitrobenzene at mass fractions W2 < 0.75 (Figure 6-19). On the other hand, separation into one crystalline and one liquid phase is observed at poly(ethylene) concentrations W2 > 0.75. In xylene, separation into one liquid and one crystalline phase always occurs, no matter what the mass fraction is. 

For large n, the right-hand side of equation (4.1) reduces to 1/n . For n = Kf, V2c equals about 0.01, a very dilute solution. The critical value of the Flory-Huggins polymer-solvent interaction parameter, Zi, is given by... 

An immiscible polymer pair is implemented in this numerical modeling. The degree of polymerization was chosen as 100, according to experimental data. The interaction parameter, Xij> in the Flory-Huggins free energy for polymers can be evaluated as a function of temperature in a binary case. In a ternary phase separation, the critical value of the interaction parameter for spinodal decomposition to occur between two polymers can be estimated as ... 

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