Flory-Huggins parameter critical

Predicting the characteristic sizes and morphologies of these nanostructures has been an intense topic of investigation from both the theoretical and experimental points of view. Critical parameters are the degree of polymerization and the volume fraction of the constituent blocks, as well as the Flory-Huggins parameter between them. More complete information about microphase separated structures in bulk block copolymers can be found in the book of Hamley [2],... 

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 advantage of expressing H in terms of the Flory-Huggins parameter Xn is that the latter is often insensitive to temperature, and may better reflect the dependence on concentration than H. Its disadvantage is that it does not apply near and above the critical temperature of the volatile component. 

A quantitative comparison between the mean field prediction and the Monte Carlo results is presented in Fig. 15. The main panel plots the inverse scattering intensity vs. xN. At small incompatibility, the simulation data are compatible with a linear prediction (cf. (48)). From the slope, it is possible to estimate the relation between the Flory-Huggins parameter, x, and the depth of the square well potential, e, in the simulations of the bond fluctuation model. As one approaches the critical point of the mixture, deviations between the predictions of the mean field theory and the simulations become apparent the theory cannot capture the strong universal (3D Ising-like) composition fluctuations at the critical point [64,79,80] and it underestimates the incompatibility necessary to bring about phase separation. If we fitted the behavior of composition fluctuations at criticality to the mean field prediction, we would obtain a quite different estimate for the Flory-Huggins parameter. 

In the insets of Fig. 15 we show binodal curves for the symmetric blend. Again, we And deviations in the immediate vicinity of the critical point but for larger incompatibilities, xN 2, the mean held predictions provide an adequate description of the phase boundary utilizing the Flory-Huggins parameter extracted from the composition fluctuations in the one-phase region, xN < 2. 

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... 

At this point, it is appropriate to critically reexamine to what extent the Flory-Huggins description (Eqs. 5-7) is valid as a phenomenological model. One important qualitative effect that is neglected completely is the correlation hole effect (14) For an effective monomer of a chain it is more likely to have other monomers in the neighborhood that belong to the same chain rather than to other chains. Of course, only interchain interactions and no intrachain interaction contribute to macroscopic phase separation, and hence the effective coordination number z that enters the Flory-Huggins parameter / gets reduced. It turns out (78) that this leads to H /N corrections in d = 3, z = z o + const/v, and similarly /c = Xoo + const/ /iV, + const/ /iV, etc. In d = 2, however, this... 

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 Flory-Huggins theory of polymer solutions has been documented elsewhere [26, 27]. The basic parameters necessary to predict polymer miscibility are the solubility parameter 6, the interaction parameter %, and the critical interaction parameter ( ) . 

Since there had not been any measurements of thermal diffusion and Soret coefficients in polymer blends, the first task was the investigation of the Soret effect in the model polymer blend poly(dimethyl siloxane) (PDMS) and poly(ethyl-methyl siloxane) (PEMS). This polymer system has been chosen because of its conveniently located lower miscibility gap with a critical temperature that can easily be adjusted within the experimentally interesting range between room temperature and 100 °C by a suitable choice of the molar masses [81, 82], Furthermore, extensive characterization work has already been done for PDMS/PEMS blends, including the determination of activation energies and Flory-Huggins interaction parameters [7, 8, 83, 84],... 

Calculate the for solutions in chlorobenzene of polystyrenes with molar weights of 2 x 105,5 x 105 and 106 g/mol the intrinsic viscosity, the critical concentration, the swelling factor, the hydrodynamic swollen volume, the second virial coefficient and the Flory-Huggins interaction parameter. 

Flory-Huggins interaction parameter critical degree of polymerization (= 2/X) phase separation not observed... 

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