Flory Huggins diagram

The so-called Flory Huggins diagrams show the vaporization pressure p in the equilibrium point GG after an indefinite time for a specified solvent concentration in the polymer at a specific temperature T (see Fig. 10.4). In these diagrams, the equilibrium concentration GG is often applied logarithmically first with respect to the vaporization pressure p, then for a constant temperature T. 

Figure 10.4 Schematic illustration of the Flory Huggins diagram...

Fig. 28 Mean-field phase diagram for ABCD tetrablock quaterpolymer melts with 0=1/4. Flory-Huggins parameters are xab = X except /Ad- Phases abbreviations MCS BC mixed centrosymmetric NCS non-centrosymmetric CS centrosymmetric. From [90]. Copyright 2004 American Chemical Society...

Figure 14. The phase diagram of the gradient copolymer melt with the distribution functions g(x) = l — tanh(ciit(x —fo)) shown in the insert of this figure for ci = 3,/o = 0.5 (solid line), and/o — 0.3 (dashed line), x gives the position of ith monomer from the end of the chain in the units of the linear chain length. % is the Flory-Huggins interaction parameter, N is a polymerization index, and/ is the composition (/ = J0 g(x) dx). The Euler characteristic of the isotropic phase (I) is zero, and that of the hexagonal phase (H) is zero. For the bcc phase (B), XEuier = 4 per unit cell for the double gyroid phase (G), XEuier = -16 per unit cell and for the lamellar phases (LAM), XEuier = 0.

The authors discuss Schroeder s paradox, referred to elsewhere in this review, and the fact that liquid water uptake increases but saturated water uptake decreases with temperature. And, at low temperature, the water uptake by membranes in contact with saturated vapor is greater than that by membranes in contact with liquid water, which suggests a fundamental difference in membrane microstructure for the two situations. An energy level diagram of thermodynamic states versus temperature was proposed, based on this Flory—Huggins-based model. 

The Flory-Huggins equation (Eq. 10) allows one to reconstruct schematic phase diagrams to express the phase separation behavior as discussed below. 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

Figure 13 Liquid-liquid equilibria phase diagrams of ternary polymer solutions (Xin et at, 2008a). Open circles the simulated results dotted lines Flory-Huggins short-dot lines RFT solid lines this work.

The main features of the polymer - solvent phase diagram can be obtained at the simple Flory - Huggins level [11,73] In effect, this theory leads to the following predictions for the dependence of the position of the critical point on the molecular mass (M —> oo) ... 

Fig. 14 Binary phase diagram for C246H494 in octacosane. The top curve shows the equilibrium liquidus for extended-chain crystals, and the bottom line the metastable liquidus for once-folded crystals. Experimental dissolution temperatures are fitted to the Flory-Huggins equation with / = 0.15 (solid lines). Vertical dotted lines (a) and (b) indicate the concentrations at which the growth rates were determined as a function of Tc in [29]. Horizontal dotted lines indicate the temperatures at which the rates were determined in [45] as a function of concentration. G(c) at Tc = 106.3 °C, measured along line (c), is shown in Fig. 12. The shading indicates schematically the crystal growth rate (black = fast), and the dashed line the position of the growth rate minimum...

Fig. 9. Phase diagram ofthe thin film with surface parameters p O.2, g=-0.5 plotted in the plane of variables % 1, for polymers of chain length N=100 and for three choices of film thicknesses D=20 (diamonds), D=60 (crosses) and D=100 (squares). Broken curve shows the bulk phase diagram of the underlying Flory-Huggins model for comparison. Remember that lengths are measured in units of the size b of an effective monomer. From Flebbe et al. [58]...

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

Blends of polystyrene/poly(2,6-dimethyl-l,4-phenylene oxide) and polystyrene/poly(vinyl methyl ether) were investigated by IGC over wide composition and temperature ranges. Flory-Huggins free energy parameters were obtained and are discussed as the criterion for thermodynamic miscibility. From the temperature variation of the free energy parameter, phase diagrams for both blends were obtained. 

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

Figure 3. Phase diagram including the hockey puck and lamellae phases. The phases are (I) bilayer lamellae, (II) monolayer lamellae, (III) bilayer hockey pucks, (IV) mono-layer hockey pucks, and (V) incomplete monolayer lamellae. Log(v3x) is plotted against X. X = <-/>/( 1 — ) where is the volume fraction of the coil, v = k/X and k = Na -ZL2 where the coil part is assumed to consist of Nsegments with a mean-square separation between adjacent segments of 6a2, and L is the rod length. % is the Flory—Huggins interaction parameter.

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