Spinodal Lines

Fig. 8 Phase diagram for PI-fc-PEO system. Only equilibrium phases are shown, which are obtained on cooling from high temperatures. ODT and OOT temperatures were identified by SAXS and rheology. Values of /AT were obtained using /AT = 65/T + 0.125. Dashed line spinodal line in mean-field prediction. Note the pronounced asymmetry of phase diagram with ordered phases shifted parallel to composition axis. Asymmetric appearance can be accounted for by conformational asymmetry of segments. Adopted from [53]...

Here Xg is the composition at the critical point. Let us now calculate the immiscibility gap and the spinodal line for a regular solution A-B ... 

Both the binodal line, defining the immiscibility gap, and the spinodal line are for a regular solution symmetrical about xA = xB = 0.5. This is shown in Figure 5.7(a), where theoretical predictions of the miscibility gaps in selected semiconductor systems are given [15],... 

Figure 5.7 (a) Theoretical predictions of the unstable regions (miscibility gap) of the solid solutions in the systems AlN-GaN, InN-GaN and AlN-InN [15]. For the system InN-GaN both the phase boundary (binodal) and spinodal lines are shown, (b) Gibbs energy of mixing for the solid solution InN-GaN at 1400 K. 

We have now derived the phase boundary between the two liquids. By analogy with our earlier examples, the two phases may exist as metastable states in a certain part of the p,T potential space. However, at some specific conditions the phases become mechanically unstable. These conditions correspond to the spinodal lines for the system. An analytical expression for the spinodals of the regular solution-type two-state model can be obtained by using the fact that the second derivative of the Gibbs energy with regards to xsi)B is zero at spinodal points. Hence,... 

The set of all points of concentrations which corresponds to the points B and C is the spinodal line and is given by the following conditions ... 

Furthermore the processes of physical gelation may be due to such a phase change across spinodal lines since micro-phase separation may induce the formation of a porous structure which could have high chemo-mechanical response. 

Fig. 2.42 Spinodal lines for a random multiblock copolymer melt of variable X (Fredrickson el al. 1992). On cooling a melt with X > AL —0.268, the first instability is predicted to be phase separation into two homogeneous liquid phases (x = %m)- On further cooling to % = the two liquid phases become unstable with respect to formation of a microphase. In contrast, a melt with X < XL first becomes absolutely unstable to the formation of microphases (x = fom)- At the critical composition of /= j, the point (AL, Xi) is an isotropic Lifshitz point.

Fig. 6.15 Spinodal lines calculated using the random phase approximation for macrophase separation (solid lines) and microphase separation (dashed lines) for blends of a PS-PI diblock (Af = 100kg mol-1./PS = 0.46) with homopolymers with M /kg mol 1 = (a) 62, (b) 200, (c) 580 (Koizumi et al. 1992). For the blends with a 1, macrophase separation occurs first on lowering the temperature (increasing jV) for most compositions.

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

The spinodal line is determined by the state at which the second derivative of the Gibbs energy with respect to composition is equal to zero. For a... 

The solutions in the region inside the spinodal domain are unstable, whereas the solid solutions in the region between the binodal and spinodal lines are metastable. The presence of a miscibility gap limits the potential usefulness of these materials in device applications. Solutions with compositions lying inside the spinodal domain cannot be grown by LPE, whereas metastable solid solutions have a tendency toward phase separation and, eventually, device degradation. 

Key words Multiple Phase Separation, Spinodal Lines, Cuprates... 

Figure 111 4 3. The phase diagrams obtained by computing the spinodal lines from Eq. 1. We can observe the occurrence of four phase separations indicated by the four spinodal lines in panel a). In the panel b), c), d), e) and f) we show the effect of the strength of the directional bonds (see text)...

We have reproduced the experimental pseudo-gap temperature using the thermodynamic model above discussed. The phase diagram has been obtained calculating the spinodal line from Eq.l. We have imposed Yo=a/Z>=19.5 meV, b= 2.00-K)-4 m3/mol, lyil=44meV, VyI= 9.25-Kr4 m3/mol, (j =Vy 16.25, ly2l=14 meV, Vy2= 4.8M04 m3/mol, a2-Vy 8.33,... 

In conclusion we have presented a model for an electronic complex system with coexistence of different electronic phases at critical densities and coexistence of different liquids described by the modified van der Waals model as proposed for supercooled water. We discuss the critical values of the anisotropic interactions for the spinodal lines. We find that this model is able to describe the evolution of the pseudo-gap temperature versus doping in different cuprate families. 

The phase diagrams, obtained by calculating the spinodal line from Eq. 1, depend on the thermodynamic parameters described above. In Fig. 3 we report same... 

Figure 111 4 3. The phase diagrams obtained by computing the spinodal lines from Eq. 1.

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