Spinodal parameter

Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within tire coexistence curve are called off-critical quenches.

Figure A3.3.5 Tliemiodynamic force as a fiuictioii of the order parameter. Three equilibrium isodiemis (fiill curves) are shown according to a mean field description. For T < J., the isothemi has a van der Waals loop, from which the use of the Maxwell equal area constmction leads to the horizontal dashed line for the equilibrium isothemi. Associated coexistence curve (dotted curve) and spinodal curve (dashed line) are also shown. The spinodal curve is the locus of extrema of the various van der Waals loops for T < T. The states within the spinodal curve are themiodynaniically unstable, and those between the spinodal and coexistence...

Above the solution treatment temperature (ca 1250°C), the alloy is single phase with a bcc crystal stmcture. During cooling to ca 750—850°C, the sohd solution decomposes spinodally into two other bcc phases a and lattice parameter composition. The matrix a-phase is rich in Ni and Al and weakly magnetic as compared with which is rich in Fe and Co. The a -phase tends to be rod-like in the (100) dkection and ca 10 nm in diameter and ca 100 nm long. As the temperature is decreased, segregation of the elements becomes mote pronounced and the difference between the saturation polarizations of the two phases increases. 

Fig. 9 Spinodal decomposition model for glass crystallization at a low temperature just above Tg. The indicated structure parameters mean (A), the average characteristic wavelength (D)y the average size of the dense domain (L), the average long period after crystallization. The numerical values are for PET crystallized at 80 °C...

The systems undergoing phase transitions (like spinodal decomposition) often exhibit scaling phenomena [ 1—4] that is, a morphological pattern of the domains at earlier times looks statistically similar to a pattern at later times apart from the global change of scale implied by the growth of L(f)—the domain size. Quantitatively it means, for example, that the correlation function of the order parameter (density, concentration, magnetization, etc.)... 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

In the same manner, with decreasing of diffusion coefficient and interaction parameter, the spinodal is reached during the evolution of the system in the pregel stage. The very low values of interfacial tension in rubber modified epoxies (interfacial tension of polymer-polymer-solvent system were reported in range of 10-4-10-1 mN/m) therefore lead to an NG mechanism for phase separation. 

In principle, all the molecular parameters in Eq. (6) can be determined independently, so that the theory can be quantitatively compared with experimental data. An example of Maxwell s construction in the dependence of x °n critical value of interaction parameter %c of charged PAAm network with the degree of ionization equals to the molar fraction of the sodium methacrylate in the chain i = xMNa = 0.012 are given in Fig. 4 (data of series D from Fig. 5). The compositions of the phases

critical value of Xc were determined by the condition that areas St and S2 defined in Fig. 4 are equal The experimental (p2e is higher and 2 determined by Maxwell s construction (Eq. 13). Thus, the experimental values of (p2e and metastable region the limits of which (p2s and (p2s are determined by the spinodal condition (two values

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Various modes of ordering are feasible. In systems with amphoteric cations (< .., Al34-), the fraction of Al3+ in tetrahedral or octahedral sites is a possible order parameter. Strain may lead to a bending of Si-O bonds. Periodic distributions of the components in space, along with elastic and/or electrostatic interactions, indicate spinodal ordering (demixing). Some examples will illustrate these general features. 

Spinodal decomposition and certain order-disorder transformations are the two categories of continuous phase transformations. Both arise from an order parameter instability in the case of spinodal decomposition, it is a conserved order parameter for continuous ordering, it is a nonconserved order parameter. 

Order-disorder transitions and spinodals were computed for linear multi block copolymers with differing sequence distributions by Fredrickson et al. (1992). This type of copolymer includes polyurethanes, styrene-butadiene rubber, high impact polystyrene (HIPS) and acrylonitrile-butadiene-styrene (ABS) block copolymers. Thus the theory is applicable to a broad range of industrial thermoplastic elastomers and polyurethanes. The parameter... 

As before, the (nonzero) vector 8p with components <5p,- gives the direction of the spinodal instability the moment densities pt and ptJ that appear in F and M are to be evaluated for the parent distribution p (cr) being studied. Note that Eq. (54) is valid for any p (cr) no specific assumptions about the parent were made in the derivation. We can thus simply drop the (0) superscript For any phase with density distribution p(external control parameters are varied, locates a spinodal instability. 

Figure 111 4 2. The pressure as a function of the reduced density at different temperatures. The phase diagrams, obtained by calculating the spinodal tine from Eq. 1, depend on the thermodynamic parameters described above. In Fig. 3 we report same representative phase diagrams with different anisotropic interactions...

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