Spinodal equilibrium

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...

The kinetics of spinodal decomposition is complicated by the fact that the new phases which are formed must have different molar volumes from one another, and so tire interfacial energy plays a role in the rate of decomposition. Anotlrer important consideration is that the transformation must involve the appearance of concenuation gradients in the alloy, and drerefore the analysis above is incorrect if it is assumed that phase separation occurs to yield equilibrium phases of constant composition. An example of a binary alloy which shows this feature is the gold-nickel system, which begins to decompose below 810°C. 

Figure 6.5 The appearence of spinodal decomposition as the temperature is lowered from a range of complete solubility, to the separation of two phases. In the range of composition between the inflection points, the equilibrium spinodal phases should begin to separate...

Fig. 1. Equilibrium phase diagram T, c)=iT/Tc,c) for the alloy model used in Ref.. Solid lines boundaries of the disordered (a) and homogeneously ordered (6) fields areas c, d and e corre.spond to the two-phase region. Dashed line i.s the ordering spinodal separating the metastable disordered area c from the. spinodal decompo.sition area d. Dot-dashed line is the conditional spinodal that separate.s the area d from the ordered metastable area e.

Another kinetic jjhenomenon where Calm s critical waves can possibly be visualized and studied is the replication of interphase boundaries (IPB) illustrated in Figs. 8-10. Similarly to the replication of APBs. it can arise after a two-step quench of an initially uniform disordered alloy. First the alloy is quenched and annealed at temperature T in some two-phase state that can be either metastable or spinodally unstable with respect to phase separation. Varying the annealing time one can grow here precipitates ("droplets ) of a suitable size /. For sufficiently large /, the concentration c(r) within A-riched droplets is close to the equilibrium binodal value C(,(T ) (thin curve in Fig. 9). 

If we imagine a line drawn on the primitive surface dividing all parts of the surface which are convex downwards in all directions from those which are concave downwards in one or both directions of principal curvature, this curve will have the equation (26), and is known as the spinodal carve. It divides the surface into two parts, which represent respectively states of stable and unstable equilibrium. For on one side A is positive, and on the other it is negative. If we assume that the tie-line of corresponding points on the connodal curve is ultimately tangent to that the direction of equations ... 

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]...

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. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

The T-p plot shown in Figure 5.11 (b) resembles the T-x plot of a binary solution. The equilibrium between the two phases is, as we have seen above, given by a similar set of equilibrium conditions in both cases. Within the spinodal regions of the... 

If the composition of a phase falls inside the spinode, the phase would undergo spontaneous decomposition into two phases. That is, heterogeneity would arise from homogeneity. From an energetic point of view, the homogeneous phase is at an unstable equilibrium state, meaning that without disturbance at all, the equilibrium could persist, but the tiniest microscopic perturbation (there would always be perturbation due to thermal motion) would grow to produce hetero-... 

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