The spinodal

The T, p coexistence curve can be calculated numerically to any desired precision and is shown in figure A2.5.10. The spinodal curve (shown dotted) satisfies the equation... 

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

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...

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

Therefore, the locus of the values ( ) with a vanishing second derivative of A delimits the region of the miscibility gap in which spinodal decomposition occurs. This locus is referred to as the spinodal (figure C2.1.10 (bl). The length scale of the concentration fluctuations at the beginning of the separation process is controlled by... 

ALnico 5, with added ductihty. The very Low Co alloys, however, require extremely long he at-treatment times because of the decreased kinetics of the spinodal decomposition. Deformation aged 23%Cr—23%Co—2%Cu exhibits (BH) of 78 kJ/m (9.75MG - Oe) (85). 

Lipson (1943, 1944), who had examined a copper-nickeMron ternary alloy. A few years ago, on an occasion in honour of Mats Hillert, Cahn (1991) mapped out in masterly fashion the history of the spinodal concept and its establishment as a widespread alternative mechanism to classical nucleation in phase transformations, specially of the solid-solid variety. An excellent, up-to-date account of the present status of the theory of spinodal decomposition and its relation to experiment and to other branches of physics is by Binder (1991). The Hillert/Cahn/Hilliard theory has also proved particularly useful to modern polymer physicists concerned with structure control in polymer blends, since that theory was first applied to these materials in 1979 (see outline by Kyu 1993). 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

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.

Fig. 5 illustrates a peculiar kinetic phenomenon which occurs when an initially disordered alloy is first annealed at temperature T corresponding to area b in Fig. 1 and then quenched to the final temperature T into the spinodal instability area d antiphase boundaries "replicate , generating approximately periodic patterns. This phenomenon reflects the presence of critical, fastest growing concentration waves under the spinodal instability (the Calm waves ). Lowering of the temperature to T < T results in lowering of the minority concentration minimum ("c-well ) within APB, while the expelled solute atoms build the c-bank adjacent to the well . Due to the... 

Then the mixture with droplets is quenched into the spinodal instability region to some T < Ta (Concentration c(r) within droplets starts to evolve towards the value C(,(T) > C(,(T ), but the evolution type depends crucially on the value Act = cj(T) — Ch(Ta). At small Act we have a usual diffusion with smooth changes of composition in space and time. But when Act is not mall (for our simulations Act O.2), evolution is realised via peculiar wave-like patterning shown in Figs. 8-10. 

Since the prefactor in Eq. (17) is a universal constant of order unity, the barrier AF / kaT is large only very close to the coexistence curve, i.e. for 5v / v /coex, while for larger 5v / the smallness of the barrier implies a very grad il transition from nucleation to spinodal decomposition.Conversely, for N x 1 where Eq. (16) holds the transition is very sharp since the barrier stays large right up to the spinodal for qo. 

If 5v //v /coex is not small, the simple description Eq. (14) in terms of bulk and surface terms no longer holds. But one can find AF from Eq. (5) by looking for a marginally stable non-uniform spherically symmetric solution v /(p) which leads to an extremum of Eq. (5) and satisfies the boundary condition v /(p oo) = v(/ . Near the spinodal curve i = v /sp = Vcoex /a/3 (at this stability limit of the metastable states both and S(0) diverge) one finds "... 

The ultimate limit of metastability is reached in reality when AF /kaTc is of order unity. Thus the width over which the spinodal singularities are rounded can be estimated from AF /kaTc oc 1 as... 

A similar treatment applies for the unstable regime of the phase diagram (v / < v /sp), where the mixture decays via spinodal decomposition.For the linearized theory of spinodal decomposition to hold, we must require that the mean square amplitude of the growing concentration waves is small in comparison with the distance from the spinodal curve. 

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 ... 

Now the plait point is on the spinodal curve, and any two corresponding points of the connodal curve adjacent to the plait point are on a part of the surface which is convex in every direction, and for which therefore... 

Thus the spinodal curve does not cut the connodal curve at the plait point, and it is simplest to assume the two curves to be tangent at that point. From (26) it follows that the direction of the tangent at any point of the spinodal curve is given by ... 

Bruder and Brenn (1992) studied the spinodal decomposition in thin films of a blend of deuterated polystyrene (dPS) and poly(styrene-co-4-bromostyrene) (PBrxS) by TOF-ERDA. They examined the effect of different substrates on the decomposition process. In one series of experiments, a solution of the polymers in toluene was spread on a silicon wafer to form a film of thickness 550 nm which was then heated in vacuum at 180°C for various times. 

Therefore, in the case of glass crystallization just above Tg we may possibly see a secondary phase separation of the spinodal decomposition (SD) type occurring inside the dense region caused by the first SD. 

Fig. 25 Annealing time evolution of the difference SAXS intensity in the induction period (a) and the crystallization period (b) for the melt crystallization of PET at 244 °C [18]. This system corresponds to crystallization from the metastable state where a nucle-ation and growth type of primary phase separation first occurs followed by the spinodal decomposition type secondary phase separation...

Finally, it should be emphasized that the spinodal-type of crystallization produces a bicontinuous structure consisting of regions with higher and lower degrees of crystallinity, which provides industrially important high-performance materials with an ideal homogeneous microstructure. 

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