Spinodal curve, points, region

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 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 8.5 Conversion vs composition transformation diagram at a constant cure temperature, showing cloud-point curves and spinodal curves that bound stable, metastable, and unstable regions , , and represent the three trajectories, starting from different initial thermoplastic concentrations and leading to different morphologies. (Pascault and Williams, 2000 -Copyright 2001. Reprinted by permission of John Wiley Sons Inc.)...

Apart from the binodal curves one may also define spinodal curves (see sec. 1.2.68). found from the conditions that dfj/d(p (for polymer and solvent) are the same In both phases. One such spinodal (for N = 100) is indicated in fig. 5.4. The region inside the spinodal is unstable, that between binodal and spinodal metastable. At the critical point the binodal and spinodal curves coincide. 

For binary mixtures, the binodal line is also the coexistence curve, defined by the common tangent line to the composition dependence of the free energy of mixing curve, and gives the equilibrium compositions of the two phases obtained when the overall composition is inside the miscibility gap. The spinodal curve, determined by the inflection points of the composition dependence of the free energy of mixing curve, separates unstable and metastable regions within the miscibility gap. 

Figure 5. Evolution of isotherms in the P - p phase diagram for the core softened potential with third critical point in metastable region. Cl - gas + hquid, C2 - LDL + HDL, and C3 - HDL + VHDL critical points. Red lines (online) are coexistence curves green lines (online) are spinodals. Critical point location rcci = 0.0064, xa = 0.1189, yci =0.0998 7tc2 = 0.1423, Xc2 = 0.3856, yc2 = 0.33 ttcs = 0.07487, xcs = 0.2398, yes =0.6856. Model parameter set a = 6.962, bh =2.094, WUa=3, b,=7.0686.

Fig. 40. Schematic description of unstable thermodynamic fluctuations in the two-phase regime of a binary mixture AB at a concentration cb (a) in the unstable regime inside the two branches tp of the spinodal curve and (b) in the metastable regime between the spinodal curve tp and the coexistence curve The local concentration c(r) at a point r = (x. y, z.) in space is schematically plotted against the spatial coordinate x at some time after the quench. In case (a), the concentration variation at three distinct times t, ti, u is indicated. In case (b) a critical droplet is indicated, of diameter 2R , the width of the interfacial regions being the correlation length Note that the concentration profile of the droplet reaches the other branch ini, of the coexistence curve in the droplet center only for weak supersaturations of the mixture, where cb -

These, and all other equations for concentration-dependent diffusion, consist of an infinite dilution diffusivity and a thermodynamic correction term. The thermodynamie correction term in all cases is equivalent to the derivative dGildx. The definition of the thermodynamic metastable limit (the spinodal curve) is the locus of points where dG2ldx = 0. This means that concentration-dependent diffusion theory predicts a diffusivity of zero at the spinodal. Thermodynamics tells us that the diffusivity goes from some finite value at saturation to zero at the spinodal. Unfortunately, it does not tell us how the diffusion coefficient declines. In addition, lack of thermodynamic data makes prediction of the spinodal difficult. We are, therefore, left with only the fact that as the concentration is increased in the supersaturated region, the diffusivity should decline towards zero but we do not know at what concentration the diffusivity becomes zero. 

Fig. Phase diagram of the mixture polystyrene and polyvinyl methyl ether, molecular weights being = 62700 (PVME) and Mw = 60(W0 (PS), as obtained from light scattering. The lower curve describes the miscibility gap binodaF, coexistence curve ), the upper curve describes the spinodal curve, which touches the coexistence curve in the critical point. The shaded region in between binodal and spinodal is believed to describe homogeneously mixed metastable one-phase states. From Snyder et al. [27]. b Phase diagram for polyisoprene-poly(ethylene-propylene) with molecular weights of 2000 and 5000, respectively. From Cumming et al [70]...

Spinodal points represent the boundary between positive and negative curvature of A-V isotherms. An equilibrium state on the spinodal curve is defined by (9p/9V)7 ,au JV = 0. Regions between the spinodal points are intrinsically unstable and violate the criterion of mechanical stability. 

Binodal points represent the points of contact of a common tangent to A vs. V at constant temperature and composition when a region of negative curvature exists between two regions of positive curvature. The locus of binodal points, known as the binodal curve or two-phase envelope, represents the experimentally observed phase boundary under normal conditions. For example, saturated liquid and saturated vapor represent states on the binodal curve. The binodal region exists between the binodal and spinodal curves, where p/ V)T,aa jv < 0. 

At the critical point, (0c, Tc), the coexistence curve (or binodal) and spinodal curve meet. It is important to note that the spinodal is never physically reached in thermal equilibrium (except at the critical point itself) since it is precluded by the first-order phase separation described in the (0, T) plane by the coexistence curve. While the coexistence curve separates the single phase and two-phase regions in thermal equilibrium, kinetic effects can allow the existence of metastable states — Le., one can have the system below the binodal curve and still observe only a single phase for a rather long time. In contrast, the region below the spinodal curve is unstable and even small, thermal fluctuations will drive the system toward equilibrium. The spinodal line represents a line of large fluctuations in the concentration 0, since the free energy cost for fluctuations of 0 away from its minimal values, as determined from Eq. (1.72), is proportional to 9 //90 when this quantity is small, the... 

Fig. 11. Binodal and spinodal curves in a conversion vs modifier volume fraction phase diagram. The critical point and the location of stable, metastable and unstable regions are shown...

In this context, it is also instructive to define the critical point as the intersection of binodal and spinodal curves. When precipitation occurs above this point, the non-solvent diffuses like a droplet through a continuum of polymer. In contrast, higher dilutions of the casting solution which enter the unstable region below the critical point cause the formation of polymer particles in a continuous liquid phase, resulting in flocculation of the polymer or formation of very weak, powdery membranes. 

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