Curve 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 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 2 (a) The pair interaction V as a function of distance PdsoRhso alloy, (b) Spinodal curve for Pdj.Rhi j alloy system. The points indicate calculated points while the solid line is the cubic spline fit through the points. 

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 "... 

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. 

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 ... 

Again, the limiting position of the line joining points of the connodal curve and the direction of the common tangent to the connodal and spinodal curves at the plait point is given by ... 

Spinning systems, wet ring, 11 615-616 Spinning/web formation, for spunbonded non woven fabrics, 17 469-474 Spinodal curves, 20 320-321 Spinodal decomposition, in polymer blends, 20 321, 322... 

The energy increase related to a compositional fluctuation resulting in a two-phase splitting may be considered as an energy threshold of activation of the de-mixing process. The spinodal curve thus defines a kinetic limit, not a phase boundary line. 

Similar to the derivation of schematic phase diagrams showing an UCST or LCSTbehavior, as shown in the previous section, it is now possible to plot a schematic phase diagram as a function of conversion, q, such as shown in Fig. 5. The binodal and spinodal curves are obtained by interconnecting all the coexisting points and inflection points of the AG -curves as a function of q similar to the procedure described in [67-69]. 

Note 1 The spinodal curve for a binary mixture is defined as the geometrical locus of all states with... 

Note 2 In the unstable region bounded by the spinodal curve, phase domains separation is spontaneous, i.e., no nucleation step is required to initiate the separation process. 

Long-range, diffusion-limited, spontaneous phase domains separation initiated by delocalized concentration fluctuations occurring in an unstable region of a mixture bounded by a spinodal curve. 

On the basis of these relationships, using expressions for the chemical potentials of the components at the level of approximation of the second virial coefficients (equations 3.17 and 3.18), the spinodal curve (equation 3.25) can be expressed mathematically in the following form (Edmond and Ogston, 1968) ... 

In practice, from a knowledge of measured values of the osmotic second virial coefficients it is rather easy to calculate the spinodal curve. It is worthy of note here to observe that reciprocal values of m, for biopolymers of rather high molecular weight (> 104 g/mol) are often comparable with the magnitude of A 24. This requires that, as well as values of the osmotic second virial coefficients, the molecular weight should also be taken into account in the prediction of the boundary conditions relating to phase separation. 

Figure 3.3 Illustration of the calculation of the phase diagram of a mixed biopolymer solution from the experimentally determined osmotic second virial coefficients. The phase diagram of the ternary system glycinin + pectinate + water (pH = 8.0, 0.3 mol/dm3 NaCl, 0.01 mol/dm3 mercaptoethanol, 25 °C) —, experimental binodal curve —, calculated spinodal curve O, experimental critical point A, calculated critical point O—O, binodal tielines AD, rectilinear diameter,, the threshold of phase separation (defined as the point on the binodal curve corresponding to minimal total concentration of biopolymer components). Reproduced from Semenova et al. (1990) with permission.

If this strain energy of the coherent a-a system is added to the other Gibbs energy contributions, both the total Gibbs energy and the spinodal curve are shifted as shown schematically in Figure 12-10. 

The spinodal curve (SC) and critical point (CP) condition may now be calculated from Eq. (80), using either the methods outlined in Section HI.A or the more traditional determinant conditions [34] (see also Appendix D). One finds... 

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.)...

Figure 17 Spinodal curves and coexistence curves of PS/cyclohexane systems (Yang et al, 2006a).

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