The Spinodal Curve

EXAMPLE 25.3 The spinodal (metastability) curve for the lattice model You can compute the spinodal curve for the lattice model by taking the s.econd derivative of Equation (15.14) and setting it equal to zero  

---

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...

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 ... 

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. 

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. 

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. 

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... 

Fig. 8. a Comparison of the spinodal curves %Nt(fA) for diblock and different ABn miktoarm copolymers with n=2,3, and 4. The experimental results from an AB2 ( ) and the two AB3 ( , Q) miktoarm stars are shown together with the investigated temperature range (vertical lines), b Critical values of %Nt plotted as a function of the number of arms of the B blocks. The dependence of the optimal composition corresponding to the minima of the spinodal curves are also shown. The line is only a guide to the eye (reproduced with permission from [78])... 

Figure 2 Typical phase diagram of an aqueous polysaccharide (l)-protein (2) dispersion showing the Gibbs free energy as a function of the volume fraction () of each, at different temperatures from Tx, where the dispersion is metastable, to the critical solution temperature (Tc), where the two components are miscible in all proportions. ABC is the spinodal curve DBE (not connected) is the binodal curve.

The locus connecting the minima ABC in Fig. 2 is the cloud point or binodal curve and DBE (not connected) is the spinodal curve (Cowie, 1991). 

Because of the complexity of the processes involved and the various methods of measuring them, phase diagrams of organogels can be envisioned by different theoretical approaches. In one option, network formation can be considered according to a spinodal decomposition mechanism. Briefly, the spinodal curve in a phase diagram represents the limit of metastability defined by the second derivative of the free energy with respect to concentration Important features of such a mechanism are (1) the phase separation in solute-rich and solute-poor... 

In the simulation of the spinodal curve, using the Equation above, the parameter can be used as an adjustable parameter to match the minimum temperature of the simulated spinodal to the minimum of the experimental cloud point temperature. Then the full spinodal curve with constant values of Xjj and can be calculated. 

Fig. 27. A plot showing the values of Xjj calculated from heat of mixing data for chlorinated paraffins with oligomeric poly-(methyl methacrylate). This is more favourable for higher chlorine content. Also shown are the values which had to be used to simulate the spinodal curves shown in the following figure...

It is possible to simulate the spinodal curves of the phase diagram of polymer pairs using the Equation-of-state theory developed by Flory and co-workers. It is only, however, possible to do this using the adjustable non-combinatorial entropy parameter, Qjj. Another problem arises in the choice of a value for the interaction parameter Xjj. This is introduced into the theory as a temperature independent constant whereas we know that in many cases the heat of mixing, and hence is strongly temperature dependent. The problem arises because Xj was never intended to describe the interaction between two polymers which are dominated by a temperature dependent specific interaction. 

If a line is supposed to be 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 ( 48.1), this curve, which is called the spinodal curve (cf. 59.1), has the equation (29), and it divides the surface into two parts, which represent, respectively, states of stable and non-stable equilibrium, A being positive on one side and negative on the other. 

---