Spinodal criterion

Appendix A Moment (Gibbs) Free Energy for Fixed Pressure Appendix B Moment Entropy of Mixing and Large Deviation Theory Appendix C Spinodal Criterion From Exact Free Energy Appendix D Determinant Form of Critical Point Criterion References... 

For completeness, we now give the explicit form [11,12,44] of the spinodal criterion (50) for truncatable systems see also Ref. 45 for an equivalent derivation using the combinatorial approach. Using Eq. (43) and abbreviating... 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

The first part of this is simply the spinodal criterion, as expected. To evaluate the second part for the moment free energy (37), we need the third derivative of sm with respect to the moment densities pt. Writing (41) as... 

To conclude this section on the dense random copolymer model, wc briefly discuss the spinodal criterion and ask whether critical or multicritical points can exist for a general parent distribution pW(a) (with p = 1). The criterion (53), applied to our one-moment free energy becomes... 

The excess part —x pj is quadratic in px and therefore only enters the spinodal criterion the additional conditions for critical points only depend on the moment entropy sm = — Ao — X px. Because of die dense limit constraint Px = 1, Ao is related to k by... 

APPENDIX C SPINODAL CRITERION FROM EXACT FREE ENERGY... 

In this appendix, we apply the spinodal criterion (50) to the exact free energy (38) and show that it can be expressed in a form identical to Eq. (55). This result has been given by a number of authors [11, 12, 44], but we include it here for the sake of completeness. 

We present an improved model for the flocculation of a dispersion of hard spheres in the presence of non-adsorbing polymer. The pair potential is derived from a recent theory for interacting polymer near a flat surface, and is a function of the depletion thickness. This thickness is of the order of the radius of gyration in dilute polymer solutions but decreases when the coils in solution begin to overlap. Flocculation occurs when the osmotic attraction energy, which is a consequence of the depletion, outweighs the loss in configurational entropy of the dispersed particles. Our analysis differs from that of De Hek and Vrij with respect to the dependence of the depletion thickness on the polymer concentration (i.e., we do not consider the polymer coils to be hard spheres) and to the stability criterion used (binodal, not spinodal phase separation conditions). 

This notation emphasizes that the chemical potentials are functions of the densities. The criterion for a spinodal at the parent phase p is then that there is an incipient instability direction 5p along which the chemical potentials do not change ... 

As expected from the general discussion in Section III. A, the criterion (57) can also be derived from the exact free energy an alternative form involving the spinodal determinant Y is given in Appendix D. Equation (57) shows that the location of critical points depend only on the moment densities p[t py, and pijk [11, 46]. For a system with an excess free energy depending only on power-law moments up to order K - 1, the critical point condition thus involves power-law moments of the parent only up to order 3 (K — 1). 

In conventional polymer notation, this result would read/m = — (a + 1) c< a/( +1) +/.] From this we can now obtain the spinodal condition, for example, which identifies the value of % where the parent becomes unstable. In our case of a single moment density the general criterion (50) simplifies to... 

The exactness statements in Section HI can also be directly translated to the constant pressure case. The arguments above imply directly that the onset of phase coexistence is found exactly from the moment Gibbs free energy All phases are in the family (A3), because one of them (the parent) is, and the requirement of equal chemical potentials is satisfied. Spinodals and (multi-) critical points are also found exactly. Arguing as in Section HI. A and using the vector notation of Eq. (53), the criterion for such points is found as... 

In this appendix, we give the form of the critical point criterion (57) that uses the spinodal determinant Y from Eq. (55) [34], At a critical point, the instabil-... 

The validity of the linear theory observed for the early stage of spinodal decomposition is chiefly related to the large size of the chain molecules. As shown above, characteristic quantities as the time t or the wavelength Am(0) of the fastest growing fluctuation are proportional to Ro and Rg, respectively. Furthermore, the Landau-Ginzburg criterion (cf. condition 2)) ensures that the mean-field regime is sufficiently extended. 

As mentioned earlier, a supersaturated solution is not in the equilibrium condition. Crystallization moves the solution toward equilibrium by relieving its supersaturation. A supersaturated solution is thus not stable. There is a maximum degree of supersaturation for a solution before it becomes unstable. The region between this unstable boundary and the equilibrium (binodal) curve is termed the metastable zone, and it is here that the crystallization process occurs. The absolute limit of the metastable zone, known as the spinodal curve (8), is given by the locus of the maximum limit of supersaturation at which nucleation occurs spontaneously. Thermodynamically, the spinodal curve within the two-phase region is defined by the criterion... 

Heermann and coworkers [296, 297] were the first to carry out simulations of spinodal decomposition in two space dimensions. In this case chains cannot penetrate into each other, so each chain can interact only with a few neighboring chains around it, and our discussion of the Ginzburg criterion (Sect. 2.5) implies that nonlinear phenomena are very important even during the early stages of the quench, and a stage where the structure factor increases exponentially fast with... 

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. 

Unstable states defined by negative curvature of Ag ixing vs. y2 at constant T and p, between the spinodal points. In this region, (9 Agmixmg/9y2)7 ,p < 0. Single-phase equilibrium states of this nature are completely disallowed even if the first stability criterion is satisfied. 

For a pure substance, such as in Figure 8.5, metastable states on an isotherm lie between stable states and unstable states. At one end of the metastable range, metastable states are separated from unstable states by a curve called the spinodal. For a pure substance, the spinodal is the locus of points at which the differential stability criterion (8.2.2) is first violated, that is, the points at which... 

Along any pure-fluid, subcritical isotherm, the spinodal separates unstable states from metastable states. At the other end of an isotherm s metastable range, metastable states are separated from stable states by the points at which vapor-Uquid, phase-equilibrium criteria are satisfied. Those criteria were stated in 7.3.5 the two-phase situation must exhibit thermal equilibrium, mechanical equilibrium, and diffusional equilibrium. Since we are on an isotherm, the temperatures in the two phases must be the same, and the thermal equilibrium criterion is satisfied. 

The middle envelope is the spinodal the set of states that separate metastable states from unstable states. Recall from 8.3 that one-phase mixtures become diffusionally unstable before becoming mechanically unstable. Therefore, the mixture spinodal is the locus of points at which the diffusional stability criterion (8.3.14) is first violated that is, it is the locus of points having... 

Figure 8.12 shows that if a mixture is mechanically unstable, then it is also diffu-sionally unstable, because the line of incipient mechanical instability lies under the spinodal, or equivalently because Kj appears in both stability criteria (8.1.30) and (8.3.13). Moreover, a one-phase mixture may be diffusionally unstable but remain mechanically stable, because the spinodal lies above the line of incipient mechanical instability, or equivalently because the mechanical criterion (8.1.30) can be satisfied while the diffusional criterion (8.3.13) is violated. Further, Figure 8.12 contains states at which no differential stability criteria are violated, but at which one-phase mixtures are metastable rather than stable. This means that a violation of any differential stability criteria (thermal, mechanical, or diffusional) is only sufficient, but not necessary, for a phase separation to occur. 

Porter s equation also provides an estimate for the spinodal, which separates unstable states from metastable ones. In terms of g, the spinodal of a binary occurs when the diffusional stability criterion is first violated, that is, when... 

Figure 8 1 g ix) and its second mole-fraction derivative computed from Porter s equation for the binary mixtures in Figure 8.20. At 60°C the diffusional stability criterion is satisfied at all compositions and the mixture is a stable single phase. However at 30°C, states between C and D violate the diffusional stability criterion and the mixture splits into two phases C and D lie on the spinodal. Filled circles at 30°C correspond to states of the same labels in Figure 8.20.

Jia] Jiang, B., Chang, M., Wei, Q., Xu, Z., Thermodynamic Criterion of Spinodal Decomposition in Ternary Systems , Acta Metall. Sin. (China), 4B(2), 75-81 (1991), translated from Acta Metall. Sin. (China), 26(5), B303-B309 (1990) (Thermodyn., Theory, 11)... 

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