Spinodal points

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

Infinite Systems The ultimate fate of infinite systems, in the infinite time limit, is quite different from their finite cousins. In particular, the fate of infinite systems does not depend on the initial density of cr = 1 sites. In the thermodynamic limit, there will always exist, with probability one, some convex cluster large enough to grow without limit. As f -4 oo, the system thus tends to p —r 1 for all nonzero initial densities. What was the critical density for finite systems, pc, now becomes a spinodal point separating an unstable phase for cr = 0 sites for p > pc from a metastable phase in which cr = 0 and cr = 1 sites coexist. For systems in the metastable phase, even the smallest perturbation can induce a cluster that will grow forever. 

The crucial question is at what value of <)> is the attraction high enough to induce phase separation De Hek and Vrij (6) assume that the critical flocculation concentration is equivalent to the phase separation condition defined by the spinodal point. From the pair potential between two hard spheres in a polymer solution they calculate the second virial coefficient B2 for the particles, and derive from the spinodal condition that if B2 = 1/2 (where is the volume fraction of particles in the dispersion) phase separation occurs. For a system in thermodynamic equilibrium, two phases coexist if the chemical potential of the hard spheres is the same in the dispersion and in the floe phase (i.e., the binodal condition). 

For sub-critical isotherms (T < Tc), the parts of the isotherm where (dp/dV)T < 0 become unphysical, since this implies that the thermodynamic system has negative compressibility. At the particular reduced volumes where (dp/dV)T =0, (Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

We have now derived the phase boundary between the two liquids. By analogy with our earlier examples, the two phases may exist as metastable states in a certain part of the p,T potential space. However, at some specific conditions the phases become mechanically unstable. These conditions correspond to the spinodal lines for the system. An analytical expression for the spinodals of the regular solution-type two-state model can be obtained by using the fact that the second derivative of the Gibbs energy with regards to xsi)B is zero at spinodal points. Hence,... 

Fet us now consider an initial composition C2. We are within the two spinodal points on the Gibbs free energy coherent curve G decomposition takes place spontaneously throughout the entire phase, with modulated compositional fluctuations whose amphtudes increase as the process advances. We always obtain pigeonite plus diopside, although theoretically the orthopyroxene plus diopside paragenesis, also in this case, corresponds to the minimum energy of the system. 

The osmotic modulus becomes zero at the spinodal point. Therefore, the spinodal temperature, Ts, is obtained by substituting K = 0 into Eq. (220)... 

Tanaka et al. found a critical point at the zero-osmotic pressure condition in ionic gels by varying the degree of ionization and diminishing the volume-discontinuity at first-order changes. At such a critical point, the first three derivatives of F with respect to V should vanish from Eqs. (2.6) and (2.26). On the other hand, the so-called spinodal point is given by K = 0, at which the volume fluctuations diverges as shown by Eq. (2.10). 

However, it remains unknown how heterogeneities affect the phase transition itself. In fact, the perturbation scheme used to derive Eq. (4.52) breaks down near the spinodal point. We can well expect that domains of a shrunken (or swollen) phase are created and pinned around heterogeneities with higher (or lower) crosslink densities. 

Let us examine the critical dynamics near the bulk spinodal point in isotropic gels, where K + in = A(T — Ts) is very small, Ts being the so-called spinodal temperature [4,51,83-85]. Here, the linear theory indicates that the conventional diffusion constant D = (K + / )/ is proportional to T — Ts. Tanaka proposed that the density fluctuations should be collectively convected by the fluid velocity field as in near-critical binary mixtures and are governed by the renormalized diffusion constant (Kawasaki s formula) [84],... 

As a new subject we have considered the effect of the frequency-dependence of the elastic moduli on dynamic light scattering. The resultant nonexponential decay of the time-correlation function seems to be observable ubiquitously if gels are sufficiently compliant. Furthermore, even if the frequency-dependent parts of the moduli are very small, the effect can be important near the spinodal point. The origin of the complex decay is ascribed to the dynamic coupling between the diffusion and the network stress relaxation [76], Further scattering experiments based on the general formula (6.34) should be very informative. 

The structure factor diverges at a spinodal point defined by 1 (xN)s = F(x, f) where x is given by eqn 2.10 with q = q. The spinodal for block copolymers is close to, but not identical to, the ODT (except for symmetric block copolymers in mean field theory) and defines the stability limit of the disordered phase. 

Fig. 7 Phase diagram of PDMS/PEMS (16.4/48.1). The cloud points that mark the binodal squares) have been obtained by turbidimetry. Pseudo-spinodal points are as explained in the text. The color encodes the modulus of the Soret coefficient. Figure from [100]. Copyright (2007) by The American Physical Society...

A completely different result comes out if one pursues the experimental course of Fig. lib. The variation of the apparent diffusion coefficient as a function of quench depth AT = T — Ts, where Ts is the spinodal point, is shown in Fig. 13. Initially, the apparent diflusity increases with increasing quench depth AT which is in accord with the mean-field theory since (% — Xs)/Xs AT. However, when the quench depth is raised further the apparent diffusion coefficient starts to decrease and finally, apparently levels off. 

Let us examine the oitical dynamics near the bulk spinodal point in isotropic gels, where K A T — 2 ) is very small, T, being the so-called spinodal... 

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