Spinodal condition

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). 

In principle, all the molecular parameters in Eq. (6) can be determined independently, so that the theory can be quantitatively compared with experimental data. An example of Maxwell s construction in the dependence of x °n critical value of interaction parameter %c of charged PAAm network with the degree of ionization equals to the molar fraction of the sodium methacrylate in the chain i = xMNa = 0.012 are given in Fig. 4 (data of series D from Fig. 5). The compositions of the phases

critical value of Xc were determined by the condition that areas St and S2 defined in Fig. 4 are equal The experimental (p2e is higher and 2 determined by Maxwell s construction (Eq. 13). Thus, the experimental values of (p2e and metastable region the limits of which (p2s and (p2s are determined by the spinodal condition (two values

[Pg.182]

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)]. 

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... 

This is still sufficiently simple that the exact spinodal condition (55)—for a system with a general density distribution p o)—can be worked out analy-... 

As promised, this is identical to the spinodal condition (55) derived from the moment free energy, with the matrix multiplications written out explicitly. 

The isotropic-to-nematic transition is defined by the characteristic equation Det M = 0 (where Det represents the determinant of a matrix). If the Van der Waals interactions were turned off (W0 = 0) so that only nematic interactions are left, then M would be the denominator of X so that X would blow up for this condition (Det M = 0). Above certain critical values of Wj s the blend forms the nematic phase. As in the case of purely flexible mixtures, the spinodal condition is ... 

The spinodal Equation can be obtained by applying the spinodal condition (Eq. 9) to the chemical potential of a polymer in a mixture, (obtained from Eq. (27) by including... 

The projections of the L(G) and Sp(L) isotherms in aP-Xcoi diagam (Fig. 3A) are almost vertical and parallel. As a consequence, the depressurization of a C02-supersaturated solution cannot perturb the fluid up to near-spinodal conditions gas exsolution will always proceed only by moderate bubble nucleations and any decompression process will be subspinodal. 

A complete series expansion of the phase equilibrium conditions, Eqs. (88X (89), with respect to ipj, ipe and Xo at the critical point generally shows [34] that the conditions for the spinodal and m-multiple critical points may be obtained in the following way. Denoting the left-hand side of Eq. (88) or Eq. (89) by the symbol FU, we can write the spinodal condition as... 

For pure liquid copolymers (typically, eg = 0.25 and z = 10 may be assumed) phase separation may be expected to occur in the usual temperature range if the interaction energy Au,p multiplied by the Avo dro constant reaches values of the order of 500 Jraol". The conditions for the spinodal and the critical point can be derived from Eqs. (115) or (116) according to Eqs. (92) and (93). The spinodal condition reads... 

Combining the spinodal condition (Eq. (10.23)) with the critical condition (Eq. (10.19)) gives the opportunity to estimate the adjustable parameter y using the critical concentration, Xgi... 

Distance from spinodal condition as = 2[(xN)s-(xN)] Thermodynamic interaction parameter B = XiaRT/Viu Mobility... 

One can easily make sure of the identity of the spinodal conditions (Equations 1.2 60, 1.3 19) and the critical point conditions (Equations 1.2 -60, 61, 1.3 19, 20) when concentrations are expressed in different units. 

The condition /d j, o = 0 means the spinodal (Equation 3.3 10), hence, according to Equation 47, / = 0 defines the spinodal as well. Then, on the basis of Equation 48, the spinodal condition is expressed by... 

An interesting situation occurs when the system is at the spinodal condition (Caneba and Soong, 1985b), wherein the determinant... 

Differentiating eqn (8.28) with respect to (j)2 gives the spinodal condition ... 

Note that in the hypothetical incompressible limit the form of both the scattering functions and spinodal condition are much simpler than the rigorous expressions of Eqs. (6.4)-(6.6). Although the IRPA can usually be fitted to low wavevector experimental scattering data, and an apparent chi-parameter thereby extracted, the literal use of the IRPA for the calculation of thermodynamic properties and phase stability is generally expected to represent a poor approximation due to the importance of density-fluctuation-induced compressibility or equation-of-state effects [2,65,66]. The latter are non-universal, and are expected to increase in importance as the structural and/or intermolecular potential asymmetries characteristic rrf the blend molecules increase. 

The conditions required for accuracy of an incompressibility assumption at the level of the scattering functions and spinodal condition are easily derived within the PRISM formalism [67]. From Eq. (6.7) a small isothermal compressibility implies that — oCw -fOl P 1, which is generally true for any dense fluid. If the related wavevector-dependent condition... 

In summary, the predictions of analytic PRISM theory [67] for the phase behavior of asymmetric thread polymer Uends display a ly rich dependence on the single chain structural asymn try variables, the interchain attractive potential asymmetries, the ratio of attractive and repulsi interaction potential length scales, a/d, and the thermodynamic state variaUes t) and < ). Moreover, these dependences are intimately coupled, which mathematically arises within the compressible PRISM theory from cross terms between the repulsive (athermal) and attractive potential contributions to the k = 0 direct correlations in the spinodal condition of Eq. (6.6). The nonuniversality and nonadditivity of the consequences of molecular structural and interaction potential asymmetries on phase stability can be viewed as a virtue in the sense that a great variety of phase behaviors are possible by rational chemical structure modification. Finally, the relationship between the analytic thread model predictions and numerical PRISM calculations for more realistic nonzero hard core diameter models remains to be fully established, but preliminary results suggest the thread model predictions are qualitatively reliable for thermal demixing [72,85]. 

The effective chi-parameter is given by the k-dependent generalization d Eq. (6.17). If a small angle approximation where )CiNc(k) is constant is invdred, then Eq. (9.S) is identical in form with Leibler s RPA result [87] and the spinodal condition is far simpler than Eq. (9.4). However, since the true chi- Kiranieter is a wavevector-dependent correlation function, not a phenomenological number, it is functionally related to all the other intramolecular and intermolecular pair correlations in the system. This non-mean-field feature has many important consequences such as the fact that k" is influenced by many chain correlations [86,88]. It must be emphasized that although Eq. (9.5) should be accurate for the hypothetical symmetric block copolymer model, since it does not properly treat compressibility effects it is expected to be inadequate for most real copolymer systems. 

By using these relations, we can express the spinodal condition as... 

The boundary between the stable and unstable states can be found by the spinodal condition... 

The critical point is the point where both binodal and spinodal conditions... 

The condition D = 0 should be calculated to find the stability limit. We find that the spinodal condition is given by... 

---