Spinodal determination

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 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-... 

Indicated in Fig. 9 are temperature ranges of supercooled, stable and superheated water at atmospherie pressure. Ibidem one can see curves representing the temperature dependenee of the logarithm of the homogeneous nucleation rate for crystallization (curve 1) and boiling-up (curve 2). The maximum rate of formation of vapor nuclei is attained at the approach of the spinodal determined by condition (3). Fig. 9 also shows how the inverse isothermal eompressibility =-v(5p/5v) changes with temperature (curve 3). An arrow shows the temperature of the spinodal of superheated water. 

There is another viewpoint on the stability of a supercooled liquid, according to which the region of metastable states of a one-component liquid does not pass into a labile region with decreasing temperature. A supereooled liquid has no spinodal determined by eondition (3). V. P. Skripov thought that at T = 228 K there was no divergenee of Pj., of supercooled water, but there was a sufficiently blurred and small normal maximum. The dashed line 3 (Fig. 9) corresponds to this point of view. 

Figure 3.72. Scholte s (1971) method of spinodal determination (I), S is the boundary of the phase separation region, 3 is the critical point. The crosses mark the conhgurative points where the scattered light intensity was measured [Reprinted from Th.G.Scholte. J. Polym. Sci. A-2 9 (1971) 1553-1577. Copyright 1971 by Wiley. Reprinted by permission of John Wiley tr. Sons, Inc.)...

The Gibbs spinodal determinant in equation (2) vanishes at the spinodal temperature T, In the vicinity of the small value of the determinant is expanded by the Debye-Scholte theory in a... 

Traditionally, turbidity or opacity has been used for detecting the cloud-point curve (CPC), which approximated the bimodal of the phase diagram. Evolution of this approach involved application of laser light scattering, which combined with small specimen size and precise temperature control led to the pulse-induced critical scattering (PICS) for spinodal determination. Unfortunately, the method is limited to the size of heterogeneity > 100 nm and the difference in refractive index of the two phases > 0.01. ° i... 

The wavelength of spinodal decomposition can be determined from the wavenumber at which the maximum in the scattering occurs. This wavelength. A, is given by... 

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.

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]

Dynamical study of the phase transition of the gels in spinodal regimes was described. The evolution of intensity of light scattered from the gels indicated the applicability of Cahn s linearized theory to the phase transition. Our work offers a basis for the determination of diffusion coefficient of gels in their spinodal regimes. 

The spinodal and the cloud point can be determined as a function of pressure and temperature (up to 150C, 1000 bar) via light-scattering measurements [41]. The intensity of the scattered light of the polymer solution is measured in a high-pressure optical cell during a pressure pulse in the polymer solution. 

These results were confirmed by Sung and Han (1995). They observed that addition of a PS-PB diblock copolymer to a blend of the corresponding homopolymers (of lower molecular weight than the diblock) leads to a retardation of phase separation in both early and late stages. Using time-resolved light scattering they determined the spinodal peak position and intensity as a function of time. Addition of copolymer was found to reduce the critical temperature of the blend. 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

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... 

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