Thermodynamics spinodal points

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, (spinodal points that correspond to those discussed for solutions in the previous section. This breakdown of the van der Waals equation of state can be bypassed by allowing the system to become heterogeneous at equilibrium. The two phases formed at T[Pg.141]

Similar to solubility, the metastable zone width and induction time of a supersaturated solution are affected by various factors, including temperature, solvent composition, chemical structure, salt form, impurities in the solution, etc. Therefore, although the spinodal point is a thermodynamic property, it is very difficult to measure the absolute value of the metastable zone width experimentally. Regardless, understanding the qualitative behavior of the metastable zone width and the induction time can be helpful for the design of crystallization processes. 

The increase of the bulk pressure at a small increment after achievement of the equihbrium density distribution allows obtaining the adsorption branch of the isotherm. If the pore is wide enough, the capillary condensation will occur, with the pressure of the condensation being corresponded to the vapor-like spinodal point. Similarly, desorption branch of the isotherm will be obtained at the decrease of pressure. In this case, the capillary evaporation will occur at a hquid-like spinodal point. The equilibrium transition pressure is obtained by comparing the grand thermodynamic potentials corresponding to the adsorption and the desorption branches of the isotherm. It corresponds to the equality of these values of the grand thermodynamic potential. 

Equations 15.5 and 15.17 can be used to establish the relationship between the reduced temperature and reduced pressure for the spinodal point (and hence for the so-called thermodynamic limit for homogeneous nucleation). Carey [4] compared data for homogeneous nucleation with this limit, and his results are illustrated in Fig. 15.6. The results generally lie between those predicted from the van der Waals equation (X = 0) and the Berthelot equation (X = 1). An empirical expression for the homogeneous nucleation temperature, which is also expressed in terms of reduced properties, is that of Lienhard [15], who relates the reduced homogeneous nucleation temperature Tr to the reduced saturation temperature Tr%at as follows ... 

Continuous thermodynamics has also been applied to derive equations for spinodal, critical point and multiple critical points. To do so with continuous thermodynamics is much easier than in usual thermodynamics. Spinodal and critical points may be calculated for very complex systems or for cases in which the segment-molar excess Gibbs energy and depends on some moments of the distribution function. In simple cases (for example, a solution of a polymer in a solvent, where the segment-molar excess Gibbs energy is independent of the distribution function) the equations of the spinodal and the critical point are known from the usual thermodynamic treatment. However, for more complex systems continuous thermodynamics has achieved real progress, for example, for polydisperse copolymer blends, the polydispersity is described by bivariant distribution functions. ... 

Thus, for phase-separated systems, the concentrations 0gi and 0g2 of the two thermodynamic stable phases are uniquely determined by the points of common tangent. The inflection points separating U- from I-shaped F(0) will accordingly represent an instability points, the so-called spinodal points given by the condition... 

Systems with concentrations between the two spinodal points will be unstable and decompose spontaneously. Polymer mixtures with overall concentrations between the binodal and the spinodal, will be metastable, and decompose following a nucleation-and-growth mechanism. For polymer blends with concentrations outside the two binodals, the system is mixed in a thermodynamically stable single phase. 

They define the thermodynamic limits of metastability. For concentrations corresponding to the spinodal points, the system is unstable and demixes spontaneously into two distinct continuous phases which form an interpenetrating system. This type of phase separation characteristic of spinodal regions, is also called spinodal decomposition. 

The spinodal represents a hypersurface within the space of external parameters where the homogeneous state of an equilibrium system becomes thermodynamically absolutely unstable. The loss of this stability can occur with respect to the density fluctuations with wave vector either equal to zero or distinct from it. These two possibilities correspond, respectively, to trivial and nontrivial branches of a spinodal. The Lifshitz points are located on the hyperline common for both branches. 

Non-Random Systems. As pointed out by Cahn and Hilliard(10,11), phase separation in the thermodynamically unstable region may lead to a non-random morphology via spinodal decomposition. This model is especially convenient for discussing the development of phase separating systems. In the linearized Cahn-Hilliard approach, the free energy of an inhomogeneous binary mixture is taken as ... 

Fig. 1.9 Calculated polymer-solvent phase diagram. The bimodal (continuous line) is the coexistence curve the points below it correspond to thermodynamically unstable states, which undergo phase separation. However, the pints between the bimodal and the spinodal (dashed line) are ki-netically stable, since there is a free-energy barrier to phase separation. C indicates the critical point the collapse temperature. The deviation of the low-concentration branch of the spinodal from the vertical axis below T is an artifact of the mean-field approximation. (From ref. [62])...

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

The presence and width of a metastable zone, in which nucleation is not spontaneous, have been discussed in Chapter 2. The thermodynamic limit of the metastable zone is a locus of points known as the spinodal curve, where spinodal decomposition replaces nucleation and crystal growth as the phase separation. In typical industrial crystallization, nucleation (and release of supersaturation) occurs at much lower supersaturations than the spinodal curve. 

From a thermodynamic point of view, phase diagrams may be constructed by changing the temperature (ii), pressure (12). or composition of a material. The present experiments are concerned with changes in composition at constant temperature and pressure, leading to a ternary phase diagram with polymer network I at one corner, monomer II at the second corner, and polymer network II at the third corner. According to classical concepts, at first there should be a mutual solution of monomer II in network I, followed by the binodal (nucleation and growth kinetics) and finally the spinodal (spinodal decomposition kinetics). 

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