Binodal equations

The binodal Equation can be derived from the diemical potentials of the components using the binodal condition quoted earlier. The resulting Equations are mudi more complicated and more difficult to solve than those for the spinodal. The simulation of binodal and spirodal curves in a practical manner is shown elsewhere... 

To calculate the binodal Equation 19.1 is used, taking into account that the sum of the mole fractions of the components in each phase must be 1 and that the chanical potential of a compound in two phases in equilibrium is the same in both of them (equilibrium condition). 

Figure 4.13. Universal binodal (Equation 4.3-129) and the data of Dobashi et al. (I980ab) [Reprinted with permission from r.Dobnshi, M.Nakala, M.Kaiieki).. 1. Chuin. Phys. 72 (1980) 6692-6697. Copyright I960 American Institute of Physics] (Figure 4.8) (Sanchez, 1985) [Reprinted with permission from I.Sanchez. J. Appl. Phys. 58 (1985) 2871 2874. Copyright 1985 American Institute of Physics]...

Figure C2.1.10. (a) Gibbs energy of mixing as a function of the volume fraction of polymer A for a symmetric binary polymer mixture = Ag = N. The curves are obtained from equation (C2.1.9 ). (b) Phase diagram of a symmetric polymer mixture = Ag = A. The full curve is the binodal and delimits the homogeneous region from that of the two-phase stmcture. The broken curve is the spinodal.

The IA phase coexistence equations could not be solved in a low E, region for the aqueous schizophyllan system, where the degree of orientation parameter a2 of the lower molecular weight polymer component 2 becomes very small. This is mainly due to poor approximations in the asymptotic expansions of p , and a,(N) at small ot2 values. The broken binodals in Panel b of Fig. 9 are not calculated results but interpolation curves. 

The binodal (or coexistence) curve, on which the compositions of the immiscible solutions (phases) lie at equilibrium, can be described by a set of equations involving equilibrium between the chemical potentials of the components in the coexisting phases (Prigogine and Defay, 1954) ... 

On the basis of these relationships, using the expressions of the chemical potentials of the components at the level of approximation of the second virial coefficients, the binodal curve can be expressed by the following set of equations (Edmond and Ogston, 1968) ... 

The occurrence of demixing morphologies characteristic for the metastable regime between the binodal and the spinodal can be understood from Fig. 17. The red dot marks the initial position of the sample with c = 0.3. Upon laser heating the temperature within the laser focus rises by AT and the distance to the binodal first increases. A stationary temperature distribution is rapidly reached and the Laplacian of the temperature field T(r,t) is obtained from the stationary solution of the heat equation (5) with the power absorbed from the laser as source term ... 

The condition of phase stability for such a system is closely related to the behavior of the Helmholtz free energy, by stating that the isothermal compressibility yT > 0. The positiveness of yT expresses the condition of the mechanical stability of the system. The binodal line at each temperature and densities of coexisting liquid and gas determined by equating the chemical potential of the two phases. The conditions expressed by Eq. (115) simply say that the gas-liquid phase transition occurs when the P — pex surface from the gas... 

According to Equation (2F-1), a liquid mixture with a total volume fraction liquid phases with binodal compositions Gibbs energy for the mixture will thus lie on the solid line between solid line is a tangent touching the predicted curve at the binodal compositions. 

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... 

McMaster simulated binodal and spinodal curves for hypothetical polymer pairs with various values of the Equation-of-state parameters We have also simulated many hypothetical spinodal curves using the equations presented in the previous section and some of these are presented in Figs. 25 and 26. Various other workers have also calculated theoretical curves. An assessment of the effect of changes in the various properties is presented below. 

The implications of this theory are as follows Eq. 4 yields a threshold weight separating a LMW, Mn < M, from HMW, Mn > M, regime. Equation 6 depicts the degree of alignment as a function of T and Mn. Equation 7 gives an approximate expression for the Hex-Nem coexistence line above Tg. When the experimental dimensions of the molecule are introduced into the theory, the Hex-Nem phase transition is predicted as a function of Mn. When T Tg, Eq. 5 predicts this transition at M 0 10 kg/mol for PF2/6 [24]. The limit in the case Mn A> M is obtained by the constant A extracted from experiment. Finally, the binodal, Eq. 7, is an interpolation between these limiting cases. 

The above equation can be solved for the interaction parameter corresponding to the phase boundary—the binodal (solid line in the bottom part of Fig. 4.8) of a symmetric blend ... 

One of the problems that we have encountered in this work is that experimental data in the literature on ApEG ai d Apx, while quite extensive, is not extensive enough (nor accurate enough) for us to thoroughly examine the trends predicted by these models under a variety of conditions. Our current and future work therefore includes some experimental measurements of binodals (and partition coefficients) and some modeling work to obtain equations for estimating Apeg and Ap)x (8). 

The binodal cnrves for all three types of phase behavior are represented by the Hand equation ... 

Now we have eight equations (7.37 through 7.44) to solve eight unknowns, Oy and Sj (i = 1, 2, 3, j = 2, 3). In principle, the solution is determined. However, these equations are not all linear the solution procedures are not straightforward. In the following, we provide the detailed iterative procedures for a simple case in which Bh is equal to -1 and Fh = 1 for the symmetric binodal curves. 

Topology of the fluid-fluid phase transitions depends on the concurrence between the repulsive and the attractive parts of EoS. The binodal location at given temperature, T, and pressure, P is a solution of the set equations ... 

Figure 6 displays a first P-x Naci diagram depicting binodal and spinodal isotherms at 623 K, 350°C. The equation of state reproduces well the tabulated data by Bischoff for the solubility curves L(G) and G(L). The pressures of the diffusion spinodal curves Sp(L) and Sp(G) decrease with increasing XNaci mole fractions (note also that the spinodal curves run through the stability field of... 

The equilibrium degree of swelling or the gel volume of shows a large change in response to external conditions. Only when the volume change is discontinuously and the coexistence of two phases is experimentally proofed, we should use the term phase transition . By applying the stability criteria on the Flory-Rehner equation one can calculate the conditions of phase transition (binodal and spinodal curve, critical point). For details see e.g. (Onuki 1993). 

The spinodal, binodal and critical point Equations derived on the basis of this theory will be discussed later. When the theory has been tested it has been found to describe the properties of polymer blends much better than the classical lattice theories 17 1B). It is more successful in interpreting the excess properties of mixtures with dispersion or weak attraction forces. In the case of mixtures with a strong specific interaction it suffers from the results of the random mixing assumption. The excess volumes observed by Shih and Flory, 9, for C6H6-PDMS mixtures are considerably different from those predicted by the theory and this cannot be resolved by reasonable alterations of any adjustable parameter. Hamada et al.20), however, have shown that the theory of Flory and his co-workers can be largely improved by using the number of external degrees of freedom for the mixture as ... 

---