Binodal limit

On a ternary equilibrium diagram like that of Figure 14.1, the limits of mutual solubilities are marked by the binodal curve and the compositions of phases in equilibrium by tielines. The region within the dome is two-phase and that outside is one-phase. The most common systems are those with one pair (Type I, Fig. 14.1) and two pairs (Type II. Fig. 14.4) of partially miscible substances. For instance, of the approximately 1000 sets of data collected and analyzed by Sorensen and Arlt (1979), 75% are Type I and 20% are Type II. The remaining small percentage of systems exhibit a considerable variety of behaviors, a few of which appear in Figure 14.4. As some of these examples show, the effect of temperature on phase behavior of liquids often is very pronounced. 

Both maximum and minimum limits exist of the solvent/feed ratio. The maximum is the value that locates the mix point M on the binodal curve near the solvent vertex, such as point Mmla on Figure 14.7(b). When an operating line coincides with a tieline, the number of stages will be infinite and will correspond to the minimum solvent/feed ratio. The pinch point is determined by the intersection of some tieline with line RnS- Depending on whether the slopes of the tielines are negative or positive, the intersection that is closest or farthest from the solvent vertex locates the operating point for minimum solvent. Figure 14.9 shows the two... 

The solutions in the region inside the spinodal domain are unstable, whereas the solid solutions in the region between the binodal and spinodal lines are metastable. The presence of a miscibility gap limits the potential usefulness of these materials in device applications. Solutions with compositions lying inside the spinodal domain cannot be grown by LPE, whereas metastable solid solutions have a tendency toward phase separation and, eventually, device degradation. 

The binodal branches do not coincide with the phase diagram axes. This means that the biopolymers are limitedly cosoluble. For instance, on mixing a protein solution A and a polysaccharide solution B a mixture of composition C can be obtained. This mixed solution spontaneously breaks down into two liquid phases, phase D and phase E. Phase D is rich in protein and E is rich in polysaccharide. These two liquid phases form a water-in-water (WIW) emulsion. Hie phase volume ratio is estimated by the inverse lever rule. The phase D/phase E volume ratio equals the ratio of the tieline segments EC/CD. Point F represents the phase separation threshold, that is, the minimal critical concentration of biopolymers required for phase separation to occur. 

Both maximum and minimum limits exist of the solvent/feed ratio. The maximum is the value that locates the mix point M on the binodal curve near the solvent vertex, such as point on Figure... 

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. 

On the basis of standard criteria for equilibrium, stability limits, and criticality yielding coexistence curve (binodal), spinodal line, and critical point, the phase behavior maybe predicted using Eq. (1) ... 

The experimental approach examines bilayers with a limited precision in depth z (8=a few nanometers) and in volume fraction < > (a few percent). It assumes that at least the central part of the analyzed profile ( >(z) describes only the internal interface between coexisting phases ([q and 2. This is not necessarily true when surface segregation regions, adjacent to both external interfaces, cannot be neglected as it is for very thin films. Related finite size effects are discussed in detail in Sect. 3.2 theoretical models and computer simulations expect that size effects modify the intrinsic profile internal interface. Therefore size effects may lead in principle to systematic errors [6] of binodal values determined for films which are very thin or are profiled at T—>TC (where the ratio D/w is also small due to the diverging w). 

As mentioned earlier, a supersaturated solution is not in the equilibrium condition. Crystallization moves the solution toward equilibrium by relieving its supersaturation. A supersaturated solution is thus not stable. There is a maximum degree of supersaturation for a solution before it becomes unstable. The region between this unstable boundary and the equilibrium (binodal) curve is termed the metastable zone, and it is here that the crystallization process occurs. The absolute limit of the metastable zone, known as the spinodal curve (8), is given by the locus of the maximum limit of supersaturation at which nucleation occurs spontaneously. Thermodynamically, the spinodal curve within the two-phase region is defined by the criterion... 

Abstract The virtual terms binodal and spinodal are equivalent to the experimental terms eoexistence curve (CXC) and metastability limit (ML), respectively, within an inherent accuracy of any semi-empirical EOS at the description of a real fluid behavior. Any predicted location of mechanical spinodal at positive pressures Psp(T)>0 merits verification because the Maxwell rule is a model (EOS)-dependent method based on the non-measurable values of chemical potential for both phases. It is not a reliable tool of CXC- and ML-prediction especially at low temperatures between the triple and normal boiling ones [Tt,Ti where the actual vapor pressures Ps(T) > 0 are quite small while... 

Keywords disorder parameter, order parameter, metastability limit, asymmetry, quasi-binodal, particle-hole-type symmetiy... 

The depth of the entry into a metastable region was limited by the action of the radiation background and easily activated boiling sites. A water superheat was accompanied by a decrease in the sound velocity (an increase in the adiabatic compressibility). Values of the sound velocity on the binodal and the line of attainable superheat (J = 10 s" m, T = const) differ on average by 8-12 %. 

To start with, let us consider a particular case of metastable states - a liquid superheated with respect to the liquid-vapor equilibrium temperature. For simplicity let us take a pure liquid at positive pressures, see Fig. 1. The region of superheated states is limited from below by the binodal Ts(p) and from above by the experimental line of attainable superheat, or, in other words, the line of spontaneous boiling-up T (p Cxp) of the liquid. An understandable limitation is imposed on the volume of superheated sample V and the time period Cxp of experiment. Naturally, the experimental time should be shorter than the life time t of the metastable state. 

FIGURE 6.19 Idealized cases of phase separation in aqueous mixtures of two polymers, concentrations c2 and c 5. (a) Segregative phase separation or incompatibility. (b) Associative phase separation or complex coacervation. The heavy lines denote the binodal (solubility limit), the thin ones are tie lines. The dots indicate critical points. 

As we have already seen (p. 127), water and succinic nitrile can form two liquid layers between the temperatures 18 5° and 55 5° while alcohol and nitrile can form two liquid layers between 13 and 31 . If, then, between these two temperature limits, alcohol is added to a heterogeneous mixture of water and nitrile, or water is added to a mixture of alcohol and nitrile, two heterogeneous ternary systems will be formed, and two binodal curves will be obtained in the triangular diagram, as shown in Fig. 97A On changing the temperature, the binodal curves will also undergo alteration, in a manner similar to that just discussed. As the temperature falls, the two curves will... 

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