Equilibrium binodal composition

Interpretation of the phase separation fluorescence results requires that accurate values of the equilibrium binodal compositions for a particular temperature be available. These are necessary in order to calculate the volume fraction of the rich phase in the two component system. This volume fraction is assumed to be constant during the early stages of spinodal decomposition. 

The binodal compositions or compositions of liquid phases a and p in equilibrium can be obtained by solving the following equations ... 

A representative plot of binodal and spinodal curves for ternary polymer/ monomer/precipitant systems (which is similar to that of a polymer/solvent/ non-solvent system) is shown in Fig. 1.1.2 at constant temperature and pressure. The phase envelope pertains to the region encompassed by the binodal curves, in which there exist two phases at equilibrium. Outside the phase envelope is the single-phase region. The so-called tie lines are straight lines that join the binodal compositions at equilibrium. If the system has an LOST, then the size of the phase envelope increases with increasing temperature. If the system has a UCST, then the size of the phase envelope decreases with increasing temperature. 

We differentiate two distinct regimes for phase separation, depending on whether the process takes place in the metastable or the unstable region of the Temperature-Composition plane (Fig.2.). These are called nucleation and growth and spinodal decomposition , respectively. These mechanisms are competitive in determining the mode of phase-separation, the dominant effect being dependent on how far from the equilibrium binodal the mixture has been thrusted. Polymer-polymer systems, because of the slow diffusion times, can be brought to well beyond the spinodal even at concentrations far from the critical point. 

After bulk thermodynamic equilibrium has been reached via a spinodal mechanism, i.e. the mixture has phase-separated to phases of composition denoted by the equilibrium binodal, a coarsening process begins. At first the size increases linearly with time, and later exponentially. This fast coarsening of the structure can occur because the high level of interconnectivity allows viscous flow of both phases. 

In an ideal stage, the extract Ex leaves in equilibrium with the raffinate Rx, so that the point Rx is at the end of the tie line through Ex. To determine the extract E2, PRi is drawn to cut the binodal curve at E2. The points R2, E3, R3, E4, and so on, may be found in the same way. If the final tie line, say ER4, does not pass through R , then the amount of solvent added is incorrect for the desired change in composition. In general, this does not invalidate the method, since it gives the required number of ideal stages with sufficient accuracy. 

Graphically, the conditions for thermodynamic equilibrium are equal to two points which have a common tangent. These points give the composition of a polymer-rich phase (I) and a solvent-rich phase (II) that can coexist in thermodynamic equilibrium. The summation of such points is also called the coexistence curve or binodal line. 

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 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. 

Tie lines of the system can be generated from the equilibrium compositions for each run and selectivities computed. The results of measurements obtained for the 5% by volume of ammonia/ethylene are represented in the binodal diagram in Fig. 3. Butene is represented as the distributed component between the solvent phase and the butadiene-rich phase. The ammonia-solvent gas mixture was considered to behave as a pseudo-solvent of fixed composition. The ratio of the integrated peaks for butene(i) and butadiene(j) was used to compute the selectivity, B (beta), defined on a solvent-free basis, as ... 

For conditions of constant pressure, or when pressure effects are negligible, binary LEE is conveniently displayed on a solubihty diagram, a plot of T vs. xi. Figure 14.12 shows binary solubility diagrams of three types. The first diagram [Fig. 14.12(a)] shows curves (binodal curves) that define an "island." They represent the compositions of coexisting phases curve UAL for the a phase (rich in species 2), and curve UBL for the P phase (rich in species 1). Equilibrium compositions jc and at a particular T are defined by the intersections of a horizontal tie line with the binodal curves. Temperature Tl is a lower consolute temperature, or... 

A system whose overall composition is represented by a point P forms one phase a system such as P separates into two phases L and ilf, while a system P" gives rise to three phases Q, R and 8. The lines such as LM are the binodals joining the phases which coexist in equilibrium. 

The binodal for binary mixtures coincides with the coexistence curve, since for a given temperature (or A%) with overall composition in the two-phase region, the two compositions that coexist at equilibrium can be read off the binodal. Any overall composition at temperature T within the miscibility gap defined by the binodal has its minimum free energy in a... 

Note that the spinodal and binodal for any binary mixture meet at the critical point (Fig. 4.8). For interaction parameters x below the critical one (for xhomogeneous mixture is stable at any composition 0 < < 1 For higher values of the interaction parameter (for x > Xc) there is a miscibility gap between the two branches of the binodal in Fig. 4.8. For any composition in a miscibility gap, the equilibrium state corresponds to two phases with compositions

coexistence curve at the same value of x-... 

The points of the phase diagram between the spinodal and binodal curves correspond to metastable mixtures. The metastable homogeneous state is stable against small composition fluctuations and requires a larger nucleation event to initiate phase separation into the equilibrium phases given by the coexistence curve. This phase separation process is called nucleation and growth. 

For binary mixtures, the binodal line is also the coexistence curve, defined by the common tangent line to the composition dependence of the free energy of mixing curve, and gives the equilibrium compositions of the two phases obtained when the overall composition is inside the miscibility gap. The spinodal curve, determined by the inflection points of the composition dependence of the free energy of mixing curve, separates unstable and metastable regions within the miscibility gap. 

This section describes how to use Hand s rule to represent binodal curves and tie lines. The surfactant-oil-water phase behavior can be represented as a function of effective salinity after the binodal curves and tie lines are described. Binodal curves and tie lines can be described by Hand s rule (Hand, 1939), which is based on the empirical observation that equilibrium phase concentration ratios are straight lines on a log-log scale. Figures 7.15a and 7.15b show the ternary diagram for a type II(-) environment with equilibrium phases numbered 2 and 3 and the corresponding Hand plot, respectively. The line segments AP and PB represent the binodal curve portions for phase 2 and phase 3, respectively, and the curve CP represents the tie line (distribntion cnrve) of the indicated components between the two phases. Cy is the concentration (volnme fraction) of component i in phase) (i or j = 1, 2, or 3), and 1, 2, and 3 represent water, oil, and microemulsion, respectively. As the salinity is increased, the type of microemulsion is changed from type II(-) to type III to type II(-i-). C, represents the total amount of composition i. 

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