Phase separation binodal line

The interception of these two curves is known as Berghmans point (BP) and defined as the point where the liquid-liquid phase separation binodal line is intercepted by the vitrification curve (9,10) see Figure 4.4. When the vitrification curve intercepts the binodal line, the development of ordinary tie lines is inhibited because molecular diffusion in the glassy state is far slower than in the liquid state see Figure 4.4. 

At concentrations above ( ), a phase separation would lead to the formation of particles dispersed in a liquid matrix. The composition of such particles should be given by the binodal line. Thus such particles will still contain enough solvent to undergo a phase separation. Indeed such an internal phase separation can be used to prepare porous polymeric particles with potential for application as chromatography beads [48]. 

If hexane is used as the low molecular weight liquid, the desired phase separation is observed when precursor mixtures containing 6-15 wt % hexane are cured isothermally at 40 °C. Further discussion of the phase separation behavior requires more detailed consideration of the schematic phase diagram, as presented in Fig. 17, which resembles the real phase diagram shown in Fig. 13. Experimentally it is found, that no phase separation occurs with hexane concentrations equal to or lower than 5 wt %. Hence the critical amount for phase separation, (j)p, is given by the intercept of the binodal line and the imaginary value of Hence no phase separation occurs if is reached before the metastable region is entered. 

The growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

Nowadays it is established that confocal microscopy observation can be a more sensitive method to assess die phase state of mixed biopolymer systems than the traditional centrifugation or viscometric methods (Alves et al., 1999, 2001 Vega et al., 2005). Indeed, microscopy can demonstrate that a system may be already phase-separated at compositions well below the apparent binodal line (as determined by these other methods). The report of Alves et al. (2001) demonstrates the relationship between specific compositional points in the phase diagram (Figure 7.1) and the observed microstructure (Figures 7.2 and 7.3) for water + gelatin + locust bean gum (LBG). The white areas in Figures 7.2 and 7.3 corre-... 

Figure 7.10 Effect of the thermodynamic incompatibility of otsi/p-casein + high-methoxy pectin (pH = 7.0, / = 0.01 M) on phase diagram of the mixed solutions and elastic modulus of corresponding casein-stabilized emulsions (40 vol% oil, 2 wt% protein), (a) (O) Binodal line for p-casein + pectin solution with critical point ( ) ( ) binodal line for asi-casein + pectin solution with critical point ( ). (b) Complex shear modulus G (1 Hz) is plotted against the pectin concentration (O) p-casein ( ) o i -casein. Dotted lines indicate the range of pectin concentration for phase separation in the mixed solutions. The pectin was added to the protein solution before emulsion preparation. Data are taken front Semenova et al. (1999a).

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. 

In contrast to the critical temperature Tc, the spinodal temperature Tsp is well below the binodal temperature for off-critical mixtures and can hardly be reached due to prior phase separation. The diffusion coefficients in the upper left part of Fig. 8 have been fitted by (23) with a fixed activation temperature determined from Dj. The binodal points in Fig. 8 mark the boundary of the homogeneous phase at the binodal. The spinodal temperatures Tsp are obtained as a fit parameter for every concentration and together define the (pseudo)spinodal line plotted in the phase diagram in Fig. 7. The Soret coefficient is obtained from (11) and (23) as... 

Phase diagram asymmetry can be evaluated by (i) the ratio of the biopolymer concentrations at a critical point, (ii) the angle made by the tie-lines with the concentration axis of one of the biopolymers and (iii) the length of the segment of a binodal curve between the critical point and the phase separation threshold. Association of macromolecules usually changes both their excluded volume and the affinity for the solvent water. This results in nonparallel tie-lines on the phase diagram. Normally, the tie-lines can be nonparallel since an increase in concentration of biopolymers is usually accompanied by their self-association. Equilibrium between the phases is not achievable when phase separation is accompanied by gelation. 

In a UCST system, when the temperature is reduced to a final value 7/ that is below the critical temperature Tc, a mixture with a concentration 0 not too far from the critical composition phase separate into two phases whose compositions lie on the opposite sides of the binodal envelope line of Fig. 9-1. The dynamics of the separation process of a single phase into these two phases is controlled by Tf, the composition , the rate of the quench dT/dt, the viscous (or viscoelastic) properties of the phases formed, and the interfacial tension F between the two phases. Although a variety of different kinds of behavior can occur, there are two generic types of phase separation, namely, spinodal decomposition (SD) and nucleation and growth (NG). SD occurs when the mixture is quenched into a part of the phase diagram where the mixture is unstable to small variations in composition, leading to immediate growth of phase-separated domains. When the quenched... 

Figure 9-1 Schematic phase diagram of a binary fluid mixture of small molecules. The two-phase region lies under the binodal line, the apex of which defines the critical temperature Tc and critical composition Between the binodal and the spinodal lines, phase separation is by nucleation and growth (NG), while under the spinodal line it is by spinodal decomposition (SD). Within the region of spinodal decomposition, near the compositional symmetry line, there is a region where the morphology is initially bicontinu-ous. Outside of this region, one of the phases is a discontinuous droplet phase. Eventually,...

Fig. 4.9. The binodal line separates the phase diagrams into a single-phase region and a two-phase region.

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. 

The entire region enclosed by the outer dome represents immiscibility. The inner dome is known as the spinodal. The outer dome is known as the binodal. In the region of composition between the binodal and spinodal lines, phase separation occurs by the nucleation and growth mechanism and leads to the formation of dispersed micro-spherical glass particles in the matrix (see also Shelby, 1997). Spinodal decomposition which takes place inside the dotted region is a special type of phase separation. In order to understand this, consider two materials A and B, melts of which... 

FIGURE 6.17 Solubility of a homopolymer according to the Flory-Huggins theory. Variables are the excluded volume parameter ft (or the polymer-solvent interaction parameter y), the net volume fraction of polymer q>, and the polymer-to-solvent molecular volume ratio q. Solid lines denote binodal, the broken line spinodal decomposition. Critical points for decomposition (phase separation) are denoted by . See text. 

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. 

The given task of enriching the raffinate phase from Pq to Pj can often be achieved in a single stage, albeit with a disproportionately large amount of solvent. The straight line Pe-Ie then coincides with a tie line. If the point M then lies on the binodal curve or outside of the miscibility gap, phase separation no longer takes place, and the solvent... 

At the critical point, (0c, Tc), the coexistence curve (or binodal) and spinodal curve meet. It is important to note that the spinodal is never physically reached in thermal equilibrium (except at the critical point itself) since it is precluded by the first-order phase separation described in the (0, T) plane by the coexistence curve. While the coexistence curve separates the single phase and two-phase regions in thermal equilibrium, kinetic effects can allow the existence of metastable states — Le., one can have the system below the binodal curve and still observe only a single phase for a rather long time. In contrast, the region below the spinodal curve is unstable and even small, thermal fluctuations will drive the system toward equilibrium. The spinodal line represents a line of large fluctuations in the concentration 0, since the free energy cost for fluctuations of 0 away from its minimal values, as determined from Eq. (1.72), is proportional to 9 //90 when this quantity is small, the... 

Figure 7.25. Phase diagram of (GVGVP)2si showing the spinodal line in addition to the usual binodal (coexistence) line that we call the T,-divide. Inset shows experimental determination of the spinodal line by extrapolation to the x-axis intercept of data on the temperature dependence of concentration fluctuations obtained at lower temperature. This means that critical data can be obtained for phase separations that would occur at elevated temperatures if denaturation did not occur. Note that spinodal and binodal lines overlap for part of the volume fraction axis. (Reproduced with permission from Manno et al. )...

In Fig. 11.3, we made a comparison between the binodals obtained from dynamic Monte Carlo simulations (data points) and from mean-field statistical thermodynamics (solid lines). First, one can see that even with zero mixing interactions B = 0, due to the contribution of Ep, the binodal curve is still located above the liquid-solid coexistence curve (dashed lines). This result implies that the phase separation of polymer blends occurs prior to the crystallization on cooUng. This is exactly the component-selective crystallizability-driven phase separation, as discussed above. Second, one can see that, far away from the liquid-solid coexistence curves (dashed lines), the simulated binodals (data points) are well consistent... 

Fig. 11.4 Comparison in the theoretical phase diagrams of phase separation and polymer crystallization in polymer solutions, and the kinetic phase diagrams of crystal nucleation. The energy parameters are set as EpIEc =1.0 and BlEc = 0.076 for the labeled Cl curves, EpjE = 1.072 and BjEc = 0.03 for C2, and EpjEc = 1.275 and BjEc = —0.1 for C3. Three solution series share the same melting points (solid lines) at polymer concentration ( = 0.125, but different depths of critical points for phase separation. The dashed lines are binodals, and the dotted lines are spinodals. The data points are the onset temperatures for the uprising of crystallite numbers on isothermal crystallization. The straight lines are drawn to guide the eyes (Zha and Hu2007) (Reprinted with permission)...

---