Binodal line

Both the binodal line, defining the immiscibility gap, and the spinodal line are for a regular solution symmetrical about xA = xB = 0.5. This is shown in Figure 5.7(a), where theoretical predictions of the miscibility gaps in selected semiconductor systems are given [15],... 

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. 

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 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... 

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,...

Figure 9.2 Schematic phase diagram of a polymer/solvent mixture, where y is the Flory chi parameter, and xe = 1/2 is x at the theta temperature. The quantity Xe X along the ordinate is a reduced temperature, and is the polymer volume fraction. CP is the critical point, and BL is the binodal line. SSL and KSL are the static symmetry line and the kinetic symmetry line, respectively. These lines define the phase-inversion boundaries during quenches. In quenches that end at the right of such a line, the polymer-rich phase is the continuous phase, while to the left of the line the solvent-rich phase is the continuous one. SSL applies at long times, after viscoelastic stresses have relaxed, while KSL applies at shorter times before relaxation of viscoelas-...

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. 

Section 4.4, the binodal curve that describes the phase boundary was defined. The highest point on the binodal line is the critical point with critical composition [Eq. (4.57)] ... 

The minimum amount of solvent is the quantity that just fulfills the given task (enrichment from Rq to Rsoii) at infinite plate number (Figure 2.3.4-6). Infinite theoretical plate number occurs when the pole P coincides with the innermost intersection of a tie line. This pole is determined by extending several tie lines. The straight line Po fmin intersects the binodal line at Lemin- The two straight lines Po-fo and Pj-Lj min then intersect at the point Mmin- The minumum feed ratio results from the lever rule as Po/Io = (Po-Mmin)/(Mmin-fo)-... 

In addition to three canonical morphologies of aggregates (S, C, and L), more complex associations of block copolymer molecules could be found in certain regions of the diagram. In particular, a recent theoretical study [24] predicts the existence of branched cylinders in the vicinity of the S-C binodal line. The latter occupy a narrow corridor and coexist with cylindrical and spherical micelles. Branched structures and networks of aggregates formed by diblock copolymer with quenched PE block were also considered in [17]. 

Fig. 18 Diagram of states in Nb, ion coordinates for diblock copolymer solution at different values of pH — p f, corresponding to Ui, =0.5 (a) and aj = 0.1 (b). Other parameters are 7=1, (p = 1, va = 0.4, Na = 50. Dashed and dotted lines correspond to asymptotic expressions for the binodal lines given by (142) (at = 0) and (149)...

Figure 5.3. Phase diagram for several elastic-contractile model proteins, showing an inverted curvature to the binodal or coexistence line (when compared with petroleum-based polymers) that is equivalent to the T,-divide, with the value of T, determined as noted in Figure 5.IB. Solubility is also inverted with insolubility above and solubility below the binodal line, that is, solubility is lost on raising the temperature whereas solubility is achieved by raising the temperature of most petroleum-based polymers in their solvents. Note that addition of a CHj group lowers the T,-divide and removal of the CH2 group raises the T,-divide. For these and the additional reason of increased ordering on increasing the temperature, the phase transitions of elastic-contractile model proteins are called inverse temperature transitions. (The curve for poly[GVGVP] is adapted with permission from Manno et al. and Sciortino et al. ).

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. )...

The common tangent rule above is the thermodynamic condition for the equilibrium between A and B phases. The temperature dependence of the concentrations at A and B states outlines the phase coexistence curve, which is called the binodal line. 

The two-phase equilibrium conditions, or a binodal line, can be found by equating the chemical potential of each component [9,10] ... 

If one of the phases lies in the microphase separated region, its chemical potential must be replaced by that of the corresponding ordered state. The chemical potential of the microphase depends on the ordered structure and its precise form is unknown for the associating polymers at this moment. Therefore, in what follows we show in the phase diagrams the binodal lines calculated on the basis of (5.6), together with the MST boundary and spinodal lines, to examine under what conditions the microphases remain thermodynamically stable. 

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