Phase diagram binodal line

A convenient way of representing the T—x y phase diagram (Eig. 14b) is by projection onto the composition triangle at the base of the figure. It is understood that the temperature varies from point to point on the projected vapor line and on the projected boiling envelope. The latter looks like an isothermal Hquid—Hquid binodal envelope, but is not. Each tie line across the boiling envelope is associated with a different boiling temperature (Eig. 15). 

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. 

Fig. 9. Phase diagrams of quasi-ternary systems containing two different molecular weight samples a PHIC-toluene with (Ni, N2) = (4.46,0.38) [73] b schizophyllan-water system with (Nt, N2) = (0.930, 0.0765) [75,76]. (O, A) experimental coexisting isotropic phase ( , , ) experimental coexisting anisotropic phase dashed segments, experimental tie lines the shadowed triangular region, the IAA triphasic region thick full curves, theoretical binodals thin full segments, theoretical tie lines...

Phase diagrams of ternary systems containing two semiflexible polymer components 1 and 2 can be calculated from thermodynamic functions given by Eqs. 26-28. The thick solid curves and thin solid segments in Fig. 9 are calculated binodals and tie lines, respectively [17]3. In the calculation, the diameter... 

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).

Figure 7.14 Complete phase diagram of the water + p-lactoglobulin + gum arabic system at 20 °C and pH = 4.2. Features indicated are , tie-lines , binodal points I, one-phase region II, two-phase region. Reproduced from Schmitt et al. (1999) with permission.

Fig. 1. Phase diagram of a polymer solution. The upper curve shows LOST and the lower curve UCST behavior. Solid lines binodal, dotted lines spinodal T. temperature, x polymer concentration...

Fig. 6.31 Results from SCFI calculations for diblock/homopolymer blends (Matsen 1995b). (a) The dimensionless Helmholtz free energy Fu() as a function of homopolymer volume fraction at y X = 12, / = 0.45 and /3 = The dashed line shows the double tangent construction used to locate the binodal points denoted with dots. The dotted line is the free energy of non-interacting bilayers, (b) Phase diagram obtained by repeating this construction over a range of %N. The dots are the binodal points obtained in (a), and the diamond indicates a critical point below which two-phase coexistence does not occur. The disordered homopolymer phase is labelled dis, and the lamellar phase lam.

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... 

Fig. 17 Phase diagram of a PDMS/PEMS (16.4/48.1) blend. The dashed lines are the binodal and the spinodal. The phase contrast micrographs show typical demixing patterns for spinodal decomposition and nucleation and growth in the respective regions. The bullets mark the initial sample positions. See text for details. Figure from [112]. Copyright (2007) by the American Chemical Society...

In Figure 3.13 the precipitation pathway enters the two-phase region of the phase diagram above the critical point at which the binodal and spinodal lines intersect. This is important because it means that precipitation will occur as a liquid droplet in a continuous polymer-rich phase. If dilute casting solutions are used, in which the precipitation pathway enters the two-phase region of the phase diagram below the critical point, precipitation produces polymer gel particles in a continuous liquid phase. The membrane that forms is then weak and powdery. 

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. 

The resultant phase diagram is shown in Fig. 1. The line connecting points at various temperatures at composition satisfying Eq. (7) is the binodal. The line connecting points at composition satisfying Eq. (9) is the spinodal. 

Fig. 6. Phase diagram calculated for the ternary system solvent (l)/rigid rod (2)/random coil (3) with X, = 1 and xj = X3 = 100. Numerals on the tie lines that extend below the lower boundary of the diagram denote vi for the conjugate anisotropic phase the binodal for which lies virtually on the 1 — 2 axis (From Ref.

Figure 5.13 Predicted phase diagrams for physical gels made from low-molecular-weight molecules with junctions of unrestricted functionality 4> is the total volume fraction of polymer, and Tr is here the reduced distance from the theta temperature, Tr = — Q/T. The parameter Aq controls the equilibrium constant among aggregates of various sizes. The outer solid lines are binodals, the inner solid lines are spinodals, and the dashed lines are gelation transitions. CP is a critical solution point, CEP is a critical end point, and TCP is a tricriti-cal point. (Reprinted with permission from Tanaka and Stockmayer, Macromolecules 27 3943. Copyright 1994 American Chemical Society.)...

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,...

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