Binodal points

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. 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.

Fig. 8 Diffusion (D) and thermal diffusion (Dj) coefficient of PDMS/PEMS (16.4/48.1) left) and Soret coefficient right) for different PDMS mass fractions given in the legends. Binodal points mark the intersection with the binodal. The dashed line segments are extrapolations into the two-phase regime. Figures from [100], Copyright (2007) by The American Physical Society...

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... 

The whole composition range, 02 = 0 1, is subdivided into five sections by the binodal compositions 0 and 0f anc the spinodal compositions 0 and 0. The binodal points A and B are defined as points of common tangent to the free energy density curve. For compositions between 0 and 0 and between 0 and 1, the free energy of the mixture increases whenever it separates into two phases of... 

There is a possible point of minor confusion here which we should clarify. The spinodes can be found from the second derivative (15.4) of the free energy curve. However, the solvus points a and b do not lie exactly on the minima of this curve, and cannot be found by setting the first derivatives equal to zero. This is apparent from the geometry of the tangent abed in Figure 15.3a. For most practical applications the differences involved here are likely to be small. A plot of (instead of G) will have the required value of zero for the same first derivatives at the binodal points (since G is zero for both pure components, and the slope of the line joining the two end-members is zero). 

Binodal points represent the points of contact of a common tangent to A vs. V at constant temperature and composition when a region of negative curvature exists between two regions of positive curvature. The locus of binodal points, known as the binodal curve or two-phase envelope, represents the experimentally observed phase boundary under normal conditions. For example, saturated liquid and saturated vapor represent states on the binodal curve. The binodal region exists between the binodal and spinodal curves, where p/ V)T,aa jv < 0. 

Binodal points represent the points of contact of a common tangent to Agmixing vs. at constant T and p when a region of negative curvature... 

When a region of negative curvature exists between two regions of positive curvature on the graph of Agmixing vs. y2, and concentration-dependent miscibility prevails, the points of contact of the common tangent (i.e., binodal points) identify two different phases, a and that are in thermodynamic equilibrium. Since chemical stability is analyzed at constant T and p, these coexisting phases exhibit the same temperature (i.e., Ta = Tp) and the same pressure (i.e., p = Pp), which are requirements for thermal and mechanical equilibrium. The requirement of chemical equilibrium for a two-phase mixture is... 

Four spinodals, four thermodynamically allowed binodal points, double phase separation (i.e., a/fi and y/5), and one minimum in Agmixing... 

Component Tanperature ( C) Binodal Points, yi Spinodal Points, yi ... 

Fig. 12.1 (a) AG ix plotted against composition for a partially miscible polymer. The points B are the binodal points, where a common tangent touches the two sections of the curve near the minima. The points S are the spinodal points, where the curve has points of inflection. M is the maximum between the spinodal points, (b) See the text. 

A particular blend of polymers A and B has binodal points Bi and B2 at compositions containing volume fractions 0.20 and 0.85 of B, respectively. If a blend containing equal volumes of A and B is made, what fraction / of the total volume will be in phases corresponding to point Bi in equilibrium ... 

Triangles denote the compositions of SAXS samples measured here, whereas circles denote binodal points obtained by polarization microscopy. 

The tie lines are defined as the lines in the two-phase region (7 = 2) on which the system intensive parameters are constant. It follows from Elquation 8 that tie lines connect the binodal points corresponding to the coexisting phases. It should be noted that any tie line is unambiguously defined by 1/ intensive parameters, according to the phase rule (Equation 112). 

Fig. 1.1.6 CPC (binodal points) of PMAA/MAA/ water system at 80 and 90° C (Reproduced with permission from Shi, 1997)...

Allowing for the fact that the 50/50 (w/w) composition is usually not a critical concentration complicates the analysis.The 50/50 cloud point may be considered to represent a binodal point (provided the samples have very narrow molar-mass distributions). The composition of the phase coexisting with that of the measured binodal point is unknown but we have an extra equation as well, two-phase equilibria being characterized by conditions (3). 

A direct link between theoretical and experimental work on depletion-induced phase separation of a colloidal dispersion due to non-adsorbing polymers was made by De Hek and Vrij [56, 109]. They mixed sterically stabilized silica dispersions with polystyrene in cyclohexane and measured the limiting polymer concentration (phase separation threshold). Commonly, one uses the binodal or spinodal as experimental phase boundary. A binodal denotes the condition (compositions, temperature) at which two or more distinct phases coexist, see Chap. 3. A tie-line connects two binodal points. A spinodal corresponds to the boundary of absolute instability of a system to decomposition. At or beyond the spinodal boundary infinitesimally small fluctuations in composition will lead to phase separation. De Hek and Vrij [56] used the pair potential (1.21) to estimate the stability of colloidal spheres in a polymer solution by calculating the second osmotic virial coefhcient B2 ... 

When two compositions can be connected through the common tangent (the thin straight lines in the figures connecting these compositions), binodal points are found the intercepts of the extrapolated lines correspond to the total pressure —P. Scenario (i) in Fig. 3.10 corresponds to gas-liquid coexistence. In situation (ii) Q(( ) are given for both the fluid state and for the solid state and the common tangent shows the compositions where fluid and solid coexist. A combination of (i)... 

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