Double critical point

Fig. 5.12 Two different 3-D representations of the phase diagram of 3-methylpyridine plus wa-ter(H/D). (a) T-P-x(3-MP) for three different H2O/D2O concentration ratios. The inner ellipse (light gray) and corresponding critical curves hold for (0 < W(D20)/wt% < 17). Intermediate ellipses stand for (17(D20)/wt% < 21), and the outer ellipses hold for (21(D20)/wt% < 100. There are four types of critical lines, and all extrema on these lines correspond to double critical points, (b) Phase diagram at approximately constant critical concentration 3-MP (x 0.08) showing the evolution of the diagram as the deuterium content of the solvent varies. The white line is the locus of temperature double critical points whose extrema (+) corresponds to the quadruple critical point. Note both diagrams include portions at negative pressure (Visak, Z. P., Rebelo, L. P. N. and Szydlowski, J. J. Phys. Chem. B. 107, 9837 (2003))...

Very large isotope effects like those shown in Fig. 5.12 seem to be limited to the hypercritical regions of phase diagrams, i.e. not too far from thermodynamic divergences of the type (dP/dT)c = 00 or (dT/dP)c = 00 (i.e. pressure-double critical points (p-DCP) or temperature-double critical points (T-DCP), respectively). 

Referring again to Figure 14.14, the isotherms at temperatures T3 and T4 are of the typical (gas + liquid) type, but at T2, a temperature below u, two critical points occur, one at f and the other at h. The one at f is a typical (liquid + liquid) critical point while the one at h is better characterized as a (gas + liquid) critical point. In most systems with type III behavior, the critical locus bh occurs over a narrow temperature range, and the double critical points occur only over this small range of temperature. 

Here, x and x" are the mole fractions in the coexisting phases Tc. pc and Xc are the temperature, the pressure and the mole fraction at the critical point PP and Pt are the critical indices Bp and Bt are the amplitudes of the equilibrium curves. For the T,x binodal at atmospheric pressure the exponent pp has a universal value close to 1/3. An analysis made for several isobars has shown that within the experimental error the quantity Pp retains its value along the line of critical points including the double critical point. The critical exponent pT for isotherms does not considerably change its value at pressures up to 70-80 MPa, with px=pp. In the vicinity of the DCP, one can observe an anomalous increase of pT. The behaviour of the exponents Pp and pT along the line of critical points at pressures from atmospheric to 200 MPa is shown in figure 3. 

Figure 1. Betapicoline/D20 mixture after Cox (1952)Ti=38.5°C, Tjj=117°C. When AT=TirTi=0 the double critical point (DPC) is reached.

Figure 1. Demixing diagrams for PS in 0-solvents and poor solvents (schematic). The variable X might be pressure, Mw D/H ratio in solvent or solute, etc. See text for a further discussion, (a, top left) PS in a 0-solvent (monodisperse approximation). For X=Mw - the X=0 intercepts of the upper and lower heavy lines drawn through the minima or maxima in the demixing curves define 0Land0u, respectively, (b, top right) PS in a poor solvent (monodisperse approximation). The heavy dot at thecenterlocates the hypercritical (homogeneous double critical) point. (c, bottom right) The effect of polydispersity. BIN=binoda] curve, CP=cloud point curve, SP=spinodal, SHDW=shadow curve. See text Modified from ref. 6 and used with permission.

Figure 4. Critical Demixing isopleths for PS/methylcyclohexane/n-heptane solutions. Parts b and c show the diagrams in the vicinity of the hypercritical (homogeneous double critical) points. Modified from ref 4 and used with permission.

IM2 Imre, A.R., Melnichenko, G., and van Hook, W.A., A polymer-solvent system with two homogeneous double critical points Polystyrene (PS)/(n-heptane + methylcyclohexane), J. Polym. Sci. PartB Polym. Phys., 37, 2747, 1999. 

A double critical point (heterogeneous double plait point) obeys, in addition to Eqs. (115) and (116), the condition... 

The double critical point is formed if stable and unstable critical points merge. 

Fig. 6. Phase diagram in the case = 1.4364 double critical point (o) (all other legends...

CPl and CP2 diminishes and finally vanishes for Ve = 1.4364, Fig. 6. Then, the stable critical point CPl and the unstable critical point CP2 merge, forming a double critical point. The cloud-point curves 1 and 2 and the shadow curves I and 2 reduce to a double critical point. If increases further, there appear only the critical point CP3 as well as the stable cloud-point curve 3 and the shadow curve 3. 

Figure 3.94 demonstratc s how two lines of the double critical points aP and 0 f can merge into one point, namely, the tricritical point of the asymmetrical type. 

At Xi i I he double cusp point coinciding with the double critical point (Figure 3.97, the point A), appears inside the initial binodal at W2a- A new binodal then arises from this point. An increase in x, within x, < X, < Xio causes a splitting of the double cusp point into two cusp points with two arcs of the new binodal between them (Figure 3.98a) (cusp- -also means the moon s horn as two arcs of the binodal look). 

The double critical point also splits into two single ones, namely, the metastable one M (to the left from the point A in Figure 3.97), which moves toward the external arc of the binodal and becomes stable, and the unstable one A, which moves towards the internal arc of the binodal (Figure 3.97, to the right from the point A to the point B). 

Every ternary system hzw a line of simple critical points, which is defined in the whole concentration range of the polymer homologues Pi tind Hence, pi, p, and wj may vary (see I able 3.6). In ternciry systems, where three-phase separation is possible, double critical points with a fixed composition W2 appccir as the extrema of the lines of simple critical points while pi and p2 remain variables. At the triple critical point of a ternary system, a certain pz/pi ratio must be preserved while pi remains a variable (see Table 3.6). 

For a ternary system, the diagram is one-dimensional with a triple criticed (tricritical) point which separates the three-phase region from the two-phzise ones with double criticed points (Figure 3.100a see also Figures 3.93 and 3.94). The location of the tricritical point r2,tc = r (pi) depends on pi, varying from r 15.645 for p = 1 to r 9.899 for pi oo (Tompa, 1949 Sole et al., 1984). 

The diagram of the critical points of a quaternary. system Ls. shown in Figure 3.100b. The existence condition of a line of double critical points is pa/pi rjrs > r (pi), i.e. a hyperbola in the rj and rs coordinates (the lower dashed line, Figure 3.100b). 

The pentary critical point results from the junction of four double critical points or of two pairs of cusp points with the critical point (Figure 3.101d). Such a point on the CPC is indistinguishable from the tricritical one in a ternary system. In the general case, the m multiple critical point gets into the CPC when the critical point merges with the (m — l)-multiple cusp point. The even-multiple critical points are always away from the CPC stable branch while an odd-multiple critical point may well appear on the CPC stable branch. 

Rg. 6.14 Molecular-weight dependence of the LCST and UCST (Shultz plot). The critical temperatures are plotted against the reciprocal of DP. The hyper critical point (HCP) and double critical point (DCP) are indicated by arrows. (Reprinted with permission from Ref. [55].)... 

The immiscibility regions of type b and d spreading from the binary subsystems are terminated by one nonvariant point in ternary systems (the double critical endpoint HN (Li = L2-G) and the tricritical point RN (Li = L2 = G), respectively). Disappearance of the immiscibility region of type c takes place only after transformation (through the tricritical or double critical points) into immiscibility region of type b or d. 

Figure 9.2d Pressure as a function of the total mass fraction cob for methylcyclohexane (A) with polydisperse poly(ethenylbenzene) (B) (M = 16500 g mol U= ) illustrating the formation of a double critical point at a a temperature...

From eq 10.59, we see that the relationship of the mole fraction x with the scaling densities tpi and (pi is independent of either or b. Hence, the theoretical expressions for the temperature dependence of the mole fraction along the two phase boundaries, developed in the previous section, remain equally valid for weakly compressible liquid mixtures. This is the physical reason why eq 10.65 yields an excellent representation of the behaviour of the mole fraction x for liquid-liquid equilibria. As an example we show in Figure 10.5 closed solubility loops in 2-butanol + waterAs we explained earlier, closed solubility loops can be represented by the expansion of eq 10.65 provided that Ar is replaced by IATulI in accordance with eq 10.66. The closed solubility loops collapse into a double critical point at P = 85.6 MPa and T = 340 K. The implications of the theory for the behaviour near such a double critical point have been elucidated by Wang et Both near the upper critical solution temperature Ju and near the lower critical solution temperature Tl, Axcxc varies as A7 il in accordance with eq 10.65a. Near the double critical point both 7 j and 7 approach the temperature I d of the double critical point. Hence, near the double critical point... 

The minimum a value required for the occurrence of a double critical point is considerably higher for v = 0.4 than for v = 0.5. A more detailed mathematical analysis [53] of (63) yields a border line for the combination of parameters, which separates the normal firom anomalous behavior. For ordinary systems, the combination of a and v values required to produce multiple critical points has so far not been observed. However, for water/PVME systems, such data may well be realistic... 

Fig. 15 Modeling of polymer solutions with anomalous phase behavior. Example of plots [53] of a as a function of ip a according to (63) at the constant v values indicated in the graph for 1. Solid curves v = 0.5, dashed curves v = 0.4. The horizontal lines mark the minimum value that a must exceed for a given v to generate an additional critical point. Squares mark anomalous double critical points...

C. M. Sorensen, G. A. Larsen, Light scattering and viscosity studies of a ternary mixture with a double critical point, J. Chem. Phys. 83 (1985) 1835-1842. 

Fig. 23 Phase diagram in the temperature-diblock copolymer plane for the (dPB PS) mixture below the Lifshitz line separating blend like from diblock-like phase behavior. The full dots and the solid line represent the critical points of a two-phase region. The hatched area indicates a crossover from Ising to isotropic Lifshitz critical behavior, and a double critical point DCP is at 7% diblock concentration. The Lifshitz line separates at high and low temperatures the disordered phases and droplet and bicontinuous microemulsion phases ( xE). Its non-monotonic shape near the DCP is caused by the strong thermal fluctuations...

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