Three-phase coexistence

The function /[0(r)] has three minima by construction and guarantees three-phase coexistence of the oil-rich phase, water-rich phase, and microemulsion. The minima for oil-rich and water-rich phases are of equal depth, which makes the system symmetric, therefore fi is zero. Varying the parameter /o makes the microemulsion more or less stable with respect to the other two bulk uniform phases. Thus /o is related to the chemical potential of the surfactant. The constant g2 depends on go /o and is chosen in such a way that the correlation function G r) = (0(r)0(O)) decays monotonically in the oil-rich and water-rich phases [12,13]. This is the case when gi > 4y/l +/o - go- Here we take, arbitrarily, gj = 4y l +/o - go + 0.01. 

A phase diagram summarizes the regions of pressure and temperature at which each phase of a substance is most stable. The phase boundaries show the conditions under which two phases can coexist in dynamic equilibrium with each other. Three phases coexist in mutual equilibrium at a triple point. 

Agrawal, R. Mehta, M. Kofke, D. A., Efficient evaluation of three-phase coexistence lines, lnt. J. Thermophys. 1994,15, 1073-1083... 

A two-dimensional illustration of three phases a, ft and % in equilibrium is shown in Figure 6.9. Two phases coexist in equilibrium in planes perpendicular to the lines indicated in the two-dimensional figure and all three phases coexist along a common line also perpendicular to the plane of the drawing. Each of the three two-phase boundaries, which meet at the point of contact, has a characteristic interfacial tension, e.g. ca for the interface, which tends to reduce the area of the... 

The point where the three lines join is called the triple point, because three phases coexist at this single value of p and T. The triple point for water occurs at T = 273.16 K (i.e. at 0.01 °C) and p = 610 Pa (0.006/ °). We will discuss the critical point later. 

The triple point on a phase diagram represents the value of pressure and temperature at which three phases coexist at equilibrium. 

Any given type of matter has a unique combination of pressure and temperature at the intersection of all three states. This pressure-temperature combination is called the triple point At the triple point, all three phases coexist. In the case of good old H2O, going to the triple point would produce boiling ice water. Take a moment to bask in the weirdness. 

Figure 4.2 Simplified phase diagram of a pure organic chemical. Note that the boundary between the solid and liquid phase has been drawn assuming the chemical s melting point (Tm) equals its triple point (T,), the temperature-pressure condition where all three phases coexist.) In reality, Tm is a little higher than T, for some compounds and a little lower for others.

Fig. 6.3 Schematic phase diagram for lamellar PS-PB diblocks in PS homopolymer (volume fraction 0h). where the homopolymer Mv is comparable to that of the PS block (Jeon and Roe 1994). L is a lamellar phase, I, and I2 are disordered phases, M may correspond to microphase-separated copolymer micelles in a homopolymer matrix. Point A is the order-disorder transition.The horizontal lines BCD and EFG are lines where three phases coexist at a fixed temperature and are lines of peritectic points. The lines BE and EH denote the limit of solubility of the PS in the copolymer as a function of temperature.

Fig. 638 Calculated constant %N (=11) phase diagram for a symmetric diblock blended with high-molecular-weight homopolymers (Janert and Schick 1997b). Biphasic regions are unlabelled. Note the large region of three-phase coexistence between the lamellar and the A-rich and B-rich disordered phases, (a) Both homopolymers have 0 = 1.5 (b) ft = 1.0, ft = 1.5.

Fig. 6.41 Calculated constant xN (=11.0) phase diagram for a blend containing equal amounts of two homopolymers and a symmetric diblock, all with equal chain length (Janert and Schick 1997a). The region of three-phase coexistence between ordered lamellar phases is shaded. Extrapolated phase boundaries are shown with dashes.

Note that our algorithm does not make use of the symmetry a — — a which could in principle be used to reduce the complexity of the three-phase coexistence problem to that of ordinary two-phase coexistence. The scenario therefore provides a genuine test of the multiphase capabilities of the algorithm. 

For example for a two-phase (p = 2) water-ice (c = 1) system we have only one degree of freedom (/ = 1). Thus, we can change the temperature of water and still have coexisting ice, but only at one given pressure. This is the melting point (temperature) A (Fig. A.2) that lies on the coexistence line. If we move to point A we are in the water phase (p = 1) and according to (A.35) we now have two degrees of freedom (/ = 2), temperature and pressure. This triple point is where all three phases coexist (p = 3) it is uniquely defined (/ = 0). This temperature is the official zero of the Celsius scale. 

The point O represents a set of conditions under which all three phases coexist in equilibrium and is called the triple point. 

This behaviour has a particular importance for the soil removal process in detergency. During the oil removal from stained fabrics or hard surfaces, ternary systems occur where three phases coexist in equilibrium. As already pointed out above, in this region the interfacial tension is particularly low. Because the interfacial tension is generally the restraining force,... 

Two other special features on the diagram are designated by black dots. The dot at point D, known as the critical point, represents the critical temperature and the critical pressure (the point at which the liquid state no longer exists, regardless of the amount of pressure). The other dot represents the intersection of the three lines, known as the triple point. The triple point represents the temperature and pressure at which all three phases coexist simultaneously. 

Figure 6 Phase diagram of the ternary mixture distearoylphosphatidylcholine (DSPQ/dioleoylphosphatidylcholine (DOPQ/cholesterol at 23° C, showing four regions of two-phase coexistencediquid crystalline and gel (La + Lp), liquid ordered and gel (Lo + Lp), liquid crystalline and liquid ordered (L + Lo), and liquid ordered and crystals of cholesterol monohydrate also one region of three-phase coexistence exists, liquid crystalline, gel and liquid ordered (La + Lp + Lo). (Reproduced from Reference 74 with permission.)...

No further information about the system may be specified and all intensive variables are fixed. Note from Figure that three phases coexist at equilibrium at only one temperature and pressure. 

Predictions of theory for rods with axial ratio x = 100 are shown by the curves in Fig. 8. Here % is plotted as ordinate against the volume fractions Vp and Vp in the coexisting phases the ordinate may alternatively, be regarded as an (inverse) measure of temperature. The narrow biphasic gap is httle affected by the interactions for negative values of %, as was noted above. If, however, % is positive, a critical point emerges at = 0.055. For values of % immediately above this critical limit, the shallow concave curve delineates the loci of coexisting anisotropic phases, these being in addition to the isotropic and nematic phases of lower concentration within the narrow biphasic gap on the left. At x = 0.070 the compositions of two of the phases, one from each of the respective pairs, reach the same value. Three phases coexist at this triple point. 

Fig. 8. Compositions in volume fractions of coexisting phases for rods of axial ratio x = 100 subject to interactions denoted by the parameter x. The binodal for isotropic phases is on the left that for anisotropic phases is on the right. The minimum of the shallow concave branch of the latter binodal is a critical point marking the emergence of two additional anisotropic phases. The cusp marks a triple point where three phases coexist. Calculations carried out according to the 1956 theory see Ref.

Figure 12.25 Phase diagram of didodecyldimethylammonium bromide (DDAB) in water and styrene at 20°C. The phases include an oil-rich isotropic phase L2, lamellar phases, and five distinct cubic strut phases, including the G, D, P, C(P), and an unknown phase C5. Above the cubic phases are regions of two- and three-phase coexistence. (From Strom and Anderson 1992, reprinted with permission from Langmuir 8 691. Copyright 1992, American Chemical Society.)...

If three phases coexist the point is called a triple point, if four, a quadruple point and so on. 

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