Upper critical point

To determine the upper critical point corresponding to the transition from laminar to turbulent jet flow, Grant and Middleman1747 derived the following empirical correlation ... 

It has been proposed to define a reduced temperature Tr for a solution of a single electrolyte as the ratio of kgT to the work required to separate a contact +- ion pair, and the reduced density pr as the fraction of the space occupied by the ions. (M+ ) The principal feature on the Tr,pr corresponding states diagram is a coexistence curve for two phases, with an upper critical point as for the liquid-vapor equilibrium of a simple fluid, but with a markedly lower reduced temperature at the critical point than for a simple fluid (with the corresponding definition of the reduced temperature, i.e. the ratio of kjjT to the work required to separate a van der Waals pair.) In the case of a plasma, an ionic fluid without a solvent, the coexistence curve is for the liquid-vapor equilibrium, while for solutions it corresponds to two solution phases of different concentrations in equilibrium. Some non-aqueous solutions are known which do unmix to form two liquid phases of slightly different concentrations. While no examples in aqueous solution are known, the corresponding... 

Binary-Liquid Option. As an alternative to this study of critical behavior in a pure fluid, one can use quite a similar technique to investigate the coexistence curve and critical point in a binary-liquid mixture. Many mixtures of organic liquids (call them A and B) exhibit an upper critical point, which is also called a consolute point. In this case, the system exists as a homogeneous one-phase solution for all compositions if Tis greater than... 

If this is an upper critical point, will be positive because of (18.84) and kg may or may not be equal to zero, although it must always satisfy... 

The upper critical point can never be exceeded. We may abandon this physically unrealistic singularity, and thus... 

As predicted by the Flory-Huggins theory, such a system shows a lower miscibility gap characterized by an upper critical point, at temperature Ta, which depends on both the oil and the amphiphile structure (Figure 3.11a). The critical composition is usually not far from the pure oil side. 

The phase diagram of a nonionic amphiphile-water binary system is more complicated (see Figure 3.12). A classic upper critical point exists, but it is usually located below 0°C. At higher temperatures most nonionic amphiphiles show a miscibility gap, which is actually a closed loop with an upper as well as a lower critical point. The lower critical point CPp is often referred to as the cloud point temperature. The upper critical point often lies above the boiling temperature of the mixture (at 0.1 MPa). The position and the shape of the loop depend on... 

Figure 1.1 Schematic view of the phase behaviour of the three binary systems water (A) oiI (B), oil (B)-non-ionic surfactant (C), water (A)-non-ionic surfactant (C) presented as an unfolded phase prism [6]. The most important features are the upper critical point cpc, of the B-C miscibility gap and the lower critical point cpp of the binary A-C diagram. Thus, at low temperatures water is a good solvent for the non-ionic surfactant, whereas at high temperatures the surfactant becomes increasingly soluble in the oil. The thick lines represent the phase boundaries, while the thin lines represent the tie lines.

The nozzle efficiency at pressure ratios above the calculated upper critical point is overestimated for nozzles with 2.34 and 3.19 nominal divergence, but is reproduced reasonably well for the nozzle of 1.38 nominal divergence. However, the calculated upper critical pressure ratio at 0.69 is 0.09 lower than the measured value for this last nozzle. Data does not exist for the upper critical pressure for the nozzle of 7.93 nominal divergence ratio, but the match between... 

The single upper critical point corresponding to particular values of temperature and system composition, is often present in two-component systems, such as tricosane - oxyquinoline. In this case one can approach the critical state from the side of two-phase system by changing the temperature in the system that has composition close to the critical one. The difference between the critical temperature and the temperature of experiment A T= Tc-- T, or the difference in the phase compositions, Ac, may be chosen as parameters that characterize the deviation of system from the critical state. 

Category VI phase behavior, shown in Fig. 10.3-3/, occurs with components that are so dissimilar that component 2 has a melting or triple point (Mj) that is well above the critical temperature of component 1. In this case there are two regions of solid-liquid-vapor equilibrium (SLVE). One starts at the triple point of pure component 2 (M ) and intersects the liquid-vapor critical line at the upper critical end point U. The second solid-liquid-vapor critical line starts below the melting point Mt and intersects the vapor-liquid critical line starting at component 1 at the lower critical end point L. Between the lower and upper critical points only solid-vapor (or solid-fluid) equilibrium exists. 

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showing stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (inner curve) meet at the (upper) critical point. A critical quench corresponds to a sudden decrease in temperature along a constant order parameter (concentration) path passing through the critical point. Other constant order parameter paths ending within the coexistence curve are called ofif-critical quenches.

The three-component model immediately makes clear that this behavior is generic [15,74]. What is unusual in, and specific to, the nonionic systems is that the triple line is traversed by varying the temperature. This is a consequence of the lower critical point that exists in these systems in addition to the usual upper critical point. That there is a lower critical point is related to hydrogen bonding, and a realistic phase diagram can be calculated only by taking this into account [75] (see Fig. 1). 

In this chapter, we describe the development of typical thermodynamic studies that have been made for quantitative predictions of phase equilibria in binary rind paucidispeise quasi-binary polymer solutions. In so doing, we are concerned only with systems which phase-separate when cooled, i.e., ones which have an upper critical point. 

Fig. 2.2-1 Miscibility gaps of several binary mixtures with upper critical point Qhs), lower critical point (center), upper plus lower critical point (rhs)...

Figure 9.2a Pressure as a function of the total mass fraction for methylcyclohexane (A) with polydisperse poly(ethenylbenzene) (B) (M = 16 500 g mol U= ) illustrating the Lower Critical Point and Upper Critical Point...

The formation of IL/O microemulsions in mixtures of [bmim][BFJ (IL) and cyclohexane, stabilized by the nonionic surfactant, TX-lOO has been proved [30]. Three-component mixtures could form IL/O microemulsions of well-defined droplet size determined by fixing the water content (mole ratio of IL to TX-lOO) [30,48,49]. An upper critical point (T) was observed in the mixture [([bmim][BFJ/ TX-lOO)-I-cyclohexane] with fixed water content (mole ratio of [bmim][BFJ to TX-lOO) [50]. The mixture separated into two microemulsion phases of different composition but with the same composition below as occurred in other systems [48]. The microemulsion system, [bmim][BF ]/TX-100 +cyclohexane, could be regarded as a pseudobinary mixture of [bmim][BF ]/TX-100 IL droplets dispersed in the cyclohexane continuous phase. Therefore, the phase behavior could be depicted in a two-dimensional diagram with concentration of droplets along the abscissa and temperature along the ordinate. A coexistence curve of temperature (T) against a concentration variable, such as volume fraction ( ), could then be drawn in the same way as it was done for pseudobinary mixtures in AOT/water/decane micro-emulsions [48]. 

It is common to encounter a liquid-liquid miscibility gap having a lower critical temperature in nonionic surfactants. The phase boundary responsible for such behavior is termed a lower consolute boundary [62] Fig. 6 displays an example. A related kind of phase behavior that is inverted with respect to temperature has also been found, mostly among zwitterionic surfactants (Fig. 7). This boundary has an upper critical point and is termed an upper consolute boundary. Phase separation occurs on heating with the lower con-... 

First of all it can be shown that a lower critical point cannot be obtained by the average potential model using only second order terms. This conclusion had already been reached from the cell model (R6wun-SON [1952], Beixemans [1953]. Therefore we shall only study here the upper critical point occurihg in the case of dispersion forces (other cases can be readily studied in the same way). The case of dispersion forces is however somewhat simpler because the excess free energy is a parabolic function in xaXb when limited to second order terms in d and p. Hence the critical mole fraction is equal to 0.5. Expressions for Te are readily obtained from (1.8.3). We find ... 

This shows clearly that an upper critical point may be due either to differences in the st s or to differences in Uj or to both effects. Now from table 9.5.1 it can be seen that the factors A are nearly the same for all three models (I), (II), (HI). This is not the case for B we have approximately... 

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