Coexistence lines applications

Meijer, E. J. Azhar, F. El, Novel procedure to determine coexistence lines by computer simulation, application to hard-core Yukawa model for charge-stabilized colloids, J. Chem. Phys. 1997,106, 4678-4683... 

Once a state point of coexistence is established, additional state points can be determined expeditiously through application of the Gibbs-Duhem integration method [48,85,86]. In this approach a differential equation for the coexistence line is used to guide the establishment of state points away from the known coexistence point. The most well known such formula is the Clapeyron equation [41]... 

The integration path is largely set by the definition of the problem one knows beforehand the plane in which the coexistence line is desired. However application of simple transformations to the field variables can be beneficial if they reshape the coexistence line into a simpler form. The advantages can be improved accuracy, precision, and stability of the integration. A very familiar example is the conversion from pressure P to ln(P) in the characterization of vapor-liquid coexistence. The Clapeyron equation in the latter instance is... 

Application of the GDI method to the coexistence lines requires establishment of a coexistence datum on each. A point on the vapor-liquid line can be determined by a GE simulation. At high temperature the model behaves as a system of hard spheres, and the liquid-solid coexistence line approaches the fluid-solid transition for hard spheres, which is known [76,77]. Integration of liquid-solid coexistence from the hard-sphere transition proceeds much as described in Section III.C.l for the Lennard-Jones example. The limiting behavior (fi - 0) finds that /IP is well behaved and smoothly approaches the hard-sphere value [76,77] of 11.686 at f = 0 (unlike the LJ case, we need not work with j81/2). Thus the appropriate governing equation for the GDI procedure is... 

The Gibbs-Duhem integration method excels in calculations of solid-fluid coexistence [48,49], for which other methods described in this chapter are not applicable. An extension of the method that assumes that the initial free energy difference between the two phases is known in advance, rather than requiring it to be zero, has been proposed by Meijer and El Azhar [51]. The procedure has been used in [51] to determine the coexistence lines of a hard-core Yukawa model for charge-stabilized colloids. 

Other methods to construct a crystal-liquid interface are possible. One example applies to systems under triple-point (three-phase) conditions where a block of crystal is sandwiched in the z direction between regions of empty space. Keeping one half of the crystal region fixed and heating the other so that it melts, a three-phase system can be constructed. If this process is done carefully enough the system should come to equilibrium so that the densities of the various phases adjust to the proper triple-point values. One advantage of this procedure is that the coexistence conditions need not be determined beforehand, but are by-products of the calculation. The method is extremely limited, however, since only one point along the crystal-liquid phase coexistence line can be studied. Also, the method is not applicable to purely repulsive potentials, such as hard spheres or inverse power interactions, which have only one fluid phase. 

Figure A2.5.28. The coexistence curve and the heat capacity of the binary mixture 3-methylpentane + nitroethane. The circles are the experimental points, and the lines are calculated from the two-tenn crossover model. Reproduced from [28], 2000 Supercritical Fluids—Fundamentals and Applications ed E Kiran, P G Debenedetti and C J Peters (Dordrecht Kluwer) Anisimov M A and Sengers J V Critical and crossover phenomena in fluids and fluid mixtures, p 16, figure 3, by kind pemiission from Kluwer Academic Publishers.

It has been pointed out however that further innovative steps are needed before single nano-tubes devices with adequate reproducibility can be made. A paper by Zhang et al. (2006) indicates parallel research lines intended to develop the various complementary aspects of nano-sciences and their applications. They observed that metallic and semi-conducting carbon nano-tubes generally coexist in as grown... 

Typically, the liquidus lines of a binary system curve down and intersect with the solidus line at the eutectic point, where a liquid coexists with the solid phases of both components. In this sense, the mixture of two solvents should have an expanded liquid range with a lower melting temperature than that of either solvent individually. As Figure 4 shows, the most popular solvent combination used for lithium ion technology, LiPFe/EC/DMC, has liquidus lines below the mp of either EC or DMC, and the eutectic point lies at —7.6 °C with molar fractions of - 0.30 EC and "-"0.70 DMC. This composition corresponds to volume fractions of 0.24 EC and 0.76 DMC or weight fractions of 0.28 EC and 0.71 DMC. Due to the high mp of both EC (36 X) and DMC (4.6 X), this low-temperature limit is rather high and needs improvement if applications in cold environments are to be considered. 

A supercritical fluid is defined as one that is beyond its critical point and thus cannot be liquefied by a change in pressure. On a P-T plot, the supercritical region lies at the end of the vapor-liquid coexistence curve and is bounded by the lines P = Pc and T = Tc. In most applications, however, the compressed liquid region above the P = Pc line and to the left of the T = Tc line is also useful. Table 1 gives critical parameters for commonly used supercritical fluids. 

On the border lines in Figure 2.5, two phases can coexist. These two-phase regions have an outstanding importance in technical applications, as they are the prerequisite of most of the thermal separation processes. The following considerations refer to the vapor-liquid equilibrium but also apply to the other phase equilibrium lines of pure substances. 

One of the representative applications of this SO2 sensor is the measurement of volcanic emissions. In this case, hydrogen sulfide gas will coexist with SO2, and its effect upon the sensor has been tested. The result is shown in fig. 55 (Adachi and Imanaka 1991). With H2S gas coexistence lower than approximately lOOOppm, the EMF value deviates to a lower value in comparison to the calculated (dashed) line. In this measurement, H2S is mixed with air with a fixed SO2 gas content. Therefore, H2S can be expected to be oxidized by the oxygen in the test atmosphere. The relation between the EMF output and the SO2 content is replotted in fig. 56 (Adachi and Imanaka 1991) as EMF vs. the total amount of SO2. The total amount corresponds to the sum of SO2 gas initially introduced and the SO2 gas produced by the oxidation, assuming that the introduced H2S gas is completely oxidized. The... 

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