Phases and phase diagrams

Phase equilibrium deals with phases, so we need a working definition of a phase. A phase is a mass of matter, not necessarily continuous, in which there are no sharp discontinuities of any physical properties over short distances. An equilibrium phase is one that (in the absence of significant gravitational, electrostatic, or magnetic effects) has a completely uniform composition throughout. In this book we will deal almost exclusively with equilibrium phases. 

All gases form one phase. All gases are miscible, so that there can be only one gas phase present in any equilibrium system at any time. 

Homogeneous solids are single phases, for example diamond, pure metals, pure mineral crystals. Some apparently simple solids are not single phases, such as cast iron, steel, wood, bacon, grass. One can observe the grain in wood, showing that it consists of layers of at least two different compositions, and can similarly observe the fat and meat 

FIGURE 1.6 Appearance and elevation-density plot for three liquid phases at equilibrium. 

FIGURE 1.7 Copper sulfate crystals dissolving slowly in an unstirred graduate cylinder. 

A phase is a part of a system that is chemically uniform and has a boundary around it. Phases can be solids, liquids and gases, and, on passing from one phase to another, it is necessary to cross a phase boundary. Liquid water, water vapour and ice are the three phases found in the water system. In a mixture of water and ice it is necessary to pass a boundary on going from one phase, say ice, to the other, water. 

The phases that are found on a phase diagram are made up of various combinations of components. Components are simply the chemical substances sufficient for this purpose. A component can be an element, such as carbon, or a compound, such as sodium chloride. The exact components chosen to display phase relations are the simplest that allow aU phases to be described. 

There is a clear difference between a single-phase system and a multiphase system. Whereas we can specify the pressure and the temperature of pure water vapor, which means there are two degrees of freedom, there is only one degree of freedom for liquid water and ice to remain in equilibrium. An arbitrary change in pressure must be accompanied by a specific change in temperature and volume. This means the number of degrees of freedom is reduced by the number of phases, p, that are in equilibrium. Therefore, / = 3 - p for a pure substance. 

Since there is a limit on the number of degrees of freedom of a system with different phases existing in equilibrium, it is possible to use a two-dimensional plot (e.g., an x-y plot) to show the possible behavior. In the beginning of this text, P-V isotherms were drawn as two-dimensional plots of the allowed behavior of a gas. We could instead have drawn V-T isobars to show the relation between volume and temperature at specific pressures. To understand and follow phase behavior, it is a plot of pressure versus temperature that is normally the most useful. This is because temperature and pressure, not volume and entropy, are most easily varied in the laboratory. Volume is not independent when the variables of interest are pressure and temperature, and so we could consider drawing constant-volume curves on a P-T plot. More interesting to follow is the pressure and temperature at which phase equilibrium is maintained, regardless of the volume. 

Normal phase transition temperatures (i.e., freezing point, boiling point) are those temperatures at which two phases coexist with the pressure at Ibar. A horizontal line drawn on a phase diagram corresponding to a pressure of 1 bar crosses the phase equilibrium curves at the normal phase transition temperatures. 

The phase diagram of water drawn on two different scales covering different temperature-pressure ranges. The triple point is shown on the expanded scale on the bottom drawing. 

P-V isotherms (T7 Tj Tj T,) of a hypothetical pure substance that can exist as a gas and as a liquid. The isotherms for the gas phase are similar to those of an ideal gas. Deviations from ideal behavior become apparent at lower temperatures. The one isotherm (at Tj) shown for the liquid state has the typical pressure-volume behavior of a liquid, a small change in volume corresponding to a large change in pressure. To change smoothly from one type of behavior to the other, there must be a P-V isotherm (Tj) with an inflection (critical) point. 

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The reeiproeal method of SCMFT requires a priori knowledge of the structure of the phases. The recently introduced real-space methods have the potential to predict the phases without prior assumptions about the structures [17-19]. It is feasible that a combination of the real-space and reciprocal-space methods will lead to a numerical platform that is capable of predicting the phases and phase diagrams for complex block copolymers. In this scheme the real-space method can be used to earry out a combinatorial search for the possible candidate structures, and the reeiprocal-space method can then be used to obtain accurate free energies of these candidate structures. 

Table 1 gives the measured data, estimates of the true values corresponding to the measurements, and deviations of the measured values from model predictions. Figure 1 shows the phase diagram corresponding to these parameters, together with the measured data. 

So far we have considered only a single component. However, reservoir fluids contain a mixture of hundreds of components, which adds to the complexity of the phase behaviour. Now consider the impact of adding one component to the ethane, say n-heptane (C7H.,g). We are now discussing a binary (two component) mixture, and will concentrate on the pressure-temperature phase diagram. 

Figure 5.20 Pressure-temperature phase diagram mixture of ethane and n-heptane...

Figure 5.21 helps to explain how the phase diagrams of the main types of reservoir fluid are used to predict fluid behaviour during production and how this influences field development planning. It should be noted that there are no values on the axes, since in fact the scales will vary for each fluid type. Figure 5.21 shows the relative positions of the phase envelopes for each fluid type. 

A volatile oil contains a relatively large fraction of lighter and intermediate oomponents which vaporise easily. With a small drop in pressure below the bubble point, the relative amount of liquid to gas in the two-phase mixture drops rapidly, as shown in the phase diagram by the wide spacing of the iso-vol lines. At reservoir pressures below the bubble point, gas is released In the reservoir, and Is known as solution gas, since above the bubble point this gas was contained in solution. Some of this liberated gas will flow towards the producing wells, while some will remain in the reservoir and migrate towards the crest of the structure to form a secondary gas cap. 

Black oils are a common category of reservoir fluids, and are similar to volatile oils in behaviour, except that they contain a lower fraction of volatile components and therefore require a much larger pressure drop below the bubble point before significant volumes of gas are released from solution. This is reflected by the position of the iso-vol lines in the phase diagram, where the lines of low liquid percentage are grouped around the dew point line. 

When oil and gas are produced simultaneously into a separator a certain amount (mass fraction) of each component (e.g. butane) will be in the vapour phase and the rest in the liquid phase. This can be described using phase diagrams (such as those described in section 4.2) which describe the behaviour of multi-component mixtures at various temperatures and pressures. However to determine how much of each component goes into the gas or liquid phase the equilibrium constants (or equilibrium vapour liquid ratios) K must be known. 

After having proved the principles a dynamic test facility has been constructed. In this facility it is possible to inject 3 tracers in a flownng liquid consisting of air, oil and water. By changing the relative amounts of the different components it is possible to explore the phase diagram and asses the limits for the measurement principle. Experiments have confirmed the accuracy in parameter estimation to be below 10%, which is considered quite satisfactorily for practical applications. The method will be tested on site at an offshore installation this summer. 

Fig. IV-17. A schematic phase diagram illustrating the condensed mesophases found in monolayers of fatty acids and lipids.

Discuss the dependence of the friction phase diagram on temperature, mono-layer density, velocity, load and solvent vapor. Explain why each of these variables will drive one to the right or left in Fig. XII-8. 

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

Phase transitions in binary systems, nomially measured at constant pressure and composition, usually do not take place entirely at a single temperature, but rather extend over a finite but nonzero temperature range. Figure A2.5.3 shows a temperature-mole fraction T, x) phase diagram for one of the simplest of such examples, vaporization of an ideal liquid mixture to an ideal gas mixture, all at a fixed pressure, (e.g. 1 atm). Because there is an additional composition variable, the sample path shown in tlie figure is not only at constant pressure, but also at a constant total mole fraction, here chosen to be v = 1/2. 

Figure A2.5.3. Typical liquid-gas phase diagram (temperature T versus mole fraction v at constant pressure) for a two-component system in which both the liquid and the gas are ideal mixtures. Note the extent of the two-phase liquid-gas region. The dashed vertical line is the direction x = 1/2) along which the fiinctions in figure A2.5.5 are detemiined.

Figure A2.5.5. Phase diagrams for two-eomponent systems with deviations from ideal behaviour (temperature T versus mole fraetion v at eonstant pressure). Liquid-gas phase diagrams with maximum (a) and minimum (b) boiling mixtures (azeotropes), (e) Liquid-liquid phase separation, with a eoexistenee eurve and a eritieal point.

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

With these simplifications, and with various values of the as and bs, van Laar (1906-1910) calculated a wide variety of phase diagrams, detennining critical lines, some of which passed continuously from liquid-liquid critical points to liquid-gas critical points. Unfortunately, he could only solve the difficult coupled equations by hand and he restricted his calculations to the geometric mean assumption for a to equation (A2.5.10)). For a variety of reasons, partly due to the eclipse of the van der Waals equation, this extensive work was largely ignored for decades. 

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

Few if any binary mixtures are exactly syimnetrical around v = 1/2, and phase diagrams like that sketched in figure A2.5.5(c) are typicd. In particular one can write for mixtures of molecules of different size (different molar volumes and F°g) the approxunate equation... 

In the absence of special syimnetry, the phase mle requires a minimum of tliree components for a tricritical point to occur. Synnnetrical tricritical points do have such syimnetry, but it is easiest to illustrate such phenomena with a tme ternary system with the necessary syimnetry. A ternary system comprised of a pair of enantiomers (optically active d- and /-isomers) together with a third optically inert substance could satisfy this condition. While liquid-liquid phase separation between enantiomers has not yet been found, ternary phase diagrams like those shown in figure A2.5.30 can be imagined in these diagrams there is a necessary syimnetry around a horizontal axis that represents equal amounts of the two enantiomers. 

Figure A2.5.30. Left-hand side Eight hypothetical phase diagrams (A through H) for ternary mixtures of d-and /-enantiomers with an optically inactive third component. Note the syimnetry about a line corresponding to a racemic mixture. Right-hand side Four T, x diagrams ((a) tlirough (d)) for pseudobinary mixtures of a racemic mixture of enantiomers with an optically inactive third component. Reproduced from [37] 1984 Phase Transitions and Critical Phenomena ed C Domb and J Lebowitz, vol 9, eh 2, Knobler C M and Scott R L Multicritical points in fluid mixtures. Experimental studies pp 213-14, (Copyright 1984) by pennission of the publisher Academic Press.

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

Figure A2.5.31. Calculated TIT, 0 2 phase diagram in the vicmity of the tricritical point for binary mixtures of ethane n = 2) witii a higher hydrocarbon of contmuous n. The system is in a sealed tube at fixed tricritical density and composition. The tricritical point is at the confluence of the four lines. Because of the fixing of the density and the composition, the system does not pass tiirough critical end points if the critical end-point lines were shown, the three-phase region would be larger. An experiment increasing the temperature in a closed tube would be represented by a vertical line on this diagram. Reproduced from [40], figure 8, by pennission of the American Institute of Physics.

Figure A3.3.2 A schematic phase diagram for a typical binary mixture showmg stable, unstable and metastable regions according to a van der Waals mean field description. The coexistence curve (outer curve) and the spinodal curve (iimer curve) meet at the (upper) critical pomt. 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 tire coexistence curve are called off-critical quenches.

Flere we discuss the exploration of phase diagrams, and the location of phase transitions. See also [128. 129. 130. 131] and [22, chapters 8-14]. Very roughly we classify phase transitions into two types first-order and continuous. The fact that we are dealing with a finite-sized system must be borne in mind, in either case. 

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