Features of phase diagrams

Two-dimensional phase diagrams for a single-substance system can be generated as projections of a three-dimensional surface in a coordinate system with Cartesian axes p, V/n, and T. A point on the three-dimensional surface corresponds to a physically-realizable combination of values, for an equilibrium state of the system containing a total amount n of the substance, of the variables p, V/n, and T. 

The concepts needed to interpret single-substance phase diagrams will be illustrated with carbon dioxide. 

The two-dimensional projections shown in Figs. 8.2(b) and 8.2(c) are pressure-volume and pressure-temperature phase diagrams. Because all phases of a multiphase equilibrium system have the same temperature and pressure, the projection of each two-phase area onto the pressure-temperature diagram is a curve, called a coexistence curve or phase boundary, and the projection of the triple line is a point, called a triple point. 

How may we use a phase diagram The two axes represent values of two independent variables, such as p and V/n or p and T. For given values of these variables, we place a point on the diagram at the intersection of the corresponding coordinates this is the system point. Then depending on whether the system point falls in an area or on a coexistence curve, the diagram tells us the number and kinds of phases that can be present in the equilibrium system. 

If the system point falls within an area labeled with the physical state of a single phase, only that one kind of phase can be present in the equilibrium system. A system containing a pure substance in a single phase is bivariant (F = 3 — 1 = 2), so we may vary two intensive properties independently. That is, the system point may move independently along two coordinates (p and V/n, or p and T) and still remain in the one-phase area of the phase diagram. When V and n refer to a single phase, the variable V/n is the molar volume Kn in the phase. 

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Eutectics and eutectoids are important. They are common in engineering alloys, and allow the production of special, strong, microstructures. Peritectics are less important. But you should know what they are and what they look like, to avoid confusing them with other features of phase diagrams. 

The normal melting, boiling, and triple points give three points on the phase boundary curves. To construct the curves from knowledge of these three points, use the common features of phase diagrams the vapor-liquid and vapor-solid boundaries of phase diagrams slope upward, the liquid-solid line is nearly vertical, and the vapor-solid line begins at P = 0 and P = 0 atm. 

Topological Features of Phase Diagrams Calculated Using... 

The roots of the CALPHAD approach lie with van Laar (1908), who applied Gibbs energy concepts to phase equilibria at the turn of the century. However, he did not have the necessary numerical input to convert his algebraic expressions into phase diagrams that referred to real systems. This situation basically remained unchanged for the next 50 years, especially as an alternative more physical approach based on band-structure calculations appeared likely to rationalise many hitherto puzzling features of phase diagrams (Hume-Rotheiy et al. 1940). 

J.7.I Topological features of phase diagrams calculated using regular solution theory... 

The defining features of phase diagrams are the phase boundaries that delineate phase domains and mark the conditions of coexistence with adjacent phases. Theoretical description of a phase diagram is therefore tantamount to finding the equations of coexistence that describe these phase boundaries. For a simple phase equilibrium between phases a and /3, as shown below, the a + /3 coexistence curve is described by an equation of the form P = P(T), whose form we now wish to determine ... 

Equilibrium Diagram The salient features of phase diagram represented in figure (2), are as follows ... 

Table 3.1 Summary of the Geometrical Features of Phase Diagrams for One- and Two-component Systems...

A phase diagram computed using self-consistent mean field theory (49,51) is shown in Figure 2. The figure shows the generic sequence of phases accessed just below the ODT temperature for diblock copolymers of different compositions. The features of phase diagrams for particular systems are different in detail, but qualitatively they are similar, and well accounted for by SCF theory. 

Figures 1 and 2 show the vertical sections atconstantAl content of 25 and 23 at.%, respectively [19980hn]. In addition to (aFe) + 2 phase field as in the case of Al-Fe system, the presence of ai + a.2 phase field in the ternary system may be seen in Figs. 1 and 2. As discussed by [19980hn], the topology of the phase boundaries involving ordered (ai, a2) and disordered phases (aFe) are consistent wifli the general features of phase diagrams associated with multicritical points [1982All].

In addition to printed compilations, more and more of the information is available on CD-ROM and latterly also on-line on the internet. This last is a feature of the service provided by MSI, Materials Science International Services in Stuttgart. This organisation, under the working name of MSIT Workplace (http //www.msiwp.com), provides information on the entire corpus of phase diagram compilations. 

Phase transitions in two-dimensional layers often have very interesting and surprising features. The phase diagram of the multicomponent Widom-Rowhnson model with purely repulsive interactions contains a nontrivial phase where only one of the sublattices is preferentially occupied. Fluids and molecules adsorbed on substrate surfaces often have phase transitions at low temperatures where quantum effects have to be considered. Examples are molecular layers of H2, D2, N2 and CO molecules on graphite substrates. We review the path integral Monte Carlo (PIMC) approach to such phenomena, clarify certain experimentally observed anomalies in H2 and D2 layers, and give predictions for the order of the N2 herringbone transition. Dynamical quantum phenomena in fluids are analyzed via PIMC as well. Comparisons with the results of approximate analytical theories demonstrate the importance of the PIMC approach to phase transitions where quantum effects play a role. 

The role of a thermodynamic approach is well known a thermodynamic check, optimization and prediction of the phase diagram may be carried out by using methods such as those envisaged by Kubaschewski and Evans (1958), described by Kaufman and Nesor (1973), Ansara et al. (1978), Hillert (1981) and very successfully implemented by Lukas et al. (1977, 1982), Sundman et al. (1985). The knowledge (or the prediction) of the intermediate phases which are formed in a certain alloy system may be considered as a preliminary step in the more general and complex problem of assessment and prediction of all the features of phase equilibria and phase diagrams. See also Aldinger and Seifert (1993). 

A schematic representation of the phase diagram for pure H20 (not to scale) is shown in Fig. 7.1. Let us examine the topological features of this diagram in terms of increasing complexity from single- to multiphase character. 

FIGURE 10.28 A phase diagram for H2O, showing a negative slope for the solid/liquid boundary. Various features of the diagram are discussed in the text. Note that the pressure and temperature axes are not drawn to scale. 

A review of some of the features of these diagrams in terms of the Phase Rule is enlightening (Findlay 1951). A system composed of two different solid forms of a substance will have one component and two solid phases. In the absence of a further definition of the system there will be one degree of freedom. In Fig. 2.7 this is either the temperature or the pressure along the I<->II line. Choosing either variable fixes a point and defines the system. However, suppose that we are interested in the situation when the two phases are in equilibrium with the liquid or the vapour. Each one of those is an additional phase, making three in total and, by virtue of the Phase Rule, rendering the system invariant. Invariance results in a triple point for each case, defined by the intersection of the I<->II curve with the I<->v. and n<->v. curves on the one hand and the intersection of the I<->II curve with the I<->1. and II<->1. curves on the other hand. 

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