or popularly known “directional solidification” (DS) can be employed to align the grain boundaries parallel to the solidification direction and hence along the long axis of the columnar grains. The resultant structure is similar to a bundle of pencils with each grain having the low modulus <001> axis aligned parallel to the long axis of the grains. This kind of texturing can be very important to material properties. In the case of nickel‐base superalloys, Duhl (1987) mentioned that VerSnyder and Guard (1960) and Versnyder and Shank (1970) demonstrated for the “first time” that by aligning the grain boundaries parallel to the principal stress axis, the stresses acting at elevated temperatures on the weak grain boundaries could be minimized, thus delaying failure initiation and enhancing creep‐rupture life.
In the extreme case of DS processes, such as in the Czochralski process, only one crystal will survive. This process is well known in the semiconductor industry to produce high‐grade silicon wafers but is also used in casting creep‐sensitive components, such as turbine blades. However, single‐crystal (SC or SX) formation can be a very expensive process.
Once initially solidified, the microstructure of a material does not remain static, and the mechanical response and physical properties can likewise change as a result of a material's history. For practical reasons, primarily to avoid sudden high‐temperature fractures, operating temperatures for most metallic solids are restricted to temperature regimes well below 0.4 T m. In nuclear and power generation industries, and for the components of gas turbine engines, exposure to temperatures higher than 0.4 T m cannot be avoided. This being said, the approaches for developing complex alloys used in aerospace and gas turbine applications have taken bold steps and the operating temperatures are now pushed upward to 0.8 T m.
In the geophysical arena, the mechanical properties of naturally occurring materials and their dependence on microstructure close to T m are paid only passing attention even though the temperatures of the lithosphere–asthenosphere boundary are extremely high.
To better understand phase transitions as well as the forces that may be at play as one approaches the transition temperature, it is necessary to have a better understanding of how a material system behaves on exploring its phase diagram.
2.3.2 Phase Diagrams
A material system is composed of both phases and components. “Phases (P)” in a system are homogeneous in chemical composition and physical state. “Components (C)” in a system represent a pure element or compound. The number of components in a system is thus the number of independent species necessary to define the composition of all the phases. Phase diagrams map the preferred or equilibrium phases of a material at different thermodynamic variables.
For a single‐component system, a simple pressure–temperature phase diagram is a useful tool to understand the phase boundaries of the material. Figure 2.3 shows the typical features of such a phase diagram. The lines on the graph map the phase boundaries that exist at equilibrium. The open spaces between the lines represent areas where a single phase exists. Phase transitions occur on the lines. These phase transitions can occur due to a change in temperature or pressure and sometimes different terminology is used to indicate that a specific parameter is changing. For example, “melting”/“freezing” are generally terms used for liquefaction/solidification phase transitions between solid and liquid due to a change in temperature.
The degrees of freedom, F, is the number of thermodynamic parameters that may vary independently while maintaining the same phase/phases. F can be derived from the number of components (C) and phases (P), according to the phase rule (Gibbs 1874–1878 ):
(2.1)
For the simple one‐component system being considered, in each single‐phase region C = 1 and P = 1 such that F = 2. Two parameters (temperature and pressure) can thus be chosen independently within this region. However, on the lines of the phase diagram, P = 2 and F = 1, and selection of one parameter influences the other. For example, on the curve between liquid and gas, if the temperature decreases, some of the gas condenses, decreasing the pressure. When three phases coexist in a single‐component system, F = 0 suggesting that this can only occur at a single temperature and pressure. This point is known as the “triple point.”
Figure 2.3 Typical pressure versus temperature of a one‐component system. T TP and P TP represent the triple point temperature and pressure, respectively, while T C and P C represent the critical temperature and pressure, respectively. Phase transitions are represented in gray and are not shown on a real phase diagram.
Another key point on such phase diagrams is the end point of a phase equilibrium curve – called a “critical point.” Figure 2.3 demonstrates a common critical point where the liquid and gaseous states become indistinguishable and are often referred to as a supercritical fluid. In the vicinity of a critical point, the physical properties of the liquid and vapor can dramatically change as they become more similar. Historically, a solid–liquid critical point has generally not been accepted (Landau and Lifshitz 1980); however, this has recently been challenged – largely through molecular dynamics simulations (Elenius and Dzugutov 2009; Mochizuki and Koga 2015).
It should be mentioned that the phase diagram of a single compound is not necessarily a single‐component system. Polymorphism can give rise to intricate phase diagrams for single compounds. For example, the phase diagram of water is quite complex and discussed further in Section 2.6.
Phase diagrams for mixtures of chemically independent components can quickly become increasingly complex. For a simple binary mixture, the open spaces in a pressure–temperature phase diagram would have C = 2 and P = 1 to give three degrees of freedom. The third degree of freedom is the composition of one of the two components. For this reason, phase diagrams for binary systems are generally expressed as temperature versus composition at a given pressure (often standard pressure), as shown in Figure 2.4. Engineers will often express the composition in terms of the weight percent of the components while chemists will often express it in terms of the mole fraction of the component.
Figure 2.4 Sample phase diagram schematic for a binary XY system. (a) Simple phase diagram demonstrating how to determine the composition present in the different phases at a point in the mixed phase region. (b) Slightly more complex phase diagram of an XY system containing a eutectic point.
Phase diagrams for binary systems contain a large amount of information. Not only do they give an indication of the phases present, but also provide information about the composition of the phases and fraction of the phases present in the mixture. For example, the point of interest highlighted in Figure 2.4a has both a liquid phase and a solid phase present. The composition of the phases can be determined by drawing a tie‐line
