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 from the point to the liquidus and solidus curves and then reading the percent composition at the intersections. The ratio of the liquid/solid present can then be determined using the two phase compositions and the overall composition, using the lever rule:
(2.2)
Figure 2.4b presents a typical phase diagram of a eutectic system. A eutectic system is a homogeneous mixture of substances that melts or solidifies at a single temperature that is lower than the melting point of any of the constituents. At a eutectic point, the liquid and two solid solutions all coexist in chemical equilibrium. Cooling through this temperature results in intricate macrostructures that can take several different forms, including lamellar, rod‐like, globular, or acicular (needle‐like) structures. Other critical points in a binary system include:
Eutectoid: Transformation of a solid phase to yield two solid phases.
Peritectic: Transformation of a liquid phase and a solid phase to yield a single solid phase.
Peritectoid: Transformation of two solid phases in an alloy system to yield a new solid phase.
Binary systems can be intricate with multiple types of transformations available to the system at different compositions. Further increasing the number of components can quickly make materials' systems very complex. Phase diagrams can thus be incredibly useful in designing materials to have specific phase compositions.
Phase diagrams can be incredibly important in engineering materials, but they do not always tell the whole story. Phase diagrams generally present equilibrium phases. Nonequilibrium phases, such as phases stabilized through quenching or even a more complex environmental history, can often be critical to achieving properties of interest. Time–temperature–transformation diagrams that plot the time required for isothermal transformation provide kinetic information about transformation processes and can be highly useful in developing synthesis processes; however, these diagrams must still be interpreted with the assistance of phase diagrams.
2.3.3 Crystal Imperfections
An individual crystal or grain in the solid phase consists of atoms aligned in space in definite patterns over long distances until it impinges on the border of another crystal. In reality, solids are not perfect. There are many different kinds of imperfections within a solid that play an important, and often determining, role in the properties of the material. Crystal imperfections thus play an important role in materials science and its use across disciplines.
Understanding imperfections and their impact on materials' properties is extraordinarily complex and dependent on the system under question. Here, we strive only to give a brief overview of some high‐level key features that will enable us to understand aspects presented later in this text.
2.3.3.1 Point Defects
Point defects are imperfections that arise from a single central site, but cause strain in a small volume around them and can impact the properties of the bulk. They include a variety of imperfections, such as those shown in Figure 2.5: (i) vacancies – empty lattice sites that can impact diffusion; (ii) interstitials – generally smaller atoms located in between normal lattice sites that impact a variety of properties, such as strength, hardness ductility, and yield strength; and (iii) substitutions – solute atoms located at a lattice site that can also impact strength and hardness.
Point defects can often occur in pairs or sets. For example, in ionic crystals, point defects can occur in pairs that maintain stoichiometry or charge neutrality, such as Schottky defects (vacancies of ions with opposite charge) or Frenkel defects (a vacancy and an interstitial). A high density of point defects, particularly vacancies, can give rise to nonstoichiometric compounds. Such compounds often consist of transition metals, lanthanides, and actinides paired with polarizable anions, such as oxides and sulfides, e.g. FexO and CuxS. Point defects can be naturally occurring, such as the substitutional impurities that give rise to beautiful gems, e.g. sapphires and rubies. Point defects can also be designed into the material to tailor its properties, such as doping semiconductors to control their electrical conductivity.
Figure 2.5 Schematic of common point defects in crystals, including a vacancy, an interstitial, and a substitution.
2.3.3.2 Line Defects/Dislocations
Line defects or dislocations are imperfections in the linear arrangement of atoms in the crystal. There are two main types of dislocations that represent the extreme forms of line defects: edge dislocations and screw dislocations. Dislocations play an important role in plastic deformation and allow deformation to occur at lower stresses than would be anticipated in a perfect crystal as only a few atoms are moved from their equilibrium positions at a given time.
Edge dislocations can be visualized as an extra partial plane of atoms. As shown in Figure 2.6, the partial plane continues into the direction of the page such that the edge dislocation line runs into the plane of the page. Creation and motion of edge dislocations across the crystal results in relative shear or slippage of planes of atoms past each other. There are generally two types of dislocation motion: slip/glide and climb.
In dislocation glide, as shown in Figure 2.6a, the slip plane is the plane in which the dislocation moves and is generally the most closely packed plane in the crystal. A family of close‐packed planes can all act as potential slip planes. As such, the crystal system of the material determines how many glide planes are possible. However, the orientation of the differential stress determines which glide planes are active or not. The edge dislocation moves in the slip plane in the direction of the shear and generally corresponds to one of the shortest lattice translation vectors in the material. The Burgers vector (b) represents the magnitude and direction of the lattice distortion resulting from a dislocation in a crystal lattice. The Burgers vector and the edge dislocation are perpendicular to each other and define the slip plane. Real dislocations will not always be in a straight line and can move along more than one slip plane.
Figure 2.6 Schematic of the basic motion or propagation of an edge dislocation. (a) Dislocation glide or slide as a result of shear stress. (b) Dislocation climb involving material transfer from the dislocation core.
In edge dislocation climb, as shown in Figure 2.6b, the dislocation moves up or down out of its slip plane to a parallel slip plane. This involves the creation or annihilation of a row of vacancies and is energetically difficult.