The Structure of Materials
1.1 STRUCTURE OF METALS AND ALLOYS
1.1.3 Point Defects
Now that the most important aspects of perfect crystals have been described, it is time to recognize that things are not always perfect, even in the world of space lattices.
This is not necessarily a bad thing. As we will see, many important materials phenom- ena that are based on defective structures can be exploited for very important uses.
These defects, also known as imperfections, are grouped according to spatial extent.
Point defects have zero dimension; line defects, also known asdislocations, are one- dimensional; and planar defects such as surface defects and grain boundary defects have two dimensions. These defects may occur individually or in combination.
Let us first examine what happens to a crystal when we remove, add, or displace an atom in the lattice. We will then describe how a different atom, called animpurity (regardless of whether or not it is beneficial), can fit into an established lattice. As shown by Eq. (1.36), point defects have equilibrium concentrations that are determined by temperature, pressure, and composition. This is not true of all types of dimensional defects that we will study.
Nd =Nexp −Ed
kBT
(1.36) In Eq. (1.36),Nd is the equilibrium number of point defects,N is the total number of atomic sites per volume or mole, Ed is the activation energy for formation of the defect, kB is Boltzmann’s constant (1.38×10−23J/atom · K), and T is absolute temperature. Equation (1.36) is an Arrhenius-type expression of which we will see a great deal in subsequent chapters. Many of these Arrhenius expressions can be derived from the Gibbs free energy,G.
When an atom is missing from a lattice, the resulting space is called avacancy (not to be confused with a “hole,” which has an electronic connotation), as in Figure 1.28. In this case, the activation energy,Ed, is the energy required to remove an atom from the lattice and place it on the surface. The activation energy for the formation of vacancies in some representative elements is given in Table 1.13, as well as the corresponding vacancy concentration at various temperatures. Note that the vacancy concentration decreases at lower temperatures. In a nonequilibrium situation, such as rapid cooling from the melt, we would not expect the equilibrium concentration to be attained. This
Self-interstitial
Vacancy
Figure 1.28 Representation of a vacancy and self-interstitial in a crystalline solid. From K. M. Ralls, T. H. Courtney, and J. Wulff,Introduction to Materials Science and Engineering.
Copyright 1976 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
STRUCTURE OF METALS AND ALLOYS 47
Table 1.13 Formation Energy of Vacancies for Selected Elements and Equilibrium Con- centrations at Various Temperatures
Nd(vacancies/cm3) Element Ed (kJ/mol)
Melting Point,
Tm(◦C) 25◦C 300◦C 600◦C Tm
Ag 106.1 960 1.5×104 1.5×1013 3.0×1016 7.8×1017
Al 73.3 660 1.0×1010 1.2×1016 2.4×1018 5.0×1018
Au 94.5 1063 1.5×106 1.5×1014 1.5×1017 1.2×1019
Cu 96.4 1083 1.1×106 1.4×1014 1.4×1017 9.0×1018
Ge 192.9 958 <1 1.3×105 1.3×1011 8.2×1013
K 38.6 63 2.1×1015 — — 1.3×1016
Li 39.5 186 4.7×1015 — — 1.4×1018
Mg 85.8 650 4.4×107 6.4×1014 3.5×1017 5.7×1017
Na 38.6 98 4.0×1015 — — 1.0×1017
Pt 125.4 1769 8.7 2.7×1011 2.0×1015 4.2×1019
Si 221.8 1412 <1 3.1×102 2.5×109 8.0×1015
Interstitial impurity atom
Substitutional impurity atom
Figure 1.29 Representation of interstitial and substitutional impurity atoms in a crystalline solid. From K. M. Ralls, T. H. Courtney, and J. Wulff,Introduction to Materials Science and Engineering. Copyright1976 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc.
is indeed the case, and vacancy concentrations in rapidly quenched metals are much closer to the liquid concentration than they are to the equilibrium solid concentration.
The second type of point defect is called an impurity. Impurities can occur in two ways: as an interstitial impurity, in which an atom occupies an interstitial site (see Figures 1.21, 1.22, and 1.29); or when an impurity atom replaces an atom in the perfect lattice (see Figure 1.29). In the first instance, either the same atom as in the lattice, or an impurity atom, can occupy an interstitial site, causing considerable lattice strain as the atomic planes distort slightly to accommodate the misplaced atom. The amount of strain created depends on how large the atom is relative to lattice atoms. It
is also possible for a lattice atom to move off of a lattice site and occupy an interstitial site. In this case, both of the defects shown in Figure 1.28 occur simultaneously, and a defect pair known as aFrenkel defect (or Frenkel disorder) occurs. In a pure Frenkel defect, there are always equal concentrations of interstitial impurities and vacancies.
The second type of impurity, substitution of a lattice atom with an impurity atom, allows us to enter the world of alloys and intermetallics. Let us diverge slightly for a moment to discuss how control of substitutional impurities can lead to some useful materials, and then we will conclude our description of point defects. An alloy, by definition, is a metallic solid or liquid formed from an intimate combination of two or more elements. By “intimate combination,” we mean either a liquid or solid solution. In the instance where the solid is crystalline, some of the impurity atoms, usually defined as the minority constituent, occupy sites in the lattice that would normally be occupied by the majority constituent. Alloys need not be crystalline, however. If a liquid alloy is quenched rapidly enough, an amorphous metal can result. The solid material is still an alloy, since the elements are in “intimate combination,” but there is no crystalline order and hence no substitutional impurities. To aid in our description of substitutional impurities, we will limit the current description to crystalline alloys, but keep in mind that amorphous alloys exist as well.
The extent to which a lattice will allow substitutional impurity atoms depends on a number of things. The factors affecting the solubility of one element in another are summarized in a set of guidelines called the Hume–Rothery rules, though they are really not rules at all. As you can imagine, atomic size plays an important role in determining solubility. The first Hume–Rothery “rule” states that if the atomic size of the host lattice and impurity atom differ by more than about 14%, the solubility of the impurity in the lattice will be small. Refer to Table 1.9 for values of atomic size.
The second rule involves electronegativity. We mentioned earlier in this chapter that electronegativity is an important concept, and it plays an important role in determining not only how soluble an impurity is, but also what type of bond will result. In general, the larger the electronegativity difference,χ, between the host atom and the impurity, the greater the tendency to form compounds and the less solubility there is. So, elements with similar electronegativities (refer to Table 1.4) tend to alloy, whereas elements with large χ tend to have more ionic bonds (see Section 1.0.3) and form intermetallics.
Intermetallics are similar to alloys, but the bonding between the different types of atoms is partly ionic, leading to different properties than traditional alloys. The third rule deals with crystal structures. One would expect like crystal structures to be more compatible, and this is generally the case. Refer to Table 1.11 for typical crystal structures, but keep in mind that the elements can have multiple structures depending on temperature, and remember that this can affect the stability of the alloy. Finally, all other things being equal, the fourth Hume–Rothery rule states that a metal of lower valency is more likely to dissolve one of higher valency than vice versa. Common valences of the elements are listed in Table 1.9. Again, elements can have multiple oxidation states.
An interesting corollary to the fourth rule is that the total number of valence electrons per atom can be used as a guideline in determining the crystal structure of the alloy.
As summarized in Table 1.14, by summing the valence electrons of the elements in the alloy and dividing by the number of types of atoms (binary=2, ternary=3, etc.), it is sometimes possible to predict the crystal structure of an alloy. The “complex cubic”
structures include cubic structures other than SC, BCC, and FCC, which we have not yet described, such as the diamond structure. As an example of this corollary, the binary
STRUCTURE OF METALS AND ALLOYS 49
Table 1.14 Common Crystal Structures of Alloys Based on Valences of Components
Valence
Electrons/Atom Structure
3/2 BCC, complex cubic, HCP
21/13 Complex cubic
7/4 HCP
alloy formed between Cu (+1 valence) and Be (+2 valence) has (1+2)/2=3/2 valence electrons/atom, and it turns out to have the BCC structure, which is different than either of the two component structures.
Cooperative Learning Exercise 1.4
Person 1: Do Cu and Ni satisfy the first and second Hume– Rothery rules for complete solid solubility?
Person 2: Do Cu and Ni satisfy the third and fourth Hume– Rothery rules for complete solid solubility?
Compare your answers. Would you predict that Cu and Ni have complete, partial, or no solid solubility in each other?
Answer :
% r
= 2.3%
(<14%
);
= χ 0.01(sm all);
botha reFCC;
Cuis lowerv alence
thanNi.
All fourare satisfied;
Cuand Ni are comp letelysoluble.
This concludes our diversion into alloys for the time being. From this point on, we will often describe metals and alloys in similar terms, and we will make distinctions between the two classes of materials only when there are substantial dissimilarities between them. Returning now to our description of point defects, we have but one type of point defect pair left to describe. Similar to a Frenkel defect in which both a vacancy and interstitial impurity must occur simultaneously, a Schottky defect (a.k.a.
Schottky disorder or imperfection) arises in ionic solids when a cation–anion vacancy pair is formed. Recall that ionic compounds occur when there is a large electronegativity difference between the components, so that a Schottky defect normally occurs in binary ionic compounds such as sodium chloride. Though the ionic compounds we will use as illustrations here are not technically metals or alloys, keep in mind that metallic solids such as intermetallics can have ionic bonding. In sodium chloride, removal of one sodium ion and one chloride ion from the lattice results in a Schottky defect (see Figure 1.30). In ionic solids where the cation (positively charged ion) and anion (negatively charged ion) have the same absolute charge (e.g.,|Na+| =1, |Cl−| =1), a Schottky defect arises from the same number of vacancies in both ions. For ionic solids in which the anion and cation have different absolute valencies (e.g., CaF2), a nonstoichiometric compound must be formed in order to maintain charge neutrality in the lattice; that is, two fluorine ions (F−)must leave for every calcium ion (Ca2+) that is removed from the lattice. Because atoms must leave the ionic lattice, Schottky
Frenkel defect
Schottky defect
Figure 1.30 Representation of Frenkel and Schottky defects in a crystalline solid. Adapted from W. G. Moffatt, G. W. Pearsall, and J. Wulff, The Structure and Properties of Materials, Vol. 1. Copyright1964 by John Wiley & Sons, Inc.
defects normally occur only at lattice perturbations such as grain boundaries or surfaces so that the removed atoms have someplace to go. We will describe Schottky defects in more detail when we come to inorganic materials, such as oxides, where binary compounds are more prevalent.