The Structure of Materials
1.2 STRUCTURE OF CERAMICS AND GLASSES
1.2.6 Defect Reactions*
As with metals, ceramic crystals are not perfect. They can contain all of the same types of defects previously described in Sections 1.1.3–1.1.5. What is unique about ceramic crystals, particularly oxide ceramics, is that the concentration of point defects, such as vacancies and interstitials, is not only determined not only by temperature, pressure, and composition, but can be influenced greatly by the concentration of gaseous species in which they come in contact (e.g., gaseous oxygen). The concentration of gaseous species affects the crystal structure, which in turn can affect physical properties such
as conductivity and thermal expansion. This opens up a multitude of applications for ceramic materials, ranging from sensors and actuators to nanoscale reactors. In order to describe some of these phenomena, we will need to accurately describe the reactions that are taking place, and quantify their effect on point defect concentrations. This is done through the development of defect reactions, in which defect concentrations are treated like any other chemical species.
1.2.6.1 Kroger – Vink Notation. We must slightly modify the notation we use for chemical reactions to avoid confusion between vacancies, which have a charge asso- ciated with them, and formal valence charges on isolated ions, like Ca2+. To do this, a system of notation has been developed called Kroger–Vink notation. Consider the generic binary crystalline compound MX. Recall that a vacancy occurs when an atom is removed from a lattice position. In the binary compound, there can be two types of vacancies: one created by removing an M atom, designated VM, the other from a missing X atom, designated VX. As in all Kroger–Vink notation, the primary symbol, in this case “V” for vacancy, indicates the type of species, and the subscript, in this case “M” or “X,” designates the lattice site. Similarly, interstitial sites are designated with a subscript “i,” and the atom occupying the interstitial position is indicated by either an M or an X. Thus, Mi represents a metal atom interstitial and Xi represents a counterion interstitial. The subscript does not tell what type of interstitial site is being occupied—for example, tetrahedral versus octahedral.
It is theoretically possible for cations to occupy anion sites, and vice versa.
Kroger–Vink notation, then, dictates that an M atom on an X site be designated as MX
and that an X atom on an M site be designated as XM. Recall that we can have defect clusters, such as a Frenkel defect. Defect clusters are enclosed in parentheses—for example, (VMVX) or (XiXM)—to indicate that the individual defects are associated with one another. Impurity atoms are also coded as to lattice position. If we introduce a metal impurity atom L into our compound MX, it might occupy a metal cation site, and is thus designated as LM. Similarly, Siis an S impurity atom on an interstitial site.
Two species we have not yet discussed are free electrons and free holes. We will use these species extensively in describing electronic properties of materials, but for now we simply note that they are dealt with like any other species. A free electron is indicated with “e” and has a charge associated with it, which is designated with a superscript prime, e, to differentiate it from a formal valance charge (−). A free electron is not localized and is free to move about the lattice. As a result, it does not occupy a specific lattice site and carries no subscript. An electron hole, which carries a positive charge, is also delocalized and is designated by hž. Here, the superscript dot indicates a positive charge in Kroger–Vink notation. We also use superscripts to indicate charges on atoms and vacancies. If, for example, we remove the ion M+from our MX lattice, the remaining vacancy has a negative charge associated with it since the original lattice was charge-neutral. The symbol VMrepresents a negatively charged metal vacancy. Similarly, if we remove X− from the lattice, a positive charge is left on the vacancy. The symbol VžX is used for a positively charged anion vacancy. For a specific compound, the symbols M and X are replaced with the actual atomic symbol.
For example, Znžži represents a Zn ion on interstitial site with a resulting 2+ charge.
The Kroger–Vink notation is summarized in Table 1.19.
It should come as no surprise that defects have concentrations—for example, [Znžži ]
—and we can write reactions with these defects. As with balancing equations, which
STRUCTURE OF CERAMICS AND GLASSES 73
Table 1.19 Summary of Kroger– Vink Notation
The Notation. . . Represents a(n). . .
V Lattice vacancy
h Free hole
e Free electron
M (e.g., Ca, Al. . .) Cation atom X (e.g., O, Cl. . .) Anion atom
Subscripts
i Interstitial lattice position
M Cation lattice position
X Anion lattice position
Superscripts
ž Positive charge
Negative charge
you learned how to do in general chemistry, there are no set rules—there is a bit of guesswork and art involved. There are a few general guidelines that should be followed in balancing defect reactions, however. The first guideline involves site relation. The number of M sites must be in correct proportion to X sites as dictated by the compound stoichiometry. For example, the ratio M:X is 1:1 in MgO and 1:2 in UO2.
The second guideline deals with site creation, and it states that defect changes that alter the number of lattice sites must not change the overall site relation (guide- line 1). Site creation is easily recognized from the subscripts: Species such as VM, VX, MM, XM, and so on, create sites, whereas the species e, hž, Mi, Li, and so on, do not create sites. As with regular reactions, the third guideline states that mass balance must be maintained; that is, any species appearing on the left side must appear on the right side of the equation. Remember that subscript symbols only indicate sites and are not involved in the mass balance. The fourth guideline in balancing defect reaction equations simply says that electrical neutrality must be maintained. Both sides of the defect equation should have the same total effective charge, but that charge need not necessarily be zero. Finally, there is no special dis- tinction for surface sites. Lattice positions at the surface are treated like every other position in the lattice. See example problem 1.5 for details on balancing defect reac- tion equations.
Now that we know how to write defect equations, let’s look at Frenkel and Schottky defects in more detail.
1.2.6.2 Defect Reaction Equilibrium Constants. Recall that a Frenkel disorder is a self interstitial–vacancy pair. In terms of defect concentrations, there should be equal concentrations of vacancies and interstitials. Frenkel defects can occur with metal atoms, as in AgBr:
Vi+AgAg Agži +VAg (1.39)
where Agži is a silver atom on an interstitial site with a+1 charge, and VAg is a silver vacancy with a−1 charge; or with anions, such as oxygen in Y2O3:
OO+Vi Oi +VžžO (1.40)
Example Problem 1.5
Write a defect reaction equation for the substitution of a CaCl2 molecule into a KCl lattice.
Solution: There are actually two ways that CaCl2 can be placed in the KCl lattice:
substitutionally and interstitially. The defect reaction equation for substitution is CaCl2(s)+2KK+2ClCl CažK+VK+2ClCl+2KCl(g)
Again, there are no set “rules” for balancing this equation, but we can describe some of the guidelines as they relate to this example.
(a)Site relation
ž KCl sites must be 1:1
ž Two K sites are used, so two Cl sites must be used. Notice that the chlorines are all equivalent, and that the Cl brought in by the CaCl2simply occupies existing Cl sites, with the removal the previous chlorine with gaseous KCl. A legitimate simplification would be to remove 2ClClfrom both sides of the defect reaction equation.
(b)Site creation
ž vacancy creation doesn’t change site balance.
(c)Mass balance
ž KCl is given off as gas. This is a common way of “getting rid” of solid species. Don’t be concerned that a solid is turning into a gas — it is definitely possible.
(d)Electrical neutrality
ž Keep in mind that we have strongly ionic species in this example; charges are involved.
ž Placing a Ca2+ion on a K+site gives a net+1 charge on the site, CaK.
ž A vacancy must be created in order to preserve charge neutrality and maintain site relation. This is a “trick” that you will have to learn.
For interstitial substitution of CaCl2in KCl, the defect reaction equation is CaCl2(s)+2KK+2ClCl Cažži +2VK+2ClCl+2KCl(g) The details of balancing this reaction are left to the reader.
STRUCTURE OF CERAMICS AND GLASSES 75
As with all reactions, defect reactions are subject to the law of mass action [see Eq. (3.4) for more details), so an equilibrium constant,KF, can be written:
KF = [Oi][VžžO]
[OO][Vi] (1.41)
We can simplify this expression by noting that defect concentrations are usually small;
that is, [Vi]≈[OO]≈1, so Eq. (1.41) becomes:
KF =[Oi][VžžO] (1.42) Frenkel defects form interstitial–vacancy pairs, so that [Oi]=[VžžO], and Equation (1.42) reduces further to
KF =[Oi]=[VžžO] (1.43) This is the general expression for the equilibrium constant of oxygen interstitials in Y2O3.
The defect concentration comes from thermodynamics. While we will discuss ther- modynamics of solids in more detail in Chapter 2, it is useful to introduce some of the concepts here to help us determine the defect concentrations in Eq. (1.43). The free energy of the disordered crystal, G, can be written as the free energy of the perfect crystal, G0, plus the free energy change necessary to create n interstitials and vacancies (ni =nv=n), g, less the entropy increase in creating the interstitials;
Scat a temperature T:
G=G0+ng−T Sc (1.44)
Equation (1.44) states that the structural energy increases associated with the creation of defects are offset by entropy increases. The entropy is the number of ways the defects (both interstitials and vacancies) can be arranged within the perfect lattice, and it can be approximated using statistical thermodynamics as
Sc=kBln
N!
(N−ni)!ni!
N! (N−nv)!nv!
(1.45) where kB is Boltzmann’s constant and N is the total number of lattice sites. Use of Stirling’s approximation (lnN!=N·lnN−N) and the fact thatni=nv =ngives
Sc=2kB[NlnN−(N −n)ln(N −n)−nlnn] (1.46) The free energy is then
G=G0+ng−2kBT
Nln N
N−n
+nln
N−n n
(1.47) At equilibrium, the free energy change with respect to the number of defects is a mini- mum, so we can obtain a relationship for the concentration of defects,n/N (assuming N−n≈N):
n N =exp
−g 2kBT
(1.48)
The free energy change is usually approximated by the enthalpy change (additional entropy changes are small). Refer back to Table 1.13 for typical defect concentrations at various temperatures, and note that the defect concentrations are orders of magnitude smaller at 100◦C, especially for large enthalpies of formation (approximated byEd in Table 1.13).
The equilibrium concentration for a Schottky disorder can be found in a similar manner. Recall that a Schottky defect is a cation–anion defect pair. For example, the migration of an MgO molecule to the surface in an MgO crystal can be described as follows:
MgMg+OO VMg+VžO+Mgsurf+Osurf (1.49) Recall that surface sites are indistinguishable from lattice sites, so we usually write
null VMg+VžžO (1.50)
where the term “null” simply means that the vacancies form from the perfect lattice, and that all cations and anions are equivalent, so that any could be used in this equilibrium expression. The Schottky equilibrium constantsKs is then
Ks=[VMg][VžžO] (1.51) and since the concentration of both types of vacancies must, by definition of the Schottky defect, be equivalent, the equilibrium constant simplifies to
Ks =[VMg]=[VžžO] (1.52) We will see in subsequent chapters how defect reactions can be used to quantitatively describe important defect-driven phenomena, particularly in ceramics.