Self-diffusion, Solute Diffusion, Diffusion in Ionic Crystals and Correlation Effects

Abstract

Self-diffusion, solute diffusion, diffusion in ionic crystals and correlation effects are the topic of this chapter. Ionic crystals are included in this chapter since many, if not all, ceramics have some iconicity in their nature. Unlike pure metals, ceramics include at least two different species, such that vacancy formation is not of a single component. Therefore a vacancy may be a cation or an anion vacancy. These vacancies define a Schottky-type defect. The enthalpy of Schottky defect formation is calculated. The Frenkel defect forms in pairs, as an atom shifted into an interstitial site simultaneously forming a vacancy. Charge neutrality in both type of defects have to be maintained. Also the enthalpy of Frenkel defect formation is calculated. Diffusion and conductivity is a topic of this chapter. Current is carried in an electric field occurs through the solid by ionic diffusion by electrons. Ionic conductors are used in various applications, such as chemical and gas sensors, but the use of solid electrolytes, such as solid oxide fuel cells (SOFC), is quite significant. Ionic conductivity and its temperature dependence is also discussed. Correlation effects are an important section in this chapter. Whereas for self-diffusion, the tracer correlation factor is a pure number, the correlation factor for solute (impurity) diffusion, is not a geometric constant. The temperature and concentration dependence of the solute correlation factor is evaluated. Binding energy, enhanced diffusion and isotope effect are also included in this chapter.

Appendix 5.1: Kröger and Vink Relation

Examples of notations

${{Al}}_{{Al}}^{x}$ :

an aluminum ion sitting on an aluminum lattice site, with neutral charge

${{Ni}}_{{Cu}}^{x}$ :

a nickel ion sitting on a copper lattice site, with neutral charge

$V_{{Cl}}^{ \bullet }$ :

a chlorine vacancy, with singular positive charge

${{Ca}}_{i}^{ \bullet \bullet }$ :

a calcium interstitial ion, with double positive charge

${{Cl}}_{i}^{\prime }$ :

a chlorine anion on an interstitial site, with singular negative charge

${{O}}_{{_{i} }}^{\prime \prime }$ :

an oxygen anion on an interstitial site, with double negative charge

e′:

an electron. A site is not normally specified

An example of MgO in a Schottky relation:

$${\o} \Leftrightarrow V_{Mg}^{\prime \prime } + V_{O}^{ \bullet \bullet } $$

(A.1)

A vacancy in a Mg sublattice is double-charged: i equals −2 and, on the O sublattice site, it has a +2 charge.

The equilibrium constant, according to the Law of Mass Action, is:

$$ k = \left[ {V_{Mg}^{\prime \prime } } \right]\left[ {V_{O}^{ \bullet \bullet } } \right] $$

(A.2)

The reaction is as follows:

$$ \left[ {V_{Mg}^{\prime \prime } } \right] = \left[ {V_{O}^{ \bullet \bullet } } \right] $$

(A.3)

The equilibrium constant may be related to the Gibbs free energy as:

$$ k = \,\,\exp - \frac{{\Delta G_{F} }}{kT} $$

(A.4)

From (A.2) and (A.4), write:

$$ \,\,\exp - \frac{{\Delta G_{F} }}{kT} = \left[ {V_{Mg}^{\prime \prime } } \right]^{2} $$

(A.5)

(A.5) is a consequence of (A.3).

Expressing $V_{Mg}^{\prime \prime }$ and noting that ΔG = ΔH − ΤΔS, for the case of Mg, one can write:

$$ \left[ {V_{Mg}^{\prime \prime } } \right] = \,\,\exp - \frac{{\Delta H_{F} }}{2kT} + \frac{{\Delta S}}{k} = A\,\,\exp\,\, \frac{{\Delta H_{F} }}{2kT} $$

(A.6)

allowing for the calculation of a Schottky concentration.

From (A.6) and (A.5), with the value of ΔG for a Schottky defect, one can write:

$$ k = \left[ {V_{Mg}^{\prime \prime } } \right]^{2} = A\,\,\exp\,\, \frac{{{\Delta H}_{F} }}{2kT} $$

$O_{i}^{\prime \prime }$ :

an oxygen anion on an interstitial site with double negative charge

$V_{Mg}^{ \bullet \bullet }$ :

a Mg interstitial ion with double positive charge

$Mg_{Mg}^{x} + O_{O}^{x} \Leftrightarrow V_{Mg}^{\prime \prime } + V_{O}^{ \bullet \bullet } Mg_{surface}^{x} + O_{surface}^{x}$ :

A Kröger-Vink representation of a Frenkel defect formation in MgO