Defects in Solids

DEFECTS IN SOLIDS

Real crystals are not perfect lattices of atoms; they always contain impurity atoms

and other defects.  These defects cause profound changes in properties.  An example

is the change in mechanical properties of metals, caused by dislocations and grain boundaries.  In this chapter lattice defects are discussed as point, line, and planar defects, emphasizing their structural features.  In later chapters their influence on properties will be considered. It is important that one not consider these defects as detrimental or bad. Many times the presence of defects, enhances the mechanical, chemical or electrical properties of a material.

Point Defects

In our previous discussions of  crystal structure in Chapters 7 and 13 it was assumed that each lattice site was filled with an atom of a particular kind.  However, it is

possible to have different no atom on a site (lattice vacancy), or an atom squeezed in between the normal lattice atoms (interstitial solid solution) or atoms on certain

lattice sites (substitutional solid solution). When this occurs, the lattice is no longer ideal, therefore vacancies, interstitials, and impurties are called "defects". Point defects are "points" which differ from the ideal lattice. Point defects are

thermodynamically stable, thus there will be a finite concentration of point defects at any given temperature. This is not true of line, planar and bulk defects which will be discussed later.

1.1.1   Vacancies

A typical sample of pure lead close to its melting point has a density of 11.3493 g/cm3. Were one to predict the density of lead, based on its lattice parameter, 0.4950nm, its crystal structure, fcc, and its molecular weight, 207.2 g/mole.

207.2    g

(4 atoms)                  mole     

m                      6.02 ´ 1023  atoms                  g

r

=      =                                        mole  = 11.35       

3                           3

V           (0.4950 ´ 10 -7 cm)          cm

The discrepancy is caused by the presence of vacancies, or missing atoms. The

fraction of vacant sites can be found from the density discrepancy,

11.35    g    - 11.3493    g

X =            cm3                    cm 3   = 6.1 ´ 10-5

11.35

g

cm3

or about 1 in every 16,000 atoms is missing. Typically at the melting point the

vacancy concentration is about 10-4. However, in any real material there is a finite vacancy concentration.

Consider the lattice with the vacancy shown in Figure 1. The formation of a vacancy from an ideal crystal can be considered to be a chemical reaction,

Ideal Lattice –> Real Lattice + Vacancy.

Note, this is not to imply that when heated the

lattice spits atoms into free space, but the reaction above can be used as a model to predict the equilibrium number of vacancies which exist at any given temperature.

Figure 1: bcc lattice with a single vacancy

The formation of a lattice vacancy in a crystal, by the reaction stated above, requires

energy as atomic bonds must be broken.To a good approximation this energy, (or enthalpy) )Hv, the formation enthalpy of the vacancy )Hv equals the bond energy of the lattice atoms times their coordination number. Thus for the lattice shown in

Figure 1, )Hv will equal eight times the Fe-Fe bond energy. Thus the stronger the bonds, the larger the enthalpy required for vacancy generation. If N is the number of vacancies formed per unit volume, the total enthalpy change is N)Hv.

However, the entropy change associated with the creation of vacancies is positive. There are two components to this entropy change. There is a vibrational entropy associated with each vacancy, )Svib, and there is the configurational change associated with the entire crystal. This can be described by the Boltzman equation,

N !

DS = k ln W = k ln                    

n !(N - n) !

k is Boltzman's Constant (R/NA), and S represents the number of ways, n vacancies

can be arranged on N lattice sites. If we make the approximation suggested by

Stirling that ln X! = XlnX-X for large values of X, then the configurational entropy becomes,

DS config

= -k[(X V ) ln (X V ) + (1 - X V ) ln (1 -  X V )]

where XV is the vacancy concentration. Note this expression describes the

configurational entropy change associated with any mixture. The algebraic

derivation has been skipped, because we want to emphasize the implications of this expression.

This means that the Gibbs Free Energy Change associated with vacancy formation

is,

DG = nDH v

- nTDS vib  - TDSconfig

DG = X V DG v

+  kT[(X V ) ln(X V ) + (1 -  X V ) ln (1 -  X V )]

Note )GV represents the "activation" energy required for vacancy formation. Several

authors neglect the vibrational component of the entropy.

The equilibrium concentration of vacancies at a given temperature can be found by finding the concentration corresponding to the minimum )G.

Duque Franky

C.I: 15.990.445

CRF

http://paws.kettering.edu/~mpalmer/ime301w2010/tutorials/defects.pdf

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