User:LeaveSleaves/Point defects
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Point defects in metals
[edit]Of the various defects the vacancy is the only species that is ever present in appreciable concentrations in thermodynamic equilibrium and increases exponentially with the rise in temperature. The vacancy is formed by removing an atom from its lattice site and depositing it in nearby atomic site where it can be easily accommodated. Favored places are free surface, grain boundary of extra half plane of edge dislocation. Such sites are called vacancy sources and the vacancy is created when sufficient energy is available to remove the atom. If Ef is the energy to remove one atom, the energy required to create ‘n’ such vacancies would be nEf. The accompanying entropy is given by the relation
where W is the number of ways of distributing n defects and N atoms in N+n sites i.e.
Thus, the free energy F of crystal with n defects relative a perfect crystal is
(1)
![{\displaystyle F=nE_{f}-kT\ln \left[{\frac {(N+n)!}{n!N!}}\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13d7dfa12d3f6d81ac1245bbf90a5f98c8a1920d)
Using Sterling’s theorem, i.e.
(2)
![{\displaystyle F=nE_{f}-kT\left[(N+n)\ln(N+n)-n\ln n-N\ln N\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca92345a71c243eeb261f71e750050d6dec08e3)
The equilibrium value of n is that for which dF/dn = 0, which defines state of min energy. Differentiating and solving (2)
Usually n<<N, so the expression can be simplified to give concentration of vacancies as
The effect of vacancies on vibrational properties of lattice also leads to inclusion of an entropy term, which is independent of temperature
The equilibrium value of vacancies rapidly due to exponential nature of the equation, rising to about 10-4 around melting point. As temperature is lowered, C is decreased in order to maintain equilibrium and to do this vacancy must migrate back to positions in the structure where they can be annihilated; these sites are called as vacancy sinks and include such places as free surface, grain boundaries and dislocations.
The defect migrates by moving through the energy maxima from one atomic site to another with a frequency
where ν0 – frequency of vibration defect in appropriate direction Sm – increase in entropy Em – increase in internal energy
The self-diffusion coefficient in a pure metal is associated with energy to form a vacancy Ef and energy to move a vacancy Em.
The dislocations are most efficient sinks due to their proximity to the center of the grain as compared to grain boundary or free surface. This process is reversible and at high temperatures, these same sites act as vacancy sources to increase concentration.
Vacancies in structure can retained at lower temperature due slow migrations. If the cooling rate for the metal is high, e.g. in quenching, high temperature vacancies are preserved.
Vacancies play significant role in behavior of metal in different processes, including annealing, creep, homogenization, precipitation, sintering, surface hardening, oxidation etc. especially by movement of atoms through structure with the help of vacancies. Vacancies also help dislocation climb, which is an important mechanism in recovery stage of annealing and softens materials under creep at higher temperatures.
The energy formation of interstitials in metals is very high. Their concentration around melting point is about 10-15 and hence are of little consequence. They are however important in ceramics, in materials where point defects are produced by jogs in screw dislocation and in irradiadiated materials.