Vacancies have been seen to be thermodynamically stable in crystalline solids above 0°K, and that their number increase exponentially with the increase of temperature, as illustrated in Fig. 4.8 for aluminium. To understand this, the usual concept of Gibbs free energy for metal-system is used.
In a perfect crystal, all lattice sites are occupied by atoms so that no vacancies are present. If Ef is the energy required to create a vacancy by removing an atom from the lattice site (Fig. 4.2) and placing it in a normal site on the crystal surface. The total increase in energy due to the formation of n such defects is nEf. Let there be N atoms on the lattice sites. Ef is given in KJmole-1, or eV/defect.
The free energy of a crystal containing vacancies will be different from that of a crystal free of vacancies.
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For most processes in solid metals at atmospheric pressure, whether or not a process occurs spontaneously depends on the function:
where, ΔS is the entropy change due to the vacancies.
The entropy of the crystal is increased in the presence of vacancies for two reasons:
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1. Vibrational Entropy:
Atoms adjacent to a vacancy are less restrained than those which are completely surrounded by the atoms and thus, can vibrate more randomly. Each vacancy contributes a small amount of vibrational entropy to the total entropy of the crystal.
If s is this contribution by one vacancy, then the total increase in entropy due to n vacancies is ns, if n is the total number of vacancies. A complete theoretical treatment of effect of vacancies requires inclusion of vibrational entropy, but as it has very secondary importance, it is normally omitted, as is being done here too.
2. Entropy of Mixing (Configurational Entropy):
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Statistical mechanics has shown that entropy may be expressed as:
where, k is the Boltzman’s constant, and w is the number of ways of distributing n defects and N atoms on N + n lattice sites. Students must have already come across such a treatment for expressing entropy for the mixing of two ideal gases, or for the making of solid solutions, where solute and solvent atoms are mixed Now, the treatment for the n vacancies can be derived.
Vacancies are generated in a crystal by expending a certain amount of work, which causes an increase in the internal energy. Since these defects can be distributed among the available lattice sites in a number of ways, the configuration entropy or entropy of mixing is positive. At any temperature above 0°K, the free energy is a minimum for a certain concentration of defects determined by the balance of the internal energy and entropy terms.
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The number of ways, w (equation 4.3), in which n defects (vacancies here) can be arranged on N + n lattice sites is:
As N is very large compared to n, and thus, atomic concentration of lattice vacancies
(we have omitted vibrational entropy term as it comes out to be independent of the temperature, and is usually written as exp [Sf/k]. The equation 4.10 becomes n/N = exp (Sf/k) exp [ – Ef/kT], The value of the term exp (Sf/k) is not accurately known but is normally taken as unity. Thus, equation 4.10 gives the concentration of vacancies)
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Fig. 4.9 illustrates that at temperature, T1 (close to melting point), the work to create vacancies increases linearly with the number of vacancies. At low concentration of vacancies, the entropy component (- TS) increases very rapidly with the increase in number of vacancies but at a decreasing rate as the number becomes large.
At n1, the two terms nEf and – T1S balance, i.e., at this point d ΔG/dn = 0, and this is the equilibrium concentration of vacancies at T1 temperature in the solid. Thus, a crystalline solid has an equilibrium concentration of vacancies at a temperature. The increase in entropy of mixing is responsible for making vacancies as stable defects in the crystalline solids.
At lower temperature T2 (T1 > T2), the nEf curve remains unchanged, but the curve -T2 S (shown as dotted) has continuously lower magnitude as T2 < T1, and thus, the minimum is shifted towards the left to n2 i.e. as the temperature decreases, the equilibrium number of vacancies become less because the entropy component (-TS) decreases.
The vacancies migrate to positions in the lattice where, they can be annihilated, i.e., they diffuse to positions called ‘vacancy sinks’, such as the free surfaces, grain boundaries and dislocations. These places are also the source of vacancies, when their number has to increase because the metal is heated to still higher temperatures.
However, below a certain temperature, the migration of vacancies becomes very slow for the equilibrium to be maintained, i.e., there are concentration of vacancies in excess of the equilibrium number in the lattice. If a heated metal is suddenly quenched, the vast majority of the vacancies which were present at high temperature can be ‘frozen-in’. The density of the metal now shall be less.
The quenched-in-vacancies play quite important role in many physical processes, such as for the formation of precipitates in age-hardening. These vacancies help in the process of diffusion of solute atoms, or even self-diffusion. In micro-electronics, for example, the specific regions of silicon must be alloyed (called doped) with a definite concentration of a soluble impurity such as boron to produce the desirable electronic properties.
It is done at 1200°C (high concentration of vacancies) for a little more than 1½ hours. Vacancies help the atoms of boron to dissolve inside the silicon by the process of diffusion. Fig. 4.10 shows the representative vacancy concentration in some metals as a function of temperature. Table 4.2 gives enthalpy of formation of vacancy in some metals.
The above discussion gives two interesting results:
1. A perfect metal (crystalline solid) is not thermodynamically stable because the free energy is lowered when the vacancies are created.
2. The number of vacancies is a strong function of temperature. Actually, most of the point defects have an equilibrium concentration whose magnitude depends primarily on the Ef of formation of the defect. Can we say that a pure metal is really thermodynamically unstable relative to impure metal? For example, the energy of formation of a copper interstitialcy is seven times of the work required to produce a vacancy.
Thus, the fraction of interstitialcy = 10 -35400/T and at 1000°C, there is one interstitialcy per 1010 grams of copper. Thus, such defects normally have not much role to play.
A similar treatment leads to the number of Schottky defects as:
where N is the number of atoms and N1 is the number of possible interstitial voids. Here in equations 4.11 and 4.12, the factor 2 appears because the energy of formation ES or EF refers to the formation of the defect pair.
Problem:
Find the equilibrium concentration of vacancies in aluminium at – 273°C, and 27°C. Given, Ef = 68 x 103 J/mole.
Solution:
