1.01.3.1 Vacancy Formation

The thermal vibration of atoms next to free surfaces, to grain boundaries, to the cores of dislocations, etc., make it possible for vacancies to be created and then diffuse into the crystal interior and establish an equi­librium thermal vacancy concentration of 4 — tsV

kBT.

given in atomic fractions. Here, 4 is the vacancy formation enthalpy, and SV is the vacancy formation entropy. The thermal vacancy concentration can be measured by several techniques as discussed in Dam­ask and Dienes,4 Seeger and Mehrer,5 and Siegel,6 and values for 4 have been reviewed and tabulated by Ehrhart and Schultz;7 they are listed in Table 2. When these values for the metallic elements are plotted versus the melt temperature in Figure 4, an approximate correlation is obtained, namely

4 ~ Tm/1067 [3]

4000 3500

2- 3000

о о

Ь 2500

0 >

2000

га

1 1500 “ 1000

500 0

0 500 1000 1500 2000 2500 3000 3500 4000

Melt temperature (K)

Figure 3 Leibfried’s empirical rule between melting temperature and the product of bulk modulus and atomic volume.

Using the Leibfried rule, a new approximate cor­relation emerges for the vacancy formation enthalpy that has become known as the cBO model8; the con­stant c is assumed to be independent of temperature and pressure. As seen from Figure 5, however, the experimental values for 4 correlate no better with BO than with the melting temperature.

It is tempting to assume that a vacancy is just a void and its energy is simply equal to the surface area 4nR2 times the specific surface energy g0. Taking the atomic volume as the vacancy volume, that is, O = 4pR 3/3, we show in Figure 6 the measured vacancy formation enthalpies as a function of 4nR2y0, using for g0 the values9 at half the melting temperatures. It is seen that 4 is significantly less, by about a factor of two, compared to the surface energy of the vacancy void so obtained. Evidently, this sim­ple approach does not take into account the fact that the atoms surrounding the vacancy void relax into new positions so as to reduce the vacancy volume 4 to something less than O. The difference

VVel = VV — O [4]

is referred to as the vacancy relaxation volume. The experimental value7 for the vacancy relaxation of Cu is —0.25O, which reduces the surface area of the vacancy void by a factor of only 0.825, but not by a factor of two.

The difference between the observed vacancy for­mation enthalpy and the value from the simplistic surface model has recently been resolved. It will be shown in Section 1.01.7 that the specific surface energy is in fact a function of the elastic strain tan­gential to the surface, and when this surface strain relaxes, the surface energy is thereby reduced. At the same time, however, the surface relaxation creates a stress field in the surrounding crystal, and hence a strain energy. As a result, the energy of a void after relaxation is given by

FC [e(R), £*] = 4-pR2g[e(R), £*] + 8nR3me2(R) [5]

The first term is the surface free energy of a void with radius R, and it depends now on a specific surface energy that itself is a function of the surface strain e(R) and the intrinsic residual surface strain e* for a surface that is not relaxed. The second term is the strain energy of the surrounding crystal that depends on its shear modulus m. The strain dependence of the specific surface energy is given by

g[e, e*]=g0 + 2(mS + 1s)(2£* + e)e [6]

Here, g0 is the specific surface energy on a surface with no strains in the underlying bulk material.

However, such a surface possesses an intrinsic, resid­ual surface strain e*, because the interatomic bonding between surface atoms differs from that in the bulk, and for metals, the surface bond length would be shorter if the underlying bulk material would allow the surface to relax. Partial relaxation is possible for small voids as well as for nanosized objects. In addition to the different bond length at the surface, the elastic constants, m and Is, are also different from the corresponding bulk elastic constants. However, they can be related by a surface layer thickness, h, to bulk elastic constants such that

+ Is = (m т i)h = mh/ (1 — 2v) [7]

where l is the Lame’s constant and v is Poisson’s ratio for the bulk solid. Computer simulations on
freestanding thin films have shown10 that the surface layer is effectively a monolayer, and h can be approxi­mated by the Burgers vector b. For planar crystal sur­faces, the residual surface strain parameter e* is found to be between 3 and 5%, depending on the surface orien­tation relative to the crystal lattice. On surfaces with high curvature, however, e* is expected to be larger.

The relaxation of the void surface can now be obtained as follows. We seek the minimum of the void energy as defined by eqn [5] by solving dFc/де = 0. The result is

(ms + ^s)e* h e*

m’R + (ms + Is) (1 — 2v)R + h and this relaxation strain changes the initially unre­laxed void volume

0

0 500 1000 1500 2000 2500 3000 3500 4000

Melting temperature (K)

Figure 4 Vacancy formation energies as a function of melting temperature.

0

0 1 2 3 4 5

Surface energy of a vacancy (eV)

Figure 6 Correlation between the surface energy of a vacancy void and the vacancy formation energy.

4

3.5 3

2.5 2

1.5 1

0.5 0

0 5 10 15 20 25 30 35 40

Bulk modulus * atomic volume (eV)

Figure 5 Vacancy formation energy versus the product of bulk modulus and atomic volume.

n O = R3 [9] consisting of n aggregated vacancies, by the amount Vrel (R) = 3nOe(R) [10]
4pR2
2 + 3mO
[12]
This equation is evaluated for Ni and the results are shown in Figure 7 as a function of the vacancy relaxa­tion volume I VVel/O I. It is seen that relaxation volumes of -0.2 to -0.3 predict a vacancy formation energy comparable to the experimental value of 1.8 eV.
and for the vacancy formation energy

Few experimentally determined values are avail­able for the vacancy relaxation volume, and their accuracy is often in doubt. In contrast, vacancy for­mation energies are better known. Therefore, we use eqn [12] to determine the vacancy relaxation volumes from experimentally determined vacancy formation energies. The values so obtained are listed in Table 3, and for the few cases7 where this is possible, they are compared with the values reported from experiments. Computed values for the vacancy relaxation volumes are between —0.2O and —0.3O for both fcc and bcc metals. The low experimental values for Al, Fe, and Mo then appear suspect.

The surface energy model employed here to derive eqn [12] is based on several approximations: isotropic, linear elasticity, a surface energy parame­ter, g0, that represents an average over different crys­tal orientations, and extrapolation of the energy of large voids to the energy of a vacancy.

Nevertheless, this approximate model provides satisfactory results and captures an important con­nection between the vacancy relaxation volume and the vacancy formation energy that has also been noted in atomistic calculations.

Finally, a few remarks about the vacancy forma­tion entropy, SV, are in order. It originates from the change in the vibrational frequencies of atoms sur­rounding the vacancy. Theoretical estimates based on empirical potentials provide values that range from 0.4k to about 3.0k, where k is the Boltzmann constant. As a result, the effect of the vacancy formation entropy on the magnitude of the thermal equilibrium vacancy con­centration, CVq, is of the same magnitude as the statisti­cal uncertainty in the vacancy formation enthalpy.