So far, when discussing the properties of materials we tacitly as-sumed that the atoms of solids remain essentially stationary. From time to time we implied, however, that the behavior of solids is af-fected by the thermally induced vibrations of atoms. The changes in properties increase even more when atoms migrate through the lattice and take new positions. In order to gain a deeper insight into many mechanical properties, we therefore need to study the “dy-namic case.” It will become obvious during our endeavor that the motion of atoms through solids involves less effort (energy) when open spaces are present in a lattice, as encountered, for example, by empty lattice sites. Thus, we commence with this phenomenon.
We have repeatedly pointed out in previous chapters that an ideal lattice is rarely found under actual conditions, that is, a lattice in which all atoms are regularly and periodically arranged over large distances. This is particularly true at high temperatures, where a substantial amount of atoms frequently and randomly change their positions leaving behind empty lattice sites, called vacancies. Even at room temperature, at which thermal motion of atoms is small, a fair number of lattice defects may still be found. The number of vacancies per unit volume, nv, increases exponentially with the absolute temperature, T, according to an equation whose generic type is commonly attributed to Arrhenius:1 nv ns exp
冢
kEBTf冣
, (6.1)6.1.1 Lattice Defects
Atoms in Motion
6.1 • Lattice Defects and Diffusion
1Svante August Arrhenius (1859–1927), Swedish chemist and founder of modern physical chemistry, received in 1903, as the first Swede, the Nobel prize in chemistry. The Arrhenius equation was originally formulated by J.J.
Hood based on experiments, but Arrhenius showed that it is applicable to almost all kinds of reactions and provided a theoretical foundation for it.
where nsis the number of regular lattice sites per unit volume, kB
is the Boltzmann constant (see Appendix II), and Efis the energy that is needed to form a vacant lattice site in a perfect crystal.
As an example, at room temperature, nv for copper is about 108 vacancies per cm3, which is equivalent to one vacancy for every 1015lattice atoms. If copper is held instead near its melt-ing point, the vacancy concentration is about 1019cm3, or one vacancy for every 10,000 lattice atoms. It is possible to increase the number of vacancies at room temperature by quenching a material from high temperatures to the ambient, that is, by freez-ing-in the high temperature disorder, or to some degree also by plastic deformation.
Other treatments by which a large number of vacancies can be introduced into a solid involve its bombardment with neutrons or other high energetic particles as they exist, for example, in nuclear reactors (radiation damage) or by ion implantation. These high en-ergetic particles knock out a cascade of lattice atoms from their po-sitions and deposit them between regular lattice sites (see below).
It has been estimated that each fast neutron may create between 100 and 200 vacancies. At the endpoint of a primary particle, a de-pleted zone about 1 nm in diameter (several atomic distances) may be formed which is characterized by a large number of vacancies.
Among other point defects are the interstitials. They involve for-eign, often smaller, atoms (such as carbon, nitrogen, hydrogen, oxy-gen) which are squeezed in between regular lattice sites. The less common self-interstitials (sometimes, and probably not correctly, called interstitialcies) are atoms of the same species as the matrix that occupy interlattice positions. Self-interstitials cause a sub-stantial distortion of the lattice. In a dumbbell, two equivalent atoms share one regular lattice site. Frenkel defects are vacancy/inter-stitial pairs. Schottky defects are formed in ionic crystals when, for example, an anion as well as a cation of the same absolute va-lency are missing (to preserve charge neutrality). Dislocations are one-dimensional defects (Figure 3.20). Two-dimensional defects are formed by grain boundaries (Figure 3.15) and free surfaces at which the continuity of the lattice and therefore the atomic bonding are disturbed. We shall elaborate on these defects when the need arises.
Vacancies provide, to a large extent, the basis for diffusion, that is, the movement of atoms in materials. Specifically, an atom may move into an empty lattice site. Concomitantly, a vacancy migrates in the opposite direction, as depicted in Figure 6.1. The prerequi-site for the jump of an atom into a vacancy is, however, that the atom possesses enough energy (for example, thermal energy) to squeeze by its neighbors and thus causes the lattice to expand momentarily and locally, involving what is called elastic strain
6.1.2
Diffusion
Mechanisms
Diffusion by
Vacancies
energy. The necessary energy of motion, Em, to facilitate this ex-pansion is known as the activation energy for vacancy motion, which is schematically represented by an energy barrier shown in Figure 6.1. Em is in the vicinity of 1eV. The average thermal (ki-netic) energy of a particle, Eth, at the temperatures of interest, how-ever, is only between 0.05 to 0.1 eV, which can be calculated by making use of an equation that is borrowed from the kinetic the-ory of particles (see textbooks on thermodynamics):
Eth32kBT. (6.2) This entails that for an atom to jump over an energy barrier, large fluctuations in energy need to take place until eventually enough energy has been “pooled together” in a small volume. Diffusion is therefore a statistical process.
A second prerequisite for the diffusion of an atom by this mech-anism is, of course, that one or more vacancies are present in neighboring sites of the atom; see Eq. (6.1). All taken, the acti-vation energy for atomic diffusion, Q, is the sum of Ef and Em. Specifically, the activation energy for diffusion for many ele-ments is in the vicinity of 2 eV; see Table 6.1
If atoms occupy interstitial lattice positions (see above), they may easily diffuse by jumping from one interstitial site to the next without involving vacancies. Interstitial sites in FCC lattices are, for example, the center of a cube or the midpoints between two corner atoms. Similarly as for vacancy diffusion, the adjacent matrix must slightly and temporarily move apart to let an inter-stitial atom squeeze through. The atom is then said to have
dif-Interstitial Diffusion
atom Em vacancy
distance E
FIGURE6.1. Schematic representation of the diffusion of an atom from its former position into a vacant lat-tice site. An activation energy for motion, Em, has to be applied which causes a momentary and local ex-pansion of the lattice to make room for the passage of the atom. This two-dimensional representation shows only part of the situation. Atoms above and below the depicted plane may contribute likewise to diffusion.
fused by an interstitial mechanism. This mechanism is quite com-mon for the diffusion of carbon in iron or hydrogen in metals but can also be observed in nonmetallic solids in which the dif-fusing interstitial atoms do not distort the lattice too much. The activation energy for interstitial diffusion is generally lower than that for diffusion by a vacancy mechanism (see Table 6.1), par-ticularly if the radius of the interstitial atoms is small compared to that of the matrix atoms. Another contributing factor is that the number of empty interstitial sites is generally larger than the number of vacancies. In other words, Ef (see above) is zero in this case.
If the interstitial atom is of the same species as the matrix, or if a foreign atom is of similar size compared to the matrix, then the diffusion takes place by pushing one of the nearest, regular lattice atoms into an interstitial position. As a result, the former interstitial atom occupies the regular lattice site that was previ-ously populated by the now displaced atom. Examples of this mechanism have been observed for copper in iron or silver in AgBr.