Unit 5 Diffusion
The oscillations of atoms are characterized by small average amplitudes up to temperatures near the melting point of the crystal. Nevertheless, some atoms may occasionally attain, even at temperatures below the melting point, a considerable energy that exceeds the mean for the whole crystal. Energy fluctuations of this kind may occur, for example, when all the atoms that are located on one side of the atom of interest simultaneously move towards this atom, thereby imparting to it a considerable momentum. If, moreover, all the atoms on the other side move away from the atom at the same instant, then it can be displaced from its normal position to another site in the structure. These thermally activated jumps of atoms or of other elements of the structure (eg. a vacancy) from one position of the structure to another constitute the fundamental mechanism of the migration of individual atoms in a crystal, known as diffusion. Here a distinction can be made between the diffusion of native atoms of the structure of a given crystal, known as self-diffusion, and the diffusion of atoms that are extraneous to the crystal structure.
The simplest mechanism of diffusion in the bulk of a crystal consists in a migration of the atom (ion) from a lattice point to its adjacent vacancy. This is referred to as the vacancy mechanism, since, in view of the very low vacancy concentration in a crystal and the indistinguishability of individual atoms (or ions) of one kind, it is advisable to consider this mechanism in terms of the movements of vacancies rather than atoms. In ionic crystals it is also reasonable to make a distiniction between the vacancy mechanism in the cationic sublattice and that in the anionic sublattice, since the cationic vacancies travel more easily in the cationic sublattice, and similarly for the anionic vacancies.
This is because in this situation the repulsion due to charges of the same sign is a minimum. Vacancies exhibit a measurable mobility at temperature equal to one fourth of the melting point of the substance, in Kelvins.
Another point defect capable of relatively easy migration in the structure is an interstitial ion (or atom). The measurable mobility of interstitials is observed just above a temperature of 273K. A typical mechanism of interstitial diffusion consists in jumps of interstitial atoms from one interstitial position to another. The jumps may also be effected in such a way that an interstitial ion knocks on an adjacent ion (atom) at a lattice point, which then jumps to an interstitial position and occupies its site. The latter mode is referred to as the interstitial mechanism with displacement.
The direct interchange of positions between neighboring ions (or atoms) with no participation of point defects is an alternative mechanism, whose variant is the ring mechanism. This interchange is accomplished here by a concerted motion of the ions of one kind along a ring-shaped circumference. A number of other variants and modifications of the mechanisms of diffusion in crystals may be conceived; however, due to lack of space, these will not be presented here. The interested reader is referred to the pertinent references listed at the end of this section.
In some crystals such as the tungsten bronzes and polyaluminates of the beta-Al2O3 type, voids are present in the perfect structure, and in concentrations that considerably exceed the equilibrium concentration of vacancies. In the zones containing voids diffusive jumps may occur with a greater facility than in the closepacked zones. Compared with crystals, glass also has a loose structure. Although the previous discussion was concerned primarily with diffusion in the crystal bulk it should also be borne in mind that in many cases, it is the high diffusivity paths that are of paramount importance. In addition to structural voids, the high diffusivity paths also comprise grain surfaces and boundaries, which are of significance for diffusion in polycrystals. For the above reasons the discussion of the diffusion in the bulk of crystals has been supplemented, wherever possible and practicable, by the data on diffusion in the high diffusivity paths.
Since atom jumps during diffusion require the surrounding structure to be strained, it is therefore intuitively evident that, for closepacked crystals, the energetically most preferred are the vacancy mechanism and the interstitial mechanisms.
Figure1.2. illustrates the jump of an atom from the interstitial position A to the adjacent position C, an element of the interstitial mechanism of diffusion. The atom here has to pass through an intermediate position B, which it can attain only if its energy is sufficient to overcome the repulsive forces exerted by the atoms surrounding this position. The changes in energy corresponding to the consecutive positions of the ion (or atom) are schematically shown in Figure 1.2. Therefore, for an atom to jump from one position to another the energy barrier has to be overcome, whose value is determined by the difference in the energy of the atom in point B and point A or C (energy minimum). Accordingly, for an atomic jump a definite activation energy, E*, is required.
Fig.1.2 Successive stages of the jump of an interstitial atom to an adjacent interstitial position (a-c); changes in energy of an atom during such (d) (after P.D shewan Diffision in Solids, McGraw-Hill, New York, 1963)
Each oscillation of an atom at an equilibrium position can actually be regarded as an attempted jump to an adjacent position. In the majority of cases, however, these attempts are unsuccessful to the energy barrier; yet once in a while a thermal fluctuation arises at a given crystal site which enables the atom to overcome this barrier. Therefore, the frequency with which atoms jump among interstitial positions is, to a first approximation, proportional to the product of the frequency of oscillation of an atom and the probability of occurrence of a fluctuation large enough for an atom (ion) to acquire an energy equal to at least E*. If Γat,i denotes the frequency of atomic jumps by the interstitial mechanism, v the frequency of oscillation of atoms (ions), and Pl the probability that a thermal fluctuation will occur that is high enough to impart to one interstitial atom an energy equal to E*, the frequency of the jumps is given by Γat,i=vPl.
Γ
(1.1)
Where k is Boltzmann's constant and T is the absolute temperature. The jump frequency tells us how often an atom abandons its normal position, but says nothing about the direction the atom will choose during its jump. As expected, the choice of direction of successive jumps of interstitial ions is random. At a temperature of 1300K, the frequency of oscillations, v, is of the order of 10[13]s-1, and the value of Γi, found from equation (1.1) for typical E* values, amounts to 10[7]-10[8]s-1. In other words, an atom (or ion) remains in the interstitial position for a long time between successive jumps. During that time an equilibrium condition is established between the atom of interest and the crystalline environment; from 10 to 50 vibrations are sufficient to effect this. As a result, the momentum of the atom in a specific direction, imparted to that atom during the jump, has been lost before a subsequent jump occurs. Additionally, the atom in an interstitial position has identical surroundings; at a moderate concentration of point defects, an atom is surrounded, for example, in a cubic structure, by six equivalent interstitial positions, and the choice of any one of these is entirely haphazard.
Selected from "Constitution and Properties of Ceramic Materials". R.Pampuch, Elservier Publ. 1991
Words and Expressions
diffusion 扩散
oscillation 振荡
fluctuation 起伏
momentum 动量
vacancy mechanism 空位机制
indistinguishability 不可区分
interstitial mechanism 间隙机制
ring mechanism 环形机制
displacement 位移
circumference 圆周
tungstun bronze 乌青铜
polyaluminate 多聚铝酸盐
void 空的
consecutive 连续的
activation energy 活化能
haphazard 偶然性
