Thermally Activated Atomic Jumping

The fundamental process in atomistic diffusion models is the thermally activated jump between neighboring sites of local minimum energy. The duration of any jump is typically very short compared to the particle s residence time in a minimum-energy site. Therefore, the average jump rate—the basis for any model of atomistic diffusive motion—is essentially inversely proportional to the average residence time. 

The residence time depends upon the probability that the local potential energy will undergo a fluctuation large enough to enable the particle to surmount the 

Kinetics of Materials. By Robert W. Balluffi, Samuel M. Allen, and W. Craig Carter. 145 Copyright 2005 John Wiley Sons, Inc. 

There are numerous approaches to modeling the jump rate.1 Below, three progressively more realistic models are presented. All three approaches produce the same basic result—the jump rate is a product of the vibration frequency in the initial stable site and a Boltzmann probability of a sufficient energy fluctuation for the jump. 

---

Of particular interest are those surfaces where AFM has provided complementary information or revealed surface stmcture which could not be obtained by STM. One obvious application is the imaging of insulators such as NaCl(OOl) [120]. In this case it was possible to observe point defects and thermally activated atomic jump processes, although it was not possible to assign the observed maxima to anion or cation. 

It has been recognized that the behavior of atomic friction, such as stick-slip, creep, and velocity dependence, can be understood in terms of the energy structure of multistable states and noise activated motion. Noises like thermal activities may cause the atom to jump even before AUq becomes zero, but the time when the atom is activated depends on sliding velocity in such a way that for a given energy barrier, AI/q the probability of activation increases with decreasing velocity. It has been demonstrated [14] that the mechanism of noise activation leads to "the velocity... 

Once temperature comes into play, the jumps of atoms between minima may be invoked prematurely, i.e., before the formation of instabilities, via thermal fluctuations. These thermally activated jumps decrease the force that is required to pull the surface atom, which leads to a decrease in the kinetic friction. The probability that a jump will be thermally activated is exponentially related to the energetic barrier of the associated process, which can be understood in terms of Eyring theory. In general, the energetic barriers are lower when the system is not at its thermal equilibrium position, which is a scenario that is more prominent at higher sliding velocities. Overall, this renders Fk rate or velocity dependent, typically in the following form ... 

Macroscopic treatments of diffusion result in continuum equations for the fluxes of particles and the evolution of their concentration fields. The continuum models involve the diffusivity, D, which is a kinetic factor related to the diffusive motion of the particles. In this chapter, the microscopic physics of this motion is treated and atomistic models are developed. The displacement of a particular particle can be modeled as the result of a series of thermally activated discrete movements (or jumps) between neighboring positions of local minimum energy. The rate at which each jump occurs depends on the vibration rate of the particle in its minimum-energy position and the excitation energy required for the jump. The average of such displacements over many particles over a period of time is related to the macroscopic diffusivity. Analyses of random walks produce relationships between individual atomic displacements and macroscopic diffusivity. 

Single-Component System with Isotropic Interfaces and No Strain Energy. This relatively simple case could, for example, correspond to the nucleation of a pure solid in a liquid during solidification. For steady-state nucleation, Eq. 19.16 applies with AQC given by Eq. 19.4 and it is necessary only to develop an expression for /3C. In a condensed system, atoms generally must execute a thermally activated jump over a... 

The validity of Coulomb s law has been verified also on the nanoscale Zworner et al. [484] showed that, for different carbon compound surfaces, friction does not depend on sliding velocity in the range between 0.1 /xm/s and up to 24 /xm/s. At low speeds, a weak (logarithmic) dependence of friction on speed was observed by Gnecco et al. [485] on a NaCl(lOO) surface and by Bennewitz et al. [486] on a Cu (111) surface. This can be modeled when taking into account thermal activation of the irreversible jumps in atomic stick-slip [487],... 

Diffusion in interstitial solid solutions occurs by interstitially dissolved atoms jumping from one interstitial site to another. For an atom to move from one interstitial site to another, it must pass through a position where its potential energy is a maximum. The difference between the potential energy in this position and that in the normal interstitial site is the activation energy for diffusion and must be provided by thermal fluctuations. The overall diffusion rate is governed by an Arrhenius-type rate equation,... 

The self-diffusivity D of an atom or ion is a measure of the ease and frequency with which that atom or ion jumps around in a crystal lattice in the absence of external forces, i.e., in a totally random fashion. Experimentally, it has long been appreciated that D is thermally activated and could be expressed as... 

Organic molecules are not so close packed as the atoms of metals or metalloids. In contrast, they are quite far apart from each other, and, in addition, are only held together by weak van der Waals forces. Thus, the electronic interaction between organic molecules is only small. Charges can only be transported by thermally activated jumping from defect to defect, and the mobility of the charge carriers is low. 

We outline briefly the physical basis of the Arrhenius equation as applied to atomic or molecular movements in a solid. All viscoelastic effects, including large strain effects (see Gtapter S), are due to thermally activated movements of segments of macromolecules under imposed mechanical stress. There is no question of the stress generating molecular movements which, in the absence of the stress, would not take place. What in fact occurs is that molecular movements or jumps occur spontaneous, and it is the function of the stress to bias them so that th no longer occur in random directions. This results in a molecular flux which leads to time-dependent mechanical strain. 

Movement of an atom or ion from one position to another is a thermally activated process. The probability Pq per mole that one atom will leave its position to jump to a neighboring available position is... 

The model of harmonic oscillations of the Mossbauer atom as described above, is quite satisfactory for solids at low temperatures. However, at higher temperatures the smaU amplitude motion of the atom around its equilibrium site may become unstable such that the atom has a flnite probability of making a sudden jump to an adjacent lattice site. Although this type of motion can in principle occur even at very low temperatures via the mechanism of quantum tunnelling, there is little evidence for this phenomenon with Mossbauer isotopes. The diffusive motion can therefore be considered in terms of thermally activated jumps... 

The practical importance of vacancies is that they are mobile and, at elevated temperatures, can move relatively easily through the crystal lattice. As illustrated in Fig. 20.21b, this is accompanied by movement of an atom in the opposite direction indeed, the existence of vacancies was originally postulated to explain solid-state diffusion in metals. In order to jump into a vacancy an adjacent atom must overcome an energy barrier. The energy required for this is supplied by thermal vibrations. Thus the diffusion rate in metals increases exponentially with temperature, not only because the vacancy concentration increases with temperature, but also because there is more thermal energy available to overcome the activation energy required for each jump in the diffusion process. 

---