Vacancy-mediated diffusion

The first qualitative observation of vacancy-induced motion of embedded atoms was published in 1997 by Flores et al. [20], Using STM, an unusual, low mobility of embedded Mn atoms in Cu(0 0 1) was observed. Flores et al. argued that this could only be consistent with a vacancy-mediated diffusion mechanism. Upper and lower limits for the jump rate were established in the low-coverage limit and reasonable agreement was obtained between the experimentally observed diffusion coefficient and a theoretical estimate based on vacancy-mediated diffusion. That same year it was proposed that the diffusion of vacancies is the dominant mechanism in the decay of adatom islands on Cu(00 1) [36], which was also backed up by ab initio calculations [37]. After that, studies were performed on the vacancy-mediated diffusion of embedded In atoms [21-23] and Pd atoms [24] in the same surface. The deployment of a high-speed variable temperature STM in the case of embedded In and an atom-tracker STM in the case of Pd, allowed for a detailed quantitative investigation of the vacancy-mediated diffusion process by examining in detail both the jump frequency as well as the displacement statistics. Experimental details of both setups have been published elsewhere [34,35]. A review of the quantitative results from these studies is presented in the next subsections. 

One of the aspects associated with vacancy-mediated diffusion that differentiates it from the hopping mechanism on this surface is the long waiting time between consecutive jumps. An example of the distribution of waiting times has been plotted for both In and Pd in Fig. 5. 

In Section 3 we derive that for the vacancy-mediated diffusion mechanism, one expects the shape of the jump length distribution to be that of a modified Bessel function of order zero. Both distributions can be fit very well with the modified Bessel function, again confirming the vacancy-mediated diffusion mechanism for both cases. The only free parameter used in the fits is the probability prec for vacancies to recombine at steps, between subsequent encounters with the same embedded atom [33]. This probability is directly related to the average terrace width and variations in this number can be ascribed to the proximity of steps. The effect of steps will be discussed in more detail in Section 4. 

In this subsection we describe a discrete model for vacancy-mediated diffusion of embedded atoms, solve it numerically for the case of In/Cu(0 0 1), and present the results. Our model is defined on a two-dimensional simple square lattice of size / x / (typically, l = 401) centered around the origin. This corresponds to the top layer of a terrace of the Cu(00 1) surface, with borders representing steps. The role of steps in the creation/annihilation of vacancies will be discussed in more detail in the next section. All sites but two are occupied by substrate atoms. At zero time the two remaining sites are the impurity (or tracer) atom, located at the origin, and a vacancy at position (1,0). This corresponds to the situation immediately after the impurity atom has changed places with the vacancy. 

In this section, we discuss the energetics of the vacancy-mediated diffusion of tracer atoms in a surface. In particular, we focus on the energetics and the profound differences that are encountered in interpreting the results from the Pd/Cu(0 01) and the In/Cu(0 01) experiments. 

We should mention here that a better way to extract the activation energies for vacancy-mediated diffusion would be to plot the tracer diffusion coefficient of the embedded atoms vs. 1/kT, rather than their jump rate. As we discussed in Section 4, the mean square jump length depends on the proximity of steps, and so does the average jump frequency. This adds non-statistical noise to the two plots in Fig. 12. However, it can be shown easily that these effects on jump length and jump rate cancel in the resulting tracer diffusion coefficient, which thus becomes independent of the distance to steps. In this way, a more accurate value for the activation energy has been obtained for the case of In/Cu(0 0 1) of 717 30meV [23]. 

Surface vacancies were shown to be responsible for the motion of embedded In and Pd atoms in the Cu(00 1) surface. The density of surface vacancies at room temperature is extremely low, but they diffuse through the surface at an extremely high rate leading to significant diffusion rates of Cu(00 1) terrace atoms. In the STM measurements the rapid diffusion of these vacancies leads to long jumps of embedded tracer atoms. Measurements of the jump length distribution show a shape of the distribution that is consistent with the model that we discussed in Section 3. In turn, this shows that the vacancy-mediated diffusion process can be accurately described with the model that is presented in Section 3, provided that the interaction between the tracer atom and the surface vacancy is properly taken into... 

ORDER-ORDER TRANSITIONS BY VACANCY-MEDIATED DIFFUSION... 

In conclusion, it is quite difiicult, both from a theoretical and an experimental viewpoint, to establish the conditions for a homogeneous first-order phase transformation. A homogeneous character is, however, natural to higher-order transforma-tions. It is all the more remarkable that in our example of an order-order transition, which should meet the requirements for homogeneity in an exemplary way, a heterogeneous structure is introduced by restricted vacancy motion. The possibility of such behaviour should therefore always be kept in mind when dealing with vacancy-mediated diffusive phase transformations. 

As discussed in Section 7.2.3, radiation can induce segregation of alloy elements at defect sinks such as grain boundaries [101]. Typically, RIS is a result of inverse Kirkendall (IK) effects in which the evolution of defect concentration field drives the evolution of alloy composition field. ID rate theory modeling [44,101] is widely used to describe the coupled evolution between defect flux and composition flux. These rate theory models considered both vacancy-mediated and interstitial-mediated solute transport, as well as point defect recombination and defect loss to dislocations. At steady state, the solute segregation direction depends on the relative diffiisivity of different species-defect coupled diffusion. In austenitic Fe-Cr-Ni alloys, the vacancy-mediated solute diffusion alone is sufficient in describing the RIS trend and the interstitial-mediated solute diffusion is usually assumed to have a neutral contribution to RIS [44]. However, in Fe-Cr F/M alloys, both interstitial- and vacancy-mediated diffusion should be considered [102]. 

Tiwary et al. [71] applied this algorithm to the vacancy-mediated diffusion in iron and the plasticity and deformation of Au nanopiUars at realistic strain rates. In both cases, good agreement with the literature is found, and for the diffusion studies, an impressive boost factor of 10 was obtained, demonstrating the usefrilness of their technique in the field of condensed matter simulations. 

Figure 12.3 Vacancy-mediated diffusion of a tracer atom in fee (100) surface, (a) Vacancy approaches tracer atom, (b) tracer atom moves into vacancy, (c) vacancy moves away, and (d) frequent visit allows tracer to move over longer distances. (From Ref [21].)...

The problem of vacancy-mediated tracer diffusion in two dimensions has been investigated for a long time [40-44] and several different methods (simulation, analytical models, enumeration of trajectories, etc.) can be used to address it. The mathematics of this type of diffusion was solved first for the simplest case [41], when the diffusion of the vacancy is unbiased (all diffusion barriers are equal the tracer atom is identical to the other atoms), the lattice is two-dimensional and infinite. There is a single vacancy present that makes a nearest-neighbor move in a random direction at regular time intervals and has an infinite lifetime, as there are no traps. The solution is constructed by separating the motion of the tracer and that of the vacancy. The correlation between the moves of the tracer atom is calculated from the probability that the vacancy returns to the tracer from a direction, which is equal, perpendicular or opposite to its previous departure. The probability density distribution of the tracer atom spreads with... 

Activation energy for vacancy-mediated tracer diffusion... 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

The class of creep mechanisms of interest here are those that are mediated by stress-biased diffusion. If we are to consider the vacancy flux in a given grain within a material that is subjected to an applied stress, it is argued that the vacancy formation energy differs in different parts of the grain, and hence that there should be a gradient in the vacancy concentration leading to an associated flux. This... 

The results were consistent with a diffusion mechanism that was mediated by vacancy-like defects in the amorphous ceramics. 

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