22.08.2015   Comprehensive nuclear materials

Transition from Atomic to Continuum Diffusion

During the migration of a point defect through the crystal lattice, it traverses an energy landscape that is schematically shown in Figure 18. The energy minima are the stable configurations where the defect energy is equal to Ef(r), the formation energy, but modified by the interactions with inter­nal and external strain fields, which in general vary with the defect location r. In order to move to the adjacent energy minimum, the defect has to be thermally activated over the saddle point that has an energy

ES(r)=Ef (r)+Em(r) [69]

where Em(r) is the migration energy. As the prop­erties of the point defect, such as its dipole tensor and its diaelastic polarizability, are not necessarily the same in the saddle point configurations as in the stable configuration, the interactions with the strain fields are different, and the envelope of the saddle point energies follows a different curve than the envelope of the stable configuration energies, as indicated in Figure 18. For a self-interstitial, we

image077

Figure 18 Schematic of the potential energy profile for a migrating defect.

must also consider the different orientations that it may have in its stable configuration. Accordingly, let Cm(r, t) be the concentration of point defects at the location r and at time t with an orientation m. For instance, the point defect could be the self­interstitial in an fcc crystal, in which case, there are three possible orientations for the dumbbell axis and m may assume the three values 1, 2, or 3 if the axis is aligned in the xb x2, or x3 direction, respec­tively. The elementary process of diffusion consists now of a single jump to one adjacent site at r + R, where R is one of the possible jump vectors.

The rate of change with time of the concentration Cm (r, t) is now given by

—C

= Cn(r — R, t)Lnm(r — R | R)

r, v

‘У ‘ Cm(r, t)Lmn(r 1 R) [70]

r, v

Here, the first term sums up all jumps from neigh­boring sites to site r thereby leading to an increase of Cm (r, t), while the second term adds up all the jumps (really the probabilities of jumps) out of the site r. The frequency (or better the probability) of a partic­ular jump from r to r + R while changing the orien­tation from m to V is denoted by Lmn(r | R). The eqn [70] applies now to each of the possible orientations, and it appears that this leads to as many diffusion equations as there are possible orientations, and these equations may be coupled if the defect can change its orientation between jumps.

To circumvent this complication, one considers an ensemble of identical systems, all having identical microstructures, and identical internal and external stress fields. The ensemble average of the defect concentration at each site, denoted simply as C(r, t) without a subscript, is now assumed to be the ther­modynamic average such that exp(-bEm)

£exp(-bEvf)

= C(r, 0exp(-b4)/N(r)

where the normalization factor N only depends on the location r as do the energies for the stable defect configurations.

Substituting this into eqn [70] on both sides constitutes another assumption. To see this, suppose that the defect concentrations CV(r — R, t) on all neighbor sites happen, at the particular instance t, to be aligned in one direction. Since their new

image078

alignments after the jump to site r is correlated with the jump vector R and the previous orienta­tion, the added defect population does not possess the equilibrium distribution of eqn [71]. However, Kronmuller et a/.33 argue that after several sub­sequent jumps of defects from the neighbors to this site r, the earlier deviation from the equilibrium distribution will have died out. Thus, introducing the thermodynamic averages on both sides of the eqn [70] is a plausible approximation. To proceed further requires a more specific form of the jump probability. For a jump from the site r to r + R, it is assumed that

 

and a drift force as

 
image079

bElv(r+1r)+bE (r)

 

[77]

 

Fi (r)

 

The components of the jump vector R are denoted by capital letters Xi, while the components of the loca­tion vector r are given by the lower case letters x. The normalization factor N(r) is replaced in the eqns [76] and [77] with an exponential function of the average defect formation energy according to

 
image080

—bf (r) [78]

 

exp

 
image081

It is important to emphasize, as Dederichs and Schroder34 first did, that the above Taylor expansion does not remove the dependence of the saddle point energy on the jump direction R. To what degree it still depends on the jump direction is a function of the crystal lattice and magnitudes ofthe elastic strains.