1.01.8.1 Effective Medium Approach

The fate of the radiation-produced atomic defects, namely self-interstitials and vacancies, is mainly determined at elevated temperatures by their diffu­sion from the places where they were created to the sinks where they are absorbed or annihilated, as in the encounter of an interstitial and a vacancy. As there are many sinks within each grain of an irradiated material, the spatial distribution of the atomic defects requires the solution of a very complex diffusion problem. Clearly, some approximations must be sought to arrive at an acceptable solution. First, we assume that the rate of defect generation, that is, the rate of displacements, is constant, and the defect concentrations between the sinks no longer changes with time or adjust rapidly when the number of sinks and their arrangement changes. Then within the regions between sinks, the diffusion fluxes are stationary, that is,

j = ~dXD, J (Г)С (r)+F'(r)C(r) [128]

is independent of time.

The task is then to divide the solid into cells, each containing one individual sink, and to solve in each diffusion equations of the following type

V • j = P — recombination [129]

for each mobile defect. Here, P is the rate of defect production per unit volume generated by the radia­tion, and the other terms represent the rates of defect disappearance by recombination with other migrat­ing defects. On the outer boundary of this cell, the defect concentration C must then match the concen­tration in adjacent cells occupied by other sinks, and its gradient must vanish. This cellular approach has been pioneered by Bullough and collaborators,51 but the drift term, the second term in eqn [128], is omit­ted when solving the diffusion equation. Its effect is subsequently taken into account by changing the actual sink boundary into another, effective boundary at which the interaction energy between the sink and the approaching defect becomes of the order of kT, k being the Boltzmann constant.

An alternate approach52,53 is to view a particular sink as embedded in an effective medium that main­tains an average concentration of mobile defects at a distance far from this particular sink, and to neglect production and losses of mobile defects nearby. In this approximation, the r. h.s. of eqn [129] is set to zero, and the outer boundary condition far from the sink is that C approaches a constant value C that remains to be deter­mined later from average rate equations. The diffusion equation is now solved with and without the drift term and the resulting defect current to the sink is evaluated. The ratio of the currents with and without the drift defines the sink bias factor. It is thereby possible to define unambiguously bias factors for each type of sink, and with these determine the net bias for a given microstructure. It is this embedding approach that we follow here to evaluate the bias of a sink.

The first attempt to determine the dislocation bias by solving the diffusion equation with drift appears to have been made by Foreman.54 He employed a cellu­lar approach retaining only the defect production term. Furthermore, anticipating small bias values, the drift term was treated by perturbation theory and numerous approximations were introduced in the derivation. The intent was to obtain rough esti­mates; nevertheless, Foreman concluded that the bias was larger than the empirical estimate. Shortly there­after, Heald55 employed the embedding approach and used the solution of Ham56 in the form presented by Margvalashvili and Saralidze.57 We shall return to this solution below, as it is the only analytical one

known. However, this solution applies only when the mobile defect is modeled as a center of dilatation (CD). For more general interactions, Wolfer and Askin58 developed a rigorous perturbation theory that includes also the interaction induced by exter­nally applied stresses. The latter was shown to result in radiation-induced creep. They also compared the bias obtained from the perturbation theory carried out to second order with the bias from Ham’s solu­tion, and demonstrated that both results agree only for weak centers of dilatation. Vacancies can be con­sidered as such weak centers, but interstitials cannot. As a result, perturbation theory to second order is in general insufficient to evaluate the dislocation bias.