Volumen II - SAM

the dumbbell is not the fundamental configuration for the SIA in bcc transition metals, but a
structure, either dumbbell or crowdion, that is expected to possess very low E m .
Regarding the interstitials clusters, the 1D glide along rows in Fe is currently well established, both,
theoretically e.g. [10-13], and experimentally [14]. The latter consisted in the observation by TEM of small
loops, nevertheless beyond the reach of current MD simulations; according to the latter, however, the general
picture is already settled at sizes much smaller: starting at 5 - 7 crowdions the clusters move in a 1D fashion
parallel to the direction, thus anticipating the behavior of larger, fully developed loops.
At this point we come to the purpose of the present study, namely, to investigate the migration mechanisms
of the first few small interstitial clusters, for two models of Mo that have been fitted to the same database
using similar procedures. One of them was derived within the (central force) EAM format, the other within
the so called embedded defect (ED) format [15] that includes angular interactions in a simplified global way;
both have been used in a previous work dealing with the primary damage in Mo [16].
The ED version was found to better agree with the phonon spectra, SIA properties, displacement threshold,
and vacancy clustering. We note however that both potentials have been fitted to obtain the (experimental)
dumbbell as least energy interstitial. The current research is not to be considered as strictly applicable
to Mo however, but as a comparison between two force models, one of which (ED) is arguably a more
consistent representation of a transition metal, and try to assess the consequences stemming from such a
difference.
For the stated purpose we apply two techniques, the common MD and a molecular statics one, so called
Monomer [17]. The latter consists in searching the potential energy surface for saddle configurations,
namely, the barriers between local energy minima according to standard transition state theory. Compared to
MD, the Monomer has certain attractive features. It is computationally cheaper; the structure of activated
states is revealed in full detail; high energy barriers may be obtained that are still macroscopically relevant
but impossible of access by MD. Last, the use of high temperatures in MD to increase statistics (generally
not relevant to the actual experimental conditions), may introduce unwanted anharmonic behavior, side
effect that is absent for the Monomer. Thus, the comparison between a dynamic and a static technique within
the present context is a second purpose of our study.
2. COMPUTATONAL METHODS
The applied MD procedure is briefly summarized next. A typical run consists of a crystallite 12x12x12 cubic
bcc cells in size, equilibrated for about 10 ps at temperatures varying between 400 K and 1600 K, at the
respective equilibrium lattice parameter. The crystallite, that contains a (previously determined) minimum
energy interstitial cluster, is then switched to microcanonical conditions maintained for at least 2 ns;
meanwhile, configurations are output every 1 ps for post-processing. A distance criterion is used to locate
interstitials and vacancies forming the cluster; the position of the geometrical center of the latter, RCM , is
recorded, where from the diffusivity is calculated according to [18,19],
2
D = < R > / 6τ
(1)
where τ is a time interval varied between 10 to 50 ps (while checking for constancy of the quotient) that
evenly partitions the total run time in sub-trajectories, and is the dispersion in the RCM displacement
(i.e., difference between RCM 's separated τ in time) with respect to the set of sub-trajectories. Finally, the so
computed D's are carried over onto an Arrhenius plot, ln(D) vs. 1/ T, to obtain the pre-exponential factor D0
and activation energy Q of the standard relationship D ~ D0 exp(-Q / kT).
Regarding the Monomer, its mathematical details are given in [17]. Briefly, it consists in driving the system
through the potential energy surface using a pseudo force. The latter is the common down hill force along all
coordinates but one, namely, the component along the eigenvector of least (and negative) local curvature is
reversed, thus pointing up hill. The method is particularly efficient at finding such eigenvector using as input
only the forces in the vicinity of the current configuration. The present implementation displaces the starting
cluster configuration along a random direction in search for a nearby saddle; once found, the system is
relaxed towards both sides of the saddle in order to find the connected minima. This is to assure that one of
them effectively is the starting configuration; the other minimum is in turn taken as new starting
configuration, and so on. Thus a sequence of local minima connected by saddles is build, which is postprocessed
to reveal the relevant saddles/mechanisms for cluster migration.
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