Chemistry Reference
In-Depth Information
necessary to directly compute the displacement of every atom to determine the
equilibrium configuration of the system. Under such circumstances, a subset of
atoms can be used to represent the whole system. The displacement of such
atoms, called representative atoms or
repatoms
, is therefore explicitly treated,
while the displacement of the remaining atoms is approximated through inter-
polation. For an interpolation scheme, the QC method makes use of the finite-
element method (FEM). Here, the solid is partitioned into a finite number of
regions (''elements''). The deformation within each element is interpolated
from the corresponding nodal displacements. A repatom is chosen for each
element, and around it a small cluster of atoms (
crystallite
) is considered.
This crystallite is distorted according to the continuum displacement field,
and the energy of the repatom is computed using the atomistic model of
choice. Figure 3 shows an example of such a coarsening mechanism. Of all
of the atoms in the system (open circles) only a few (solid circles) are chosen
as repatoms. In some QC formulations, the repatoms coincide with the FEM
nodes. The density of the repatoms is adjusted according to the severity of the
deformation gradient (in the example, there are many more repatoms near the
dislocation core than away from it).
Mathematically,
if a semiempirical model
is used,
then the total
energy of the system,
E
tot
ð
u
Þ
, can be written as the sum of the energies of
each atom:
X
N
E
tot
ð
u
Þ¼
E
i
ð
u
Þ
½
3
i
¼
1
Figure 3
Example of repatom selection near a dislocation core. The repatoms (solid
circles) become the nodes of a linear triangular mesh for the FEM. The density of the
repatoms increases close to the dislocation core because the deformation gradient is
larger.
