Chemistry Reference
In-Depth Information
that would be found for the same observable in a fully atomistic and canonical
system at equilibrium:
ðf
q
r
ðf
q
r
ðf
q
r
h
A
gÞi
CG
¼h
A
gÞi
N
r
;
V
;
T
¼h
A
gÞi
N
;
V
;
T
½
52
Using this methodology, the dynamical behavior of the representative
atoms is obtained by deriving the equation of motion from the coarse-grained
Hamiltonian. Moreover, because the interest is on simulating the system under
constant-temperature conditions, the coupling to a thermal reservoir is simu-
lated using a Nos´-Poincar´ thermostat.
207,208
More specifically, such a ther-
mostat is applied only to the set of representative atoms, not to all the atoms in
the system. By following such a procedure, the validity of Eq. [52] is preserved.
Lastly, computational efficiency needs to be discussed. However complete
the formulation of the coarse-grained alternative to MD methodology is up to
here, additional approximations are required to make it computationally effi-
cient. To begin with, it is assumed that the thermally averaged positions of
the constrained atoms can be expressed as a finite-element interpolation of the
positions of the representative atoms, i.e., using finite-element shape functions.
This is analogous to the procedure followed in the standard QC method to
determine the instantaneous positions of the nonrepresentative atoms.
Moreover, the computation of
V
CG
is noticibly expedited when both the local
harmonic approximation and the Cauchy-Born rule are taken into account.
Under such circumstances,
V
CG
becomes
X
ðf
q
r
ðf
q
r
g;
b
Þ¼
gÞ
V
CG
E
i
i
2
NL
"
#
½
53
X
n
e
2
ln
Det
ð
D
CB
ð
F
e
ÞÞ
ð
þ
n
e
E
CB
ð
F
e
Þþ
b
3
2
p
=
b
Þ
e
where NL indicates the nonlocal representative atoms,
n
e
and
n
e
are the total
number of atoms and the number of constrained atoms, in element
e
, respec-
tively, and
E
i
ðf
q
r
is the energy of the
i
th nonlocal representative atom,
calculated exactly as it would be in a standard MD simulation. However,
unlike standard MD simulations,
E
CB
gÞ
are the potential
energy and the determinant of the dynamical matrix of an atom embedded in
an infinite perfect crystal subject to a uniform deformation gradient,
F
e
. The
CG potential given in Eq. [53] is reasonably fast to compute, and is an accu-
rate approximation for temperatures up to about half the melting temperature.
ð
F
e
Þ
and Det
ð
D
CB
ð
F
e
ÞÞ
Applications
Dupoy et al. applied this methodology to the study of the
temperature dependence of the threshold for dislocation nucleation during
nanoindentation.
205
