Environmental Engineering Reference
In-Depth Information
In these systems, it is problematic to define thermal conductivity
according to Fourier's law, due to large TJ at the interface of
two different materials. Instead, because of the asymmetry of
the heterogeneous materials, people are more interested in the
rectification effect of heat flow in such materials [46], namely
the difference between the heat flux in different directions. In the
following section, we extend our study to heterogeneous materials
and discuss the effect of heat bath parameter on the thermal
rectification.
We use Si/Ge nanojunction as an example. It has a fixed cross-
section of 3
×
3 unit cells, 5 unit cells of Si and 5 unit cells of Ge in
the longitudinal direction. Three layers of heat bath are applied at
each end, with heat bath temperature set at 310 K and 290 K. Here
wedefine
J
+
(
J
−
)tobetheheatfluxofnon-equilibriumsteadystate
when Si (Ge) end is attached to the high-temperature heat bath. We
define the rectification e
ciency
RE
to be
J
+
−
J
−
J
−
RE
=
.
(1.36)
Figure 1.10a,b shows the dependence of heat current on heat
bathparameters.Ithasbeencheckedthattemperatureprofileswith
these parameters are similar to that shown in Fig. 1.10c for both
NH and Langevin heat bath. With NH heat bath,
J
+
and
J
−
have the
distinctdependenceon
τ
.Inthesmall
τ
limit,
J
+
ismuchlargerthan
J
−
. With the increase of
τ
,
J
+
drops rapidly and finally converges
to a small value, while
J
−
first increases, then decreases, and finally
converges to a value that is slightly larger than
J
+
. As a result, there
exists a large value of
RE
in the small
τ
limit as shown in Fig. 1.10d.
Moreover,
RE
changes from positive to negative when 0.05
≤
τ
≤
100. Thus NH heat bath fails to give a consistent result. In contrast,
with Langevin heat bath, Fig. 1.10b shows the same dependence on
λ
for both
J
+
and
J
−
, and there is only a small difference between
them. This results in a small value of
RE
as shown in Fig. 1.10d
for Langevin heat bath. More importantly, Langevin heat bath can
produce a consistent result (
RE
<
0) that
J
−
is always larger than
J
+
, regardless of the heat bath parameter
λ
. Si-Ge nanowire is a
mass graded nanojunction. The results calculated with Langevin
heat bath and NH heat bath with large
1) suggest that the
heat flux runs preferentially along the direction of decreasing mass.
τ
(
τ>
