Environmental Engineering Reference
In-Depth Information
Therefore, thermal conductivity can be calculated from the
spectral density of heat current as [55]
κ
=
κ
(
ω
)
|
ω
=
0
=
2
ω
=
0
. (1.63)
Strictly speaking, thermal conductivity can only be obtained
from this method in the static limit,
ω
→
0, which is in practice
infeasibleduetothefinitetimesimulation.Therefore,thestaticlimit
is simply approximated by extrapolating the high-frequency data to
zero frequency [55]. Later, this spectral method was improved by
Volz
et al.
[58] with the assumption of single exponential decay of
HCACF in Eq. 1.54. Based on this assumption, the spectral thermal
conductivity is then fitted at highfrequency according to [58]
κ
(
ω
)
=
1
3
k
B
T
2
V
|
S
(
ω
)
|
κ
(0)
ωτ
0
, (1.64)
where
κ
(0) and
τ
0
are two fitting parameters, corresponding to the
static thermal conductivity and single exponential decay constant,
respectively. In this method, the static thermal conductivity is
obtainedas the fitting parameter, thus no extrapolation isinvolved.
+
1
i
1.6.3
Determination of Cut-Off Time
In EMD simulation, the accuracy of HCACF is limited by the total
simulation time, which corresponds to the maximum ensemble
average of the correlation function at
t
=
0 (Eq. 1.52). HCACF
becomes less accurate over time because of the smaller ensemble
average one can get from a finite time simulation. Therefore,
numericalerror(noise)isinevitablyintroducedintothecalculation,
and eventually will contaminate HCACF when it decays to a small
value. Consequently, HCACF is only reliable up to a finite time (cut-
offtime).Thus,thermalconductivitycanonlybecalculatedfromthe
truncated HCACF.
Previous study suggested to determine cut-off time based on the
first dip (FD) method when the tail of HCACF first decays to zero
[56]. This corresponds to the time when the accumulative thermal
conductivity, defined as
t
t
0
3
k
B
T
2
V
κ
a
(
t
)
=
Cor
(
t
),
(1.65)
=
t
0
