Biomedical Engineering Reference
In-Depth Information
8
(a)
(b)
GNS 30 nm
7
6
GNS 150 nm
hv
GNS
5
4
Au
3
GNP
2
1
Water
-200
-150
-100
-50
0
Distance r (nm)
50
100
150
200
FIGURE 18.5
GNP heats up the immediate surrounding (assume one GNP in infinite medium) (a) Schematic of gold nanoparticle and its imme-
diate surrounding; (b) The steady state temperature profile around the nanoparticle for two NPs, 30 nm spherical GNP and 150 nm gold nanoshell
with 120 nm Si core, laser fluence 10
4
W/cm
2
.
treated as a continuum. However, the mean free path in gold
(31 nm) is comparable to the size of a GNP, and one cannot
use the heat equation to describe the heat transfer within the
particle. Here GNP with radius
r
p
is treated as a lumped system
with the same temperature, and the heat equation for the sur-
rounding medium (
r
>
r
p
) is given by:
internal heating of the GNP. Assuming water as the surrounding
medium, and the particle sizes of 10 nm and 100 nm, the char-
acteristic time
nan
τ
is on the order of 1 ns and 100 ns, respec-
tively. This is also related to the
thermal confinement
concept
introduced later and is schematically shown in Figure 18.6. Note
that the characteristic time is only a rough estimate of the time
scale, and one has to solve the governing equation to obtain
more accurate estimations. Experimental measurements of heat
dissipation from GNPs in aqueous solution show the same trend
(i.e., the thermal relaxation time is proportional to the square of
the particle radius (Hu 2002)).
After a short time (1−100 ns calculated before), the tempera-
ture distribution around a single continuously heated GNP in an
infinite medium reaches steady state and can be easily obtained.
Applying the constant heat flux boundary condition at the GNP
surface (
r
=
) and negligible temperature change far away from
the particle
Tr
∂
∂
T
t
1
∂
∂
rk
T
r
∂
∂
2
ρ
C
=
.
(18.7)
2
rr
The complete solution to the transient heating is available
elsewhere (Goldenberg 1952, Keblinski 2006). Here the nomen-
clature of Keblinski et al. (Keblinski 2006) is used, and the char-
acteristic time of the transient heating process is described by:
2
τ=
α
r
p
(18.8)
nano
→∞ =
∞
, the surrounding medium tempera-
ture due to the heated GNP is given by
(
)
T
where α is the thermal diffusivity. This characteristic time,
t
nano
,
is a measure of the time for the temperatures of the GNP surface
and surrounding medium to reach steady state given a uniform
−=
π
Q
Tr
()
T
nano
,
rr
≥
(18.9)
∞
p
4
kr
TABLE 18.3
Diffusion Length and Knudsen Number Scaling
Diffusion Length Knudsen Number
H
2
O Au H
2
O Au
10
−15
(femto second) 0.012 nm 0.36 nm 16.67 86.11
10
−12
(pico second) 0.38 nm 11 nm 0.53 2.82
10
−9
(nano second) 12 nm 0.36 µm 0.02 0.09
10
−6
(micro second) 0.38 µm 11 µm 5.26 × 10
−4
2.82 × 10
−3
10
−3
(milli second) 12 µm 360 µm 1.67 × 10
−5
8.61 × 10
−5
1 (second) 380 µm 11 mm 5.26 × 10
−7
2.82 × 10
−6
60 (minute) 3 mm 87 mm 6.67 × 10
−8
3.56 × 10
−7
3600 (hour) 23 mm 68 cm 8.70 × 10
−9
4.56 × 10
−8
Note:
(1) difusion length = square root of (difusivity ×time); (2) the characteristic length for
Knudsen number is the diffusion length (L); (3) continuum assumption holds for Kn << 1, i.e.,
t
≥ 10
−9
s (nano second). Mean free paths (λ) for water and gold are listed in Table 18.4.
Time Scale (s)
