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where
ε
p
is the dielectric function of the particles and
L
is their depolariza-
tion factor. Spheres are characterized by
L
1
/
3
. A random distribution of
ellipsoids can be represented by a sum over three components - corre-
sponding to the symmetry axes - to yield
=
in the dilute limit (Niklasson
and Granqvist, 1984). The calculations presented below pertain to prolate
('cigar-shaped') as well as oblate ('pancake-shaped') spheroids with depo-
larization factors obeying
α
1. The depolarization factors can be
expressed in terms of the major (
a
) and minor (
c
) axes of the spheroidal
particles through known formulas (Landau
et al.
, 1984).
Wavelength-integrated transmittance values, specifi cally
T
lum
(
Σ
L
i
=
τ
,
t
) and
T
sol
(
τ
,
t
), were computed according to Eq. [11.1]. Data on
ε
p
were obtained
from the literature (Mlyuka
et al.
, 2009b), and
ε
m
was set to 2.25 which is
applicable to a matrix of a typical glass or polymer. Furthermore, the spher-
oids were taken to be oriented at random. The free electrons in the high-
temperature phase of VO
2
have an exceedingly short mean free path (Allen
et al.
, 1993; Okazaki
et al.
, 2006; Gentle
et al.
, 2007) and, perhaps surprisingly,
this innocent-looking oxide is not well understood in terms of fundamental
theories. Notwithstanding this situation, the very small value of the mean
free path makes it unnecessary to incorporate any size dependence on
ε
p
(as there is, for example, in coinage-metal nanoparticles (Granqvist and
Hunderi, 1977) and for the non-uniform metal fi lms mentioned above in
Section 11.2.4 (Norrman
et al.
, 1978)).
The upper panel of Fig. 11.8 shows the experimental arrangement wherein
light with photon energy
¯
ω
(with
¯
being Planck's constant divided by 2
π
and
being angular frequency) is incident onto the nanoparticles; this
panel also defi nes an 'aspect ratio' as a relationship between the axial
lengths by
m
ω
c
/
a
for oblate spheroids.
The data in Fig. 11.8 can be directly compared with those for a VO
2
fi lm
with
t
=
a
/
c
for prolate spheroids and
m
=
m in Fig. 11.7. The middle and lower panels of Fig. 11.8 prove
that the nanoparticle composites have much higher values of
=
0.05
μ
T
sol
and
T
lum
than the fi lms, which clearly shows the superior properties of the nanother-
mochromic composites. Spherical particles yield the highest transmittance.
Still better properties can be obtained with VO
2
hollow nanoshells (Li
et al.
, 2011a). In terms of the underlying physics, the infrared optical absorp-
tion of the VO
2
-based nanoparticles is governed by a plasmon resonance
that is present in the metallic state at
Δ
τ
>
τ
c
but absent in the insulating state
at
τ
c
(Bai
et al.
, 2009; Li
et al.
, 2010).
Nanoparticles based on VO
2
can be produced in numerous ways as briefl y
reviewed in recent papers by Li
et al.
(2010, 2011a, 2012). More or less sym-
metrical particles can be prepared by wet chemical methods, molten salt
synthesis, confi ned-space combustion, etc. There are also many methods to
make nanorods (prolate spheroids with large aspect ratio) and nanosheets
(oblate spheroids with small aspect ratio) by these techniques. A metastable
τ
<
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