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Heating of composite solid aerosol particles by
laser radiation
Liudmila G. Astafyeva,
Nikolai V. Voshchinnikov and
Lawrence B.F.M. Waters
Submitted to Applied Optics
L.G. Astafyeva is with Stepanov Institute of Physics, Belarus Academy of Sciences, Sco-
rina pr. 68, Minsk, 220072 Belarus. N.V. Voshchinnikov is with Sobolev Astronomical
Institute, St. Petersburg University, Bibliothechnaya pl. 2, St. Petersburg, 198504 Russia.
L.B.F.M. Waters is with Astronomical Institute \Anton Pannekoek", University of Ams-
terdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands and Institute of Astronomy,
Catholic University of Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium.
Received .
c
2001 Optical Society of America
1

Abstract
The heating of lidar-irradiated multilayered particles is analyzed theoretically and
numerically by solution to the heat-conduction equation. The internal intensity and
temperature distributions are presented for particles composed of air, quartz and
carbon. It is shown that the heating time of such particles substantially depend
on particle radius, layer position and thickness. In particular, the decrease of the
thickness of surface carbon layer can result in the reduction of the heating time of
multilayered particles.
Key words: atmospheric aerosol particles, multilayered particles, heating, laser
radiation
1. Introduction
Investigation of laser radiation interaction with solid aerosol particles is one of important
practical problems especially in connection with the lidar remote sensing of atmospheric
aerosol. Such particles can absorb laser radiation and get warm. Most of theoretical studies
on the heating of solid particles by laser radiation are related to homogeneous spheres 1 or
core-mantle (two-layered) spheres. 2 But the solid atmospheric particles usually have het-
erogeneous and composite structure 3 . Examples are particles with trapped air bubbles or
composites of mineral aerosols.
In this paper, the heating of composite solid aerosol spherical particles under the action
of intense laser radiation is investigated. We simulate such particles assuming that they
are three-layered spheres and compare the results with those for two-layered spheres. The
particles consist of concentric spherical layers of the two materials: quartz or carbon, each
with a specied volume fraction. The innermost layer (core) is assumed to be lled with
the air. The volume fraction of the core remains constant in dierent multilayered particles
and occupies one third of the particle volume. Quartz and carbon were chosen because their
important role in physics of terrestrial atmosphere. The mathematical formulation of the
2

problem of heating of three-layered particles consists in the solution to the heat-conduction
equation in spherical coordinates with corresponding initial and boundary conditions taking
into account the nonuniform heat release inside the particles and temperature dependencies
of optical and thermophysical properties of particle material.
2. Theory
Let a continuous nonpolarized plane-wave laser beam with irradiance I interacts with a
solid three-layered particle of outer radius R 3 . The particle absorbs radiation and becomes
warmer. The surrounding medium is assumed to be vacuum. The process of heating and the
temperature variation inside the three-layered sphere with time t is described by the system
of the equations in a spherical coordinate system (r; ; ') with the origin at the particle
center (i = 1; 2; 3)
c i
(T i
) i
(T i
) @T i
@t
= 1
r 2
@
@r

K i
(T i
)r 2 @T i
@r
!
+ 1
r 2 sin 
@
@

K i
(T i
) sin 
@T i
@
!
+ Q i
(r; ; T i ; t): (1)
Here, index 1 corresponds to the core with radius R 1 , index 2 corresponds to the interme-
diate layer with radius R 2 and index 3 is related to the outer layer. T i (r; ; t) is the local
temperature inside the particle, c i (T i ),  i (T i ) and K i (T i ) are the material heat capacity,
density, and thermal conductivity, respectively. The internal temperature is independent of
the azimuthal angle ' because of the symmetry. The power density of heat sources at a
given point inside the particle Q i (T i ) is determined by equation
Q i (T i ) = IB i
4n i  i

; (2)
where
B i
= jE (i)
r
j 2 + jE (i)

j 2
E 2
0
: (3)
In Eqs. (2)-(3), E (i)
r
and E (i)

are the components of the electric eld inside dierent layers,
E 0 is the electric eld of the incident laser beam, m i = n i i i the complex refractive index
3

of the corresponding material,  the radiation wavelength. The components of the electric
elds at given point in the core can be written as:
8 > > > > > > > > <
> > > > > > > > :
E (1)
r
= E 0 cos '
k 2
1 r 2
1
X
l=1
l(l + 1) l (k 1 r)
l
(k 1 R 1 ) C (1)
l
Q l () sin ;
E (1)

= E 0 cos '
k 1 r
1
X
l=1
l (k 1 r)
l (k 1 R 1 )
h
C (1)
l
D l (k 1 r)S l () + iB (1)
l
Q l ()
i
:
(4)
The expressions for components of the electric elds inside second and third layer have the
similar form
8 > > > > > > > > > > > > > > > > > <
> > > > > > > > > > > > > > > > > :
E (2);(3)
r
= E 0 cos '
k 2
2;3 r 2
1
X
l=1
l(l + 1)
"
l (k 2;3 r)
l (k 2;3 R 2;3 ) C (2);(3)
l
+  l (k 2;3 r)
 l (k 2;3 R 2;3 )
~
C (2);(3)
l
#
Q l () sin ;
E (2);(3)

= E 0 cos '
k 2;3 r
1
X
l=1
("
l (k 2;3 r)
l (k 2;3 R 2;3 ) C (2);(3)
l
D l (k 2;3 r) +  l (k 2;3 r)
 l (k 2;3 R 2;3 )
~
C (2);(3)
l
G l (k 2;3 r)
#
S l ()
+ i
"
l (k 2;3 r)
l (k 2;3 R 2;3 ) B (2);(3)
l
+  l (k 2;3 r)
 l (k 2;3 R 2;3 )
~
B (2);(3)
l
#
Q l ()
)
:
(5)
In Eqs. (4)-(5), k i = 2m i = is wave number in the corresponding layer, l (z) Riccati-
Bessel function,  l (z) Riccati-Hankel function, D i (z) = 0
l
(z)= l (z) and G l (z) =  0
l
(z)= l (z)
are logarithmic derivatives of the corresponding functions. The angular functions Q l () and
S l
() are dened with the aid of the Legendre polynomials P l
(cos ) and their derivatives.
The coeфcients B i
l
, C i
l
and ~
B i
l
, ~
C i
l
in Eqs. (4)-(5) are (m s
= 1:0 i0:0 is the refractive index
of the surrounding medium, k = 2= is the wave number in vacuum)
4

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <
> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > :
B (1)
l
= i l
2l + 1
l(l + 1) m 1s
1
l (k 2 R 2 ) l (k 2 R 1 )
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
1

1
;
C (1)
l
= i l
2l + 1
l(l + 1)
1
l (k 2 R 2 ) l (k 2 R 1 )
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
1

2
;
B (2)
l
= i l 1 2l + 1
l(l + 1) m 2s
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
A 2

1
;
~
B (2)
l
= i l 1 2l + 1
l(l + 1) m 2s
l (k 2 R 1 )
l (k 2 R 2 )
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
A 1

1
;
C (2)
l
= i l 1 2l + 1
l(l + 1)
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
~
A 2

2
;
~
C (2)
l
= i l 1 2l + 1
l(l + 1)
l (k 2 R 1 )
l (k 2 R 2 )
1
l (k 3 R 3 ) l (k 3 R 2 )
1
 l (kR 3 )
~
A 1

2
;
B (3)
l
= i l
2l + 1
l(l + 1) m 3s
"
l (k 2 R 1 )
l (k 2 R 2 )
 l (k 2 R 2 )
 l (k 2 R 1 ) A 1 H 2 A 2 L 2
#
1
 l (kR 3 )
1

1
;
~
B (3)
l
= i l
2l + 1
l(l + 1) m 3s
l (k 3 R 2 )
l (k 3 R 3 )
"
A 2 H 1
l (k 2 R 1 )
l (k 2 R 2 )
 l (k 2 R 2 )
 l (k 2 R 1 ) A 1 L 1
#
1
 l (kR 3 )
1

1
;
C (3)
l
= i l
2l + 1
l(l + 1) m 3s
"
l (k 2 R 1 )
l (k 2 R 2 )
 l (k 2 R 2 )
 l (k 2 R 1 )
~
A 1
~
H 2
~
A 2
~
L 2
#
1
 l (kR 3 )
1

2
;
~
C (3)
l
= i l
2l + 1
l(l + 1) m 3s
l (k 3 R 2 )
l (k 3 R 3 )
"
~
A 2
~
H 1
l (k 2 R 1 )
l (k 2 R 2 )
 l (k 2 R 2 )
 l (k 2 R 1 )
~
A 1
~
L 1
#
1
 l (kR 3 )
1

2
;
(6)
where
5

8 > > > > > > > > > > > > > > > > > > > > > > > > > > <
> > > > > > > > > > > > > > > > > > > > > > > > > > :

1 = A 1
l (k 2 R 1 )
l (k 2 R 2 )
 l (k 2 R 2 )
 l (k 2 R 1 )
"
l (k 3 R 2 )
l (k 3 R 3 )
 l (k 3 R 3 )
 l (k 3 R 2 ) F 2 L 1 F 1 H 2
#
+ A 2
"
F 1 L 2
l (k 3 R 2 )
l (k 3 R 3 )
 l (k 3 R 3 )
 l (k 3 R 2 ) F 2 H 1
#
;

2 = ~
A 1
l (k 2 R 1 )
l
(k 2 R 2 )
 l (k 2 R 2 )
 l
(k 2 R 1 )
"
l (k 3 R 2 )
l
(k 3 R 3 )
 l (k 3 R 3 )
 l
(k 3 R 2 )
~
F 2
~
L 1
~
F 1
~
H 2
#
+ ~
A 2
"
~
F 1
~
L 2
l (k 3 R 2 )
l (k 3 R 3 )
 l (k 3 R 3 )
 l (k 3 R 2 )
~
F 2
~
H 1
#
(7)
and
8 > > > > > > > > > > > > > > > > > > > > > > > > > > <
> > > > > > > > > > > > > > > > > > > > > > > > > > :
A 1 = D l (k 2 R 1 ) m 12 D l (k 1 R 1 );
A 2 = G l (k 2 R 1 ) m 12 D l (k 1 R 1 );
L 1 = D l (k 3 R 2 ) m 23 G l (k 2 R 2 );
L 2 = G l (k 3 R 2 ) m 23 D l (k 2 R 2 );
H 1 = D l (k 3 R 2 ) m 23 D l (k 2 R 2 );
H 2 = G l (k 3 R 2 ) m 23 G l (k 2 R 2 );
F 1 = G l
(kR 3 ) m 3s D l
(k 3 R 3 );
F 2 = G l (kR 3 ) m 3s G l (k 3 R 3 ):
(8)
The expressions for ~
A 1 , ~
A 2 , ..., ~
F 2 can be obtained from the corresponding expressions for
A 1 , A 2 , ...., F 2 (see Eq. (8)) by the replacement of m ij
for m ji
, where m ij
= m i =m j
.
At the center, interfaces and on the surface of the particle the following conditions must
be satised:
6

8 > > > > > > > > > > > > > > > > > > > > > > > > > > > <
> > > > > > > > > > > > > > > > > > > > > > > > > > > :
jT 1 (0; ; t)j < 1; 0    ; t > 0;
T 1 (R 1 ; ; t) = T 2 (R 1 ; ; t); T 2 (R 2 ; ; t) = T 3 (R 2 ; ; t);
K 1 (T 1 ) @T 1 (R 1 ; ; t)
@r
= K 2 (T 2 ) @T 2 (R 1 ; ; t)
@r
; K 2 (T 2 ) @T 2 (R 2 ; ; t)
@r
= K 3 (T 3 ) @T 3 (R 2 ; ; t)
@r
;
K 3 (T 3 ) @T 3 (R 3 ; ; t)
@r
= "  T 4 ;
@T i (R i ; ; t)
@
     =0
= @T i (R i ; ; t)
@
     =
= 0 i = 1; 2; 3;
(9)
where  is the Stefan-Boltzmann constant and "  is the spectral coeфcient of blackness.
The solution of system (1) with the conditions (9) is possible only numerically. For
this purpose, the local one-dimensional iterative scheme was created. The iteration cycle is
continued up to the achievement of the required accuracy. The temperature dependencies of
thermophysical properties of quartz and carbon were taken into account in the calculations.
Using the approximate formulas obtained from the data presented in Refs. 4 { 6, the values
of thermal conductivity K(T ), heat capacity c(T ), and density (T ) were corrected at any
time according to the temperature of every mesh point.
3. Numerical results
A computer code to solve the coupled set of equations discussed above have been developed.
Numerical results are presented for the cases of three-layered non-rotating spheres of air-
quartz-carbon and air-carbon-quartz irradiated by lidar beam of wavelength =1.06 m with
laser uxes 10 6 W/cm 2 assuming the initial temperature T 0 =210 K. At this wavelength the
refractive indices of m quartz = 1:5 0:0077i and m carbon = 2:8 0:533i.
The intensity distribution inside multilayered particles has a substantially inhomogeneous
character. The maximum is located near the particle surface in the illuminated or shadow
7

hemisphere in dependence of the particle structure and size. In the case of two-layered
particle consisting of the hole and quartz shell (Fig. 1), the region of the maximum intensity
lies in the shadow hemisphere near the particle surface that is the result of the focusing of
laser radiation by the quartz shell.
If the outer layer is carbonaceous, the intensity distribution is essentially changed (see
Fig. 2). The maximum intensity occurs within the thin layer on the particle surface located
in the illuminated hemisphere of the particle. The secondary maximum lies in the shadow
hemisphere. It vanishes with increasing the particle radius.
If the carbon is in the intermediate layer of the three-layered particle, the intensity
distribution (see Fig. 3) resembles those in Fig. 1 but its maximum value is one-fth only
as the intensity in Fig. 1. The enhanced intensity is concentrated in the thin, rather small
in magnitude, layer in the illuminated hemisphere for particle of given size (Fig. 3). With
increasing the particle size this layer grows and becomes dominating. For larger particles,
the energy evolves in the thin shell on the particle surface in the shadow hemisphere, the
values of intensity in the remainder part of the particle are very small..
Figures 4,5 show the internal temperature distribution of multilayered particles. In the
case of air-quartz particle (Fig. 4), the primary and secondary temperature peaks appear at
the places of maxima of internal intensity. The presence of carbon layer independent of its
position (as an example, for the air-quartz-carbon particle, see Fig. 5) leads to the smoothing
temperature distribution. At the same time, there exists a small dierence between the
temperature distributions inside dierent (air-quartz-carbon or air-carbon-quartz) three-
layered particles. The core region in air-carbon-quartz particle seems to look smaller and
the temperature distributions are more smoother than in air-quartz-carbon particle (Fig. 5)
although in both cases the core volume fraction is the same.
It is interesting to compare the radii dependencies of heating time for multilayered par-
ticles. They have rather complex character. As follows from Fig. 6 where the results for
air-quartz-carbon particles are shown, the heating time grows with particle radius. It reaches
a maximum which value depends on the thickness of carbon layer and then decreases. For
8

comparison in the case of two-layered (air-carbon) particles (curve 1 in Fig. 6) the heat-
ing time is maximal for R 3  20 m. When the outer carbon volume fraction inside the
air-quartz-carbon particle is reduced, the position of maximum heating time shifts to the
region of larger radii and decreases. When the volume occupied by the carbon shell is the
one-fth part of the particle volume, the heating time is maximal for R 3  28 m and falls
approximately 40 percent.
The value of the heating time is the greatest in the case of the two-layered particles. It
decreases when the carbon layer becomes thinner. This behaviour can be explained by the
fact that in the case of thick carbonaceous shells the radiation does not penetrate inside
the particle. As a result, the particle surface only is heated and the process of the heat-
conductivity slows down. Note that when the carbon volume fraction becomes smaller 20%
the dependence t(R 3 ) begins to oscillate that re ects the strongly oscillating behaviour of
the internal intensity.
Figure 7 shows the size dependence of the heating time of air-quartz and air-carbon-
quartz particles. If R 3  35 m, the heating time of the core-mantle spheres is smaller than
that of three-layered particles but the situation reverses if R 3 > 35 m. This is connected
with the behaviour of the maximum internal intensity in three-layered particles when the
particle radius and the thermophysical properties of the components change.
Comparing the heating of air-quartz-carbon and air-carbon-quartz particles (Fig. 8), we
can conclude that it is much more diфcult to warm the particles with the outer carbon
layer. For example, the heating time of the air-quartz-carbon particles is 10{15 times larger
than for air-carbon-quartz particles if R 3 < 20 m. When increasing the particle radii
these dierences become smaller. For R 3 ' 50 m, for example, the heating time of air-
quartz-carbon is approximately twofold higher than the heating time of the air-carbon-quartz
particle.
Conclusion
Complex structure of aerosol particles has substantially in uence on the internal intensity
distributions and, as a result, on their heating time and possible destruction. Firstly, the
9

reduction of strongly absorbed outer carbon layer on the multilayered particle surface leads
to the decrease of the heating time. Secondly, the heating time of the particles with outer
quartz layer is always lower than in the case of the outer carbon shell. In the size particle
region R 3 < 20 m the heating time of the air-quartz-carbon spheres is more than ten times
higher than in the case of air-carbon-quartz particles. In the interval R 3  30 m these
dierences become smaller and consist 2{7 times.
The research described in this publication was made possible in part by grant N 99/652
from the INTAS.
10

References
1: R.L. Armstrong and A. Zardecki, \Propagation of high energy laser beams through
metallic aerosols," Appl. Opt. 29, 1786-1792 (1990); A.P. Prishivalko, L.G. Astafyeva,
and S. T. Leiko, \Heating and destruction of metallic particles exposed to intense laser
radiation," Appl. Opt. 35, 965-972 (1996); L.G. Astafyeva and A.P. Prishivalko, \Heating
of solid aerosol particles to intense optical radiation," Int. J. Heat Mass Transfer 41, 489-
499 (1998).
2: L.G. Astafyeva and A.P. Prishivalko, \Heating of aluminum particles with oxide covers
by intense laser radiation," Fiz. Khim. Obrab. Mater. N4, 18-27 (1993); L.G. Astafyeva
and A.P. Prishivalko, \Heating of metallized particles by high-irradiance laser radiation,"
Inzh. Fiz. J. 66, 340-344 (1994); L.G. Astafyeva and A.P. Prishivalko, \Heating of ho-
mogeneous and hollow particles of aluminum oxide by intense laser radiation," Teploz.
Vys. Temp. 32, 230-235 (1994); L.G. Astafyeva, A.P. Prishivalko, and S.T. Leiko, \Dis-
ruption of hollow aluminum particles by intense laser radiation," J. Opt. Soc. Am. B 14,
432-436 (1997); L.G. Astafyeva, A.P. Prishivalko, and S.T. Leiko, \Heating and destruc-
tion of hollow aluminum oxide particles by laser radiation," Fiz. Khim. Obrab. Mater.
N5, 27-32 (1997).
3: W. Fett, Der Atmospharische Staub (Wissenschaften, Berlin, 1958); R.D. Cadle, Sus-
pended Particles in Lower Atmosphere, in Chemistry of Lower Atmosphere (Mir, Moscow,
1976, p.90-154); X. Green and V. Lane, Aerosols are dust, smoke, mist (Khimiya,
Leningrad, 1978); K.Ya. Kondrat'ev, N.I. Moskalenko and D.B. Pozdnyakov, Atmo-
spheric Aerosol (Gidrometeoizdat, Leningrad, 1983); Aerosol and Climate (Gidrome-
teoizdat, Leningrad, 1991).
4: R.E. Krzhizhanovsky and Z.Yu. Shtern, Thermophysical Properties of Nonmetallic Ma-
terials (Oxides) (Energy, Leningrad, 1973).
5: V.S. Chirkin, Thermophysical Properties of Materials of Nuclear Technics (Atomizdat,
11

Moscow, 1968).
6: Thermodynamical Properties of Individual Matter. Handbook, (Nauka, Moscow, 1979).
12

FIGURES
Fig. 1. Irradiance distribution inside the two-layered air-quartz sphere, V air =V 3 = 1=3,
V quartz =V 3 = 2=3, R 3 = 5 m,  = 1:06 m. The arrow shows the direction of the propagation
of incident lidar beam.
Fig. 2. Irradiance distribution inside the three-layered air-quartz-carbon
sphere, V air =V 3 = V quartz =V 3 = V carbon =V 3 = 1=3, R 3 = 5 m,  = 1:06 m. The arrow shows
the propagation direction of incident lidar beam.
Fig. 3. The same as in Fig. 2 but for air-carbon-quartz sphere.
Fig. 4. Temperature distribution inside the two-layered air-quartz sphere, V air =V 3 = 1=3,
V quartz =V 3 = 2=3, R 3 = 5 m,  = 1:06 m. The irradiance of lidar radiation is 10 6 W/cm 2 .
The arrow shows the propagation direction of the incident lidar beam.
Fig. 5. Temperature distribution inside the three-layered air-quartz-carbon sphere,
V air =V 3 = V quartz =V 3 = V carbon =V 3 = 1=3, R 3 = 5 m,  = 1:06 m. The irradiance of lidar
radiation is 10 6 W/cm 2 .
Fig. 6. The heating time of the air-quartz-carbon spheres from T = 210 K to T = 400 K as
a function of the outer particle radius, V air =V 3 = 1=3. The eect of variation of carbon fraction
is illustrated. 1 { V carbon =V 3 = 2=3, V quartz =V 3 = 0; 2 { V carbon =V 3 = 1=3, V quartz =V 3 = 1=3;
3 { V carbon =V 3 = 0:25, V quartz =V 3 = 0:42; 4 { V carbon =V 3 = 0:20, V quartz =V 3 = 0:47; 5 {
V carbon =V 3 = 0:15, V quartz =V 3 = 0:52; The irradiance of lidar radiation with  = 1:06 m is
10 6 W/cm 2 .
13

Fig. 7. The heating time of the air-quartz-carbon spheres from T = 210 K to T = 400 K as
a function of the outer particle radius, V air =V 3 = 1=3. The eect of variation of carbon fraction
is illustrated. 1 { V carbon =V 3 = 2=3, V quartz =V 3 = 0; 2 { V carbon =V 3 = 1=3, V quartz =V 3 = 1=3;
3 { V carbon =V 3 = 0:25, V quartz =V 3 = 0:42; 4 { V carbon =V 3 = 0:20, V quartz =V 3 = 0:47; 5 {
V carbon =V 3 = 0:15, V quartz =V 3 = 0:52; The irradiance of lidar radiation with  = 1:06 m is
10 6 W/cm 2 .
Fig. 8. The heating time of the air-quartz-carbon spheres from T = 210 K to T = 400 K as
a function of the outer particle radius, V air =V 3 = 1=3. The eect of variation of carbon fraction
is illustrated. 1 { V carbon =V 3 = 2=3, V quartz =V 3 = 0; 2 { V carbon =V 3 = 1=3, V quartz =V 3 = 1=3;
3 { V carbon =V 3 = 0:25, V quartz =V 3 = 0:42; 4 { V carbon =V 3 = 0:20, V quartz =V 3 = 0:47; 5 {
V carbon =V 3 = 0:15, V quartz =V 3 = 0:52; The irradiance of lidar radiation with  = 1:06 m is
10 6 W/cm 2 .
14