Chemical Physics 308 (2005) 103–108
www.elsevier.com/locate/chemphys
Theoretical study of heating of spherical nanoparticle in media
by short laser pulses
Viktor K. Pustovalov
*
Department of Innovation Research, Belarussian Institute of System Analysis, Masherov Pr., 7, Minsk 220004, Belarus
Received 29 April 2004; accepted 18 August 2004
Available online 15 September 2004
Abstract
Theoretical investigation is made of heating of a small solid (metal) spherical particle in a liquid or gaseous medium on exposure
to intense laser radiation pulse. The solutions are obtained for quasi-stationary distributions of temperature inside and around of a
particle and conductive heat transfer between the particle and the surrounding medium with allowance for the temperature dependence of the transfer coefficients. The heating of spherical nanoparticle in medium on exposure to laser radiation pulse and following
cooling is considered on the base of analytical solutions. The time dependencies of particle temperature are obtained. Comparison of
some predicted results of the heating of gold spherical nanoparticle in liquid media with experimental data is given and agreement of
theoretical results with experimental data validates the model and theory developed.
2004 Elsevier B.V. All rights reserved.
Keywords: Laser pulses; Nanoparticle; Heating
1. Introduction
In the past several years the interaction between solid
(metal) nanoparticles and laser light and heat dissipation
in nanoparticles has become a great interest and an
increasing important topic [1–10] (also see the references
in these papers). There are several reasons for this situation. Small solid and metal (gold, silver and other) particles are used in many different fields including
catalysis, microelectrodes and nonlinear materials. It
has recently been shown that laser-induced heating can
be used to control size, shape and structure of nanoparticles [1–4]. The use of metal particles for biomolecule
detection, optical limiting application has continued to
make the spectroscopy of small metal particles an
important area of research [5–8]. Laser action is accomplished by using laser pulses at a wavelength that is
*
Tel.: +375 17 2233485; fax: +375 17 2233540.
E-mail address: [email protected].
0301-0104/$ - see front matter 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemphys.2004.08.005
strongly absorbed by the particles but not by surrounding media, and with a pulse duration that is sufficiently
short to minimize heat flow away from the absorbing
particles. In laser medicine new method for selective cell
targeting based on the use of light-absorbing nanoparticles that are heated by short laser pulses to create highly
localized cell damage was proposed [9]. Some perspectives are connected with application of small absorbing
particles for laser treatment and optoacoustic imaging
of different tumors [10].
The important feature of heating of solid (metal) particles by laser radiation is the possibility of heating a
particle up to the temperature T0 > Tm, and in some
cases up to T0 > Tev, T0 is the temperature of particle,
Tm the melting temperature of particle material, Tev
the value of evaporation (boiling) temperature of particle material. Values of Tm, Tev are much higher than the
initial temperature of particle and surrounding media
T1:Tm, Tev T1 (for example, Tm = 1336 K, Tev =
2933 K for gold [11], T1 300 K). This leads to the
104
V.K. Pustovalov / Chemical Physics 308 (2005) 103–108
necessity of taking into account the actual temperature
dependence of transfer (conduction) coefficients for the
particle material and surrounding media in a wide temperature range [11,12], considering phase changes, investigating non-linear and non-isothermal heat transfer of a
particle with surrounding medium. A number of problems of heating and evaporation of small solid particles
by optical radiation were previously considered theoretically [13–15]. The results [15] were used for investigation of heating of gold nanoparticle by short laser
pulse in [9]. But this result has integro-differential
expression and needs additional computer calculation
for real use and estimations of temperature and also it
was not used the real dependence of heat conduction
coefficient on temperature.
It should be noted that in experiments [1–8] 10–50 nm
particles are not visible under light microscope and attempts to image the processes of laser action on particles
were not successful. In this case carrying out theoretical
estimations and interpretation of experimental results
are very important. To gain some insights into the mechanisms of interaction for individual nanoparticles theoretical study should performed for investigation of the
process characteristics and mechanisms of laser-particle
interaction, detailed treatment (time and space distributions) and description of some characteristics for estimation and correct interpretation of experimental
parameters and results. It follows from the above that
of great importance is the theoretical study of heating,
phase changes of metal nanoparticles in some medium
(continuum) exposed to intense laser radiation pulse,
taking a more correct account of the actual temperature
dependence of thermophysical parameters that will be
carried out in this paper.
2. Methods and results
Suppose laser radiation with wavelength k and intensity I0 be incident starting from time t = 0, on a spherical solid (metal) particle with the radius r0 and initial
temperature T1 (equal to the surrounding medium temperature). The particle absorbs the radiation energy and
becomes heated. Within the range T1 < T0 < 0.8Tev we
can neglect the process of evaporation of metal particle
because the vapor pressure of particle material (gold) is
less than 1% of vapor pressure value under T = Tev and
particle gives up its heat to the surrounding media by
conduction. The processes of heating of spherical particle, which absorb laser radiation energy, and of heat
transfer between particle and the surrounding medium
with no mass transfer are described by equation:
oT
¼ divðk 0 ðT Þ grad T Þ þ q0 ;
q0 c0
ot
with the initial and boundary conditions
ð1Þ
T ðr 6 r0 ; t ¼ 0Þ ¼ T 1 ;
ðk 0 ðT Þ grad T Þsp ¼ J e ;
T ðr ¼ r0 Þ ¼ T 0 ;
ð2Þ
where t is the time, q0 and c0 are the density and the specific heat capacity of the particle material, T the temperature, r the radius with the origin at the center of
particle, k0 the coefficient of thermal conductivity of
the particle material, Je is the energy flux density removed from the particle surface, low index ‘‘sp’’ means
that expression in brackets is calculated at the surface
of particle. The laser pulse energy is absorbed by the
electrons in metal particle and after that energy is transferred from electron gas into lattice. The electron–
phonon coupling process has a time constant of about
1 ps [2]. The power density of energy generation in
the particle q0 due to radiation energy absorption can
be generally non-uniform throughout the particle volume, with the non-uniformity being dependent on size
and optical constants of the particle material. In our
case it is used the approximation 2pr0/k < 1 and it will
be assumed that q0 is virtually uniform (constant)
throughout the particle volume.
There are few characteristic times in this case. The
characteristic time t0 required for the formation of a quasi-stationary temperature profile inside the particle is estimated from the formula: t0 r20 =4v0 , where v0 is the
coefficient of thermal diffusivity for particle material.
For characteristic particle sizes r0 10–30 nm and
v0 1.2 · 104 m2 s1 for gold, t0 2 · 1012 s. In many
experimental cases t0 is much smaller than the characteristic times of laser pulse duration and heating of a particle, and the non-stationary Eq. (1) can be considered
in quasi-stationary approximation at (oT/ot) = 0. The
importance of taking into account the dependence of
thermophysical parameters of particle and surrounding
medium on temperature during laser treatment was noted
in [16]. Analytical quasi-stationary solutions of Eqs. (1)
and (2) describing the distribution of T inside of a particle
with regard to power law temperature function of conduction coefficient of particle material k0 = k01(T/T1)b,
where k01 = k0(T = T1) index b = const., (for example,
for gold in temperature range 273 < T < 2000 K,
b 0.5 [16]) were obtained in [13]. These solutions have
next form in a spherical coordinate system with the origin
at the particle center at q0 = const
1=bþ1
q0 ðr20 r2 Þðb þ 1ÞT b1
;
b 6¼ 1; T ¼ T 0 1 þ
6k 01 T 0bþ1
q ðr2 r2 Þ
b ¼ 1; T ¼ T 0 exp 0 0
:
6k 01 T 1
ð3Þ
1ÞT b1 r20 Þ=ð6k 01 T 0bþ1 Þ
and ðq0 r20 Þ=
The terms ðq0 ðb þ
ð6k 01 T 1 Þ in Eq. (3) under r = 0 characterize the non-uniformity of the temperature distribution inside of the particle and the difference between the temperature at the
V.K. Pustovalov / Chemical Physics 308 (2005) 103–108
center T(r = 0) and at the particle surface T0. The estimation for characteristic values of the laser and material
parameters showed that these expressions are much smaller than 1, and the approximation to the temperature T0,
which is uniform over the particle volume and which coincides with temperature of its surface, can be used.
After integration over the volume for the spherically
symmetric case and transition to a uniform temperature
T0 over the particle volume the equation which describes
the heating of a particle and which results from Eq. (1)
will have the form
dT 0 1
q0 c 0 V 0
¼ I 0 ðtÞK abs S 0 J e S 0
ð4Þ
4
dt
with the initial condition
T0
ðt ¼ 0Þ ¼ T 1 ;
ð5Þ
where
Z r0
1
q0 ðtÞ4pr2 dr ¼ I 0 ðtÞK abs S 0 ;
4
0
4
V 0 ¼ pr30
3
is the volume and S 0 ¼ 4pr20 is the surface area of a
spherical particle of radius r0, Kabs(r0, k) is the efficiency
absorption factor of radiation with wavelength k by a
spherical particle of radius r0 [17]. Mie theory gives integral characteristics of interaction of plane radiation
wave with spherical particle – the efficiency factors for
absorption Kabs, scattering Ksca, and extinction Kext of
radiation by particle [17]. The computer calculation of
efficiency factors Kabs, Ksca and Kext for spherical gold
nanoparticles in wide range of radiuses and for some laser wavelengths was performed in [18].The optical
absorption of real metal nanoparticles can be modeled
by using an extension of MieÕs theory taking into account the shape of nanoparticles (spherical, ellipsoidal,
cylindrical and others), the electron mean free path
(25 nm) and electron scattering at the boundary of
particle with diameter smaller than free path [17,19].
For small particles under condition 2pr0k < 1 for efficiency absorption factor Kabs can be used analytical
expression with accuracy of a few percent [17]
2
8pr0
m 1
K abs ¼ Im
;
ð6Þ
m2 þ 2
k
where Im is the imaginary part of expression in brackets
and m is the complex index of radiation refraction. Eq.
(4) can take into account phase changes that occur during heating and cooling of the particle. The quantity Je
near the particle surface is composed of energy losses
due to heat conduction Jc and radiation cooling Jr
J e ¼ J c þ J r;
dT
J c ¼ kðT Þ
;
dr sp
ð7Þ
J r ¼ erðT 40 T 41 Þ;
ð8Þ
where k = k1(T/T1)a is the dependence of the coefficient of thermal conductivity of a surrounding medium
105
on temperature, k1 = k(T = T1), index a = const.,
(for example, for air a 0.75 in temperature range
273 < T < 2 · 103 K, for water vapor under atmospheric
pressure a 1.5, for water under saturation pressure
a 0 in temperature range 273 < T < 600 K with precision 20% [12]), e is the particle surface emissivity, r the
Stefan–Boltzmann constant r = 5.67 · 108 J/m2 s K4.
Of interest is the study of the integral energy parameters that characterize the interaction of laser radiation
with the particle – its heating, melting, cooling from
the onset of irradiation t = 0 to the time considered t:
the quantity of radiation energy Qabs absorbed by the
particle, the quantities of heat spent for particle melting
Qm and removed from the particle by heat conduction
Qc and thermal radiation Qr, and also the thermal energy of the particle ET
Z t
Z t
2
2
Qabs ¼ pr0
I 0 ðtÞK abs dt; Qc ¼ 4pr0
J c dt;
0
0
Z t
J r dt; ET ¼ q0 c0 V 0 T 0 ; Qm ¼ q0 V 0 Lm ;
Qr ¼ 4pr20
0
ð9Þ
where Lm is the heat of unit mass melting. Integration
over the time t = 0 to t with regard for Eq. (4) and the
energies (9) will give the energy conservation law for
the particle
Qabs þ ET1 ¼ ET þ Qc þ Qr þ Qm ;
ð10Þ
ET1 = q0c0V0T1 is the initial thermal energy of a
particle, energy Qm = 0 under T < Tm. Eq. (10) can be
used for control of the energy conservation law of the
processes of laser action on particle.
The characteristic time for heat exchange between a
single particle and surrounding medium and formation
of quasi-stationary distribution of temperature in medium is tT r20 =4v, where v is the coefficient of thermal diffusivity of the medium around particle, for r0 = 30 nm
and v = 2.2 · 105 m2/s (air) [12] tT 1· 1011 s and for
v = 1.53 · 107 m2/s (water) [12] tT 1.4 · 109 s. For
short pulses, such that tp tT there is practically no heat
exchange between a particle and its environment during a
pulse, as confirmed by direct numerical calculations. If the
pulse duration is tp > tT the approximation of quasi-stationary heat exchange between a particle and its environment is justified.
The quasi-stationary temperature distribution
around of the particle in a spherical coordinate system
with the origin at the particle center under condition
t > tT was obtained in [13]
"
!#1=aþ1
aþ1
r0
T0
a 6¼ 1 T ðrÞ ¼ T 1 1 þ
1
;
r
T1
r0 =r
T0
;
a ¼ 1 T ðrÞ ¼ T 1
T1
ð11Þ
106
V.K. Pustovalov / Chemical Physics 308 (2005) 103–108
radius r P r0. A strong temperature gradient in the surrounding media is followed from (11). This dependence
describes the temperature distribution for t > tT. At the
distance of r = 2r0 from the particle surface the temperature reaches value of about 50% of the temperature of
the particle surface.
The heat flux Jc due to nonlinear conduction from the
particle in the quasi-stationary approximation is described by the following expression taking into account
(8) and (11):
" #
aþ1
k1 T 1
T0
a 6¼ 1 J c ¼
1 ;
ða þ 1Þr0
T1
ð12Þ
k1T 1
T0
a ¼ 1 J c ¼
ln
;
r0
T1
Heat transfer is rapid across the interface and there is
substantial heat loss from the particle into surrounding
medium. The heat flux because of radiation cooling Jr
is much smaller than Jc for temperature interval
T1 < T0 < 5 · 103 K (see estimation lower) and in our
investigation Jr will be neglected.
In some cases can be heating of a particle placed in
liquid medium over the temperature of explosive ebullution (vaporisation) of liquid (water) near the particle
surface and the vapor blanket (bubble) can be formed
around particle [14]. As soon as the bubble has been
formed which insulates the particle from laser radiation
(if bubble has been formed before the end of laser radiation action) and surrounding water. In this case the
particle will be contacted with water vapor under some
pressure and will be changed the value of heat conduction coefficient.
Eq. (4) in combination with expression (12) has analytic solution which describes the heating of a particle by
radiation of intensity I0 = const. since t = 0 till t = tp given below for two values of a:
I 0 K abs r0
½1 expðBtÞ;
a ¼ 0; T 0 ¼ T 1 þ
4k 1
ð13Þ
A þ 1 ðA 1Þ exp ðABtÞ
;
a ¼ 1; T 0 ¼ T 1 A
A þ 1 þ ðA 1Þ exp ðABtÞ
where
1=2
I 0 K abs r0
3k 1
A¼
þ1
; B¼
:
2k 1 T 1
c0 q0 r20
If the condition Btp 1(tp tT) is obeyed and the loss
of heat from the particle by heat conduction during the
time tp can be ignored, we find from expression (13) with
a = 0 that expansion of the exponential function gives
maximal value of T for t = tp
T max ¼ T 1 þ
3I 0 tp K abs
:
4q0 c0 r0
ð14Þ
Upon the attainment of the melting temperature Tm
the particle is being melted for a certain period of time
tm. Before and during particle melting there is no evap-
oration and the heat removed from particle by the mechanism of heat conduction with energy flux Jc. The
interval tm can be estimated from (4)
tm ¼
4r0 q0 Lm
:
3ðI 0 K abs 4J c Þ
ð15Þ
In many cases it is possible to neglect Jc in (15)
as compared with I0Kabs and tm can be estimated
from
tm ¼
4q0 r0 Lm
:
3I 0 K abs
ð16Þ
On melting or solidification of the particle in the
process of heating and cooling such parameters as q0,
c0, Kabs, radius r0, volume V0 undergo variations. When
T0 < Tm and T0 > Tm the parameters for a solid (index s)
or liquid (index l) state will be, respectively, used in
equations. For example, the density of gold in transition
from the solid to liquid state decreases noticeably: from
q0s = 1.93 · 104 kg m3 to q0l = 1.736 · 104 kg m3 [11].
In the absence of evaporation this leads to an increase in
the gold particle radius after melting according to the
particle mass conversation law: r0l = 1.035r0s. Energy
Qc can be calculated using Eq. (9) and expressions for
Jc (12), T (13) for period of time [0, tp] and (18) for
t > tp taking into account Tmax = T(t = tp), tmax = tp under a = 0
exp ðBtÞ 1 exp Btp
Qc ¼ Qabs 1 þ
:
ð17Þ
Btp
The cooling of particle after laser action can be described by expressions, which can be derived from (4)
under I0 = 0, T(t = tmax) = Tmax and t > tmax
a ¼ 0;
T 0 ¼ T 1 þ ðT max T 1 Þ expðcðt tmax ÞÞ;
a ¼ 1;
T0 ¼ T1
c1 þ exp ð2cT 1 ðt tmax ÞÞ
;
c1 exp ð2cT 1 ðt tmax ÞÞ
ð18Þ
where
c¼
3k 1
;
q0 c0 r20 ða þ 1ÞT a1
c1 ¼
T max þ T 1
;
T max T 1
Tmax is the maximal value of particle temperature usually at the end of laser pulse tp = tmax. In some experiments it were used repetitive laser pulses (laser pulse
train) with varying number of pulses ranging up to
103–105. In this case between pulse actions there are
periods of time when particle will be cooled. The value
of Tmax can be derived from (4) under condition dT0/
dt = 0 when heat generation inside particle and heat
conductive loss from particle is equal each other during
laser pulse action for t < tp
T max ¼ T 1
I 0 K abs ða þ 1Þr0
1þ
4k 1 T 1
1=aþ1
:
ð19Þ
V.K. Pustovalov / Chemical Physics 308 (2005) 103–108
The high peak temperature exists only for short period of time as the particle rapidly cools after the laser
pulse. The characteristic time for particle cooling tc is
proportional tc 1/c(a = 0) or tc 1/2cT1 (a = 1), as
follows from (18), and using the expression for c we have
tc r20 =3v1 for two values of a, v1 = k1/q0c0. So, characteristic cooling time is approximately equal characteristic time tT for heat exchange between particle and
surrounding media and formation of quasi-stationary
temperature distribution around of a particle.
3. Discussion and conclusion
The phenomenon of laser heating of small particles in
different experimental publications [1–8] is imaged as
follows. During the laser irradiation pulse the particle
is absorbing the radiation energy and the particle temperature is growing. Long laser pulses with pulse duration tp that exceed the thermal relaxation time tT,
tp tT, cause heating of both the particle and the surrounding medium as heat diffuses across the particle
boundary. If the laser pulse durationtp is much less than
tT then the laser energy can be thermally confined within
the particle for some period of time, causing rapid heating of the particle itself. Under power laser action temperature approaches the melting point and particle
melts. In the case of further heating temperature can
achieve the value of boiling (evaporation) of particle
material and particle can evaporate.
We can make of some estimations concerning experimental results [1,4,6] on the base of Eqs. (9)–(19). The
approach of reshaping and size reduction of nanoparticles in solution by pulsed laser irradiation was developed
in [1]. It was used next experimental parameters – wavelength 532 nm, rectangular shape laser pulse a 7 ns
duration, laser pulse energy between 5 and 60 mJ, intensity between 1.87 · 1010 and 2.26 · 1011 W/m2. Gold
particles with radiuses of between 7 and 20 nm were
placed in liquid solutions. Thermal relaxation times for
particle with r0 = 7, 20 nm are equal tT 7· 1011 and
5.7 · 1010 s, respectively, and much less than tp = 7
ns. Consequently the heat diffuses from particle during
laser action and we should take into account a quasi-stationary conduction flow from particle. But in [1] it was
neglected of heat conduction (convective) flow from particle during laser action for estimations. The temperature rise of particle under action of laser pulse with
energy 60 mJ calculated without heat loss from a particle was about 3 · 104 K [1]. This value is totally disregarded the real situation and we should take into
account the conductive heat flow from particle during
laser pulse action and in this case using (13) with a = 0
we have, DTmax = Tmax T1 = 5 · 102 and 4.6 · 103
K for r0 = 7 and 20 nm, respectively. When the laser
pulse energy was 5 mJ almost no change as a result of
107
laser action on particles was observed [1]. In this case
the maximal temperature at the end of single laser pulse
action is DTmax = 3.9 · 102 K for particle with r0 = 20
nm and smaller values for particles with smaller radiuses. When the energy was 28 mJ the change was clearly
observed [1] and this value can be viewed as threshold
one, first of all, for particles with maximal values of radius. In this case DTmax = 2.1 · 103 K for particle with
r0 = 20 nm and achieved temperature is greater than
melting temperature Tm = 1336 K for gold. The heated
particles were melted and became spherical. It was really
noted in [1] that the shape of the particles originally nonspherical became spherical. For example, for r0 = 20 nm
at I0 = 2.26 · 1011 W/m2 Eq. (16) yields tm = 1.4 · 1010
s. As a result of action of laser pulse train during 5–60
min with repetition rate 10 Hz can be the heating of particle with initial r0 = 20 nm up to temperature of evaporation with decreasing of radius in 2–3 times. It should
be noted that for these calculations of Tmax it was used
the value of thermal conductivity coefficient for water
surrounding particle. But as a result of laser action on
particle surrounded by liquid (water) can be produced
vapor bubble (blanket) around particle and the conditions of thermal transfer were changed. The use of the
thermal conductivity coefficient for water vapor under
pressure leads to increasing of particle temperature in
comparison with values noted above.
In [4] it was experimentally investigated the heating of
spherical gold particles with radius between 25 and 50
nm under action of laser pulses with next parameters:
wavelength 400 nm, spot size 3 · 108 m2, pulse duration 0.1 ps, pulse energy up to 3 lJ. Characteristic
times tT for r0 = 25, 50 nm are equal 9 · 1010 and
3.6 · 109 s and tp is much smaller than tT. In this case
we can neglect by conduction heat losses from particle
during laser pulse action and use Eqs. (14) and (16)
for estimations. For pulse energy 3 lJ maximal values
of temperature Tmax for particles with r0 = 25 nm are
equal 2.1 · 103 K. Consequently, particle with
r0 P 25 nm can be melted under laser action. This possibility confirmed by experimental results [4]. The melting time tm (16) for r0 = 25 nm is approximately equal
0.02 ps and the process of melting (fusion) can be finished during period of laser action. The estimation of
characteristic cooling time tc gives value tc 900 ps
for r0 = 25 nm and 3.6 ns for r0 = 50 nm.
Formation of gold small nanoparticles by action of
intense pulsed laser radiation on initial gold nanoparticles was investigated in [6]. It was used next experimental parameters – wavelength 532 nm, pulse duration
tp 10 ns and fluence 1 · 104 J/m2 and more, repetition
rate 10 Hz, initial radius of gold particles 10 nm. Thermal relaxation time for particle with radius 10 nm is
equal about tT 6 · 1010 s tp and can be used quasi-stationary approximation for estimations. Under laser
irradiation with fluence 1 · 104 J/m2 gold nanoparticle
108
V.K. Pustovalov / Chemical Physics 308 (2005) 103–108
was heated up to 4.6 · 103 K, see (13). In this case
T0 > Tev and heat losses from particle are determined
by evaporation and conduction. Estimations of Jc and
Jr on the base of (8) using the characteristic temperature
value T0 = 3 · 103 K give J rmax rT 40 4:6 106 W=m2
(e = 1, value T1 was neglected), Jc k1 (T0 T1)/
r0 3 · 1010 W/m2 (value k1 1 · 104 W/m K for
water vapor under some pressure and greater for water)
and Jc Jr. In [6] it was estimated that Jr > Jc, see also
[1], that disagrees with real situation. The absorbed photon energy per one gold nanoparticle having the radius
of 10 nm Qabs is given by (9), Qabs ¼ pr20 I 0 tp K abs at
I0 = const and for laser fluence I0tp = 1 · 104 J/pulse m2
Qabs = 3.75 · 1012 J.
A few attempts to develop theoretical models for
description of the processes of laser pulse action on
nanoparticles were made during last years. Empirical
model of laser heating of nanoparticles under very
rough assumptions, such as – the characteristic thickness
of temperature boundary layer is equal to the particle
radius, absorption of laser radiation energy by spherical
particle is described by BeerÕs law not Mie theory and
so on, was developed in [7]. But comparison their theoretical results with own experimental ones showed disagreement. Qualitative model of laser heating based on
assumption of strong balance between energy absorption and heat losses by particle was proposed in [8]. This
model can not be used for description of non-stationary
stage of heating processes before the achievement of
Tmax during period of time since t = 0 till tmax. Heat flow
from particle in [8] is determined by simple difference between particle temperature and temperature of liquid
near the particle surface without taking into account
the temperature gradient around particle. It was numerically solved the heat transfer equation describing heat
dissipation from sphere after the ending of laser radiation action in [5].
In this paper, theoretical study of the heating of
spherical solid (metal) small particle by laser pulse is
developed. Consideration is made of heating, phase
changes and cooling of a spherical particle in some liquid or gaseous medium under the action of laser pulse.
The solutions are obtained for quasi-stationary distributions of temperature inside and around of a particle and
for conductive heat transfer of a particle with surrounding media taking into account nonlinear dependence of
heat conduction coefficients on temperature. Analytical
expressions for particle temperature depending on time,
parameters of particle, laser radiation and surrounding
medium are obtained. Comparison of some predicted results of the heating of gold spherical nanoparticle in liquid media with experimental data [1,4,6] is given and
agreement of theoretical results with experimental data
validates the model and theory developed. Our theoretical results can be directly related to experimental investigations of laser pulsed action on metal nanoparticles
and can be applied for estimations and interpretation
of experimental results.
Acknowledgement
This investigation was supported by BMBF (German
Ministry of Education and Research) WTZ Project BLR
03/003.
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