# Ultrashort Laser Pulse Heating of Nanoparticles: Comparison of Theoretical Approaches

Authors: Renat R. Letfullin; Thomas F. George; Galen C. Duree; Brett M. Bollinger
Journal: Advances in Optical Technologies
 (2008)

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## Abstract

The interaction between nanoparticles and ultrashort laser pulses holds great interest in laser nanomedicine, introducing such possibilities as selective cell targeting to create highly localized cell damage. Two models are studied to describe the laser pulse interaction with nanoparticles in the femtosecond, picosecond, and nanosecond regimes. The first is a two-temperature model using two coupled diffusion equations: one describing the heat conduction of electrons, and the other that of the lattice. The second model is a one-temperature model utilizing a heat diffusion equation for the phonon subsystem and applying a uniform heating approximation throughout the particle volume. A comparison of the two modeling strategies shows that the two-temperature model gives a good approximation for the femtosecond mode, but fails to accurately describe the laser heating for longer pulses. On the contrary, the simpler one-temperature model provides an adequate description of the laser heating of nanoparticles in the femtosecond, picosecond, and nanosecond modes.


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## Body

## 1. Introduction

The application of ultrashort laser pulse thermal-based killing of
abnormal cells (e.g., cancer cells) targeted with absorbing nanoparticles (e.g.,
solid gold nanospheres, nanoshells, or nanorods) is becoming an extensive area
of research [1–3]. High laser
energies in ultrashort pulses can be concentrated on an extremely small
biological mass infused with metallic nanoparticles, and the heat gained by the
particles used as a protein denaturizing agent in targeted cells. Photons
emitted from the ultrashort laser pulse are absorbed by free electrons within
the metal via inverse Bremsstrahlung and transferred to the lattice subsystem,
and then to the surrounding medium. Injected into biological media,
nanoparticles provide a highly selective method for ablating target cells when
coupled with appropriate laser pulse duration.Traditional knowledge of laser-nanoparticle interactions has necessitated
specialized models for each case, dependent upon the laser pulse duration.
Previous research has recognized the existence of these conditions and
dual-temperature models for the ultrashort laser pulse mode calculating
electron, and lattice subsystem temperatures are readily available (see, e.g.,
[4, 5]). Pulses of
longer duration are modeled using a uniform heating model [6–9], an
appropriate approximation for pulse durations greatly exceeding the
electron-phonon coupling time. On a dimensional scale, this approximation is
reasonable for particles sizes not much larger than a laser wavelength, which
is completely applicable for the heating of nanoparticles.Ultrashort pulses, specifically those in the femtosecond and picosecond
ranges, impose several challenges in modeling material response. Free electrons
with minimal capacity for heat are the first to absorb energy, rapidly
attaining high temperatures, and transferring thermal energy to the material
lattice. These processes do not occur instantaneously: time must be allowed for
the cooling of the electrons and the heating of the lattice. In our work, electron
cooling and lattice heating have time delays on the order of femtoseconds and
picoseconds, respectively. Ultrashort laser pulses end before the transfer of
energy to the lattice is complete, requiring two-temperature models in order to
describe the further conversion of energy from electron excitation to heat
within the lattice system.In the present paper, we demonstrate that a simpler, one-temperature
model (OTM) utilizing the uniform heating approximation is appropriate for
understanding ultrashort laser pulse interactions with metal nanoparticles. The
approximation may be used in this situation due to the extremely small size of
nanoparticles in comparison to the wavelength of laser radiation. Using this
idea, the time delay between the electron and lattice interactions will be
bypassed, and the simpler model will ultimately yield results similar to the
two-temperature model (TTM). Comparative simulations of the two modeling
approaches are performed in this paper to confirm OTM as an appropriate approximation
for nanoparticle heating in the femtosecond, picosecond, and nanosecond
regimes, thus providing an effective modeling method for further nanomedicine
research to explore.


## 2. Theoretical Background

### 2.1. One-Temperature Model

During the interaction of a laser pulse of intensityI0 and pulse duration τL with a metal nanoparticle of radius r0,
the laser energy is absorbed by free electrons due to the inverse
Bremsstrahlung, and then transferred from the electron gas into the lattice. In
OTM, it is assumed that the electron heat transfer into phonon subsystem is
very fast, that is, the electron and lattice temperatures are equal Te=Ts at any moment of the time. In this approximation, we can
limit our description to only one lattice temperature distribution, Ts(t,r),
which could be found by numerical solution of a heat-mass transfer equation
between the particle and the surrounding medium: (1)dTs(t,r)dt=μs(Ts)ρsCs(Ts)ΔTs(t,r)+Q(t,r)ρsCs(Ts)−jD(Ts)S0+3Lr0Cs(Ts)dr0dt, where Δ=∂2/∂x2+∂2/∂y2+∂2/∂z2 is a Laplace operator; μs(Ts),C(Ts),L,ρs, and r0 are,
respectively, the heat conductivity, specific heat, evaporation heat, density,
and radius of the nanoparticle; Q(t,r) is a heat source; jD(Ts) is the heat lost from the surface of the
nanoparticle into the surrounding medium; S0=4πr02 is the particle surface area.The power density of energy generation in the particleQ(t,r) due to radiation energy absorption is generally nonuniform throughout the
particle volume, with the nonuniformity being dependent on the size and optical
constants of the particle. Since for nanoparticles 2πr0/λ<1, we can assume that Q(t,r) is uniform throughout the particle volume [6–9], and it can be
described by the equation (2)Q(t,r0)=3Kabs(r0,λ)I0f(t)4r0, where Kabs(r0,λ) is the absorption efficiency of the
nanoparticle as a function of laser wavelength λ and particle radius r0, and f(t) is a time profile of a laser pulse.Heat exchange between the nanoparticle surface and the surrounding medium
is rapid, and heat loss becomes substantial for relatively long laser pulses.
Assuming that the heat lost from the surface of nanoparticle occurs only due to
heat diffusion into surrounding medium, the energy flux densityjD(Ts) removed from the particle
surface can be expressed as a nonlinear function of temperature [7–9]: (3)jD(Ts)=μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1], where μ∞ is the
heat conductivity of the surrounding medium at normal temperature T∞,
and the power exponent s=const depends on the thermal properties of
the surrounding medium.After integration over the volume for the spherically symmetric case and
transition to a uniform temperature over the particle volume, the equation
which describes the kinetics of laser heating of the nanoparticle and results
from (1) takes the following form [7–9]: (4)dTsdt=3KabsI0f(t)4r0Cs(Ts)ρs−μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1]+3Lr0Cs(Ts)dr0dt. Here, the first term on the right
side of the equation describes the heat generation into the spherical volume
due to laser energy absorption by the nanoparticle. The second term describes
the energy losses from the surface of the particle into the surrounding medium
due to the heat diffusion process. The last term describes the energy losses
due to the evaporation of the particle. This evaporation depends on the laser
pulse characteristics and particle properties, and it can be realized in five
different regimes (see, e.g., [8, 9]): free-molecular, convective, diffusive, gas-dynamic, and explosive [3] modes of
evaporation. For example, within the approximation of free-molecular flow, the
evaporation term in (4) can be written as (5)4πr02dr0dtρs=−η4πr02Vs(Ts)ρv(Ts), where η is
the accommodation coefficient, and Vs(Ts) and ρs(Ts) are,
respectively, the average velocity and density of the vapor at temperature Ts.
If the heating of the nanoparticle occurs below the temperature of phase
transition in the particle material, the third term on the right side of (4)
can be neglected.


### 2.2. Two-Temperature Model

In
TTM, the temperature relaxation in time and sample depth can be modeled by
two-coupled diffusion equations: one describing the heat conduction of
electrons and the other that in the lattice. Both equations are connected by a
term that is proportional to the electron-phonon coupling constantγ and
to the temperature difference between electrons and lattice, originally
proposed by Anisimov et al. [10]. We modify this original set of equations,
adding a heat exchange term between the surface of the particle and the
surrounding medium. Also, we are taking into account the dependence of the
thermophysical parameters of electrons, particle, and surrounding medium on
temperature during the laser treatment, as presented in Table 1. The modified
set of equations describing the heating of electron and lattice subsystems and
the energy transfer between particle and surrounding medium has the form (6)Ce(Te)∂Te∂t=−∂Q(z)∂z−γ(Te−Ts)+S,Ci(Ts)∂Ts∂t=γ(Te−Ts)−μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1], where Q(z)=−ke(∂Te/∂z) is the heat flux; z is
the direction perpendicular to the target surface; ke is the
electron thermal conductivity; S=I0f(t)α⁢exp⁡(−αz) is the laser heating source
term; α is the material absorption coefficient; Ce(Te) and Ci(Ts) are the temperature-dependent
heat capacities (per unit volume) of the electron and lattice subsystems. The
expressions for the temperature dependence of the heat capacities, as well as
the values for the electron-phonon coupling constant γ and material
data, are listed in Table 1.Table 1
Input parameters used for simulations in both models.
ParametersMagnitude and unitsReferenceLaser pulse shapef(t)=exp[−(at−b)2−(at−c)2],a,b,c=constEnergy densityE=10−3J/cm2Volume density of the goldρ0=0.0193kg/cm3[11]Electron heat capacity (gold)Ce(Te)=aTe4−bTe3+cTe2−dTe+e,(JK2cm3)a=6.0×10−17,b=3.0×10−12,c=5.07×10−8,d=6.7589×10−5,e=7.207×10−2,[12]Electron-phonon coupling
constant for goldγ=2.27×1010WKcm3[13]Gold specific heatC=129JKkgSpecific heat of waterCw(Ts)=a(1+b(Ts−293K)),(JKkg)a=4182.0,b=1.016×10−4.Radius of gold nanoparticler0=20nmAbsorption efficiency of gold nanoparticleKabs=4.0195(r0=20andλ=528nm)[2, 14]Absorption coefficient of gold nanoparticleα=8.736×105cm−1(r0=20nmandλ=528nm)[14]Power exponents=1.0Thermal conductivity of waterμ0(Ts)=a(1+b(Ts−T∞)),(WKcm)a=0.00597,b=1.78×10−3.


## 2.1. One-Temperature Model

During the interaction of a laser pulse of intensityI0 and pulse duration τL with a metal nanoparticle of radius r0,
the laser energy is absorbed by free electrons due to the inverse
Bremsstrahlung, and then transferred from the electron gas into the lattice. In
OTM, it is assumed that the electron heat transfer into phonon subsystem is
very fast, that is, the electron and lattice temperatures are equal Te=Ts at any moment of the time. In this approximation, we can
limit our description to only one lattice temperature distribution, Ts(t,r),
which could be found by numerical solution of a heat-mass transfer equation
between the particle and the surrounding medium: (1)dTs(t,r)dt=μs(Ts)ρsCs(Ts)ΔTs(t,r)+Q(t,r)ρsCs(Ts)−jD(Ts)S0+3Lr0Cs(Ts)dr0dt, where Δ=∂2/∂x2+∂2/∂y2+∂2/∂z2 is a Laplace operator; μs(Ts),C(Ts),L,ρs, and r0 are,
respectively, the heat conductivity, specific heat, evaporation heat, density,
and radius of the nanoparticle; Q(t,r) is a heat source; jD(Ts) is the heat lost from the surface of the
nanoparticle into the surrounding medium; S0=4πr02 is the particle surface area.The power density of energy generation in the particleQ(t,r) due to radiation energy absorption is generally nonuniform throughout the
particle volume, with the nonuniformity being dependent on the size and optical
constants of the particle. Since for nanoparticles 2πr0/λ<1, we can assume that Q(t,r) is uniform throughout the particle volume [6–9], and it can be
described by the equation (2)Q(t,r0)=3Kabs(r0,λ)I0f(t)4r0, where Kabs(r0,λ) is the absorption efficiency of the
nanoparticle as a function of laser wavelength λ and particle radius r0, and f(t) is a time profile of a laser pulse.Heat exchange between the nanoparticle surface and the surrounding medium
is rapid, and heat loss becomes substantial for relatively long laser pulses.
Assuming that the heat lost from the surface of nanoparticle occurs only due to
heat diffusion into surrounding medium, the energy flux densityjD(Ts) removed from the particle
surface can be expressed as a nonlinear function of temperature [7–9]: (3)jD(Ts)=μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1], where μ∞ is the
heat conductivity of the surrounding medium at normal temperature T∞,
and the power exponent s=const depends on the thermal properties of
the surrounding medium.After integration over the volume for the spherically symmetric case and
transition to a uniform temperature over the particle volume, the equation
which describes the kinetics of laser heating of the nanoparticle and results
from (1) takes the following form [7–9]: (4)dTsdt=3KabsI0f(t)4r0Cs(Ts)ρs−μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1]+3Lr0Cs(Ts)dr0dt. Here, the first term on the right
side of the equation describes the heat generation into the spherical volume
due to laser energy absorption by the nanoparticle. The second term describes
the energy losses from the surface of the particle into the surrounding medium
due to the heat diffusion process. The last term describes the energy losses
due to the evaporation of the particle. This evaporation depends on the laser
pulse characteristics and particle properties, and it can be realized in five
different regimes (see, e.g., [8, 9]): free-molecular, convective, diffusive, gas-dynamic, and explosive [3] modes of
evaporation. For example, within the approximation of free-molecular flow, the
evaporation term in (4) can be written as (5)4πr02dr0dtρs=−η4πr02Vs(Ts)ρv(Ts), where η is
the accommodation coefficient, and Vs(Ts) and ρs(Ts) are,
respectively, the average velocity and density of the vapor at temperature Ts.
If the heating of the nanoparticle occurs below the temperature of phase
transition in the particle material, the third term on the right side of (4)
can be neglected.


## 2.2. Two-Temperature Model

In
TTM, the temperature relaxation in time and sample depth can be modeled by
two-coupled diffusion equations: one describing the heat conduction of
electrons and the other that in the lattice. Both equations are connected by a
term that is proportional to the electron-phonon coupling constantγ and
to the temperature difference between electrons and lattice, originally
proposed by Anisimov et al. [10]. We modify this original set of equations,
adding a heat exchange term between the surface of the particle and the
surrounding medium. Also, we are taking into account the dependence of the
thermophysical parameters of electrons, particle, and surrounding medium on
temperature during the laser treatment, as presented in Table 1. The modified
set of equations describing the heating of electron and lattice subsystems and
the energy transfer between particle and surrounding medium has the form (6)Ce(Te)∂Te∂t=−∂Q(z)∂z−γ(Te−Ts)+S,Ci(Ts)∂Ts∂t=γ(Te−Ts)−μ∞Ts(s+1)r02Cs(Ts)ρs[(TsT∞)s+1−1], where Q(z)=−ke(∂Te/∂z) is the heat flux; z is
the direction perpendicular to the target surface; ke is the
electron thermal conductivity; S=I0f(t)α⁢exp⁡(−αz) is the laser heating source
term; α is the material absorption coefficient; Ce(Te) and Ci(Ts) are the temperature-dependent
heat capacities (per unit volume) of the electron and lattice subsystems. The
expressions for the temperature dependence of the heat capacities, as well as
the values for the electron-phonon coupling constant γ and material
data, are listed in Table 1.Table 1
Input parameters used for simulations in both models.
ParametersMagnitude and unitsReferenceLaser pulse shapef(t)=exp[−(at−b)2−(at−c)2],a,b,c=constEnergy densityE=10−3J/cm2Volume density of the goldρ0=0.0193kg/cm3[11]Electron heat capacity (gold)Ce(Te)=aTe4−bTe3+cTe2−dTe+e,(JK2cm3)a=6.0×10−17,b=3.0×10−12,c=5.07×10−8,d=6.7589×10−5,e=7.207×10−2,[12]Electron-phonon coupling
constant for goldγ=2.27×1010WKcm3[13]Gold specific heatC=129JKkgSpecific heat of waterCw(Ts)=a(1+b(Ts−293K)),(JKkg)a=4182.0,b=1.016×10−4.Radius of gold nanoparticler0=20nmAbsorption efficiency of gold nanoparticleKabs=4.0195(r0=20andλ=528nm)[2, 14]Absorption coefficient of gold nanoparticleα=8.736×105cm−1(r0=20nmandλ=528nm)[14]Power exponents=1.0Thermal conductivity of waterμ0(Ts)=a(1+b(Ts−T∞)),(WKcm)a=0.00597,b=1.78×10−3.


## 3. Results and Discussion

Comparative simulations
using the models described above have been performed for the laser heating of a
gold nanoparticle with radiusr0=20 nm in a surrounding
medium of water. The same set of input data presented in Table 1 was used for
the calculations in both models. The temperature dependences of the electron
heat capacity for the gold, specific heat, and thermal conductivity for the
water were obtained by interpolating the experimental data available in the
literature (references are listed in Table 1). As can be seen from the table,
the electron heat capacity is much less than the lattice heat capacity, and
therefore electrons can be heated to very high transient temperatures. Then,
the evolution of the electron temperature involves energy transfer to the
lattice and energy losses due to the electron heat transport into the target.
The electron-phonon coupling process has several characteristic time scales:
electron thermalization time τe, electron cooling time τc, lattice heating time τi, and duration of the laser pulse τL. The relationship between them defines three different regimes of the
laser-metal interaction—femtosecond, picosecond, and nanosecond modes
of heating.


### 3.1. Femtosecond Pulses

TTM and OTM have been
solved numerically to predict the time dependence of the electron and lattice
temperatures in the femtosecond mode when the laser pulse duration is shorter
than the electron thermalization and cooling times,τL≪τe,τc. The calculations were
performed for a laser pulse energy density of E=1.0 mJ/cm2 and pulse duration of τL=60 femtoseconds. The time profile of the femtosecond laser pulse given in
Table 1 and shown in Figure 1(b) (solid curve) corresponds to the experimentally
observed output from an amplified Ti: sapphire laser (Legend-HE from Coherent Inc., Santa Clara, Calif, USA). The laser flux is chosen at the
level 1.0 mJ/cm2 to provide the cell lethality during a single laser
pulse. The results of the simulations for the heating of a gold nanoparticle by
a femtosecond laser pulse are shown in Figures 1 and 2. Figure 1(b) displays
the results obtained by TTM, and Figure 1(b) presents the results of OTM
simulations. A comparison of these two models is shown in Figure 2.(a) Electron (solid curve) and lattice (dashed curve) temperature evolutions on
the femtosecond time scale for a gold nanoparticle predicted by TTM. (b)
Evolution of the nanoparticle temperature (dashed curve) after the femtosecond
laser pulse, predicted by OTM, and laser pulse shape (solid curve).
(a)(b)Figure 2
Comparison of nanoparticle temperature evolutions after the 60 femtoseconds
laser pulse-predicted TTM (solid curve) and one-temperature model (dashed
curve).As follows from Figure1(a),
thermal equilibrium among the excited electrons with equilibrium temperature Te≈3300K for a given laser fluence is
established within 175 femtoseconds. We should note that the equilibrium
temperature for electrons is reached long after the end of a laser pulse, which
had a duration of 60 femtoseconds. The electrons remain in the thermal
equilibrium state from several hundred femtoseconds up to 1 picosecond (see Figure 3(a)). Then, the electrons cool exclusively by coupling to the lattice,
resulting in a linear decay of the electron temperature during the first 10 picoseconds (Figure 3(a)). Our simulations agreed with the electron relaxation
time measured in [15] for femtosecond pulse excitation of a DNA-modified gold
nanoparticle.(a) Electron temperature relaxation on the picosecond time scale. (b)
Temperature time distributions for a gold nanoparticle predicted by OTM (solid
curve) and TTM (dashed curve) after the 60 picoseconds laser pulse.
(a)(b)The slow rate of electron
heat diffusion into the phonon subsystem on the femtosecond time scale results
in a delay of about 100 femtoseconds in the heating of the bulk sample (dashed curve in Figure1(a)). Once the electron thermal equilibrium is established, a
hot electron bath raises the temperature of the cold lattice up to 1090 K for a
given laser energy density E=1 mJ/cm2.The results of heating the
gold nanoparticle obtained by OTM are demonstrated in Figure1(b). This figure
also shows the femtosecond laser pulse time profile used in the calculations.
Comparative simulations for the evolution of the nanoparticle temperature using
both models under the same conditions are presented in Figure 2. It follows
from these simulations that both models demonstrate the same scenario in the
heating kinetics of a metal nanoparticle by a femtosecond laser pulse. Both
models reveal approximately a 100-femtosecond time delay in the heating of the
particle, followed by a maximum lattice temperature of around 1090 K within 175 femtoseconds after the end of a laser pulse. Even the maximum values of the
particle temperature predicted by both models are the same (see Figure 2). The
saturation parts in the lattice temperature curves are explained by negligibly
small heat diffusion from the surface of the nanoparticle into the surrounding
medium on the femtosecond time scale. A slight difference in the slopes of the
temperature curves within the first 100 femtoeconds of heating occurs due to the
assumption made in OTM that the electron heat transfer into the lattice
subsystem is very fast. Because of this, the particle temperature in OTM
promptly follows electron thermal behavior. A comparison of these two models
shows that the simpler OTM gives the same results as the more precise TTM.
Thus, OTM provides an adequate description of the laser heating of
nanoparticles in the femtosecond regime.


### 3.2. Picosecond Pulses

In this mode, the
constantsa,b, and c in the time profile for the laser
pulse have been chosen to provide the laser pulse width of 60 picoseconds at
FWHM with the same pulse shape listed in Table 1 and shown in Figure 1(b). As
can be seen from Figures 1(b) and 3(b), in the picosecond regime the electron
thermalization time τe and electron cooling time τc are much less than the duration τL of the 60 picoesonds laser
pulse. Hence, during the first 10 picoseconds, the electron subsystem has
already been completely cooled (Figure 3(a)), and the electron temperature
above the ambient no longer exists for the considered picosecond mode of
heating. This is confirmed by our calculations presented on Figure 3(b) (dashed
curve). Thus, TTM provides a very good approximation for the femtosecomd mode
as soon as the electron temperature exists, but it fails to describe the laser
heating of nanoparticles for longer pulse durations in the picosecond and
nanosecond regimes.Opposite to TTM, OTM
describes the picosecond heating kinetics very well. A typical time evolution
of the particle temperature predicted by OTM on the picosecond time scale is
displayed in Figure3(b) (solid curve) for the same material constants listed
in Table 1. The main feature is the appearance of heat lost from the surface of
the nanoparticle into the surrounding medium on the picosecond time scale.
After about 200 picoseconds, cooling of the nanoparticle begins due to heat
diffusion into the water. The maximum temperature reached by a 20-nm gold
particle for the given laser pulse is 995 K. This temperature is sufficient to
initiate any thermal killing mechanisms in cancer cells.


### 3.3. Nanosecond Pulses

For laser heating of
metal nanoparticles in the nanosecond regime, the characteristic lattice
heating timeτi is much smaller than the laser pulse
duration: τL≫τi. This means that,
the temperature inside the nanoparticle is nearly uniform over the whole
particle at the time scale of the laser pulse duration τL. In
this case, the electron and lattice temperatures are equal, Te=Ts, so that the homogeneous heating of the particle and
quasisteady heat exchange with the surrounding medium can be described by just
OTM. The characteristic lattice heating time τi required for
the formation of a quasistationary temperature profile across the nanoparticle
can be estimated from the formula τi=r02/4χ, where r0 is the particle radius and χ the
thermal diffusivity of the particle material. For gold nanoparticle (χ=1.18×10−4m2/s) with radii r0= 20–30 nm, the lattice
heat diffusion time is τi∼2×10−12s≪τL∼10−8s.Sample calculations have
been carried out using OTM for gold nanoparticles with radiir0= 30–35 nm in
different surrounding biomedia for an incident laser pulse of energy E=10 mJ/cm2 and pulse duration τL=8 nanoseconds with the time
profile shown in Figure 1(b). The laser pulse profile and duration 8 nanoseconds have been chosen to be close to those used in previous experiments
[2]. The laser flux chosen is ten times higher than in the regimes considered
above to provide approximately the same maximum nanoparticle temperature as
observed for femtosecond and picosecond laser heating. We should note that 10 mJ/cm2 is comparable to the laser fluence currently used in the
photothermolysis of cancer cells [1, 3]. The kinetics of heating and cooling the
gold nanoparticle are demonstrated in Figure 4, where (a) illustrates the time
dynamics of laser heating of a 30-nm gold particle in different biological
media: blood, human prostate, tumor, and fat. The thermophysical
characteristics of gold and biological surrounding media for different
temperatures are listed in Table 2. Figure 4(b) shows results of thermal
calculations for a 35-nm gold particle, which is heated and cooled in water at
different heat transfer rates s.Table 2
Thermophysical characteristics of the gold particle and surrounding biological tissue.
MaterialSpecific heatC(J/Kkg)Interval ofT (K)Thermal conductivity μ0 (W/m K)Thermal diffusivityχ(m2/s)Gold129273–3733181.18×10−4Water4181.6−4215.6C(T)=4182(1+1.016×10−4(T−293K))273–3730.597−0.682μ(T)=0.597(1+1.78×10−3(T−T∞))1.43×10−7Human prostate37403100.529Blood3645–3897273–3730.48–0.61.6×10−7Fat2975273–3730.185–0.233Tumor31603100.561Skin273–3730.210–0.410Kinetics of heating and cooling of a gold nanoparticle by a nanosecond laser
pulse of energy density 10 mJ/cm2 and duration 8 nanoseconds.
Calculations have been made by using OTM. (a) Illustration of the time dynamics
of laser heating of a 30-nm gold particle in different biological media: fat
(dashed-dotted curve), blood (dashed curve), tumor (solid curve), and prostate
(dotted curve). (b) Results of thermal calculations for a 35-nm gold particle:
heating and cooling in the water surrounding medium at different heat transfer
rates s:s=1.0 (solid curve), s=1.25 (dashed curve),
and s=1.5 (dashed-dotted curve).
(a)(b)It follows from these
calculations that during the laser pulse duration the transfer of heat from the
nanoparticle into the surrounding media is slight, and the particle rapidly
reaches a high temperature. The heating rate is about 1012 Ks-1. The temperature of the particle continues to rise even after the end of the
laser pulse. The highest temperature, 770 K, for a given laser pulse fluence is
observed for the heating time of 13.5 nanoseconds, when the laser pulse has
already degraded (see Figure 4(a)). After that time, the transfer of heat from
the particle to the surrounding medium becomes increasingly important, since
the energy source is no longer present in the system. The temperature of the
particle and surrounding medium remains high (~ 400 K) up to 20 nanoseconds, exceeding
the laser pulse duration by 2.5 times. The total time for one cycle (heating
from the initial temperature 300 K to maximum temperature, followed by cooling
back to the initial temperature) is about 30 nanoseconds.We have also examined the
effect of different biological surroundings on the laser heating dynamics of 30 nm gold particles. Four biomedia were used: namely, blood, human prostate,
tumor, and fat. Results of computer simulations of the time-temperature
profiles of gold nanoparticles in various biological media, performed by using
OTM, are plotted in Figure4(a). As follow from our calculations, the laser
heating and temperature behavior of the gold nanoparticles in blood, prostate,
and tumor are comparable to the water surrounding medium case, since the
thermodynamic properties of those media are very close to each other (see Table 2). This means that for the thermal calculations of laser heating of biological
media, the thermal properties of water can be used if the water content in the
media is high. But the heating of a gold nanoparticle in fat is substantially
different from the water case, since the fat has low-thermal characteristics.
Here, we observe higher overheating of the particle at the same energy level
and duration of the laser pulse due to the relatively low-thermal conductivity
of fat as compared to other biomedia.The temperature dynamics
of the particle is sensitive to the power exponents used in the
temperature dependence of the heat lost from the surface of the nanoparticle
into the surrounding medium, that is, (jD(Ts) in (3) (see Figure 4(b)). The value of s=1 better corresponds to the real biological surrounding. The power s>1 describes the medium with high thermophysical characteristics, like the
cooling liquids and metals. The medium with s<1 has a low thermal
conductivity and can be used as a thermal isolator.It is interesting to
investigate the effect of the particle’s radius on the temperature dynamics of
the nanoparticle heated by the nanosecond laser radiation in the biological
surroundings. There are two competitive factors here. On one hand, according to
the Mie diffraction theory, the absorption efficiency of the gold nanoparticle
drops with the decreasing size of the particle. On the other hand, the heating
rate increases for smaller particles as follows from (4). To find which factor
has a stronger effect on the effective laser heating of a gold nanoparticle, we
have calculated the maximal temperature profile for different nanoparticle
radii in blood and compared it to the gold nanoparticle absorption curve. The
results of these simulations are performed by using OTM and presented in Figure 5. It follows from this figure that the optical effect is much stronger than
the thermal effect when the radius of nanoparticle is less than 35 nm. In the
radii range 1–35 nm, the
overheating effect of the particle behaves according to the absorption
efficiency. For a radius of 35 nm, the thermal processes dominate over the
optical properties. For large radii (≥35nm), the maximal temperature profile
is saturated with oscillations, repeating the maxima and minima of the
absorption curve. The saturation of the maximal temperature curve for large
particle radii is explained by the balance between heating of the particle due
to absorption of laser energy and energy losses from the surface of the
particle due to heat diffusion into the surrounding biological medium.Figure 5
Nondimensional (a.u.) absorption efficiencyKabs and maximal temperature Tmax curves as a function of the particle’s radius for the gold nanoparticles in
blood heated by a laser pulse of energy density 10 mJ/cm2 and duration 8 nanoseconds.


## 3.1. Femtosecond Pulses

TTM and OTM have been
solved numerically to predict the time dependence of the electron and lattice
temperatures in the femtosecond mode when the laser pulse duration is shorter
than the electron thermalization and cooling times,τL≪τe,τc. The calculations were
performed for a laser pulse energy density of E=1.0 mJ/cm2 and pulse duration of τL=60 femtoseconds. The time profile of the femtosecond laser pulse given in
Table 1 and shown in Figure 1(b) (solid curve) corresponds to the experimentally
observed output from an amplified Ti: sapphire laser (Legend-HE from Coherent Inc., Santa Clara, Calif, USA). The laser flux is chosen at the
level 1.0 mJ/cm2 to provide the cell lethality during a single laser
pulse. The results of the simulations for the heating of a gold nanoparticle by
a femtosecond laser pulse are shown in Figures 1 and 2. Figure 1(b) displays
the results obtained by TTM, and Figure 1(b) presents the results of OTM
simulations. A comparison of these two models is shown in Figure 2.(a) Electron (solid curve) and lattice (dashed curve) temperature evolutions on
the femtosecond time scale for a gold nanoparticle predicted by TTM. (b)
Evolution of the nanoparticle temperature (dashed curve) after the femtosecond
laser pulse, predicted by OTM, and laser pulse shape (solid curve).
(a)(b)Figure 2
Comparison of nanoparticle temperature evolutions after the 60 femtoseconds
laser pulse-predicted TTM (solid curve) and one-temperature model (dashed
curve).As follows from Figure1(a),
thermal equilibrium among the excited electrons with equilibrium temperature Te≈3300K for a given laser fluence is
established within 175 femtoseconds. We should note that the equilibrium
temperature for electrons is reached long after the end of a laser pulse, which
had a duration of 60 femtoseconds. The electrons remain in the thermal
equilibrium state from several hundred femtoseconds up to 1 picosecond (see Figure 3(a)). Then, the electrons cool exclusively by coupling to the lattice,
resulting in a linear decay of the electron temperature during the first 10 picoseconds (Figure 3(a)). Our simulations agreed with the electron relaxation
time measured in [15] for femtosecond pulse excitation of a DNA-modified gold
nanoparticle.(a) Electron temperature relaxation on the picosecond time scale. (b)
Temperature time distributions for a gold nanoparticle predicted by OTM (solid
curve) and TTM (dashed curve) after the 60 picoseconds laser pulse.
(a)(b)The slow rate of electron
heat diffusion into the phonon subsystem on the femtosecond time scale results
in a delay of about 100 femtoseconds in the heating of the bulk sample (dashed curve in Figure1(a)). Once the electron thermal equilibrium is established, a
hot electron bath raises the temperature of the cold lattice up to 1090 K for a
given laser energy density E=1 mJ/cm2.The results of heating the
gold nanoparticle obtained by OTM are demonstrated in Figure1(b). This figure
also shows the femtosecond laser pulse time profile used in the calculations.
Comparative simulations for the evolution of the nanoparticle temperature using
both models under the same conditions are presented in Figure 2. It follows
from these simulations that both models demonstrate the same scenario in the
heating kinetics of a metal nanoparticle by a femtosecond laser pulse. Both
models reveal approximately a 100-femtosecond time delay in the heating of the
particle, followed by a maximum lattice temperature of around 1090 K within 175 femtoseconds after the end of a laser pulse. Even the maximum values of the
particle temperature predicted by both models are the same (see Figure 2). The
saturation parts in the lattice temperature curves are explained by negligibly
small heat diffusion from the surface of the nanoparticle into the surrounding
medium on the femtosecond time scale. A slight difference in the slopes of the
temperature curves within the first 100 femtoeconds of heating occurs due to the
assumption made in OTM that the electron heat transfer into the lattice
subsystem is very fast. Because of this, the particle temperature in OTM
promptly follows electron thermal behavior. A comparison of these two models
shows that the simpler OTM gives the same results as the more precise TTM.
Thus, OTM provides an adequate description of the laser heating of
nanoparticles in the femtosecond regime.


## 3.2. Picosecond Pulses

In this mode, the
constantsa,b, and c in the time profile for the laser
pulse have been chosen to provide the laser pulse width of 60 picoseconds at
FWHM with the same pulse shape listed in Table 1 and shown in Figure 1(b). As
can be seen from Figures 1(b) and 3(b), in the picosecond regime the electron
thermalization time τe and electron cooling time τc are much less than the duration τL of the 60 picoesonds laser
pulse. Hence, during the first 10 picoseconds, the electron subsystem has
already been completely cooled (Figure 3(a)), and the electron temperature
above the ambient no longer exists for the considered picosecond mode of
heating. This is confirmed by our calculations presented on Figure 3(b) (dashed
curve). Thus, TTM provides a very good approximation for the femtosecomd mode
as soon as the electron temperature exists, but it fails to describe the laser
heating of nanoparticles for longer pulse durations in the picosecond and
nanosecond regimes.Opposite to TTM, OTM
describes the picosecond heating kinetics very well. A typical time evolution
of the particle temperature predicted by OTM on the picosecond time scale is
displayed in Figure3(b) (solid curve) for the same material constants listed
in Table 1. The main feature is the appearance of heat lost from the surface of
the nanoparticle into the surrounding medium on the picosecond time scale.
After about 200 picoseconds, cooling of the nanoparticle begins due to heat
diffusion into the water. The maximum temperature reached by a 20-nm gold
particle for the given laser pulse is 995 K. This temperature is sufficient to
initiate any thermal killing mechanisms in cancer cells.


## 3.3. Nanosecond Pulses

For laser heating of
metal nanoparticles in the nanosecond regime, the characteristic lattice
heating timeτi is much smaller than the laser pulse
duration: τL≫τi. This means that,
the temperature inside the nanoparticle is nearly uniform over the whole
particle at the time scale of the laser pulse duration τL. In
this case, the electron and lattice temperatures are equal, Te=Ts, so that the homogeneous heating of the particle and
quasisteady heat exchange with the surrounding medium can be described by just
OTM. The characteristic lattice heating time τi required for
the formation of a quasistationary temperature profile across the nanoparticle
can be estimated from the formula τi=r02/4χ, where r0 is the particle radius and χ the
thermal diffusivity of the particle material. For gold nanoparticle (χ=1.18×10−4m2/s) with radii r0= 20–30 nm, the lattice
heat diffusion time is τi∼2×10−12s≪τL∼10−8s.Sample calculations have
been carried out using OTM for gold nanoparticles with radiir0= 30–35 nm in
different surrounding biomedia for an incident laser pulse of energy E=10 mJ/cm2 and pulse duration τL=8 nanoseconds with the time
profile shown in Figure 1(b). The laser pulse profile and duration 8 nanoseconds have been chosen to be close to those used in previous experiments
[2]. The laser flux chosen is ten times higher than in the regimes considered
above to provide approximately the same maximum nanoparticle temperature as
observed for femtosecond and picosecond laser heating. We should note that 10 mJ/cm2 is comparable to the laser fluence currently used in the
photothermolysis of cancer cells [1, 3]. The kinetics of heating and cooling the
gold nanoparticle are demonstrated in Figure 4, where (a) illustrates the time
dynamics of laser heating of a 30-nm gold particle in different biological
media: blood, human prostate, tumor, and fat. The thermophysical
characteristics of gold and biological surrounding media for different
temperatures are listed in Table 2. Figure 4(b) shows results of thermal
calculations for a 35-nm gold particle, which is heated and cooled in water at
different heat transfer rates s.Table 2
Thermophysical characteristics of the gold particle and surrounding biological tissue.
MaterialSpecific heatC(J/Kkg)Interval ofT (K)Thermal conductivity μ0 (W/m K)Thermal diffusivityχ(m2/s)Gold129273–3733181.18×10−4Water4181.6−4215.6C(T)=4182(1+1.016×10−4(T−293K))273–3730.597−0.682μ(T)=0.597(1+1.78×10−3(T−T∞))1.43×10−7Human prostate37403100.529Blood3645–3897273–3730.48–0.61.6×10−7Fat2975273–3730.185–0.233Tumor31603100.561Skin273–3730.210–0.410Kinetics of heating and cooling of a gold nanoparticle by a nanosecond laser
pulse of energy density 10 mJ/cm2 and duration 8 nanoseconds.
Calculations have been made by using OTM. (a) Illustration of the time dynamics
of laser heating of a 30-nm gold particle in different biological media: fat
(dashed-dotted curve), blood (dashed curve), tumor (solid curve), and prostate
(dotted curve). (b) Results of thermal calculations for a 35-nm gold particle:
heating and cooling in the water surrounding medium at different heat transfer
rates s:s=1.0 (solid curve), s=1.25 (dashed curve),
and s=1.5 (dashed-dotted curve).
(a)(b)It follows from these
calculations that during the laser pulse duration the transfer of heat from the
nanoparticle into the surrounding media is slight, and the particle rapidly
reaches a high temperature. The heating rate is about 1012 Ks-1. The temperature of the particle continues to rise even after the end of the
laser pulse. The highest temperature, 770 K, for a given laser pulse fluence is
observed for the heating time of 13.5 nanoseconds, when the laser pulse has
already degraded (see Figure 4(a)). After that time, the transfer of heat from
the particle to the surrounding medium becomes increasingly important, since
the energy source is no longer present in the system. The temperature of the
particle and surrounding medium remains high (~ 400 K) up to 20 nanoseconds, exceeding
the laser pulse duration by 2.5 times. The total time for one cycle (heating
from the initial temperature 300 K to maximum temperature, followed by cooling
back to the initial temperature) is about 30 nanoseconds.We have also examined the
effect of different biological surroundings on the laser heating dynamics of 30 nm gold particles. Four biomedia were used: namely, blood, human prostate,
tumor, and fat. Results of computer simulations of the time-temperature
profiles of gold nanoparticles in various biological media, performed by using
OTM, are plotted in Figure4(a). As follow from our calculations, the laser
heating and temperature behavior of the gold nanoparticles in blood, prostate,
and tumor are comparable to the water surrounding medium case, since the
thermodynamic properties of those media are very close to each other (see Table 2). This means that for the thermal calculations of laser heating of biological
media, the thermal properties of water can be used if the water content in the
media is high. But the heating of a gold nanoparticle in fat is substantially
different from the water case, since the fat has low-thermal characteristics.
Here, we observe higher overheating of the particle at the same energy level
and duration of the laser pulse due to the relatively low-thermal conductivity
of fat as compared to other biomedia.The temperature dynamics
of the particle is sensitive to the power exponents used in the
temperature dependence of the heat lost from the surface of the nanoparticle
into the surrounding medium, that is, (jD(Ts) in (3) (see Figure 4(b)). The value of s=1 better corresponds to the real biological surrounding. The power s>1 describes the medium with high thermophysical characteristics, like the
cooling liquids and metals. The medium with s<1 has a low thermal
conductivity and can be used as a thermal isolator.It is interesting to
investigate the effect of the particle’s radius on the temperature dynamics of
the nanoparticle heated by the nanosecond laser radiation in the biological
surroundings. There are two competitive factors here. On one hand, according to
the Mie diffraction theory, the absorption efficiency of the gold nanoparticle
drops with the decreasing size of the particle. On the other hand, the heating
rate increases for smaller particles as follows from (4). To find which factor
has a stronger effect on the effective laser heating of a gold nanoparticle, we
have calculated the maximal temperature profile for different nanoparticle
radii in blood and compared it to the gold nanoparticle absorption curve. The
results of these simulations are performed by using OTM and presented in Figure 5. It follows from this figure that the optical effect is much stronger than
the thermal effect when the radius of nanoparticle is less than 35 nm. In the
radii range 1–35 nm, the
overheating effect of the particle behaves according to the absorption
efficiency. For a radius of 35 nm, the thermal processes dominate over the
optical properties. For large radii (≥35nm), the maximal temperature profile
is saturated with oscillations, repeating the maxima and minima of the
absorption curve. The saturation of the maximal temperature curve for large
particle radii is explained by the balance between heating of the particle due
to absorption of laser energy and energy losses from the surface of the
particle due to heat diffusion into the surrounding biological medium.Figure 5
Nondimensional (a.u.) absorption efficiencyKabs and maximal temperature Tmax curves as a function of the particle’s radius for the gold nanoparticles in
blood heated by a laser pulse of energy density 10 mJ/cm2 and duration 8 nanoseconds.


## 4. Conclusions and Summary

The
comparative analysis of OTM and TTM for heating of a metal nanoparticle in the
femtosecond, picosecond, and nanosecond regimes has shown that(i)
in the femtosecond mode, the thermal equilibrium among the
excited electrons is established within the first 175 femtoesonds, long after
the end of the laser pulse duration;(ii)
the electrons remain in the thermal equilibrium state up to 1 picosecond;(iii)
both models demonstrate the same scenario in the heating
kinetics of a metal nanoparticle by a femtosecond laser pulse: about a 100-femtosecond
time delay in the heating of the particle is observed, until reaching a maximum
lattice temperature and saturation in temperature curves after 175 femtoeconds;(iv)
the electron cooling time due to coupling to the lattice is
about 10 picoecond, which imposes an upper time limit for TTM application;(v)
TTM gives a very good approximation for the femtosecomd mode
while an electron temperature exists, but it fails to describe the laser
heating of nanoparticles for longer pulse durations in the picosecond and
nanosecond regimes;(vi)
OTM shows that the heat lost from the surface of the
nanoparticle into the surrounding medium becomes noticeable after 200 picoseconds;(vii)
the heating of a metal
nanoparticle by a nanosecond laser pulse in fat provides higher particle overheating
than in blood, prostate, and water as surrounding media due to the thermally isolating
property of the fat;(viii)
the optical properties of the
nanoparticle have a much stronger effect on the heating dynamics in the
nanosecond mode than the thermal effects when the radius of the particle is
less than 35 nm. For larger particles, the thermal processes dominate the
optical properties, and the temperature curve is determined by the balance
between heating of the nanoparticle and energy losses from the surface of the
particle due to heat diffusion into the surrounding biological medium. Thus, the comparison of the two models
shows that OTM provides an adequate description of the laser heating of
nanoparticles in the femtosecond, picosecond, and nanosecond regimes.



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*Source: 251718-2008-05-11.xml*