Supporting Information
Computational Modeling of Pulsed Laser-Induced Heating and Evaporation of
Gold Nanoparticles
Michael Strasser, †, ‡ Kenji Setoura, † Uwe Langbein ‡ and Shuichi Hashimoto †, *
†
Department of Optical Science and Technology, The University of Tokushima,
Tokushima 770-8506, Japan,
and
‡
Department of Physics, RheinMain University of Applied Sciences,
D-65428 Russelsheim, Germany.
* Corresponding author: [email protected]
1
S1. Two-temperature model
In the following we give equations for the energy deposition and heat loss term, as well as
temperature-dependent material properties. The energy deposition term is described by:
S (t ) =
Cabs ( λ ) ⋅ P ( t )
V
(S1)
where Cabs is the absorption cross-section of an Au NP, P(t) is the time-dependent laser power
density, and V is the particle volume. One can describe P(t) in the center of the Gaussian power
distribution by:
P (t ) =
2 ln ( 2 )
π ⋅τ p
 4 ln ( 2 ) 2 
⋅ I ⋅ exp  −
t 


τ 2p


(S2)
Here, the power density function is defined by the pulse duration τp (FWHM) and laser fluence I.
The heat loss term F was given by Plech and co-workers1 by introducing an interfacial thermal
conductance g, experimentally determined as 105 MW m-2 K-1 for small gold NPs of 35 nm
diameter in aqueous solution. The implication of g is given by the surface wetting of nanoscale
objects. Taking the results into account, F is given by
F≈
(
g
Tl − Tma
a
)
(S3)
The heat loss term is determined by the temperature gradient of Tl and the medium temperature at
the NP-water interface Tma (r = a), with a being the NP radius. Based on past studies2-4 the
temperature dependent conductance parameter is given by:
g=3
Cm k m
a ⋅ Cl
(S4)
Here the factor 3 and a originates from the ratio of surface to volume. Furthermore, in the solid state
the linear thermal expansion coefficient of Au is given by 1.24135 × 10 −5 + 5.0786 × 10 −9 ⋅ Tl .5 For
2
the liquid state, on the other hand, no thermal expansion coefficients were found, thus Koike’s
experimental determined density function ρ Au (Tl ) = 1.74 ×104 − 1.44 ⋅ (Tl − 1337 K ) was applied.6 It
is known that the volume of Au increases of approximately 3.8 % due to the phase transition from
solid to liquid, which is in good agreement with the above function.7 As results are based on
experiments performed in vacuum and atmospheric pressure, one can assume that the behavior may
differ for Au NPs under high pressures. Moreover, we assume that thermal expansion stops at the
b.p. of Au. The electron-phonon coupling factor and electron heat capacity were adopted
from the calculations of Lin and co-workers,8 based on the real electron density of state (DOS)
instead of the frequently used linear FEG model.9-11 For temperatures sufficiently above the Debye
temperature, the specific heat capacity of gold can be calculated by cl (Tl ) = (119 + 3.061× 10−2 ⋅ Tl )
in the solid phase and is assumed to reach a constant value of 149 J kg −1 K −1 in the liquid
phase.7,12 At high pressure, physical properties of water such as refractive index,13 thermal
conductivity, density and heat capacity4 are accurately known at limited temperatures. For
temperatures above 800 K (Ref. 13) and 2000 K (Ref. 9), where the temperature at the NP-water
interface exceeds the area of experimental accuracy, the parameters were fitted or assumed as
constant.
Numerical procedure
To solve the PDE of medium system, we applied MATLAB, particularly the pdepe algorithm which
is suitable to solving parabolic-elliptic PDE’s in one dimension in the form of,
∂u  ∂u
∂  ms

c  x, t , u , 
= x − ms
x
∂x  ∂t
∂x 

∂u   
∂u 

f  x, t , u ,   + s  x, t , u , 
∂x   
∂x 

(S5)
Here , , , ⁄ is a flux term and , , , ⁄ a source term. In a spherical system,
the symmetry parameter ms equals two. With the assumption that the temperature gradient in the
electron and lattice system is negligible compared to the medium system ( , ≫ ), the flux
term is assumed to be zero. Furthermore, the source term in the medium equation is zero, because
the heat transfer across the NP-water interface is satisfied by the boundary condition. A convenient
method to solve the coupled differential equations is a matrix in the form of:
3
0
 Ce   dTe / dt  
 C  ⋅  dT / dt  = 
0
 l  l
 

−2
2
Cm   ∂Tm / ∂t   r ⋅ ∂ / ∂r ⋅ km ⋅ r ⋅ ∂Tm / ∂r
(
)
  −G ⋅ (T − T ) + S 
e
l
 

 +  G ⋅ (Te − Tl ) − F 
 
0

 
(S6)
Based on temperatures of the last process-step, values were recalculated for the present step of
procedure.
Initial and boundary conditions
To solve the PDE of the medium system we have to define both, initial and boundary conditions in
the form of:
T ( r , 0 ) = T0 ( r )
(S7)
∂T

p ( r, t, T ) + q ( r, t ) f  r, t, T ,
∂r


=0

(S8)
Assuming the three systems as in thermal equilibrium for t = 0, the initial conditions simplifies to
T ( r,0 ) = T0 . Moreover, thermal radiation on the NP surface and medium convections are
negligible.14,15 To keep the electron and lattice temperature uniform, it is necessary to define heat
barriers for the ODE’s.
∂Te ,l
∂r
=0
(S9)
It has been demonstrated that an interface resistivity between NP and medium exists. The effect is
attributed to surface wetting of nanoscale objects.16 Because of the finite interface conductivity ,
the temperature between NP and medium drops suddenly. The boundary conditions at the NP-water
interface (r = a) and the water-water interface (r → ∞) are written as:1
∂Tma
g
⋅ Tl − Tma +
=0
km
∂r
(S10)
Tm∞ − T0 = 0
(S11)
(
)
4
It is necessary to set the start value of r equal to the particle radius, a. When r is set to zero the right
boundary condition is extinguished.
S2. Spatio-temporal evolution of water temperature
medium temperature / K
100
2500
80
1800
t / ns
60
40
1050
20
300.0
0
40
60
80
100
120
140
radial coordinate / nm
Figure S2. Temperature evolution of water as functions of radial distance and time. The NP-water
interface and heat sink are defined at r = 25 and r = 150 nm. Simulation conditions are equivalent to
those of Figure 3a.
5
S3. Diameter-dependent behavior of Tmax.
We compared the laser-induced size reduction of Au NPs on excitation of the interband with the
occurrence of intraband excitation. Previously, consideration was given to the problem by
discussing the ratio of Cabs to particle volume, V as function of particle diameter.14 One can see the
ratio directly connected with the energy deposition term, S(t) (see Supporting Information, S1).
However, Cabs at LSPR-band region suffered from the temperature-induced bleaching (Figure 3a).
Hence, the maximum particle temperature is shown as a function of size at two excitation
wavelengths, 532 and 355 nm, in Figure S3. At both excitation wavelengths, the curves exhibit a
concave shape with a peak at different diameters. If the effects of the temperature-dependent
dielectric function of Au on Cabs are not considered, the curves monotonically increase with
decreasing particle size. For laser excitation at 532 nm, the effect of LSPR band bleaching exceeds
the increase at around 80 nm. In the interband region, far less bleaching occurs. However, an
intrinsic size effect is known to occur from surface scattering of electrons for particle radii of order
equal to the mean free path; the mean free path is about 20 nm at room temperature.17 Hence, the
maximum at about 50 nm and subsequent decrease for excitation at 355 nm is attributed to the
intrinsic size effect. Suppose we perform laser-induced size reduction starting from
100-nm-diameter Au NP, Figure S3 suggests that Tmax increases with decreasing particle diameter
depending on the excitation wavelength. The peak value of Tmax is reached at approximately 80 nm
with a 532-nm excitation and 50 nm for a 355-nm excitation. As size reduction due to photothermal
evaporation depends on particle temperature and proceeds with increasing Tmax, controlled size
reduction is not feasible for particle diameters between 100–80 nm (532 nm excitation) and 100–50
nm (355 nm excitation) because photothermal evaporation is possible before the peak in Tmax is
reached. Nevertheless, after passing through Tmax, fast particle cooling occurs that leads to a
cessation in size reduction. This assumption is well supported by the experimental results by the
Werner group.18
6
8000
50 mJ cm-2
355 nm
532 nm
Tmax / K
b.p.
6000
4000
100
90
80
70
60
50
40
30
diameter / nm
Figure S3. Calculated maximum particle temperature as a function of particle diameter for two
excitation wavelengths, 355 nm (red dashed line) and 532 nm (black solid line). A pulse width of 5
ns (FWHM), an external pressure of 100 MPa and a laser fluence of 50 mJ cm−2 were used for the
calculation.
7
S4. Particle evaporation under pulsed-laser-irradiation
To obtain the amount of evaporated atoms under pulsed-laser-irradiation, the Kelvin equation and
kinetic gas theory was applied. As experimentally demonstrated as well as theoretically
formulated,19,20 the vapor pressure over a curved surface is given by
ln
Pr  M   2γ s 
=


P∞  TR ρ   a 
(S12)
Where Pr is a vapor pressure over the curved surface, P∞ the equilibrium vapor pressure21 over a
flat surface, M the atomic weight, R the gas constant, ρ the density, a the particle radius and the
surface energy. Given that the surface energy represents the energy which is needed to break the
bonds, the parameter is temperature-dependent. Egry and coworkers investigated the surface energy
of liquid gold and found a linear relationship between surface energy and gold temperature, Tl.22
γ s = 1.15 − 0.14 ×10−3 (Tl − 1337 K )
(S13)
By applying the kinetic gas theory, one can express the number of atoms which leaves the particle
surface nD by the number of atoms in the vapor phase nV and the evaporation coefficient αs.19
1/ 2
1
 8 RTl 
nD = α s nV 

4
 πM  .
(S14)
When the vapor leaving the NP is described as a perfect gas, the rate of particle size reduction is
expressed by:
1/ 2
 M  α s Pr
dr
=

dt a  2π R ρ 2  Tl1/2
(S15)
To obtain the amount of evaporated atoms per pulse θ, the differential equation is applied on
temperature curves in ∆t = 10 ps time-steps. Within the time-step, the temperature is assumed as
constant. For the steepest temperature rise, the approximation leads to an inaccuracy of ± 1 K.
8
Scheme S4. Pictorial temperature evolution for an Au NP under ns pulsed-laser irradiation as a
function of time delay. The bars indicate the integration areas.
We assumed that the evaporation takes place only in the liquid phase. Thus, the integration area was
restricted to the particle temperatures above the m.p. of gold, Scheme S4. Accordingly, we can
describe the change of particle radius as a sum over the integration area.
1/2
 M  α s Pr ,i
∆a = ∑ 
∆ti
2 
Ti1/2
i =1  2π R ρ i 
n
(S16)
Note that ρ, P, and γs are the functions of temperature and have to be recalculated in every step. The
evaporation coefficient has previously been determined for particles in vacuum to 0.5.19 It is
possible that both the ambient pressure and light intensity affects the evaporation coefficient. The
latter may arise from light pressure. Hence, αs has to be determined experimentally. In this work,
however, αs was not determined because of experimental difficulties. To obtain θ, we calculated the
reduction of particle volume (from Eq. S16) and transfer it into the evaporated mass through the
density of Au at room temperature. The amount of evaporated atoms is given by the ratio of
evaporated mass to the atomic mass of Au.
θ=
evaporated mass
atomic mass × atomic mass unit
9
(S17)
S5. Refractive index gradient.
(a)
(b)
refractive index
100
80
1.3
300 K
1000 K
2000 K
3000 K
water
NP
1.2
1.244
60
t / ns
refractive index
1.349
40
1.134
NP
1.1
20
1.0
0
50
100
150
1.024
0
200
40
radial coordinate / nm
60
80
100
120
140
radial coordinate / nm
Figure S5. (a) Steady-state Tp-dependent refractive index of water as a function distance from the
NP center at 100 MPa. The multi-core-shell model is schematically depicted as inset, where the Au
NP is surrounded by multi- layers of water with decreasing temperatures (particle diameter: 50 nm).
(b) Radial distance- and time-dependent refractive index distribution for the pulsed laser heating.
The particle diameter is 50 nm. The laser parameters: fluence, 50 mJ cm−2; excitation wavelength,
532 nm; FWHM pulse width, 5 ns. The external pressure is set to 100 MPa.
10
S6. Dielectric function of gold
The dielectric function includes the response of intraband and interband transitions. Furthermore,
the intrinsic size effect which contributes to the damping frequency through electron collision on
the surface of the particle needs to be considered for particle radii in the range of the mean free path
of electrons. Taking these effects into account, one can calculate the temperature and size dependent
dielectric function of gold.
Figure S6-1. Real part of the refractive index of Au as a function of wavelength. The spectra at two
temperatures, 300 K and 1500 K, fitted to Otter’s experimental data23,24 were shown together with
the spectra by Johnson & Christy25 measured at ambient temperatue.
11
Figure S6-2. Imaginary part of the refractive index as a function of wavelength. The curves fitted to
Otter's experimental data23,24 were compared with experimental data from Johnson & Christy25 at
ambient temperature.
12
REFERENCES
1.
Plech, A.; Kotaidis, V.; Gresillon, S.; Dahmen, D.; Plessen, G. Laser-induced heating and
melting of gold nanoparticles studied by time-resolved x-ray scattering. Phys. Rev. B 2004, 70,
195423.
2.
Siems, A. Weber, S. A. L.; Boneberg, J.; Plech, A. Thermodynamics of nanosecond
nanobubble formation at laser-excited metal nanoparticles. New. J. Phys. 2011, 13, 043018.
3.
Ge, Z.; Cahill, D. G.; Braun, P. V.; AuPd Metal Nanoparticles as Probes of Nanoscale Thermal
Transport in Aqueous Solution. J. Phys. Chem. B 2004, 108, 18870-18875.
4.
Haar, L.; Gallagher, J. S.; Kell, G. S. NBS/NRC Steam Tables, McGraw-Hill, New York, 1976.
5.
Nix, F. C.; MacNair, D. The Thermal Expansion of Pure Metals: Copper, Gold, Aluminum,
Nickel, and Iron. Phys. Rev. 1941, 60, 597.
6.
Paradis, P. F.; Ishikawa, T.; Koike, N. Density of liquid gold measured by a non-contact
technique Gold Bull. 2008, 41, 242-245.
7.
Iida, T.; Guthrie, I. L. The Physical Properties of Liquid Metals, Clarendon Press, Oxford,
1993.
8.
Lin, Z.; Zhigilei, L. V.; Celli, V. Electron-phonon coupling and electron heat capacity of metals
under conditions of strong electron-phonon nonequilibrium. Phy. Rev. B 2008, 77, 075133.
9.
Grua, P.; Bercegol, H. Dynamics of electrons in metallic nanoinclusions interacting with an
intense laser beam. Proc. SPIE 2000, 4347, 579.
10. Grua, P.; Morreeuw, J. P.; Bercegol, H. Electron kinetics and emission for metal nanoparticles
exposed to intense laser pulses. Phys. Rev. B 2003, 68, 035424.
11. Rashidi-Huyeh, M.; Palpant, B. Thermal response of nanocomposite materials under pulsed
laser excitation. J. Appl. Phys. 2004, 96, 4475.
12. Green, D. W.; Perry, R. H. Perry’s Chemical Engineers’ Handbook 8th Ed, McGraw-Hill, New
York, 2007.
13. Harvey, A. H.; Gallagher, J. S.; Levelt Sengers J. M. H. Revised Formulation for the Refractive
Index of Water and Steam as a Function of Wavelength, Temperature and Density. J. Phys.
Chem. Ref. Data 1998, 27, 761.
14. Werner, D.; Hashimoto, S. Improved Working Model for Interpreting the Excitation
Wavelength- and Fluence-Dependent Response in Pulsed Laser-Induced Size Reduction of
Aqueous Gold Nanoparticles. J. Phys. Chem. C 2011, 115, 5063-5072.
15. Donner, J. S.; Baffou, G.; McCloskey, D.; Quidant, R. Plasmon-Assisted Optofluidics. ACS
Nano 2011, 5, 5457-5462.
13
16. Baffou, G.; Rigneault, H. Femtosecond-pulsed optical heating of gold nanoparticles. Phys. Rev.
B 2011, 84, 035415.
17. Hartland, G. V. Optical Studies of Dynamics in Noble Metal Nanostructures. Chem. Rev. 2011,
111, 3858-3887.
18. Werner, D.; Hashimoto, S. Controlling the Pulsed-Laser-Induced Size Reduction of Au and Ag
Nanoparticles via Changes in the External Pressure, Laser Intensity, and Excitation Wavelength.
Langmuir 2013, 29, 1295-1302.
19. Sambles, J. R. An Electron Microscope Study of Evaporating Gold Particles: The Kelvin
Equation for Liquid Gold and the Lowering of the Melting Point of Solid Gold Particles. Proc.
Roy. Soc. Lond: A 1971, 324, 339-351.
20. Asoro, M. A.; Kovar, D.; Ferreira, P. J. In situ Transmission Electron Microscopy Observations
of Sublimation in Silver Nanoparticles. ACS Nano 2013, 7, 7844-7852.
21. Alock, C. B.; Itkin, V. P.; Horrigan, M. K. Vapour Pressure Equations for the Metallic
Elements: 298–2500K. Canadian Metal. Quar. 1984, 23, 309-313.
22. Egry, I.; Lohoefer, G.; Jacobs, G. Surface Tension of Liquid Metals: Results from
Measurements on Ground and in Space. Phys. Rev. Lett. 1995, 75, 4043.
23. Johnson, P. B.; Christy, R. W. Optical Constants of the Noble Metals. Phys. Rev. B 1972, 6,
4370.
24. Setoura, K.; Werner, D.; Hashimoto, S. Optical Scattering Spectral Thermometry and
Refractometry of a Single Gold Nanoparticle under CW Laser Excitation. J. Phys. Chem. C
2012, 116, 15458-15466.
25. Otter, M. Temperaturabhängigkeit der optischen Konstanten massiver Metalle. Z. Phys. 1961,
161, 539-549.
14