Supplementary information for:
Direct observation of blocked nanoscale surface evaporation on SiO2
nanodroplets
Neng WAN 1,3,$, Jun XU 2, Li-Tao SUN 1,*, Matteo MARTINI 3, Qing-An HUANG 1,
Xiao-Hui HU 1, Tao XU 1, Heng-Chang BI 1, Jun SUN 1
1
SEU-FEI Nano-Pico Center, Key Laboratory of MEMS of Ministry of Education,
School of Electrical Science and Engineering, Southeast University, 210096 Nanjing,
China
2
National Laboratory of Solid State Microstructures, School of Electronic Science
and Engineering and School of Physics, Jiangsu Provincial Key Laboratory of
Advanced Photonic and Electronic Materials, Nanjing University, 210093 Nanjing,
China
3
Laboratoire de Physico-Chimie des Matériaux Luminescents, Université Claude
Bernard Lyon 1, UMR 5620 CNRS–UCBL, 69622 Villeurbanne Cedex, France
$ wanneng007@yahoo.com.cn
* slt@seu.edu.cn
1
Kelvin effect in evaporation dynamics
Systematic observation on the variation in evaporation temperature Tonset (e-beam
intensity) with respect to tip diameter was unsuccessful because of the variation in tip
diameter during the evaporation processes. In addition, the observation was performed
only from the SiO2 tips with different diameters, and no evident evaporation was
observed from the sidewall. Therefore, the Kelvin effect should be involved in the
tip-dominated evaporation. The Kelvin effect indicates that small-sized nanoparticles
tend to evaporate at lower temperatures compared with their bulk counterpart. This
result is attributed to the large curvature in small-sized nanoparticles that increases the
balance pressure on the surface following the equation
pn
4 M
 exp( n ) . In this
pb
 p RTd
equation, pn and pb are the balanced pressure for the nanoparticles and their bulk
counterpart, respectively,  n is the surface energy of the nanoparticle, which can be
higher than their bulk counterpart [18], M is the molar mass,  p is the density of
vapor, R is the gas constant, T is temperature, and d is the nanoparticle diameter.
The increased curvature decreases the evaporation entropy by H n  H b 
and the onset temperature for evaporation by
4M n
d
Tonset
16n rs3
, where
 1
Tonsert ,b
3EB d
H b =359 kJ mol-1 is the evaporation entropy for bulk SiO2, E B =22 eV is the
1/ 3
 3M 
cohesive energy of SiO2 [13], rs  

 4 N A 
 0.2959 nm is the radius of the
molecular, and Tonset and Tonsert ,b are the onset temperatures for the nanoparticles and
their bulk counterpart, respectively. For SiO2 nanoparticles with sizes of several tens
2
of nanometers, the change in evaporation entropy is ~1% of that in bulk SiO2 [14].
Considering a surface energy of 0.98 J m-2 and Tonsert ,b =2103 K for the SiO2 bulk,
typically the onset temperature of boiling could be lowered by ~7 and 14 K in 50 and
20 nm SiO2 nanoparticles, respectively. Further assuming a possible two to six times
increase in surface energy in nano-sized SiO2 [18], the lowered temperature should be
in the order of ~100 K.
Following the Kelvin effect, the evaporation from the sidewall of the silica
nanowire should be thermally and dynamically prohibited considering a negative
curvature, which causes an increased onset temperature for evaporation. This can be
the reason why only tip evaporation was found.
Thermal equivalent and estimation of temperature distribution using the finite
element method (FEM)
Temperature distribution is related to heat gain and loss at each part that may
includes several aspects. First, e-beam irradiation heats the SiO2 nanowire in a
uniform manner, providing an energy of QEH=VH=VQeJ/e, where V is the irradiated
volume, Qe=Qc+Qr, with Qc being the collision stopping power (0.51 eV nm−1 in this
case) and Qr being the radiative stopping power (1.57×10−3 eV nm−1) due to
bremsstrahlung, J is the electron current density, and e is the elementary charge
(1.6×10−19 C) [12]. The heat provided by e-beam irradiation is the only source of
temperature increase. It gives a uniform temperature distribution for an isolated object
without considering other channels of thermal loss.
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Heat dispassion includes several channels. The first one is the heat loss caused by
dQEV
dV
is the evaporation
 jH n , where jm  
dt
dt
dV
velocity at the tip, with jV 
being the volume evaporation velocity and
dt
evaporation, which is given by
 being the density. Evaporation causes a temperature gradient given by
dQEV
dT
, where  is the thermal conductivity and S is the cross-section area.
 S
dt
dx
Considering a typical evaporation rate of 40 nm3 ms-1 and a tip diameter of 50 nm
under an e-beam intensity of approximately 20 A cm-2, the heat loss is estimated to be
~10-13 W. Thus, the temperature gradient generated by evaporation is roughly ~102 K
m-1 (~10-7 K nm-1). This value is nearly negligible in the current research scale,
indicating that evaporation causes a very limited temperature gradient along the
evaporation tip.
Second, if the nanowire is heated to a high temperature, thermal radiation from the
surface should be also considered according to
 g =0.93 is
the
grey
index
of
SiO2,
dQSTR
  g s S (T 4  T04 ) , where
dt
 s  5.67 108 Wm 2 K 4 is
the
Stefan–Boltzmann constant, and the environment temperature is set to be T0 =300K .
Based on the viscosity flow and evaporation processes observed during the in situ
TEM evaporation, a temperature low-boundary higher than the glass transition
temperature and a high-boundary lower than the boiling point (as bulk evaporation
was not observed) are anticipated, giving a temperature region of 1450 K< T <2500 K
(for SiO2, the glass transition temperature is Tg=1450 K, melting point is Tm=2000 K,
and boiling point is Tb=2500 K). In this range, the thermal radiation is estimated to be
10-11 to 10-8 W under the studied geometry, which is evidently larger than the heat loss
4
caused by tip evaporation. The large thermal radiation in the nanostructure can be
attributed to their large specific surface area. Although thermal radiation takes away
considerable heat, its total effect is to decrease the thermal gradient because the higher
temperature region radiates more heat and cools down more rapidly.
Finally, for the SiO2 tip heated under the current geometry, thermal conduction
through its length direction causes heat dispassion, thereby causing a temperature
gradient given by
dQTR
dT
, where QTR is the heat transfer through a
  S
dt
dx
cross-sectional area parallel to the r-axis.
Therefore, the temperature distribution along the evaporation tip can be estimated
roughly using FEM. Calculations were done from the first element on the tip, where
the “heat out” is caused by the evaporation from the tip and thermal radiation. The
surface thermal radiation is calculated by assuming a temperature (in the range of 500
K to 2200 K in different calculations), and the heat given by e-beam is calculated by
the volume under different e-beam intensities (5 to 100 A cm-2). After the
above-mentioned three kinds of heat were calculated, the thermal gradient at the right
side of the unit was calculated according to
dQTL
  S  Tgradient , where
dt
dQTL
 WTL  WEH  WSTR  WTR , with WTL being the total power, WEH being the e-beam
dt
heating power, WSTR being the surface thermal radiation power, and WTR being the
heat transportation power, and the temperature gradient Tgradient 
dT
. The
dx
temperature of the next element can be obtained by d  Tgradient . By repeating this
process, the temperature distribution can be determined from element after element.
In a detailed FEM process, the tip was divided into several trapezium-shaped units
5
(truncated cone in three-dimensional) by planes perpendicular to the axis of the
evaporation tip, as shown in Figure s1(a). The heat related to this element includes the
heat provided by the e-beam (relates with the e-beam intensity), heat loss due to
surface thermal radiation (STR, relates to the temperature), and heat transportation
from the right (heat in) and left side cross-sections (heat out). The evaporation tip has
a bottom radius of 86.10 nm, a tip diameter of 17.95 nm, and a length of 116.29 nm (a
dimension adopted from experimental). The tip was divided into elements with a
width of d  5.2857 nm . Calculations were conducted from the first element on the
tip, where “heat out” was caused by tip evaporation and thermal radiation. Surface
thermal radiation was calculated by assuming a temperature (in the range of 500 K to
2200 K, in different calculations). The e-beam heat was calculated by the volume
under different e-beam intensities (5 A cm-2 to 100 A cm-2).
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Figure 3(a) A unit used in FEM calculations for estimating the e-beam heating effect. STR:
surface thermal radiation. (b) Calculated STR power distribution at different (pre-assumed)
temperatures (in Kelvin), as indicated by labels. Power distribution due to e-beam heating is
also shown. Insert: An enlarged region showing the comparison of WEH and WSTR. (c)
Estimated dependence of tip temperature on e-beam intensity. Insert shows the
correspondence log-log plot, which exhibits a linear relationship. Note the error bars are
added by rough estimation. (d) Calculated temperature gradient distribution at different
(pre-assumed) temperatures.
Typical results of the surface thermal irradiation power and temperature gradient
distribution at an e-beam intensity of 20 A cm-2 are shown in Figures s1(b) and s1(d),
7
respectively. Figure s1(d) shows that Tgradient values are normally in the order of
10-21 K nm-1. These small temperature gradients provide in fact a uniform temperature
distribution along the tip at every pre-given temperature, which accords to previous
studies [S1]. The highest achievable temperature under e-beam irradiation was also
estimated from the calculations by comparing the e-beam heating power (WEH) and
surface thermal irradiation (WSTR) in all elements, as shown in Figure s1(b).
Considering WEH > WSTR gives reasonable results. The highest achievable temperature
was realized when WSTR approached WEH. A pre-given temperature of 1750 K showed
that WSTR > WEH in the 10 nm position, indicating that a temperature of T>1750 K is
not reasonable (Figure 3(b)). Meanwhile, a pre-given temperature of 1700 K showed
that WEH > WSTR in all positions. Thus, the highest achievable temperature should be
between 1700 and 1750 K. A similar conclusion can be obtained from the
Tgradient calculated at different pre-given temperatures (Figure s1 (d)) when a "tip
cooling" effect is involved under stable evaporation, indicating that Tgradient  0
should be fulfilled. This conclusion was fulfilled at T ≤ 1750 K, which qualitatively
accords to the judgment made by comparing the power (but less strict than the power
criteria).
The temperature upper-boundary at the tip was obtained at different e-beam
intensities, as shown in Figure s1 (c), which shows a region of ~1200 K to 2500 K for
the e-beam intensity used in the current study. It is also found that a log (T)-log (Iebeam)
plot provides a linear relationship, indicating that an experience relationship of
T   I ebeam can be used to describe the dependence of temperature and e-beam
8
intensity. In addition, the actual temperature should be lower than the estimated
upper-boundary when the heat loss caused by evaporation and thermal transportation
is not considered. Heat loss caused by evaporation has a positive dependence on
temperature (or e-beam intensity, evaporation at higher temperature is more rapid,
thus larger), and thermal transportation may be enhanced under higher temperature
because of increased temperature gradient. Therefore, the deviation from the
upper-boundary should be larger at higher e-beam intensities. Given a temperature
range of 1450 K< T <2500 K under an e-beam intensity of 5 A cm-2 to 100 A cm-2, a
possible pre-melting (where melting is already observable) effect must be considered
when approaching the glass transition point Tm at Tm- ΔT with ΔT of ~50 K to 200 K.
Therefore, the temperature dependence shown in Figure s1(c) provides a quite
reasonable estimation. Moreover, Figure s1(c) provides a considerably believable
estimation because the evaporation and thermal transportation caused only a small
temperature gradient, as discussed above. Typically, taking a temperature of 1600 K
for an e-beam intensity of ~20 A cm-2 is safe for the calculation of the evaporation
velocity as a low boundary.
Shape of the evaporation tip
At the estimated temperature range, a considerable decrease in the viscosity of SiO2
is anticipated according to  (T )  0 (T  Tc )  , where  0 is the viscosity at a
reference temperature Tc and  is a constant dependent on different materials. For
SiO2, when approaching the melting temperature, it has a small viscosity of ~106 Pa∙s.
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Therefore, the change in the shape of the evaporation tip may be attributed to the
surface energy minimum.
Considering an angle dependence of the surface tension  ( ) , with  being the
angle between the surface normal and the x-axis (see inset in Figure s2), then the
equilibrium
takes
the
form
by
minimizing
the
total
surface
energy
[17] ES    ( )dS  2 0  r 1  x '2 dr , which obtains x(r )  c1 cosh 1 (r / c1 )  c2 ,
S
r
with c1 and c2 being two constants [17]. The data can be well fitted with this
relationship, as shown in Figure s1, indicating that the surface energy dominated the
shape evolution of the SiO2 tip.
Furthermore, given an area at any place S   r 2   r02 cosh 2 ( x / r0 ) , without
considering the change in density during mass diffusion at different places (due to the
small thermal expansion coefficient of silica), at arbitrary time and x-place, the mass
conservation is given by dVx  S x vx dt , with stable evaporation
dVx jm

. As a result,
dt

the diffusion velocities varied at different places:
vx 
dVx
dVx
jm


2
2
2
 S  dt  r0 cosh ( x / r0 )  dt  r0 cosh 2 ( x / r0 )
(1)
This relationship is considered in the experimental determination and normalization
of evaporation velocity in tips with different diameters at different places.
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Figure s2. Fitting of the surface profile of the evaporation tip with different tip radii under
different e-beam intensities. Insert: coordinates and track used for fitting the evaporation SiO2
tip.
Evaporation velocity
The theoretical evaporation velocity (number of molecular escape from a surface)
is given by j 
pS
, where p is the pressure, S is the evaporation area, m is
2 mk BT
the molecular weight, and k B is the Boltzmann constant. Using a vapor pressure given
by the Antoine relationship ln P  52.23  a  T 1  b , where a=506 and b=13.43
[s2], the evaporation velocities at different temperatures under different evaporation
surface areas are calculated and plotted in Figure s3.
Normally, evaporation from a surface can be influenced strongly by several factors,
such as impurities, surface contamination, and local geometry. Therefore, an
evaporation coefficient  =0–1 should be introduced, which provides an actual
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evaporation velocity of j 
  pS
.
2 mk BT
The change in the onset evaporation temperature may also induce the change in
evaporation dynamics (Kelvin effect). The estimated temperatures (Figure 3(c))
showed errors comparable with the temperature changes estimated using the Kelvin
effect (see Figure s1). Hence, we did not add such a variation in Figure s2.
Figure s3. Calculated evaporation velocity at different temperatures with different evaporation
surface areas (different radii ranging from 10 nm to 35 nm with 5 nm step). The figure is
plotted in a log (jv) ~ 1/T manner.
Evaporation of nanoparticles at different e-beam intensities
Treating evaporation as a thermally activated process with an energy barrier G ,
the evaporation rate can be described as j  C exp[
G
] , where C is a constant and k is
kT
the Boltzmann constant [1, 2]. Considering T   I ebeam and tecs  Vr j 1 , with
12
Vr  4 r 3 / 3
being
the
have tecs  Vr j 1  Vr C exp[
volume
of
the
nanoparticle
inclusion,
we
G
].

k I ebeam
The time (time of escape, tecs ) for the evaporation of the nanoparticles (normally
2 nm to 6 nm) pinned onto the surface was determined directly from in situ TEM
observations. The determination of the particle diameter has an uncertainty of ~±2 nm.
Thus, a constant average diameter of ~4 nm ( Vr  V0 ) is safe to assume in determining

the dependence of tecs on I ebeam . This process obtains tecs ~ exp[k0 I ebeam ] . Therefore, the
plot of Ln( Ln(tecs )) with respect to Ln( I ebeam ) provides a linear dependence. Figure s4
shows that the data can be well fitted using this relationship, indicating that the
evaporation of nanoparticle inclusions was also determined by the tip temperature. In
addition, the accelerated evaporation of small-sized nanoparticle inclusions at higher
temperature is beneficial for a more rapid evaporation of SiO2.
Figure s4 plot of Ln( Ln(tecs )) with respect to Ln( I ebeam ) . Blue line gives the linear fit.
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References
[S1] G. Baffou, C. Girard, R. Quidant, Mapping heat origin in plasmonic structures,
Phys. Rev. Lett. 104(13), 136805-8 (2010).
[S2] I. Wichterle and J. Linek, Antoine Vapor Pressure Constants of Pure
Compounds, (Academia, Praha, 1971).
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