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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 16n 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. 3 Heat dispassion includes several channels. The first one is the heat loss caused by dQEV dV is the evaporation jH 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 5.67 108 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). 6 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 ebeam 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. 9 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. 10 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 11 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 ebeam 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. 13 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). 14