In situ nanoparticle size measurements of gas-borne silicon nanoparticles by time-resolved laser-induced incandescence
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Abstract
This paper describes the application of time-resolved laser-induced incandescence (TiRe-LII), a combustion diagnostic used mainly for measuring soot primary particles, to size silicon nanoparticles formed within a plasma reactor. Inferring nanoparticle sizes from TiRe-LII data requires knowledge of the heat transfer through which the laser-heated nanoparticles equilibrate with their surroundings. Models of the free molecular conduction and evaporation are derived, including a thermal accommodation coefficient found through molecular dynamics. The model is used to analyze TiRe-LII measurements made on silicon nanoparticles synthesized in a low-pressure plasma reactor containing argon and hydrogen. Nanoparticle sizes inferred from the TiRe-LII data agree with the results of a Brunauer–Emmett–Teller analysis.
List of symbols
- c_{g,t}
Thermal molecular speed of the gas at equilibrium (m s^{−1})
- c_{o}
Speed of light in a vacuum (2.998 × 10^{8} m s^{−1})
- c_{p}
Specific heat of the nanoparticle (J kg^{−1} K^{−1})
- c_{v,t}
Thermal speed of evaporating atoms (m s^{−1})
- d_{p}
Nanoparticle diameter (nm)
- E(m_{λ})
Complex absorption function
- h
Planck’s constant (6.626 × 10^{−34} J s)
- ΔH_{v}
Heat of vaporization (J mol^{−1})
- I_{b,λ}
Spectral blackbody intensity (W)
- J_{evap}
Evaporating mass flux (kg s^{−1})
- J_{λ}
Spectral incandescence (a.u.)
- k_{B}
Boltzmann constant (1.38 × 10^{−23} J molecule^{−1} K^{−1})
- m_{v}
Mass of vaporized atoms (kg)
- m_{g}
Molecular mass of the gas (kg)
- m_{λ}
Complex index of refraction
- N_{g}″
Incident number flux of gas molecules
- N_{v}″
Number flux of vaporized atoms
- n_{g}
Number density of gas molecules
- n_{v}
Number density of evaporated vapor
- P(d_{p})
Probability density of particle diameters
- p_{g}
Gas partial pressure (Pa)
- p_{v}
Vapor pressure (Pa)
- Q_{abs,λ}
Spectral absorption efficiency
- q_{cond}
Conduction heat transfer (W)
- q_{evap}
Evaporation heat transfer (W)
- q_{rad}
Radiation heat transfer (W)
- R
Universal gas constant (8.314 J mol^{−1} K^{−1})
- R_{s}
Specific gas constant (J kg^{−1} K^{−1})
- T_{cr}
Critical temperature of liquid silicon (K)
- T_{eff}
Pyrometrically defined effective temperature (K)
- T_{g}
Gas temperature (K)
- t_{i}
Discrete time (ns)
- T_{i}
Initial temperature (K)
- T_{m}
Melting temperature of silicon (K)
- T_{p}
Nanoparticle temperature (K)
- T_{s}
Surface temperature (K)
- U_{ij}
Interatomic potential between atoms i and j (eV)
- v_{1}
Incident gas velocity (m s^{−1})
- v_{2}
Scattering gas velocity (m s^{−1})
- v_{xy}
Gas atom velocity parallel to surface (m s^{−1})
- v_{z}
Gas atom velocity perpendicular to surface (m s^{−1})
- x
Particle size parameter
- X
Uniformly distributed random number
- α
Thermal accommodation coefficient
- δ
Tolman length (nm)
- γ
Specific heat ratio
- γ_{s}
Surface tension of silicon (N m^{−1})
- λ
Wavelength (nm)
- μ
Ratio of gas atom mass to surface atom mass
- ρ
Nanoparticle density (kg m^{−3})
- ξ
Sticking coefficient
1 Introduction
The unique electromagnetic properties of silicon nanoparticles have led to a multitude of existing and emerging roles in diverse areas of science and engineering. In medicine, for example, silicon nanoparticles may be used for biomedical diagnostics, targeted drug delivery, cancer therapy, cell tracking and labeling, and tissue engineering [1]. Photovoltaic device performance has undergone a paradigm shift with the introduction of nanoscale films containing silicon quantum dots, which can greatly increase photoelectric conversion efficiency [2]. Silicon nanoparticles also enhance the performance of other electronic equipment, including lithium-ion batteries [3], solid-state devices, LEDs, and printable electronics [4]. Gas-phase synthesis is the most economical route for mass production of silicon nanoparticles, but since the electromagnetic properties of silicon nanoparticles depend strongly on their size, these reactors must be designed and operated to produce nanoparticles having a highly controlled size distribution. Accordingly, there is a pressing need for an instrument that can make temporally and spatially resolved size measurements within the reactor to elucidate nanoparticle formation and growth mechanisms, pinpoint production problems, and eventually provide feedback for closed-loop control.
Time-resolved laser-induced incandescence (TiRe-LII), a combustion diagnostic normally used to measure the volume fraction and size of soot primary particles, is a promising candidate to fulfill this need. In this technique, a laser pulse heats the nanoparticles within a sample volume of aerosol to incandescent temperatures. Following the laser pulse, the spectral incandescence is measured as the nanoparticles equilibrate with their surroundings. Since larger nanoparticles cool more slowly than smaller nanoparticles, in principle the average size, and to some extent the size distribution, can be inferred from the observed spectral incandescence decay.
While TiRe-LII was initially conceived to measure the size and concentration of primary soot particles (e.g. [5, 6, 7, 8]), several studies have investigated the feasibility of using this technique to size synthetic nanoparticles. Vander Wal et al. [9] first showed this approach could be viable for sizing metallic nanoparticles based on observed incandescence decay data, but they did not use the technique to recover particle sizes. Subsequent studies assessed the ability to extend TiRe-LII to size metal aerosols containing Ag [10], Fe [11, 12, 13, 14, 15], Mo [16, 17], and Ni [18] nanoparticles and oxide aerosols containing MgO [19], TiO_{2} [20, 21], Fe_{2}O_{3} [22], and SiO_{2} [23] with varying success. The unique challenges associated with TiRe-LII measurements on synthetic nanoaerosols are due to the following: lower vaporization temperatures compared to carbonaceous nanoparticles, resulting in comparably weak signals [15]; high-temperature chemistry that may change the chemical composition of the nanoparticles [22]; and non-incandescent laser-induced emission from excited fragments and potential plasmas [9]. In this context, silicon is a promising material due to its high boiling point and because elemental silicon is chemically stable at high temperatures.
Despite the growing interest in gas-phase-synthesized silicon nanoparticles with well-defined properties, there have only been two prior attempts to size silicon nanoparticles using TiRe-LII [24, 25]. While strong TiRe-LII signals were obtained from the laser-heated Si nanoparticles in low-pressure plasmas, the subsequent analysis neglected the 1/λ dependence of emission and absorption efficiency in the Rayleigh regime [26], as well as the contribution of evaporation to nanoparticle cooling, which has been shown to be very important in low-pressure aerosols [27].
This paper presents pioneering experimental and theoretical work aimed at extending the capabilities of TiRe-LII to silicon nanoparticles. The paper describes the procedure and instrumentation used to collect the TiRe-LII data and briefly introduces the gas-phase synthesis process. The following section presents the heat transfer model required to analyze the TiRe-LII data, including the thermal accommodation coefficient, α, which is obtained by molecular dynamics for Si/Ar and Si/He, starting from first principles by using a combination of ab initio calculations and atomistic MD simulations to realistically model the gas/surface scattering that underlies α. Nanoparticle sizes inferred from the TiRe-LII data are found to be generally consistent with those found using Brunauer–Emmett–Teller (BET) analysis [28] and transmission electron microscopy (TEM) [29] on material from the same synthesis process.
2 Experimental procedure
Time-resolved laser-induced incandescence measurements are carried out 20 cm downstream from the plasma zone using the Artium 200M TiRe-LII system shown in Fig. 2. The instrument consists of a transmitter module containing a 1,064-nm Nd:YAG laser and optics, a receiver module containing collection optics and two photomultiplier tubes, and a computer for instrument control and data acquisition. Optical access to the aerosol is obtained through three quartz windows in the reactor walls. Inert gas flushing prevents particle deposition on the windows and allows continuous operation of the reactor for several hours. A laser pulse is shone across the reactor chamber through two opposite windows. The laser was operated with a repetition rate of 10 Hz. A nearly uniform “top-hat” beam profile with a square 2.8 mm × 2.8 mm cross section was generated by relay imaging an aperture into the measurement location where fluences were in the 0.12–0.16 J cm^{−2} range. The resulting incandescence signal of the laser-heated nanoparticles is detected through the third quartz window, perpendicular to the laser pulse; the probe volume is defined by intersection of the laser beam and the detector solid angle. The incandescence signal is split by a dichroic mirror, passed through two band-pass filters centered at 442 and 716 nm (full width at half maximum of 50 nm), and imaged onto the photomultiplier tubes. Further details of this procedure are provided in Ref. [6].
The in situ size measurements of the silicon nanoparticles by TiRe-LII are complemented with the measurement of an average nanoparticle size calculated from their specific surface area as measured by BET analysis of silicon powder collected via a filter behind the reactor [28]. BET infers the specific surface area of nanoparticles from the physisorption of N_{2} by a sample of nanoparticle powder, which was kept at 150 °C and under vacuum overnight to remove residual water. Assuming that the nanoparticles are monodisperse spheres, the specific surface area can be converted to a representative nanoparticle diameter (corresponding to the Sauter mean diameter for a polydisperse powder) based on the knowledge of the sample mass and density. This technique is often used to determine the size of non-aggregated nanoparticles and has also been applied to size silicon nanoparticles produced from the reactor in previous studies [29]. Typical measurements carried out at CENIDE show a repeatability with <1 % variation in nanoparticle size, while a previous study that compared BET measurements of reference powders made independently by several laboratories showed a variation in the results below 5 % [32]. Although the BET analysis is done on the nanoparticles leaving the reactor, one would not expect the nanoparticle morphology to differ considerably between the TiRe-LII probe volume and the nanoparticle exit due to the strong Columbic repulsive forces between the Si nanospheres.
3 Heat transfer modeling
The nanoparticle mass lost due to evaporation is found by integrating J_{evap} over the temperature decay. Since laser heating is excluded from the heat transfer model, mass loss was estimated by assuming the peak temperature for prevaled for 10 ns of laser heating as a worst-case scenario. This calculation showed that the nanoparticle mass would decrease less than 15 % (5 % reduction in d_{p}) for all nanoparticles larger than 10 nm and only 4 % (2 % reduction in d_{p}) for 30 nm nanoparticles. This is considered low enough that nanoparticle size was assumed to be static during the cooling process.
4 Predicting the thermal accommodation coefficient using molecular dynamics
One of the main obstacles in extending TiRe-LII to new aerosols is that the thermal accommodation coefficient is unknown. Most published values for α (e.g. [43]) are found under conditions different from those encountered in TiRe-LII and thus cannot be applied directly to TiRe-LII analysis. Consequently, in most TiRe-LII experiments, α is inferred by comparing incandescence decay data to nanoparticle sizes found using ex situ analysis, such as TEM [11, 15, 44, 45, 46]. These accommodation coefficients are then used to infer nanoparticle sizes from other TiRe-LII data, a somewhat circular process.
It has been shown, however, that molecular dynamics (MD) can be used to predict this parameter for various gas–surface systems [47, 48, 49]. In the present work, this approach is used to predict α for the Si/Ar system over a range of surface and gas temperatures. We also carry out the same procedure for Si/He to investigate how the reduced gas molecular mass and potential well depth influences the accommodation coefficient.
Parameterization for the Stillinger–Weber potential for silicon [50]
ε | 2.17 eV |
σ | 2.01 Å |
a | 1.80 |
λ | 21.0 |
γ | 1.20 |
cos(θ_{b}) | −1/3 |
A | 7.049556277 |
B | 0.6022245584 |
p | 4 |
q | 0 |
Pairwise Morse potential parameters between the gas and silicon atoms
Si/Ar | Si/He | |
---|---|---|
D | 4.669 (meV) | 1.130 (meV) |
r_{e} | 4.647 (Å) | 4.534 (Å) |
λ | 1.256 | 1.398 |
The above parameterization forms the basis of MD simulations of gas molecules scattering from the laser-heated silicon nanoparticle. Simulations were carried out using LAMMPS [55]. The silicon surface is modeled using 544 silicon atoms, initially arranged in a diamond crystal lattice. Periodic boundary conditions are applied on the lateral surfaces (perpendicular to the free surface) of the computational domain. The silicon is initially brought to the specified temperature by applying the Nosé–Hoover thermostat [56, 57] (NVT ensemble) with a damping constant of 0.1 ps for 30 ps. The simulation is continued for 5 ps under the NVE ensemble to ensure that the system has reached equilibrium conditions at the desired temperature, during which time the surface density is also tracked. At the conclusion of this simulation, the silicon atom positions and velocities are stored in a restart file.
Figure 10 also shows that the average energy increase is zero when T_{s} = T_{g}, in accordance with the 2nd Law of Thermodynamics. The individual normal and tangential components of translational energy also appear to follow the same rule, suggesting that the normal and tangential modes of the gas molecule are uncoupled.
5 Analysis of TiRe-LII data
The MD-derived accommodation coefficients are then used to interpret the experimental TiRe-LII data. Silicon nanoparticle diameters are initially found by nonlinear regression of the experimental pyrometrically defined effective temperature, T_{eff}, to the same effective temperature derived from a simulated incandescence signal found by solving Eq. (1). Since the procedure for inferring the nanoparticle size distribution from TiRe-LII data requires an initial condition of uniform nanoparticle temperatures (taken to be the peak temperature in the case of a “top-hat” beam profile), and because of a smoothing effect lasting several nanoseconds around the peak temperature, it is necessary to extrapolate a hypothetical peak nanoparticle temperature that would be compatible with the conduction and evaporation cooling models. Accordingly, instead of the experimentally observed peak temperature of 3,075 K, a somewhat higher initial temperature, T_{i} = 3,100 K, was chosen as an initial condition to account for the smoothing effect at the peak.
A thermocouple was inserted in the central gas flow slightly above the TiRe-LII probe volume. After correcting for radiation losses from the probe, it indicated a gas temperature of 1,300 K, which is consistent with temperatures found through planar laser-induced florescence measurements carried out in a similar reactor [59].
Accounting for free molecular heat conduction by the H_{2} molecules requires knowledge of the \(\alpha_{{{\text{H}}_{2} }}\), which was not quantified using MD due to the complexity of deriving ab initio potentials for a polyatomic molecule. Since previous work has shown that the mass ratio and the thermal accommodation coefficient are closely related [47], H_{2} was modeled as monatomic and \(\alpha_{{{\text{H}}_{2} }}\) was assigned the value of α_{He} = 0.11 due to similar mass of the two gases. Uncertainty introduced by this assumption is addressed later in the paper.
Most probable nanoparticle size distribution parameters and credible intervals
d_{p,g} (nm) | 24.2 | (22.8, 25.7) |
σ_{g} | 1.43 | (1.39, 1.46) |
d_{p,32} (nm) | 33.2 | (32.6, 33.8) |
We must also consider, separately, how model parameter uncertainty affects the recovered nanoparticle size distribution parameters. As noted above, the gas temperature within the probe volume is difficult to measure precisely due to the limited access afforded by the reactor geometry, but is approximately 1,300 K based on a thermocouple measurement in near the probe volume. An uncertainty of ±200 K is assigned as a conservative estimate of this uncertainty, primarily due to uncertainty in laser position with respect to the thermocouple location. The extrapolated initial nanoparticle temperature used in the sizing analysis is assigned an uncertainty of ±25 K, based on the difference between the experimentally observed peak temperature (3,075 K) and the assumed value (3,100 K). The thermal accommodation coefficient for H_{2} is assigned a conservative uncertainty of ±50 %. Uncertainties in ρ, c_{p}, p_{g}, γ_{s}, T_{cr}, and the MD-derived thermal accommodation coefficients are taken to be 10 % of their nominal values.
Relative sensitivity coefficients and estimated error for inferred nanoparticle size distribution parameters due to model parameter uncertainty
d_{p,g} (nm) | σ_{g} | d_{p,32} (nm) | ||||
---|---|---|---|---|---|---|
RSC | ε | RSC | ε | RSC | ε | |
T_{i} | −148.0 | ±1.2 | 3.90 | ±0.03 | −75.0 | ∓0.6 |
T_{g} | 73.4 | ±11.3 | −2.38 | ∓0.37 | 8.7 | ±1.3 |
p_{g} | −33.8 | ∓3.4 | 1.10 | ±0.11 | −1.7 | ∓0.2 |
c_{p} | −21.7 | ∓2.2 | −0.05 | − | −31.9 | ∓3.2 |
ρ | −24.3 | ∓2.4 | − | − | −33.4 | ∓2.4 |
α_{Ar} | −21.8 | ∓2.2 | 0.70 | ±0.07 | −1.1 | ∓0.1 |
\(\alpha_{{{\text{H}}_{2} }}\) | −11.7 | ∓5.9 | 0.37 | ±0.19 | −0.6 | ∓0.3 |
γ_{s} | 2.6 | ±0.3 | −0.05 | – | 1.5 | ±0.2 |
T_{cr} | 47.9 | ±4.8 | −0.70 | ∓0.07 | 38.7 | ±3.9 |
Ex situ measurements made by BET analysis on a nanoparticle powder consisting of spherical, non-aggregated particles loosely connected by point contacts give an approximate diameter of 33.3 nm. This diameter can also be interpreted as the Sauter mean diameter since the nanoparticle size is derived from the ensemble volume of nanoparticles (found from the mass of the sample and bulk density of Si), divided by the specific surface area, which is inferred from N_{2} adsorption. This value is in excellent agreement with the TiRe-LII-derived Sauter mean of 33.2 nm and lies well within the error bars generated by measurement noise and model parameter uncertainty.
6 Conclusions
This paper describes the application of TiRe-LII to size silicon nanoparticles within a low-pressure microwave plasma reactor. Inferring nanoparticle sizes from TiRe-LII data requires a model of the free molecular heat conduction and evaporation through which the laser-heated nanoparticles equilibrate with the surrounding gas. Free molecular heat conduction depends on the thermal accommodation coefficient, which is calculated using molecular dynamics with DFT-derived gas/surface potentials. A parametric study elucidated how the thermal accommodation coefficient is expected to vary over a range of gas and particle temperatures.
The nanoparticle diameters were then inferred from the experimentally observed temperature decay, using a conduction model with separate terms for H_{2} and Ar and an evaporation model using Watson’s equation for the heat of vaporization. Maximum likelihood estimates of lognormal distribution parameters were found through Bayesian analysis; the geometric mean (and median) diameter of 24.2 nm is generally consistent with expected nanoparticle diameters, while the TiRe-LII-derived geometric standard deviation of 1.43 is larger than what is typically observed in TEM micrographs of silicon nanoparticle powders produced by the reactor. On the other hand, the Sauter mean diameter inferred from the TiRe-LII data matches the diameter found from BET analysis on a nanoparticle powder.
Ongoing work is focused on improving the evaporation model by including the influence of nanoparticle size on the surface energy used in the Kelvin equation. Uncertainty estimates will also be made more robust by combining the model parameter uncertainty and signal noise within an extended Bayesian framework. This lays the way for a more rigorous statistical treatment of TiRe-LII data and the possibility of obtaining more robust estimates of nanoparticle morphology by combining TiRe-LII data with other techniques that provide complementary information.
Notes
Acknowledgments
This research was supported by grants from the Natural Science and Engineering Council of Canada (NSERC) and the Deutsche Forschungsgemeinschaft (DFG). One of the authors (TA Sipkens) was also supported by a scholarship from the Government of Ontario. Compute Canada and SharcNet (www.sharcnet.ca) provided the computational resources.
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