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Abstract
In many high-temperature gas-phase nanoparticle synthesis processes, freshly nucleated particles are liquid and solidify during growth and cooling. This study presents an approach to determine the location of the liquid-to-solid phase transition and the volume fraction and number density of particles of both phases within a gas phase reactor. Spectrally-resolved line-of-sight attenuation (LOSA) measurements are applied to a silicon nanoparticle aerosol generated from monosilane in a microwave plasma reactor. A phantom-based analysis using particle number density, particle size, and temperature distribution from direct numerical simulation (DNS) of the reacting flow indicates that the contributions from the two particle phases can be decoupled under practical conditions, even with noisy data. The approach was applied to analyze spatially and spectrally resolved LOSA measurements from the hot gas flow downstream of the plasma zone where both solid and liquid silicon particles coexist. Extinction spectra were recorded along a line perpendicular to the flow direction by a spectrometer with an electron-multiplying charge-coupled device (EMCCD) camera, and two-dimensional projections were deconvolved to obtain radial extinction coefficient distributions of solid and liquid particles across the cross-section of the flow. Particle number densities of both particle phases were retrieved simultaneously based on the size-dependent extinction cross-sections of the nanoparticles. The particle-size distribution was determined via thermophoretic sampling at the same location with subsequent transmission electron microscopy (TEM) analysis. The particle temperature distribution was determined from the particle’s thermal radiation based on line-of-sight emission (LOSE) measurements. The approach for phase-selective data analysis can be transferred to other materials aerosol systems as long as significant differences exist in extinction spectra for the related different particle classes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nanoparticles of specific sizes, elemental composition, and structure can be produced by a variety of methods. In the case of gas-phase synthesis, suitable precursors thermally decompose in a high-temperature environment or via photolytic processes to form supersaturated vapors of condensable species, followed by particle nucleation and growth [1,2]. Gas-phase processes are particularly well-suited for producing high-purity materials with well-controlled properties in continuous, potentially large-scale flows [2–5]. Among gas-phase synthesis techniques, plasma-based approaches have the advantage of forming non-aggregated and spherical particles, producing oxygen-free particles, and suppressing the formation of particle agglomerates [2,6–8].
In order to understand, optimize, and control gas-phase processes, diagnostic techniques are needed to characterize the localized properties of the particle phase, which include particle composition, size distribution, temperature, as well as the reaction conditions such as species concentration and gas-phase temperature. Often, laser-based methods are used to derive such quantities in situ [9,10]. In gas-phase synthesis processes, the particle system may contain nanoparticles from different materials [11–16], for instance, hematite (α-Fe2O3), maghemite (γ-Fe2O3), magnetite (Fe3O4), and wüstite (FeO) were observed to coexist in flame synthesis of iron oxide nanoparticles [11,15,16]. The aerosol may also contain materials of different phases. During the plasma synthesis of silicon nanoparticles, Si atoms from the thermal decomposition of monosilane (SiH4) initially nucleate forming liquid silicon nanoparticles, and then solidify as the temperature of the carrier gas drops below the size-dependent melting point or as the size increases at a given temperature [17,18]. Asif et al. [17] exploited the distinct optical properties of Si particles between the liquid phase representing metallic properties and the solid phase representing properties of a semiconductor via line-of-sight attenuation (LOSA) in the UV/Vis wavelength region. They presented an empirical approach to locate the transition between solid and liquid particles in the off-gas of a plasma reactor. Building on this work, this paper presents a theory-driven phase-selective model that is combined with a tomographic algorithm to quantitatively determine the number density, thus the volume fraction (with the prior knowledge of particle size distribution), distributions of solid and liquid Si nanoparticles at different radial positions in the plasma reactor flow. One crucial challenge when applying LOSA to complex systems containing multiple kinds of particles is that one needs to deal with overlapping extinction spectra from each particle class. The difficulty is to decouple the overlapped extinction, therefore, achieve the desired physical quantities of each phase. This model then provides the not-previously exploited opportunity to detect volume fractions of liquid and solid Si particles simultaneously in a spatially-resolved fashion in the hot off-gas of a plasma reactor.
Several optical diagnostics techniques may be used to interrogate the particle phase within a nanoparticle aerosol. Among the laser-based methods, phase-selective laser-induced breakdown spectroscopy (PS-LIBS) elucidates the elemental composition of the particle by selectively evaporating/ionizing nanoparticles while avoiding the breakdown of the surrounding gas [11,19,20]. The technique is highly sensitive towards the particle phase, but it cannot distinguish between solid and liquid phases. In time-resolved laser-induced incandescence (TiRe-LII), a short laser pulse heats the particles within a probe volume and the volume fraction and particle size are then inferred from the peak amplitude of the incandescence signal and subsequent decay, respectively [10,19,21–23]. The fact that for many materials the optical properties can be phase- and thus temperature-dependent is a challenge for quantitative analysis of LIBS and LII as the interaction with the heating laser as well as the emission depends on the optical properties of the target material. Raman scattering can provide information about the particle phase [24,25] and angle-resolved elastic light scattering provides information about particle size and morphology [26], but the related weak signals are often disturbed by interference and secondary scattering.
Line-of-sight attenuation/emission (LOSA/LOSE) spectroscopy has been successfully used to measure particle size [27,28], temperature [29,30], and volume fraction distributions [31,32], as well as optical properties [17,33] of nanoparticles in aerosols. LOSA is widely used to quantitatively determine nanoparticle volume fractions from monochromatic extinction information for particles that absorb within the Rayleigh limit of Mie theory. Particles emit incandescence, based on which LOSE may infer a representative particle temperature along the line-of-sight. LOSA and LOSE can be combined to carry out spatially-resolved reconstruction of particle temperature and volume fraction distributions [34–37], which is less sensitive to uncertainties in the particle refractive index compared to using LOSA by itself. The LOSA/LOSE methods can also be applied to mixtures of phases/components [37,38]. In such cases, retrieving the desired physical quantities from the emission/extinction signals is a complex and spatially ill-posed inverse problem, and the recovered result is easily disturbed by measurement noise and model errors [39,40], which makes it difficult to apply these models to experiments.
Generally, solving the inverse problem in LOSA is much easier than that in LOSE because the wavelength-dependent extinction coefficient and temperature are coupled in LOSE. In this study, we use LOSA to distinguish the distribution of solid and liquid Si particles in a plasma reactor. Spectrally-resolved extinction spectra in the UV/VIS wavelength range can potentially also provide information about the particle size because of the size dependence of the extinction cross-section. Of particular interest in this work is the spatially-resolved determination of the volume fraction of the liquid and solid Si nanoparticle phase. Further insight into the phase transition is obtained from the temperature profiles determined from spectrally-resolved LOSE measurements. The refractive index of solid Si is temperature-dependent, which results in the extinction spectra changing as a function of temperature. Therefore, the temperature distribution from the LOSE measurement also gives guidance for choosing the refractive index to better fit the extinction spectra, thus obtaining the required parameters with higher accuracy. It should be noted that the accuracy of the approach is influenced by the differences in extinction efficiency between the two phases; generally, it is easier to distinguish the difference between solid and liquid phase for larger particles.
This article is organized as follows: The optical properties of solid and liquid Si nanoparticles will be described in Section 2, and the spectroscopic phase-selective reconstruction model will be explained in the same section. Section 3 shows how this model can be incorporated into an inversion technique to infer the quantities-of-interest (particle number density distributions of solid and liquid Si, and thus volume fraction distributions) from extinction spectra simulated based on direct numerical simulation (DNS) data. The model will then be applied in Section 4 to recover these quantities from experimental spectra measured across a silicon nanoparticle aerosol stream generated in a plasma-heated flow reactor.
2. Theoretical background
2.1 Optical properties of solid and liquid Si nanoparticles
Liquid and solid silicon have significantly different optical properties because of the dominating metallic and semiconducting (intraband transitions) character of the respective phase. Figure 1 displays the refractive indices of solid (left part) and liquid (right part) Si nanoparticles; the latter were obtained in our previous study based on Drude theory [18], while the former were measured at different temperatures by Šik et al. [41]. It should be noted that the refractive index of liquid Si was not obviously affected by the temperature in the investigated wavelength range. This is because the conductivity and the mean collision time are the temperature-dependent parameters in the Drude Theory, and the temperature dependences occur principally in the wavelengths around 250 nm.
Fig. 1. Refractive index of solid and liquid silicon as a function of wavelength. The refractive index of solid Si changes with temperature. nλ is the real part and kλ is the imaginary part of the refractive index mλ, i.e., mλ= nλ+ikλ.
Figure 2 shows the extinction efficiency and the single scattering albedo (the ratio of scattering and extinction efficiencies) of liquid and solid Si for three particle sizes (10, 30, 60 nm diameter) calculated using Mie theory [42,43] and the refractive index data in Fig. 1. In general, the extinction efficiency and the single scattering albedo of liquid Si decreases monotonically with wavelengths in the range from 300 to 600 nm independent on particle size, while for solid Si, a peak in the extinction efficiency appears in the near UV/Vis spectral range corresponding to resonances of bound oscillators [17] where valleys occurs in the single scattering albedo. With particle sizes smaller than 10 nm, the extinction efficiency of liquid Si is much higher in the respective wavelength range than that of solid Si, and the (temperature-dependent) resonance absorption of solid Si is not significant. As particle sizes increases to 30 nm, the resonance peaks gradually become more distinct with decreasing temperature. When the particle size reaches 60 nm, the resonance peaks become sharper and more prominent, and shift to shorter wavelengths with lower temperature. These structured extinction spectra provide the means for distinguishing liquid and solid nanoparticle fractions in the LOSA optical probe volume. For wavelengths above 500 nm, the extinction efficiency of liquid Si is much higher than that of solid Si irrespective of particle size (Fig. 2). Therefore, when determining particle temperature distributions in the LOSE measurement, the emission signals were considered only from “hot” liquid Si in the wavelength range beyond 550 nm when determining particle temperature distributions. The single scattering albedo for Si with a size of 30 nm is less than 0.12 at a temperature of 1123 K in the respective wavelength range. Considering the insignificant forward scattering for small detection angles, the high temperature, and the fact that most particles have sizes below 30 nm in the plasma synthesis, the scattering effect was neglected in this study.
Fig. 2. Extinction efficiency and single scattering albedo of solid and liquid Si nanoparticles at various temperatures with particles diameters at 10, 30, and 60 nm. S and L represents solid and liquid phase, respectively. The temperature dependence was only considered for solid particles.
In the following section, we will reconstruct number density and volume fraction distributions of the solid and liquid state Si based on phantom studies. In measurements presented later, the temperature distribution determined by the LOSE measurement provides the required information to choose the appropriate refractive index of solid Si.
2.2 Radiative transfer equation (RTE)
As shown in Fig. 3, in an absorbing, emitting, and scattering medium [44,45] (neglecting the in-scattering term) the RTE is given by
After some manipulation, Eq. (1) can be solved to yield
Fig. 3. Radiative heat transfer schematics in an absorbing, emitting, and scattering medium. The light beam with a direction of ŝ is attenuated by absorption and scattering, and enhanced by emission.
2.3 Line-of-sight attenuation/extinction (LOSA) measurement
In LOSA diagnostics, four individual measurements (cases) need to be done sequentially [32]. The RTE for these cases can be respectively described based on Eq. (2) as follows:
Case 1: Nanoparticles absent and light source on
Case 2: Nanoparticles present and light source on
Case 3: Nanoparticles present and light source off
Case 4: Nanoparticles absent and light source off
where $I_{\lambda ,\textrm{ext}}^{\textrm{off},0}$ is the background radiation intensity.According to Eqs. (3–6), the pathlength relevant for optical attenuation, i.e., opacity, along the projection j can be expressed as
2.4 Phase-selective reconstruction model
The phase-selective model is used to determine the number density distributions of an axisymmetric aerosol flow consisting two particle phases. With additional information about the particle-size distribution, the volume fraction distributions of the two particle phases can also be obtained. For reconstruction, the cross-section of an axisymmetric target is subdivided into several evenly-spaced annular elements, where physical parameters (e.g., particle number density, volume fraction, temperature) in each segment are assumed uniform. The reconstruction scheme is shown in Fig. 4.
Fig. 4. Line-of-sight reconstruction scheme of a horizontal cross-section in an axisymmetric target with a radius R. The fields of volume fraction, particle number density, and temperature are schematically discretized. A projection line j crosses through the cross-section, and xj is the intersection point between the projection line j with the x axis.
Based on Eq. (7), one obtains the opacity along a projection line j in a continuous form
In the model, the cross-section was equally divided into M ring-shaped segments and N = M lines cross through one half of the cross-section. Equation (8) is a Volterra integral equation of the first kind, and the deconvolution is an ill-posed problem. After discretization of Eq. (7), N detection lines form Eq. (9)
With the knowledge of ${\boldsymbol L}$ from geometric analysis and τλfrom the LOSA measurement, the local βλ can be deconvolved by Tikhonov regularization [46], which can be solved by
The discretized local extinction coefficients βλ at each measurement wavelength is related to the unknown aerosol properties by
Introducing the local parameter Wλ associated with the extinction cross-section, namely the average extinction cross-section, and defining
To reduce the complexity of the inverse problem, the particle-size distributions of liquid and solid particles in a certain segment are assumed as known. The particle-size distribution in a segment i was assumed to be log-normal with a particle median diameter µi and a standard deviation of σi [8]. The effects of the assumption of the particle size distributions on the reconstruction accuracy are discussed in section 3 (Phantom study). With the knowledge of the particle-size distribution (i.e., σi and µi) in Eq. (17)
Then, one can obtain the volume fraction distributions of solid and liquid particle phase in segment i as
In summary, the input parameter is the spatially-resolved opacity τλ at multiple wavelengths from λ1 to λv, while the output parameters are the particle number density, the volume fraction distributions of liquid and solid phases. Solving the inverse problem consists of three steps, as shown in Fig. 5: (1) Retrieve the extinction coefficient βλfrom the integrated value τλ based on Eq. (10) by Tikhonov regularization; (2) Retrieve the particle number density of both phases in each segment separately with prior knowledge of assumed size distributions; and (3) determine the volume fraction distributions of liquid and solid particles based on the retrieved number density via Eqs. (18) and (19).
Fig. 5. Algorithm for solving the inverse problem of retrieving spatially resolved particle number density and volume fraction distributions of the solid and liquid phases in a nanoparticle aerosol.
2.5 Line-of-sight emission (LOSE) measurement
The particle temperature is one important indicator for the phase transition. The temperature reconstruction strategy was adapted from our previous study [30,36] for extracting temperature and volume fraction from an incandescing germanium or silicon nanoparticle stream leaving the plasma region of the same reactor utilized in this study. The spatially resolved temperature distribution of the luminous nanoparticle zone was determined by LOSE measurements consisting of the two cases with radiation energy detected with and without the luminous aerosol, respectively:
The temperature reconstruction scheme is based on the following assumptions or considerations: (1) The nanoparticle cloud was assumed as optically thin so that effects of radiation self-absorption and multiple scattering can be neglected; (2) The emitted intensity at wavelengths longer than 550 nm is due entirely to incandescence from liquid particles. This is because: For particles no larger than 30 nm, the absorption efficiency ratio of liquid-to-solid is larger than 5 for wavelengths longer than 550 nm; the ratio of scattering to extinction efficiency of liquid Si is smaller than 0.033 in the emission measurement wavelength range.
The integrated radiation intensity from liquid particle emission along a detection line from 0 to smax traversing the particle stream cross-section can be derived as
The local absorption coefficient of the aerosol κλ is calculated as
The emission intensity along a projection line j in a continuous form
Discretizing Eq. (24) and considering multiple detection lines yields
Introducing Eq. (23) into Eq. (25) and conversing into a matrix at a wavelength of λ then results in
Based on Wien’s approximation, the blackbody radiation intensity Iλ,b is expressed as
Multiplying λ5 inside the logarithm terms of Eq. (28) gives
3. Phantom study
We demonstrate the reconstruction method for distinguishing solid and liquid Si and quantitatively retrieving the number density and volume fraction distributions of both phases using a direct numerical simulation (DNS) of a microwave plasma reactor operating at 800 W. The calculations were performed with an in-house solver based on the open source finite volume library OpenFOAM [47]. The DNS model has been applied to spray flames of the gas phase synthesis of nanoparticles, and the experiments and simulations are in good agreement with the centerline velocity and particle size distribution [48]. The method was further adapted for simulating the Si stream in the plasma reactor. A mesh with a resolution of 300 µm featuring 26 million computation cells was utilized, and the mesh consists of 99% hexagonal cells as well as polyhedral cells close to geometry features. The volume of the plasma torch is modeled as a strong spherical homogenous heat source. All swirl inlets have a Dirichlet boundary condition for the magnitude velocity and temperature, and the center jet has a Dirichlet temperature, velocity profile, and composition sets. The nanoparticle dynamics model features a full phase coupling of the gas and particle phase by homogeneous and heterogeneous condensation, as well as evaporation. The simulation also includes transport phenomena, buoyancy, and thermophoresis. Details are available in Ref. [49].
Figure 6 shows the DNS results of total number density, the probability density distribution of particle diameter along one radial direction, total volume fraction, and temperature distributions of the cross-section of the Si stream. The DNS does not distinguish between solid and liquid particles, and the phase fractions were obtained from Eq. (30) in post-processing. Here, we use total number density and total volume fraction to stand for the quantities of both phases. The total number density in Fig. 6(a) shapes a solid sphere structure with high values in the center, while the total volume fraction in Fig. 6(c) forms a hollow annular structure with high values at radial locations from 4 to 6 mm. It is noted that the distinct four-fold symmetry is because the flow of the swirl inlets interacts with the central jet at the height of approx. 2 cm above the nozzle, and the structure remains throughout the reactor due to the high Schmidt number of the particle phase. The probability density of particle size distribution in Fig. 6(b) indicates the particles with small sizes dispersed in the center, which attributes to the low values of volume fractions in that region. The temperature distribution forms annular structures with the lowest temperature in the center.
Fig. 6. Simulation data from DNS of a horizontal cross-section in Si stream. (a) Total number density (b) Probability density of particle size distribution at various radial positions along the specific radial direction in (a) (dash line with white color) (c) Total volume fraction (d) Temperature distributions. Total volume fraction and total number density containing the quantities with both phases of liquid and solid Si. r is the distance from the centerline.
The DNS data was further post-processed to determine the location of the liquid-to-solid phase transition, and the quantities of interest in individual phases. The phase transition depends on particle size and temperature, and the melting point of particles increases with particle size. We used the particle size dependent melting temperature Tm function Eq. (30) to distinguish solid and liquid phases [50,51],
Based on the relationship between phase transition temperature and particle size, the individual quantities of liquid and solid states of Si are separated in Fig. 7. For calculating the extinction spectrum in the direct problem, the mesh was refined to 20 µm via linear interpolation. Each projection was obtained by summing the data on the mesh crossed through by the projection. Figure 7(e) shows the simulated extinction spectrum with the prior knowledge of the number density distributions of liquid and solid Si and the particle size distributions, and the refractive index of solid Si at the temperature of 1157 K in Fig. 1 is applied since it is closest to the simulated temperature.
Fig. 7. Individual quantities of solid and liquid Si post-processed from simulation. (a) Volume fraction of liquid Si (b) Volume fraction of solid Si (c) Number density of liquid Si (d) Number density of solid Si (e) Simulated spectrally- and spatially-resolved extinction spectrum.
In the following phantom study, the volume fraction and number density distributions of solid and liquid Si will be retrieved from the simulated extinction spectrum with the known location-depended particle size distributions from simulations. It should be noted that the simulation-based ground truth of volume fraction and number density distributions are not axisymmetric for both phases, as shown in Fig. 7. The cross-section with a radius of 10 mm was discretized with 500 rings same resolution as the direct problem. The quantities of interest are reconstructed based on the left and right sides of the extinction spectrum, and compared with the ground data in Fig. 8. The reconstructed results are in good agreement with the ground truth, even at the locations with high gradients in the number density and volume fraction distributions of the solid phase. However, in the center region, most particle sizes are very small with diameters less than 5 nm, which results in weak extinction, at the same time, the systematic errors introduced by the inversion algorithm happen in the center region. Therefore, the number density of the liquid phase close to the centerline within 3 mm could not be properly reconstructed.
Fig. 8. Comparison with reconstructed and simulated ground truth of number density and volume fraction distributions of liquid and solid Si. It is not axisymmetric for the ground truth of volume fraction and number density distributions of solid and liquid Si, and the quantities extracted from two-dimensional distributions in Fig. 7 with various polar angle form clusters. The reconstruction data is obtained based on the left and right sides of the simulated extinction spectrum, respectively.
In the DNS, the particle-size distribution varies with the radial position. With the knowledge of spatially-resolved temperatures and particle sizes from DNS simulation and the relationship between phase transition temperature and size in Eq. (30), the particles were separated as solid and liquid phases. In the experiments, the particle-size distribution was measured by thermophoretic sampling on carbon-coated copper grids and subsequent TEM analysis with limited spatial resolution. As liquid particles solidify on the cold grids, the total particle-size distribution is determined independent of the particle phase.
In the following, we will reconstruct the quantities of interest by using the particle size distribution (independent of location) from a realistic experiment operating at the same condition and same probed height from the plasma zone as the DNS study. The particle size for an 800 W microwave power was fitted with a log-normal size distribution with a CMD of 22.8 nm and a standard deviation of 1.34 nm [17]. The difference in extinction spectra between simulation and experiment may be caused by several factors, such as electronic/photonic shot noise, flow field discretization, and the assumption of axisymmetric target. Here, we artificially add Gaussian distributed noise that is representative of the measured extinction spectra. The noise extracted from experiments obeys an approximately unbiased normal distribution with a standard deviation of 8.13 × 10−4, which was derived by fitting a normal probability density function (PDF) to a noise histogram.
Figure 9 shows the reconstructed results with and without consideration of the interfering noise and compared with ground truth data. For simplifying purposes, the ground truth and reconstructed profiles are averaged data for both sides. The reconstructed results of the number density distribution of solid Si and volume fraction distributions of both phases are consistent with the ground truth data, while the number density distribution of liquid Si could not be retrieved at locations within 5 mm from the centerline. The temperature decreases from 1604 K at the centerline to 1568 K at 5 mm from the centerline, which corresponds to the reduced melting size point from 53 nm to 36 nm. From the ground truth in Fig. 6(b), most particles are smaller than 10 nm at locations within 5 mm from the centerline. The inaccuracy of the liquid Si number density distribution is speculated as the large deviations in the particle size between the experimental log-normal distribution and the ground truth. The relative reconstruction errors of the peaks in volume fraction distributions are 0.17 and 0.042 without and with noise interference for liquid Si, 0.23 and 0.3 for solid Si, and in number density distributions of solid Si are 0.18 and 0.24. The results with the interference of noise follow the ones retrieved from the ideal noise-free condition, except for the locations with high gradients in volume fractions and number densities of solid Si.
Fig. 9. Comparison of ground truth and reconstructed data of particle number density and volume fraction distributions of both phases. The reconstructed data is retrieved using a log-normal distribution independent on locations based on extinction spectrum with and without consideration of noise interference.
4. Experimental study
4.1 Experimental procedure
Figure 10 displays the optical layout of the diffuse LOSA/LOSE measurements. A laser-driven light source (LDLS, Energetiq EQ-99) provides broadband radiation from 200 to 850 nm, whose divergent light cone is collimated with an off-axis paraboloid and steered into an integrating sphere to generate a homogeneous diffuse irradiance. The standard deviation of the light source emission was calculated over 200 images, which is from 0.16% to 0.74% of the light source emission depending on the wavelengths. The extinction by the particles should be higher than the fluctuation caused by the instability of the light source. An aperture is placed at its exit, which is open and closed in the LOSA and LOSE measurements, respectively. A pair of UV-grade fused-silica plano-convex lenses (focal lengths: 150 and 300 mm, diameter: 50.8 mm) image the output aperture into the probe region (particle stream inside flow reactor). A second pair of UV-grade fused-silica plano-convex lenses (focal lengths: 300 and 100 mm, diameter: 50.8 mm) then image the transmitted light as well as the emission of the incandescing particles from the center of the reactor onto the entrance slit of a spectrometer (Acton SP-150, slit width: 100 mm, focal length: 150 mm, 150 grooves/mm grating) connected to an EMCCD camera (Andor, iXon DV887, 512 × 512 pixels, sensor size 8.2 × 8.2 mm2) at the exit port. The spectral resolution is 5 nm full-width at half maximum (FWHM) as measured by a mercury Pen-Ray lamp. The spatial dispersion (along the imaged slit length) is approx. 0.048 mm/pixel. The imaging spatial resolution was determined by placing a negative 1951 USAF target in the object plane, and the target with line pairs was back-illuminated by filtered light (488 nm with 10 nm FWHM) from a laser-driven light source (Energetiq EQ-99). It shows that 1.59 lp/mm (line pair per mm) can still be discerned with the contrast value of 0.18. The LOSA measurements were performed in the wavelength range from 316–554 nm, whereas due to low signal-to-noise ratios and scattering for LOSE measurements, the spectrum with the strongest emission signal between 550 and 699 nm was utilized. For all measurements, 100 extinction spectra (each with 300 ms exposure time), and three emission spectra (each with 5 s exposure time) were averaged to reduce shot noise.
Fig. 10. Experimental set-up for LOSA/LOSE spectrally resolved line-imaging measurements in the Si nanoparticle stream of the microwave plasma reactor. The aperture between integrating sphere and source imaging lenses is totally open in LOSA measurements while totally closed in LOSE measurements. The inserted image shows the luminous Si nanoparticle stream.Fig. 11. Spatially-resolved extinction spectra along a horizontal line (40 cm above the nozzle exit). Microwave powers of 800 W (left) and 600 W (right).
An aerosol-containing Si nanoparticle stream was generated in the exhaust flow of a microwave plasma reactor (more details are given in Ref. [54]). Due to the swirl-stabilized gas flow in this reactor, the generated particle stream is shaped into a hollow thin-walled cylindrical structure with a thickness of ∼1.5 mm. Particle solidification is located in the observation region and the solid-to-liquid ratio changes as a function of distance from the discharge region and the experimental conditions (i.e. particle-size distribution and temperature). We varied the solid-to-liquid volume fraction ratio at a given position downstream of the plasma zone by changing the microwave power. Decreasing the power reduced the temperature of the particle-laden gas flow, in which part or all of the liquid particles solidify before passing the probe section. The experiments were carried out with microwave powers of 800 or 600 W at a total pressure of 100 mbar. A precursor flow of 0.03 slm (standard liter per minute) of monosilane (SiH4) mixed with argon at 2 slm and hydrogen at 0.2 slm was injected at the bottom of the reactor into a vertical fused-silica tube (passing through the plasma region) through a nozzle with an inner diameter of 4 mm. A co-annular swirl flow of Ar/H2 at 6.6/0.5 slm stabilized the nozzle flow such that particles were confined to a hollow cylindrical flow. Two optical ports at opposite sides of the reactor chamber fitted with fused silica windows located approx. 300 mm downstream from the plasma zone allowed the LOSA/LOSE measurements. Using thermophoretic sampling on carbon-coated copper grids and subsequent TEM analysis, the particle size for a 800 W microwave power was fitted with a log-normal size distribution with a CMD of 22.8 nm and a standard deviation of 1.34 nm [17]. At the condition of 600 W, the CMD increased to 25.4 nm and the standard deviation to 1.44 nm. The particle-size distribution of liquid particles is assumed the same as the solid particles, and particle sizes between 1 and 100 nm were considered in the reconstruction model. A lower microwave power leads to a lower temperature, which entails a lower flow velocity (for a constant incoming mass flow rate), thus leading to a correspondingly longer residence time [3]. Therefore, particles have more time to coalesce, which results in larger particles.
4.2 Experimental procedure
Figure 11 depicts the maps of the extinction data τλ calculated based on Eq. (15) along each projection line (i.e., wavelength vs. position) at microwave powers of 600 and 800 W. With the decrease in microwave power, the extinction significantly increases in the ultraviolet. This may be attributed to (1) the increase in particle size (2) the increase in particle number density (3) a larger fraction of liquid particles solidifying. Extinction is still observed in the outer zone in the map of 600 W, which is attributed to recirculation of cold particles in the reactor outside the hot stream that contains the initially formed particles. The corresponding spatially-resolved spectra in the emission measurements are shown in Fig. 12. With the decrease in microwave power, the emission intensities dramatically decrease, which may be due to the decrease in particle temperature and number density.
Fig. 11. Spatially-resolved extinction spectra along a horizontal line (40 cm above the nozzle exit). Microwave powers of 800 W (left) and 600 W (right).
Fig. 12. Spatially-resolved emission spectra along a horizontal line (40 cm above the nozzle exit). Microwave powers of 800 W (left) and 600 W (right).
The structure of the Si nanoparticle stream in the experiment is not ideally axisymmetric, as depicted in Figs. 11 and Fig. 12, and the peak of the (wavelength-averaged) extinction data along the radial axis at the left side of the flow is approximately 1.4 times as high as at the right side no matter of microwave power. One approach to deal with this asymmetric structure is to tomographically measure extinction and emission signals using multiple detectors at different angles; however, the enclosure surrounding the stream limits the detection angle range in the microwave plasma reactor. In this study, the Si stream was assumed as axisymmetric, and we used the spatial variation in (time-averaged) signals at the left and the right side of the flow as an indication of expected symmetry deviation and evaluated the data independently to get information about the effect of the asymmetry on the analysis. Moreover, the effect of the asymmetric structure of the extinction spectra on the accuracy of the reconstruction results are analyzed in the Supplement 1.
Figure 13 shows the particle number density, and volume fraction distributions of liquid- and solid-phase Si reconstructed from the measured extinction spectra on the experiment condition for a microwave power of 800 W with an uncertainty area, in comparison to the DNS simulation data (Fig. 6). As for the reconstructed distributions, similar trends in number density and volume fraction distributions were observed and only a small fraction of solid particles were detected for both sides of the particle stream at this condition. The simulated volume fraction distributions for both phases have a comparable tendency to the ones reconstructed from the experiment, but with higher intensities in the peaks. And the radial locations of the peaks shift from approx. 7 mm in the simulation profiles to approx. 5 mm in the reconstruction profiles. Meanwhile, the reconstructed number density distribution of liquid Si could not match the simulated one at radial positions less than 5 mm, as discussed in the above phantom study part. The uncertainty areas were obtained by varying the prior information of CMD within ±5 nm, because errors may exist in the measured particle-size distribution due to sampling errors and inaccurate size determination from TEM analysis [55]. As shown in Fig. 13, the uncertainty in the particle-size distribution has a larger impact on the retrieval results of the number densities of both phases than the retrieval results of volume fractions.
Fig. 13. Reconstructed quantities of number density and volume fraction distributions from LOSA diagnostics, and temperature distribution from LOSE diagnostics of the Si nanoparticle flow downstream of the plasma zone (800 W microwave power). “rs”, “ls” and “avg” represent the right and the left side of the particle stream, and the average of both sides, respectively.
Figure 13 also displays the reconstructed temperature distribution from the line-of-sight integrated particle emission via the LOSE measurement, with the comparison of DNS data. Because the accuracy of the temperature determined by the thermal radiation-based method is limited to regions of high temperature and high liquid particle volume fractions [56], the temperature is only shown in Fig. 13 in regions with particle volume fraction larger than 50% of its maximum. The uncertainty areas for the temperature profiles are also shown in the figure using the same method as that for particle number density and volume fraction. It is observed that the uncertainty in the CMD only has a slight influence on the temperature distribution. It should be noted that the temperature determined by the LOSE measurement is based on the radiation emission from liquid Si, while neglecting the emission contribution from solid Si. With the assumption of the local temperature of liquid Si being the same as that of solid Si, the retrieval temperature gives the guidance to choose the refractive index of solid Si. Among the five temperatures for which the index of refraction of solid Si is given in the literature [41], the one at 1123 K was chosen since it is the closest one to the temperature expected in the LOSE measurements. For the present feed gas flows and microwave power operating conditions, the expected liquid volume fraction should be much larger than that of the solid. The number density and volume fraction profile peaks of the accumulated liquid Si are located at approx. 7 mm from the centerline for both sides, where the particle temperature is 1530 K for the right side of the particle stream and 1631 K for the other side.
With 600 W microwave power, the correspondingly lower temperature causes the synthesized particles to solidify before they reach the probe volume. Figure 14 shows reconstructed data for this operating condition. The retrieved number density and volume fraction distributions of both phases on the right side of the particle stream still keep similar trends as on the left side.
Fig. 14. Reconstructed quantities of the axial distribution of number density and volume fraction from LOSA diagnostics, and temperature distribution from LOSE diagnostics of the Si nanoparticle flow downstream of the plasma zone (600 W microwave power). The indication of the legends is the same as Fig. 14.
The integral value along the number-density or volume-fraction profile of solid Si is comparable to that of liquid Si. At the 600 W condition, the number density and volume fraction peaks of the accumulated liquid particles are still located at approx. 7 mm from the centerline with the temperature of 1300 K for both sides, while the solid fraction peaks at approx. 6.2 mm for both sides for which the temperature is 1372 K for the right and 1423 K for the left side, respectively. The temperature at the location of the maximum volume fraction of liquid Si is lower than that at the position where the maximum concentration of solid Si is observed, which is contrary to the expectation. This may be attributed to uncertainties in the particle-size distribution, for example, the particle size could be different between the solid and liquid phase particles and vary along the radial position.
Our spectroscopic line measurements were restricted to an area of 12.3 mm around the centerline of the flow. This is the dimension considered in the 2D reconstruction algorithm. While the hot gas flow of freshly generated particles only extents up to ∼8 mm around the axis and is thus fully covered by the measurement, additional extinction is caused by the recirculation of cold particles along the line-of-sight in the reactor. Because of the much longer path length (400 mm distance between the windows), this contribution is not negligible and visible at radii > 8 mm in Fig. 11. Since the 2D reconstruction is active only within the measured domain of ±12.3 mm (i.e., a pathlength of 24.6 mm), the off-center extinction is “compressed” into the remaining zone and thus overestimates the local extinction. We corrected for this effect assuming homogeneous distribution of the recirculated particles across a pathlength of 38 mm.
We reconstructed extinction spectra based on the retrieved data of number density distributions of solid and liquid Si. The experimental and simulated extinction spectra at 6 and 7 mm from the centerline for the left side of the particle flow are compared in Fig. 15. The uncertainty areas along with the simulated extinction spectra are also determined by varying the prior information of CMD within ±5 nm. In general, the simulated extinction spectra are consistent with the measured ones. At 800 W microwave power, these extinction spectra decrease monotonically with wavelength and no obvious resonant peak from solid Si appears. At 600 W microwave power, broad smooth peaks from solid Si appear at around 400 nm. The deviation between the extinction spectra may also be introduced by the uncertainty in the refractive index of solid Si.
Fig. 15. Experimental and reconstructed extinction spectra at specific off-axis positions of 6 and 7 mm at 600 and 800 W microwave power.
5. Conclusions and future perspectives
We developed a reconstruction method to quantitively distinguish and retrieve the fractions of solid and liquid particles in a silicon nanoparticle aerosol from spatially- and spectrally-resolved line-of-sight attenuation (LOSA) measurements exploiting spectroscopic information of both phases. The reconstruction method was firstly tested using DNS data, and the effects of the Gaussian noise and the particle size distribution on the reconstructed results were analyzed in phantom studies. The approach was further tested with data from an experimental system with an unknown mix of solid/liquid particle ensemble synthesized in a microwave-heated flow of SiH4 diluted in H2 and Ar. Modifying the microwave power from 800 to 600 W resulted in varying temperature at the measurement location and thus a modification of the solid-to-liquid ratio. Corresponding particle temperatures were determined by line-of-sight emission (LOSE) measurements.
It is an inverse problem to retrieve the number density distributions of two nanoparticle phases from the coupled extinction data, and the reconstructed results are affected by measurement noise and model error, such as the assumption of an axisymmetric shape of the particle flow, the uncertainty in the radiative materials properties, the particle-size distribution, as well as potential differences in particle-size distribution between both phases. Using Bayesian statistics will further facilitate the uncertainty analysis in future work. Combining with other measurement techniques, e.g., Mie scattering, for obtaining spatially-resolved particle size distribution would also be a helpful strategy to increase the robustness of the data analysis. In the future, it is expected to include a spectroscopic model of the temperature dependence for the optical properties of solid Si to improve the reconstruction accuracy. This strategy will support the determination of the temperature of solid Si particles based on the wavelength position of the resonance absorption peaks.
The approach demonstrated in this paper shows promising results for decoupling the extinction data of solid and liquid Si from the measured spectra and it is expected to be applicable also to other complex nanoparticle aerosol systems as long as significant differences exist in the extinction spectra between two particle phases.
6. Nomenclature
Funding
Alexander von Humboldt-Stiftung; Deutsche Forschungsgemeinschaft (FOR 2284, project number 262219004).
Acknowledgments
The authors thank Khadijeh Mohri, Muhammad Asif and Paolo Fortugno from the University of Duisburg-Essen for valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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