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Characterization of oscillation modes in levitated droplets using image and non-image based techniques

Characterization of oscillation modes in levitated droplets using image and non-image based... www.nature.com/npjmgrav ARTICLE OPEN Characterization of oscillation modes in levitated droplets using image and non-image based techniques 1✉ 1 1 1 2 2 Nevin Brosius , Jason Livesay , Zachary Karpinski , Robert Singiser , Michael SanSoucie , Brandon Phillips and Ranga Narayanan The dynamics of levitated liquid droplets can be used to measure their thermophysical properties by correlating the frequencies at which normal modes of oscillation most strongly resonate when subject to an external oscillatory force. In two preliminary works, it was shown via electrostatic levitation and processing of various metals and alloys that (1) the resonance of the first principal mode of oscillation (mode n = 2) can be used to accurately measure surface tension and (2) that so-called “higher-order resonance” of n = 3 is observable at a predictable frequency. It was also shown, in the context of future space-based experimentation on the Electrostatic Levitation Furnace (ELF), a setup on the International Space Station (ISS) operated by Japan Aerospace Exploration Agency (JAXA), that while the shadow array method in which droplet behavior is visualized would be challenging to identify the n = 3 resonance, the normal mode n = 4 was predicted to be more easily identifiable. In this short communication, experimental evidence of the first three principal modes of oscillation is provided using molten samples of Tin and Indium and it is subsequently shown that, as predicted, an “image-less" approach can be used to identify both n = 2 and n = 4 resonances in levitated liquid droplets. This suggests that the shadow array method may be satisfactorily used to obtain a self-consistent benchmark of thermophysical properties by comparing results from two successive even-mode natural frequencies. npj Microgravity (2023) 9:3 ; https://doi.org/10.1038/s41526-023-00254-7 INTRODUCTION studying materials via electrostatic levitation that are challenging to levitate on Earth. In addition to being able to test a wider Levitation can be used to study physical phenomena in a variety of materials, a microgravity environment yields a more contactless, relatively contamination-free environment. This is spherical droplet and thus is more theoretically tractable with notably important for providing a benchmark measurement analytical models that assume a spherical geometry. This setup is process for thermophysical properties, permitting the observation shown in Fig. 2 and uses a laser to cast the droplet’s shadow on a of meta-stable states of matter and highly chemically reactive photodetector as a function of time during processing. The materials, and studying fundamental material behavior . As the advantage of this detection scheme, as opposed to using a high- name suggests, electrostatic levitation consists of a charged speed camera, for example, is that it takes less processing time to droplet held between two electrodes via an electrostatic field .In analyze the droplet’s movement when oscillated, a priority in typical processing, a material is loaded into the chamber, charged, operations on the space station. levitated by means of a control system, and melted. After this, a In a previous work by Brosius et al. , it was predicted that, due variety of processing techniques can take place; in the measure- to the inherent nature of this indirect method of characterizing ment of surface or interfacial tension, for example, the droplet is the droplet, it would be challenging to identify odd modes of subjected to an oscillatory electric field and its behavior is oscillation in levitated droplets on the JAXA setup—this could also recorded, via a high-speed camera or other optical sensors. Using be deduced from work by Egry et al. . This prediction was the formula derived by Rayleigh for the oscillations of an inviscid, spherical droplet , one can predict the frequency at which a given supported by an experiment where high-speed video data was waveform, or mode, will oscillate at its surface: used to capture the oscillation and resonance of mode n = 3. Following the experiment, the magnitude of mode n = 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn  1Þðn þ 2Þγ resonance was quantified by projecting the corresponding (1) f ¼ Legendre polynomial on the droplet’s outline. Measuring the 3πM magnitude of this projection over a frequency range resulted in a clear maximum and therefore the identification of mode n = 3 In the above equation, γ is the surface or interfacial tension, M is resonance. Following this, the image data from the experiment the mass, and n represents the mode of oscillation, which can be were fed into an image processor to calculate the area of the an integer from 2 to infinity. Each mode represents a unique droplet’s shadow to determine whether or not resonance could be spherical harmonic and, when the droplet is axisymmetric observed using the area data alone (without the knowledge of (independent of azimuthal angle ϕ), can be reduced to a the droplet outline itself), using a mean squared deviation of the Legendre polynomial with an argument of cosðθÞ, where θ is droplet’s area for the same range of frequencies. The results of this the polar angle. Shown in Fig. 1 is a graphical representation of experiment from ref. are shown in Fig. 3 and indicate that the modes n = 2, n = 3, and n = 4. reduction from video data to area data removes (or at the least, In 2016, the Japan Aerospace Exploration Agency (JAXA) reduces) the ability to identify/quantify the resonance of mode launched the Electrostatic Levitation Furnace (ELF) to the KIBO module on the International Space Station for the purpose of n = 3. For completeness, it is shown, by contrast, that the 1 2 University of Florida Department of Chemical Engineering, Gainesville, FL 32611, USA. NASA Marshall Space Flight Center, Huntsville, AL 35812, USA. email: nbb5056@ufl.edu Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; N. Brosius et al. resonance of mode n = 2 could still be identified using non- followed by mode n = 3 at a corresponding step size, and finally imaging-based data. including the mode n = 4 frequency sweep. In the following sections, a set of proof-of-concept experiments The other marked difference between the previous work and show that (1) the visualization of mode n = 4 is observed at the experiments conducted for this work is the fact that the temperature measurement was not absolutely accurate for the expected frequencies and quantified using a spectral analysis of samples due to limitations of the pyrometer that was used. Both the droplet’s outline and (2) an “image-less" technique using only Indium and Tin have very low melting points (156 °C and 231 °C, the area of the droplet can be used to identify mode n =4in respectively) which were difficult to measure using the existing addition to mode n = 2 and agrees with the result obtained using equipment. It was observed that the apparent melting tempera- the outline of the droplet as the ground truth. ture (that is, the temperature reading from the pyrometer at which the droplet was observed to melt) was much higher than realistically possible. Therefore, the sensor temperature reading METHODS was simply used to infer that the droplet maintained a constant Experimental methods temperature during testing—see Fig. 4. The liquid state was All experiments in this work took place at NASA Marshall Space doubly verified by exciting the droplet in mode n =2before Flight Center (MSFC) in Huntsville, AL on the Electrostatic testing began. Levitation (ESL) Laboratory. The materials tested were Tin and The sample mass was also recorded both before and after Indium, chosen because of their low surface tension-to-density processing. Both Tin and Indium have very low vapor pressures ratios. A detailed explanation of the process of levitation and in the liquid state and therefore minimal mass loss (<1%) was subsequent sample processing (melting, subcooling, then observed. imposed frequency sweeps) is given in ref. . The droplet behavior is characterized via a high-speed camera at 5000 fps at a Analysis methods resolution of 512 by 512 pixels for a duration sufficient to get at The method in which the droplet’s shape is characterized is least 100 cycles of oscillation. Each sample was forced at 63 described in detail in ref. . In summary, it is assumed that the distinct oscillation frequencies, the number being limited by the droplet’s deformations are axisymmetric and thus can be described storage capacity of the camera. by Legendre polynomials as a function of polar angle θ.That is, the The first difference between the cited work and this work is the outline of the droplet given by r(θ, t)isdefined as: inclusion of a frequency sweep for finding mode n = 4 resonance, which followed the same regimen as modes n = 2 and n = 3. A rðθ; tÞ¼ R þ A P ðcosðθÞÞ cosð2πf tÞ (2) 0 n n n frequency sweep is done for mode n = 2 at a prescribed step size, n¼2 where R is the resting (non-disturbed) radius, A is the amplitude 0 n of normal mode n, P is the nth Legendre polynomial, and f are n n defined in Eq. (1). In principle, the magnitude of each normal mode can be found by projecting the corresponding Legendre polynomial in polar coordinates. However, for modes of small amplitude and with slight off-axis tilt, it was more practical in this work to perform a spatial analog of a Fourier transform on the droplet outline, referred to throughout the duration of this work as the Spatial Discrete Fourier Transform, or Spatial DFT. This process allows for a slightly improved sensitivity to the identification of modes at Fig. 1 The first three modes of oscillation in an axisymmetric small magnitudes. spherical droplet. Each mode represents the nth spherical harmonic and starts at n = 2 to satisfy the conservation of mass. The solid line Image-based processing. The image-based processing of the represents the droplet at t = 0 and the dashed line represents the mode at t = T/2, where T is the oscillation period. experimental data begins with the conversion of each video’s Fig. 2 An illustration of the area array setup used to characterize the behavior of the oscillating droplet at JAXA and on ISS KIBO. The setup uses a laser to measure the amount of radiation that is blocked by the cross-sectional area of the droplet via a photosensor to infer the change in droplet shadow area from a baseline and therefore monitor droplet deformation over time (reproduced with permission from Springerⓒ (2022) from the work of ref. ). npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; N. Brosius et al. Fig. 4 The pyrometer readings for each of the samples during experimentation. This does not include the transients involved in the melting or solidification processes at the beginning and end of processing, respectively. Recall that the pyrometer reading was used as a measure of the relative temperature difference across trials rather than an absolute measurement. The time-averaged amplitude of each mode is then computed for every frequency tested on the droplet to fully capture the so- called “resonance curve” for a given mode. Denoting the amplitude for each peak as α (t), and assuming its behavior can be described by α ðtÞ¼ a sinð2πf tÞ (where f is given by n n n n Eq. (1)), one can approximate the time-averaged modal amplitude a given N frames with the below formula: !1 !1 T 2 2 2 pffiffiffi X ðα ðtÞÞ dt 2 (3) a ¼ 2  a n R T n;i dt i¼1 where T is the total period of recording time for a given forcing frequency. It has previously been shown in the density measurement of levitated materials that selection of the centroid as the origin can result in reduced stability of fit and (in the case of this work) variation in the absolute amplitude of the spatial coefficients α . Fig. 3 Comparison of frequency sweep analysis results obtained Since the relative amplitude is of concern, this will not impact the using an image-based method versus the simulation of a non- findings of this analysis (i.e., what is deemed the natural image-based technique for the first two oscillation modes. The frequency for a given mode). Nevertheless, it would be a resonance of a n = 2 can be easily identified via both image-based valuable investigation to understand the implications of the and non-image-based means, while b n = 3 resonance was only placement of the origin. detected using the image-based technique. The sample is 57.345 mg Inconel 625 at 1350 °C. Borrowed with permission Generation and subsequent analysis procedure of simulating area from ref. . array data using video data. Following the procedure of ref. , the video data is transformed to area data to simulate the frames into a series of single-pixel-width outlines of the droplet, observation of the drop with a shadow array rather than a which are subsequently transposed into polar coordinates about camera. Image analysis software was used to convert the video to the droplet’s calculated center of mass (centroid). The polar form binary pixel images and calculate the area of the droplet for each of the droplet outline is then analyzed by taking a periodic frame. These data are subsequently analyzed to quantify excitation of the mode of interest. Due to the significantly extension (in this case 20 total cycles, or “rotations" about the reduced amplitude of oscillation upon mode n = 4 resonance, the center of mass, was deemed sufficient) and computing its area oscillations immediately about the forcing frequency were corresponding Spatial Fourier spectrum, with each peak effec- isolated by means of a DFT. This general procedure is outlined tively representing discrete polar wavelengths. In other words, graphically in Fig. 6. the DFT is used to decompose the droplet’s outline (r(θ)) into a To summarize Fig. 6, each of the videos acquired during the series of sinusoidal functions of nθ (akin to a rectangular Fourier mode n= 4 frequency sweep are batch processed to transform transform decomposing a function into sinusoidal functions of nπx from image data to droplet area versus time. A DFT is then applied ,where L is the domain of the function of interest). Therefore, to each dataset, and the spectrum immediately surrounding the amplitude peaks resulting from the DFT correspond to each n and, since this process is repeated for each frame of video, are forcing frequency is integrated to find the approximate response recorded as a function of time. The overall process for each frame amplitude. This approach effectively isolates the droplet’sarea is shown in Fig. 5. change as it relates to the frequency at which it was forced and Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 3 N. Brosius et al. Fig. 5 A graphic representation of the process involved in the Spatial DFT method of analyzing the outline of the droplet to identify resonance. For each image of each video, a the droplet image is converted into b an outline via image analysis software, c transposed from cartesian to polar coordinates about its centroid. d The polar form of the droplet’s outline is periodically extended 20 times and e a DFT is performed on the data. The amplitude of the peak corresponding to the mode of interest (in this case, mode n = 2) is used as the metric to quantify droplet resonance and time-averaged for each video to form the final resonance curve. therefore reduces the noise imparted by other non-resonance- testing on the ISS in the shadow array setup shown in Fig. 2), it based oscillations. Plotting the amplitude of response versus the was shown herein that it is still feasible to isolate and quantify the frequency of forcing for a frequency sweep yields the familiar resonance of mode n = 4 by performing DFT analysis on the resonance curve using the area data alone. corresponding area versus time data. To this end, the so-called “benchmarking" method described in ref. can be performed even if odd modes (n = 3) are difficult to observe with a shadow Reporting summary array setup. Further information on research design is available in the Nature The proof of concept is shown in Fig. 8.Itcan be observed Research Reporting Summary linked to this article. that the frequency at which the measured area oscillation response is a maximum corresponds directly with the RESULTS AND DISCUSSION frequency at which the video data showed the strongest n = 4 oscillation amplitude. In other words, the quantification Observation of n = 2, n = 3, and n = 4 modes using the image- methods for both image-less and an image-based approach based approach agree in the case of n = 4, as it had been shown in prior works A summary of the experiments is shown in Table 1.Six total with n = 2. samples (five Tin, one Indium) were successfully processed; that is, clear resonance was observed for each of the n= 2, n= 3, and n= 4 modes. Furthermore, it can be observed that the ratio of the Accuracy and precision of measurement modal frequencies are consistent with what is predicted by Eq. (1). A thorough commentary on precision and accuracy as it A graphical representation of this self-consistency is shown in Fig. 7, concerns the general procedure of reporting natural frequency where the amplitude of response for each mode is plotted as a 4 is documented in ref. . To summarize, the precision (that is, the function of the ratio ,where f is the experimentally observed f n reporting on the natural frequency for a given sample and resonant frequency of mode n. mode) is primarily driven by the step size taken during the experiment. In other words, if the step size is 1 Hz and the Proof of concept using simulated area array data amplitude clearly peaks at 185 Hz, the reported value is Using video data to create a simulation of the data obtained using 185 ± 1 Hz. Therefore, the experiment can be carried out with a shadow array approach (a method to be employed in future theoretically arbitrary precision. npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA N. Brosius et al. Fig. 6 A graphic representation of the process of simulating and subsequently analyzing non-image-based data using pre-existing image-based data. In this process, a the videos of the droplet’s oscillation behavior at set forcing frequencies are b converted into area vs. time data, c broken down into frequency components via a DFT and d each dataset is quantified by integrating a frequency “band" centered around the forcing frequency. e The resonance curve reflects this amplitude versus the forcing frequency and shows a clear peak, in this case for mode n = 4. Table 1. A summary of the experimental results, identifying the frequency at which the droplet was observed to oscillate most strongly for modes n = 2, n = 3, and n = 4 for purposes of comparing with their theoretical frequency ratios predicted by Eq. (1). Sample Material Mass (mg) f (Hz) f (Hz) f (Hz) f /f f /f f /f 2 3 4 3 2 4 2 4 3 1 Tin 50.3 85 ± 2 172 ± 2 270 ± 2 2.02 ± 0.05 3.18 ± 0.08 1.57 ± 0.02 2 Indium 39.8 104 ± 2 195 ± 2 305 ± 1 1.88 ± 0.04 2.93 ± 0.06 1.56 ± 0.02 3 Tin 50.4 91 ± 1 174 ± 1 273 ± 1 1.91 ± 0.02 3.00 ± 0.03 1.57 ± 0.01 4 Tin 66.7 78 ± 2 149 ± 2 234 ± 1 1.91 ± 0.06 3.00 ± 0.08 1.57 ± 0.02 5 Tin 60.4 80 ± 2 156 ± 2 245 ± 0.5 1.95 ± 0.05 3.06 ± 0.08 1.57 ± 0.02 6 Tin 60.3 84 ± 2 158 ± 2 245 ± 1 1.88 ± 0.05 2.92 ± 0.07 1.55 ± 0.02 Theoretical values 1.94 3 1.55 The accuracy of the measurement (which is defined as the the droplet is spinning while being forcibly oscillated. During experimentation, extreme care was taken to monitor the spin of possible shift in resonant frequency from the ‘actual’ natural the droplet by monitoring high-definition live video with which frequency of the mode) can be impacted by several things that one could easily infer the rotation of the droplet during the depend on experimental parameters, material properties, and melting process and subsequently afterward by watching operating environment. An impressively thorough analysis of the recorded high-speed video of the forced oscillations. If the imaging-related errors in the levitated droplet environment is droplet was rotating during the oscillation for any of the trials, presented in ref. and includes uncertainty propagation due to that trial was discarded and another oscillation at that frequency edge detection, pixel volume, and aspect ratio. The primary drivers was attempted. The error associated with droplet rotation can of experimental error associated with reporting the resonant also be reported in the form of a frequency shift and has been frequencies of a forcibly oscillated droplet are (1) the electric field, thoroughly investigated by many researchers, notably in ref. , (2) the gravitational field, and (3) the amplitude of deformation wherein the natural frequency of a rotating droplet through linear while the droplet has forcibly oscillated. Each of these sources of stability analysis was presented as error are managed by either minimizing their effect or referring to theoretical works that quantify the related error, which has been 2 19Ω f  f 1 þ (4) postulated to be on the order of 1% but realistically no more than 21 ω 5% of the reported natural frequency . where f is the natural frequency of a non-rotating droplet Notably, in addition to these aforementioned primary sources according to Eq. (1), Ω is the rotational angular frequency, and of measurement error, there is also a possible impact to ω = 2πf . The rotational frequency of the droplets tested in this measurement accuracy (there is a resonant frequency shift) if 0 0 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 3 N. Brosius et al. multiple resonance points can be calibrated with theoretical predictions to add additional precision to the measurement of surface tension in microgravity environments. DATA AVAILABILITY All relevant data are available from N.B. CODE AVAILABILITY All relevant codes for image analysis protocols are available from N.B. Received: 8 July 2022; Accepted: 10 January 2023; Fig. 7 The resonance curves for each normal mode as a function of the ratio of the forcing frequency to the experimentally REFERENCES observed resonant frequency. Sample is 60.4 mg Tin. 1. Brillo, J. Thermophysical Properties of Multicomponent Liquid Alloys (Walter de Gruyter GmbH & Co KG, 2016). 2. Rhim, W.-K. et al. An electrostatic levitator for high-temperature containerless materials processing in 1-g. Rev. Sci. Instrum. 64, 2961–2970 (1993). 3. Rayleigh, L. On the capillary phenomena of jets. Proc. R. Soc. Lond. 29,71–97 (1879). 4. Brosius, N. et al. Benchmarking surface tension measurement method using two oscillation modes in levitated liquid metals. npj Microgravity 7,1–8 (2021). 5. Egry, I. Surface tension measurements of liquid metals by the oscillating drop technique. J. Mater. Sci. 26, 2997–3003 (1991). 6. Hyers, R. W. Computer-aided experiments in containerless processing of materials. In Solidification of Containerless Undercooled Melts (eds Herlach, D. M. Matson, D. M.) 31–49 (Wiley‐VCH Verlag GmbH & Co, 2012). 7. Yoo, H., Park, C., Jeon, S., Lee, S. & Lee, G. W. Uncertainty evaluation for density measurements of molten ni, zr, nb and hf by using a containerless method. Metrologia 52, 677 (2015). 8. Kitahata, H. et al. Oscillation of a rotating levitated droplet: analysis with a mechanical model. Phys. Rev. E 92, 062904 (2015). 9. Ishikawa, T. & Paradis, P.-F. Electrostatic levitation on the ISS. In Metallurgy in Space Fig. 8 The comparison between the resonance quantification of (eds Fecht, H.-J. Mohr,M.)65–92 (Springer,Cham 2022). mode n = 4 using an image-based method and an image-less method. Sample is 60.4 mg Tin. ACKNOWLEDGEMENTS N.B. and R.N. acknowledge support from NASA NNX17AL27G and NASA CAN work did not exceed ≈1 Hz as evidenced by high-speed video 80MSFC20M0001. N.B. acknowledges support from the NASA Space Technology analysis and constant monitoring of the live-stream high- Research Fellowship under grant NASA 80NSSC18K1173 and the University of Florida definition video feed of the experiment. For the most Office of Research. The authors thank Trudy Allen and Glenn Fountain at NASA conservative (low frequency) case of mode n = 2witha natural Marshall Space Flight Center for experimental support. frequency of 78 Hz, a 1 Hz rotation frequency would result in an error of ≈0.01%. AUTHOR CONTRIBUTIONS Concluding remarks N.B. designed the experimental plan and developed the image analysis procedure. N.B., J.L., Z.K., R.S., B.P., and M.S. conducted experiments at NASA MSFC. N.B. and Z.K. In summary, the n = 2 and n = 4 oscillation modes can be conducted image analysis at the University of Florida. N.B. wrote the manuscript with detected via an area array setup as is available on the ISS KIBO the assistance of J.L., Z.K., C.S., R.N., B.P., and M.S. module’s ELF facility. This assertion is supported by the agreement between the actual image-based data and the simulated area array data. The Spatial DFT method in which the droplet’s shape COMPETING INTERESTS was analyzed varies slightly from the previously used Projection The authors declare no competing interests. Method but is robust in its ability to detect oscillating modes that are slightly off-axis and small amplitude. The so-called “odd"-modes of oscillation remain to be elusive in ADDITIONAL INFORMATION an image-less approach, due to the fact that, for a nearly Supplementary information The online version contains supplementary material axisymmetrical oscillating droplet, the odd Legendre polynomials available at https://doi.org/10.1038/s41526-023-00254-7. integrate to zero (or, in reality, a vanishingly small value) when Correspondence and requests for materials should be addressed to Nevin Brosius. compared to their even counterparts. This can potentially be resolved, as alluded to in ref. , by viewing the droplet from Reprints and permission information is available at http://www.nature.com/ different viewing angles with respect to the direction of the reprints oscillating electric field. Ultimately, this work will serve to facilitate the benchmarking Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims objectives outlined in ref. where a sample can be levitated and in published maps and institutional affiliations. npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA N. Brosius et al. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http:// creativecommons.org/licenses/by/4.0/. © The Author(s) 2023 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 3 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png npj Microgravity Springer Journals

Characterization of oscillation modes in levitated droplets using image and non-image based techniques

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www.nature.com/npjmgrav ARTICLE OPEN Characterization of oscillation modes in levitated droplets using image and non-image based techniques 1✉ 1 1 1 2 2 Nevin Brosius , Jason Livesay , Zachary Karpinski , Robert Singiser , Michael SanSoucie , Brandon Phillips and Ranga Narayanan The dynamics of levitated liquid droplets can be used to measure their thermophysical properties by correlating the frequencies at which normal modes of oscillation most strongly resonate when subject to an external oscillatory force. In two preliminary works, it was shown via electrostatic levitation and processing of various metals and alloys that (1) the resonance of the first principal mode of oscillation (mode n = 2) can be used to accurately measure surface tension and (2) that so-called “higher-order resonance” of n = 3 is observable at a predictable frequency. It was also shown, in the context of future space-based experimentation on the Electrostatic Levitation Furnace (ELF), a setup on the International Space Station (ISS) operated by Japan Aerospace Exploration Agency (JAXA), that while the shadow array method in which droplet behavior is visualized would be challenging to identify the n = 3 resonance, the normal mode n = 4 was predicted to be more easily identifiable. In this short communication, experimental evidence of the first three principal modes of oscillation is provided using molten samples of Tin and Indium and it is subsequently shown that, as predicted, an “image-less" approach can be used to identify both n = 2 and n = 4 resonances in levitated liquid droplets. This suggests that the shadow array method may be satisfactorily used to obtain a self-consistent benchmark of thermophysical properties by comparing results from two successive even-mode natural frequencies. npj Microgravity (2023) 9:3 ; https://doi.org/10.1038/s41526-023-00254-7 INTRODUCTION studying materials via electrostatic levitation that are challenging to levitate on Earth. In addition to being able to test a wider Levitation can be used to study physical phenomena in a variety of materials, a microgravity environment yields a more contactless, relatively contamination-free environment. This is spherical droplet and thus is more theoretically tractable with notably important for providing a benchmark measurement analytical models that assume a spherical geometry. This setup is process for thermophysical properties, permitting the observation shown in Fig. 2 and uses a laser to cast the droplet’s shadow on a of meta-stable states of matter and highly chemically reactive photodetector as a function of time during processing. The materials, and studying fundamental material behavior . As the advantage of this detection scheme, as opposed to using a high- name suggests, electrostatic levitation consists of a charged speed camera, for example, is that it takes less processing time to droplet held between two electrodes via an electrostatic field .In analyze the droplet’s movement when oscillated, a priority in typical processing, a material is loaded into the chamber, charged, operations on the space station. levitated by means of a control system, and melted. After this, a In a previous work by Brosius et al. , it was predicted that, due variety of processing techniques can take place; in the measure- to the inherent nature of this indirect method of characterizing ment of surface or interfacial tension, for example, the droplet is the droplet, it would be challenging to identify odd modes of subjected to an oscillatory electric field and its behavior is oscillation in levitated droplets on the JAXA setup—this could also recorded, via a high-speed camera or other optical sensors. Using be deduced from work by Egry et al. . This prediction was the formula derived by Rayleigh for the oscillations of an inviscid, spherical droplet , one can predict the frequency at which a given supported by an experiment where high-speed video data was waveform, or mode, will oscillate at its surface: used to capture the oscillation and resonance of mode n = 3. Following the experiment, the magnitude of mode n = 3 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn  1Þðn þ 2Þγ resonance was quantified by projecting the corresponding (1) f ¼ Legendre polynomial on the droplet’s outline. Measuring the 3πM magnitude of this projection over a frequency range resulted in a clear maximum and therefore the identification of mode n = 3 In the above equation, γ is the surface or interfacial tension, M is resonance. Following this, the image data from the experiment the mass, and n represents the mode of oscillation, which can be were fed into an image processor to calculate the area of the an integer from 2 to infinity. Each mode represents a unique droplet’s shadow to determine whether or not resonance could be spherical harmonic and, when the droplet is axisymmetric observed using the area data alone (without the knowledge of (independent of azimuthal angle ϕ), can be reduced to a the droplet outline itself), using a mean squared deviation of the Legendre polynomial with an argument of cosðθÞ, where θ is droplet’s area for the same range of frequencies. The results of this the polar angle. Shown in Fig. 1 is a graphical representation of experiment from ref. are shown in Fig. 3 and indicate that the modes n = 2, n = 3, and n = 4. reduction from video data to area data removes (or at the least, In 2016, the Japan Aerospace Exploration Agency (JAXA) reduces) the ability to identify/quantify the resonance of mode launched the Electrostatic Levitation Furnace (ELF) to the KIBO module on the International Space Station for the purpose of n = 3. For completeness, it is shown, by contrast, that the 1 2 University of Florida Department of Chemical Engineering, Gainesville, FL 32611, USA. NASA Marshall Space Flight Center, Huntsville, AL 35812, USA. email: nbb5056@ufl.edu Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; N. Brosius et al. resonance of mode n = 2 could still be identified using non- followed by mode n = 3 at a corresponding step size, and finally imaging-based data. including the mode n = 4 frequency sweep. In the following sections, a set of proof-of-concept experiments The other marked difference between the previous work and show that (1) the visualization of mode n = 4 is observed at the experiments conducted for this work is the fact that the temperature measurement was not absolutely accurate for the expected frequencies and quantified using a spectral analysis of samples due to limitations of the pyrometer that was used. Both the droplet’s outline and (2) an “image-less" technique using only Indium and Tin have very low melting points (156 °C and 231 °C, the area of the droplet can be used to identify mode n =4in respectively) which were difficult to measure using the existing addition to mode n = 2 and agrees with the result obtained using equipment. It was observed that the apparent melting tempera- the outline of the droplet as the ground truth. ture (that is, the temperature reading from the pyrometer at which the droplet was observed to melt) was much higher than realistically possible. Therefore, the sensor temperature reading METHODS was simply used to infer that the droplet maintained a constant Experimental methods temperature during testing—see Fig. 4. The liquid state was All experiments in this work took place at NASA Marshall Space doubly verified by exciting the droplet in mode n =2before Flight Center (MSFC) in Huntsville, AL on the Electrostatic testing began. Levitation (ESL) Laboratory. The materials tested were Tin and The sample mass was also recorded both before and after Indium, chosen because of their low surface tension-to-density processing. Both Tin and Indium have very low vapor pressures ratios. A detailed explanation of the process of levitation and in the liquid state and therefore minimal mass loss (<1%) was subsequent sample processing (melting, subcooling, then observed. imposed frequency sweeps) is given in ref. . The droplet behavior is characterized via a high-speed camera at 5000 fps at a Analysis methods resolution of 512 by 512 pixels for a duration sufficient to get at The method in which the droplet’s shape is characterized is least 100 cycles of oscillation. Each sample was forced at 63 described in detail in ref. . In summary, it is assumed that the distinct oscillation frequencies, the number being limited by the droplet’s deformations are axisymmetric and thus can be described storage capacity of the camera. by Legendre polynomials as a function of polar angle θ.That is, the The first difference between the cited work and this work is the outline of the droplet given by r(θ, t)isdefined as: inclusion of a frequency sweep for finding mode n = 4 resonance, which followed the same regimen as modes n = 2 and n = 3. A rðθ; tÞ¼ R þ A P ðcosðθÞÞ cosð2πf tÞ (2) 0 n n n frequency sweep is done for mode n = 2 at a prescribed step size, n¼2 where R is the resting (non-disturbed) radius, A is the amplitude 0 n of normal mode n, P is the nth Legendre polynomial, and f are n n defined in Eq. (1). In principle, the magnitude of each normal mode can be found by projecting the corresponding Legendre polynomial in polar coordinates. However, for modes of small amplitude and with slight off-axis tilt, it was more practical in this work to perform a spatial analog of a Fourier transform on the droplet outline, referred to throughout the duration of this work as the Spatial Discrete Fourier Transform, or Spatial DFT. This process allows for a slightly improved sensitivity to the identification of modes at Fig. 1 The first three modes of oscillation in an axisymmetric small magnitudes. spherical droplet. Each mode represents the nth spherical harmonic and starts at n = 2 to satisfy the conservation of mass. The solid line Image-based processing. The image-based processing of the represents the droplet at t = 0 and the dashed line represents the mode at t = T/2, where T is the oscillation period. experimental data begins with the conversion of each video’s Fig. 2 An illustration of the area array setup used to characterize the behavior of the oscillating droplet at JAXA and on ISS KIBO. The setup uses a laser to measure the amount of radiation that is blocked by the cross-sectional area of the droplet via a photosensor to infer the change in droplet shadow area from a baseline and therefore monitor droplet deformation over time (reproduced with permission from Springerⓒ (2022) from the work of ref. ). npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; N. Brosius et al. Fig. 4 The pyrometer readings for each of the samples during experimentation. This does not include the transients involved in the melting or solidification processes at the beginning and end of processing, respectively. Recall that the pyrometer reading was used as a measure of the relative temperature difference across trials rather than an absolute measurement. The time-averaged amplitude of each mode is then computed for every frequency tested on the droplet to fully capture the so- called “resonance curve” for a given mode. Denoting the amplitude for each peak as α (t), and assuming its behavior can be described by α ðtÞ¼ a sinð2πf tÞ (where f is given by n n n n Eq. (1)), one can approximate the time-averaged modal amplitude a given N frames with the below formula: !1 !1 T 2 2 2 pffiffiffi X ðα ðtÞÞ dt 2 (3) a ¼ 2  a n R T n;i dt i¼1 where T is the total period of recording time for a given forcing frequency. It has previously been shown in the density measurement of levitated materials that selection of the centroid as the origin can result in reduced stability of fit and (in the case of this work) variation in the absolute amplitude of the spatial coefficients α . Fig. 3 Comparison of frequency sweep analysis results obtained Since the relative amplitude is of concern, this will not impact the using an image-based method versus the simulation of a non- findings of this analysis (i.e., what is deemed the natural image-based technique for the first two oscillation modes. The frequency for a given mode). Nevertheless, it would be a resonance of a n = 2 can be easily identified via both image-based valuable investigation to understand the implications of the and non-image-based means, while b n = 3 resonance was only placement of the origin. detected using the image-based technique. The sample is 57.345 mg Inconel 625 at 1350 °C. Borrowed with permission Generation and subsequent analysis procedure of simulating area from ref. . array data using video data. Following the procedure of ref. , the video data is transformed to area data to simulate the frames into a series of single-pixel-width outlines of the droplet, observation of the drop with a shadow array rather than a which are subsequently transposed into polar coordinates about camera. Image analysis software was used to convert the video to the droplet’s calculated center of mass (centroid). The polar form binary pixel images and calculate the area of the droplet for each of the droplet outline is then analyzed by taking a periodic frame. These data are subsequently analyzed to quantify excitation of the mode of interest. Due to the significantly extension (in this case 20 total cycles, or “rotations" about the reduced amplitude of oscillation upon mode n = 4 resonance, the center of mass, was deemed sufficient) and computing its area oscillations immediately about the forcing frequency were corresponding Spatial Fourier spectrum, with each peak effec- isolated by means of a DFT. This general procedure is outlined tively representing discrete polar wavelengths. In other words, graphically in Fig. 6. the DFT is used to decompose the droplet’s outline (r(θ)) into a To summarize Fig. 6, each of the videos acquired during the series of sinusoidal functions of nθ (akin to a rectangular Fourier mode n= 4 frequency sweep are batch processed to transform transform decomposing a function into sinusoidal functions of nπx from image data to droplet area versus time. A DFT is then applied ,where L is the domain of the function of interest). Therefore, to each dataset, and the spectrum immediately surrounding the amplitude peaks resulting from the DFT correspond to each n and, since this process is repeated for each frame of video, are forcing frequency is integrated to find the approximate response recorded as a function of time. The overall process for each frame amplitude. This approach effectively isolates the droplet’sarea is shown in Fig. 5. change as it relates to the frequency at which it was forced and Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 3 N. Brosius et al. Fig. 5 A graphic representation of the process involved in the Spatial DFT method of analyzing the outline of the droplet to identify resonance. For each image of each video, a the droplet image is converted into b an outline via image analysis software, c transposed from cartesian to polar coordinates about its centroid. d The polar form of the droplet’s outline is periodically extended 20 times and e a DFT is performed on the data. The amplitude of the peak corresponding to the mode of interest (in this case, mode n = 2) is used as the metric to quantify droplet resonance and time-averaged for each video to form the final resonance curve. therefore reduces the noise imparted by other non-resonance- testing on the ISS in the shadow array setup shown in Fig. 2), it based oscillations. Plotting the amplitude of response versus the was shown herein that it is still feasible to isolate and quantify the frequency of forcing for a frequency sweep yields the familiar resonance of mode n = 4 by performing DFT analysis on the resonance curve using the area data alone. corresponding area versus time data. To this end, the so-called “benchmarking" method described in ref. can be performed even if odd modes (n = 3) are difficult to observe with a shadow Reporting summary array setup. Further information on research design is available in the Nature The proof of concept is shown in Fig. 8.Itcan be observed Research Reporting Summary linked to this article. that the frequency at which the measured area oscillation response is a maximum corresponds directly with the RESULTS AND DISCUSSION frequency at which the video data showed the strongest n = 4 oscillation amplitude. In other words, the quantification Observation of n = 2, n = 3, and n = 4 modes using the image- methods for both image-less and an image-based approach based approach agree in the case of n = 4, as it had been shown in prior works A summary of the experiments is shown in Table 1.Six total with n = 2. samples (five Tin, one Indium) were successfully processed; that is, clear resonance was observed for each of the n= 2, n= 3, and n= 4 modes. Furthermore, it can be observed that the ratio of the Accuracy and precision of measurement modal frequencies are consistent with what is predicted by Eq. (1). A thorough commentary on precision and accuracy as it A graphical representation of this self-consistency is shown in Fig. 7, concerns the general procedure of reporting natural frequency where the amplitude of response for each mode is plotted as a 4 is documented in ref. . To summarize, the precision (that is, the function of the ratio ,where f is the experimentally observed f n reporting on the natural frequency for a given sample and resonant frequency of mode n. mode) is primarily driven by the step size taken during the experiment. In other words, if the step size is 1 Hz and the Proof of concept using simulated area array data amplitude clearly peaks at 185 Hz, the reported value is Using video data to create a simulation of the data obtained using 185 ± 1 Hz. Therefore, the experiment can be carried out with a shadow array approach (a method to be employed in future theoretically arbitrary precision. npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA N. Brosius et al. Fig. 6 A graphic representation of the process of simulating and subsequently analyzing non-image-based data using pre-existing image-based data. In this process, a the videos of the droplet’s oscillation behavior at set forcing frequencies are b converted into area vs. time data, c broken down into frequency components via a DFT and d each dataset is quantified by integrating a frequency “band" centered around the forcing frequency. e The resonance curve reflects this amplitude versus the forcing frequency and shows a clear peak, in this case for mode n = 4. Table 1. A summary of the experimental results, identifying the frequency at which the droplet was observed to oscillate most strongly for modes n = 2, n = 3, and n = 4 for purposes of comparing with their theoretical frequency ratios predicted by Eq. (1). Sample Material Mass (mg) f (Hz) f (Hz) f (Hz) f /f f /f f /f 2 3 4 3 2 4 2 4 3 1 Tin 50.3 85 ± 2 172 ± 2 270 ± 2 2.02 ± 0.05 3.18 ± 0.08 1.57 ± 0.02 2 Indium 39.8 104 ± 2 195 ± 2 305 ± 1 1.88 ± 0.04 2.93 ± 0.06 1.56 ± 0.02 3 Tin 50.4 91 ± 1 174 ± 1 273 ± 1 1.91 ± 0.02 3.00 ± 0.03 1.57 ± 0.01 4 Tin 66.7 78 ± 2 149 ± 2 234 ± 1 1.91 ± 0.06 3.00 ± 0.08 1.57 ± 0.02 5 Tin 60.4 80 ± 2 156 ± 2 245 ± 0.5 1.95 ± 0.05 3.06 ± 0.08 1.57 ± 0.02 6 Tin 60.3 84 ± 2 158 ± 2 245 ± 1 1.88 ± 0.05 2.92 ± 0.07 1.55 ± 0.02 Theoretical values 1.94 3 1.55 The accuracy of the measurement (which is defined as the the droplet is spinning while being forcibly oscillated. During experimentation, extreme care was taken to monitor the spin of possible shift in resonant frequency from the ‘actual’ natural the droplet by monitoring high-definition live video with which frequency of the mode) can be impacted by several things that one could easily infer the rotation of the droplet during the depend on experimental parameters, material properties, and melting process and subsequently afterward by watching operating environment. An impressively thorough analysis of the recorded high-speed video of the forced oscillations. If the imaging-related errors in the levitated droplet environment is droplet was rotating during the oscillation for any of the trials, presented in ref. and includes uncertainty propagation due to that trial was discarded and another oscillation at that frequency edge detection, pixel volume, and aspect ratio. The primary drivers was attempted. The error associated with droplet rotation can of experimental error associated with reporting the resonant also be reported in the form of a frequency shift and has been frequencies of a forcibly oscillated droplet are (1) the electric field, thoroughly investigated by many researchers, notably in ref. , (2) the gravitational field, and (3) the amplitude of deformation wherein the natural frequency of a rotating droplet through linear while the droplet has forcibly oscillated. Each of these sources of stability analysis was presented as error are managed by either minimizing their effect or referring to theoretical works that quantify the related error, which has been 2 19Ω f  f 1 þ (4) postulated to be on the order of 1% but realistically no more than 21 ω 5% of the reported natural frequency . where f is the natural frequency of a non-rotating droplet Notably, in addition to these aforementioned primary sources according to Eq. (1), Ω is the rotational angular frequency, and of measurement error, there is also a possible impact to ω = 2πf . The rotational frequency of the droplets tested in this measurement accuracy (there is a resonant frequency shift) if 0 0 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2023) 3 N. Brosius et al. multiple resonance points can be calibrated with theoretical predictions to add additional precision to the measurement of surface tension in microgravity environments. DATA AVAILABILITY All relevant data are available from N.B. CODE AVAILABILITY All relevant codes for image analysis protocols are available from N.B. Received: 8 July 2022; Accepted: 10 January 2023; Fig. 7 The resonance curves for each normal mode as a function of the ratio of the forcing frequency to the experimentally REFERENCES observed resonant frequency. Sample is 60.4 mg Tin. 1. Brillo, J. Thermophysical Properties of Multicomponent Liquid Alloys (Walter de Gruyter GmbH & Co KG, 2016). 2. Rhim, W.-K. et al. An electrostatic levitator for high-temperature containerless materials processing in 1-g. Rev. Sci. Instrum. 64, 2961–2970 (1993). 3. Rayleigh, L. On the capillary phenomena of jets. Proc. R. Soc. Lond. 29,71–97 (1879). 4. Brosius, N. et al. Benchmarking surface tension measurement method using two oscillation modes in levitated liquid metals. npj Microgravity 7,1–8 (2021). 5. Egry, I. Surface tension measurements of liquid metals by the oscillating drop technique. J. Mater. Sci. 26, 2997–3003 (1991). 6. Hyers, R. W. Computer-aided experiments in containerless processing of materials. In Solidification of Containerless Undercooled Melts (eds Herlach, D. M. Matson, D. M.) 31–49 (Wiley‐VCH Verlag GmbH & Co, 2012). 7. Yoo, H., Park, C., Jeon, S., Lee, S. & Lee, G. W. Uncertainty evaluation for density measurements of molten ni, zr, nb and hf by using a containerless method. Metrologia 52, 677 (2015). 8. Kitahata, H. et al. Oscillation of a rotating levitated droplet: analysis with a mechanical model. Phys. Rev. E 92, 062904 (2015). 9. Ishikawa, T. & Paradis, P.-F. Electrostatic levitation on the ISS. In Metallurgy in Space Fig. 8 The comparison between the resonance quantification of (eds Fecht, H.-J. Mohr,M.)65–92 (Springer,Cham 2022). mode n = 4 using an image-based method and an image-less method. Sample is 60.4 mg Tin. ACKNOWLEDGEMENTS N.B. and R.N. acknowledge support from NASA NNX17AL27G and NASA CAN work did not exceed ≈1 Hz as evidenced by high-speed video 80MSFC20M0001. N.B. acknowledges support from the NASA Space Technology analysis and constant monitoring of the live-stream high- Research Fellowship under grant NASA 80NSSC18K1173 and the University of Florida definition video feed of the experiment. For the most Office of Research. The authors thank Trudy Allen and Glenn Fountain at NASA conservative (low frequency) case of mode n = 2witha natural Marshall Space Flight Center for experimental support. frequency of 78 Hz, a 1 Hz rotation frequency would result in an error of ≈0.01%. AUTHOR CONTRIBUTIONS Concluding remarks N.B. designed the experimental plan and developed the image analysis procedure. N.B., J.L., Z.K., R.S., B.P., and M.S. conducted experiments at NASA MSFC. N.B. and Z.K. In summary, the n = 2 and n = 4 oscillation modes can be conducted image analysis at the University of Florida. N.B. wrote the manuscript with detected via an area array setup as is available on the ISS KIBO the assistance of J.L., Z.K., C.S., R.N., B.P., and M.S. module’s ELF facility. This assertion is supported by the agreement between the actual image-based data and the simulated area array data. The Spatial DFT method in which the droplet’s shape COMPETING INTERESTS was analyzed varies slightly from the previously used Projection The authors declare no competing interests. Method but is robust in its ability to detect oscillating modes that are slightly off-axis and small amplitude. The so-called “odd"-modes of oscillation remain to be elusive in ADDITIONAL INFORMATION an image-less approach, due to the fact that, for a nearly Supplementary information The online version contains supplementary material axisymmetrical oscillating droplet, the odd Legendre polynomials available at https://doi.org/10.1038/s41526-023-00254-7. integrate to zero (or, in reality, a vanishingly small value) when Correspondence and requests for materials should be addressed to Nevin Brosius. compared to their even counterparts. This can potentially be resolved, as alluded to in ref. , by viewing the droplet from Reprints and permission information is available at http://www.nature.com/ different viewing angles with respect to the direction of the reprints oscillating electric field. Ultimately, this work will serve to facilitate the benchmarking Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims objectives outlined in ref. where a sample can be levitated and in published maps and institutional affiliations. npj Microgravity (2023) 3 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA N. Brosius et al. 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