arising from Y. Qian et al. Communications Chemistry https://doi.org/10.1038/s42004-022-00674-8 (2022)
Aerosol science would benefit greatly from understanding the interfacial structure of airborne particles. However, measuring the surface structure in situ is challenging because of the weak second-order nonlinearity needed to generate a surface response, the dispersive/scattering nature of the sample, the dilution of particles (~106/mL), and the size range of the majority of them (10–300 nm, in diameter)1. Indeed, vibrational surface SFS spectra, as well as non-resonant surface second harmonic scattering (SHS, Fig. 1a, which is comparable) measured from particles dispersed in solution of this size range, have been reported with samples containing ~1011 particles/mL2,3. There is no principle difference in the mechanism and rules that govern nonlinear scattering for particles that are airborne or dispersed in solution4, with the difference in surrounding medium being incorporated by the difference in refractive index between the particle and surrounding medium. Since the 1980’s numerous groups have worked on developing and verifying nonlinear light scattering theories, and these theories, like their linear counterpart, generally agree with nonlinear light scattering experiments5. Indeed, the single difference between linear and nonlinear light scattering is the induced polarization of the material that is used as a source term. Therefore, the results of the Article this contribution is arising from1 raise questions as to the nature of the source of the SF scattering object(s), as there is a difference on the order of ~107 between the expected scattering power and the detected spectra.
Here, we first explicitly determine the number of particles that can be reasonably measured for a certain size in SHS/SFS experiments. To do so, we combine theoretical predictions per particle with experiment. We will first consider the throughput of the SFS/SHS experiment and consider its size and number density dependence, then provide a signal-to-noise ratio analysis of various comparable experiments. Using this analysis in combination with SFS and SHS experiments performed on the same samples, we determine the detection limit in terms of particle density for a certain size. Finally, we discuss several explanations for the difference in terms of detected intensity and expected aerosol density.
Size dependence
Vibrational SFS and non-resonant SHS are second-order nonlinear optical techniques. The intensity of the generated photons obeys the following expression:
with Np the particle density, \({\Gamma }^{\left(2\right)}(R,\,\theta ,{\chi }^{\left(2\right)})\) the effective (single) particle susceptibility, which depends on the radius R, the scattering angle θ, the (surface) susceptibility of the particle χ(2), Ei the pulse energies of the participating beams, τ the pulse duration, A the overlap area and f the repetition rate. The effective particle susceptibility that determines the response of a single particle \({\Gamma }^{\left(2\right)}(R,\,\theta ,{\chi }^{\left(2\right)})\) is determined by the single particle light-matter interaction process and is highly size dependent. \({\Gamma }^{\left(2\right)}\) also contains effects of absorption/linear scattering as it is also a function of the electromagnetic field functions6. For R~<200 nm, \({\Gamma }^{\left(2\right)}\propto \,{R}^{3}\) \((I({\omega }_{0})\propto {R}^{6})\), and for larger particles, this size dependence levels off, reaching \(I({\omega }_{0})\propto {R}^{3}\) at R~1000 nm. For R <200 nm, the Rayleigh-Gans-Debye approximation, one of the approximate solutions to the Maxwell equations, works well to describe the data. For larger sizes, it becomes more approximate, and depending on the conditions nonlinear Mie theory needs to be used4. Nonlinear Mie theory offers an exact solution for both linear and nonlinear scattering, assuming the single scattering particles are spherical. Dipolar and quadrupolar scattering both lead to identical size-dependent behaviors4,7. The scattering pattern is also highly size dependent with scattering maxima appearing between 90° and 55° for R < 50 nm, which gradually move forward. For water droplets in the air, however, the refractive index contrast ensures that for micron-sized or larger particles scattering light is emitted in every direction4. Particles in the air have a bigger linear refractive index difference between the bulk and the particle medium compared to solid/liquid dispersions, which generally have a smaller refractive index contrast and therefore exhibit less losses due to linear scattering of the incident beams. We can expect that the estimations based on the systems used in this work are over-estimating the actual scattering efficiencies.
Signal-to-noise ratio
To relate the experimental throughput of the experiments presented in the Article this contribution is arising from1, we first compare their signal-to-noise ratio (SNR) to previously published vibrational SFS and non-resonant SHS data, and then perform vibrational SFS and non-resonant SHS measurements on hexadecane nanodroplets. Table 1 shows the experimental parameters that relate the 3 published experiments, as well as the parameters used to collect the data in Fig. 1c, d. The SFS experiment reported in the Article this contribution is arising from1 has a comparable SNR compared to previously published SFS data and a 1.4–2.4 smaller SNR to non-resonant SHS. Because SHS and SFS have comparable SNRs, we retrieve the particle vs size dependence from SHS. The primary reason to do so is that the SHS intensity of any particle surface can be compared both experimentally and theoretically to the known incoherent bulk response of neat water, which therefore represents a calibration benchmark. This incoherent SH light is known as hyper Rayleigh scattering, and is subtracted from the raw data, to obtain the pure particle scattering (Eq. S1). With a known hyperpolarizability tensor of water, it is therefore possible to explicitly compute the theoretical response that matches 10% of the magnitude of the intensity recorded with SNR = 1. We take this as the detection limit of the SFS and non-resonant SHS experiments of particles. Thus, we compute using the theory in Refs. 8,9, as a function of particle radius, which number density of particles is needed to generate an intensity that matches 10% of the incoherent neat bulk water intensity in the SSS polarization combination (all beams polarized perpendicular to the scattering plane). We insert typical values for the surface susceptibility (χ(2) = 10−22 m2/V) and the surface potential (Φ0 = 100 mV), which represent the surface properties of the particles.
Estimation of particle density vs size dependence
The line in Fig. 1b shows the detection limit in terms of # particles/mL as a function of the radius of the particles, based on extrapolating the various size-dependent models4. For a 20 nm radius, this means 2.5 × 1013 particles/mL are needed to generate the desired SF intensity. Moving to larger particles, we arrive at 5.6 × 1010 particles/mL (50 nm radius), 4.2 × 107 (500 nm radius), and finally, extrapolating to non-resonant SH imaging, 1 for objects in the size range of 5–10 microns10. For diameters close to 100, 200, and 300 nm, data points from angle-resolved non-resonant second harmonic scattering measurements from the surface of silica particles dispersed in water11,12 are shown. Based on this result, we estimate that 106 particles/mL can only generate a detectable response if R > 1000 nm. Alternatively, 40 nm particles would have to be present at a density of 2.5 × 1013 particles/mL, as indicated by the dashed arrow. Neither of these criteria are met in the Article this contribution is arising from1.
Fig. 1c shows SFS spectra of deuterated hexadecane droplets with a 109 ± 1 nm average hydrodynamic radius stabilized with 8 mM sodium dodecyl sulfate (SDS), following the protocol of Chen et al.13 and summarized in the Supplementary Methods. SFS spectra were recorded with particle densities between 1011/mL and 1013/mL, achieved by diluting a stock emulsion (size distribution is shown in the inset). In agreement with Fig. 1b, the recorded SF intensity vanishes below a particle density of 1012/mL. Figure 1d shows data measured by non-resonant SHS of the same sample diluted with 0.8 mM SDS solution. In both experiments, no detectable signal can be obtained below 4.9 × 109 particles/mL, in agreement with predictions of Fig. 1b for identically-sized objects.
Discussion and conclusions
Although the values of χ(2), the scattering angle range, the precise value of the refractive indices, or the surface potential may change, given the difference examined here (a factor of 107 for a particle of 40 nm in diameter) these details are relatively unimportant as they would create a maximum difference in the scattering intensity in the range 0.1–10. Since the instrument used in the Article this contribution is arising from1 is not outperforming the instruments used here (Table 1), this brings about the question what actually generates the unexpectedly strong SFS response. Based on nonlinear light scattering theory summarized by Eq. (1), there are only two candidates for increasing the scattered intensity by ×107: The size of the particles, with the scattering objects being much larger than reported, or χ(2), which should then increase by ~103. A single particle of a few microns in size could easily overpower the emission of all other particles and generate the measured SF intensity, for example \(\frac{I(1\,\mu m)}{I(10\,{nm})} \sim ({10}^{2})^{3}\). To get insight here, one would have to measure the SF scattering patterns, as was done in Ref. 2. A time-dependent intensity trace would provide information about the stability of the intensity14. The second option might revolve around the crystallization of surfactant in a semi-crystalline layer on the surface of the particles. Extremely dilute crystallites have also shown to produce large SFS intensities15. It is additionally known in aerosol literature that due to differences in evaporation rates, particles size, morphology and crystallinity can drastically change once they are released in the aerosol chamber16,17,18.
Summarizing, performing vibrational surface SFS from aerosols with a diameter centered near 40 nm and a density of 106 particles/mL1 is at odds with understanding based upon nonlinear light scattering theory as well as state of the art experiments conducted by numerous labs, on (metal/plasmonic) particles, droplets, harmonophores, liposomes and other objects with vastly different compositions in both liquid and solid media. The nonlinear light scattering data and theory obtained since the 1980’s generally agrees well with one another, and in that sense shows the same type of confidence as one has with linear light scattering19. That the Article this contribution is arising from1 shows data that deviates by a factor of ~107 from current state of the art data and understanding is therefore extremely puzzling to us.
Data availability
The data sets generated and analyzed during the current study are available from the corresponding author upon reasonable request.
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Acknowledgements
A.M. and S.R. thank the Julia Jacobi Foundation. A.M. thanks the Swiss National Science Foundation (Ambizione grant number PZ00P2_174146). T.W.G., A.S.C., and T.W. acknowledge funding from the Novo Nordisk Foundation (Facility Grant NanoScat, No. NNF18OC0032628). T.W.G. thanks the Lundbeck Foundation for a postdoc fellowship (postdoc grant R322-2019-2461).
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S.R. conceived and designed the work. A.M. carried out the theoretical calculations. T.W.G and A.S.C. performed the SFS/SHS experiments. A.M., T.W.G., and S.R. wrote the manuscript. T.W. and S.R. directed the work. All authors discussed the results and contributed to the manuscript.
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Marchioro, A., Golbek, T.W., Chatterley, A.S. et al. A discrepancy of 107 in experimental and theoretical density detection limits of aerosol particles by surface nonlinear light scattering. Commun Chem 6, 114 (2023). https://doi.org/10.1038/s42004-023-00903-8
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DOI: https://doi.org/10.1038/s42004-023-00903-8
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