Estimation of spectral absorption function range via LII measurements of flame-synthesized TiO₂ nanoparticles

Abstract

The Laser-Induced Incandescence (LII) technique is widely used for the study of soot production in flames. More recently, it has also drawn attention also for the characterization of metal-oxide flame synthesis. To retrieve the particle volume fraction from the LII signal, information on the effective particle temperature and the absorption function $E(m_{\lambda })$ depending on the wavelength are needed. In this study, a new approach is proposed to determine these parameters from LII measurements at a given laser fluence by accounting for the spectral shape of $E(m_{\lambda })$ obtained from the estimated effective particle temperature. The feasibility of the method is first demonstrated on a carbon black nanoparticle aerosol in a non-reactive cold environment. A good agreement with the literature data for $E(m_{\lambda })$ is found. The developed approach is then applied to TiO$_2$ nanoparticles produced by flame synthesis. The obtained spectral response of $E(m_{\lambda })$ is in line with the literature results for TiO$_2$. The proposed approach represents an essential step toward in-situ estimation of particle volume fraction from the LII signal, which can be a valuable tool for further characterization of metal-oxide flame synthesis.

Postprocessing procedure to estimate \(T_{\text {eff}}\) from the LII spectral emission

The proposed methodology relies on the estimation of a possible range for the effective temperature. However, the range of the possible effective particle temperatures, i.e., $T_{\text {eff}}^{\text {min}}$ and $T_{\text {eff}}^{\text {max}}$ may depend on the postprocessing procedure used to estimate them from the LII spectral emission.

First of all, the LII signal $I_{SLII}$ at $\lambda _0$ and $\lambda _0+\Delta \lambda$ have to be known to calculate $T_{\text {eff}}^{\lambda _0}$. As an example, the prompt spectral emission for carbon black particles at laser fluence F = 0.06 J/cm$^2$ is visualized in Fig. 10. The presented signal exhibits a certain degree of noise. A postprocessing procedure is then needed to obtain an estimation of $I_{SLII}$ for very close wavelengths. A piecewise linear reconstruction is first considered here. For this, the spectral emission is averaged over a 20 nm interval as visualized by the bars in Fig. 10a. The signal emission is then reconstructed by supposing a linear behavior of $I_{SLII}$ between the sampling points (represented by the symbols). The reconstructed signal is visualized by the blue line in Fig. 10a. Alternatively, a polynomial fitting of the fifth order is also considered. The obtained fitted signal is visualized by the red line in Fig. 10b.

The two reconstructed signals are then used to calculate $T_{\text {eff}}^{\lambda _0}$ using Eq. (2) for $\lambda _0 \in [560,700]$ nm. Results are visualized in Fig. 11a. As expected, the estimation of $T_{\text {eff}}^{\lambda _0}$ varies with the retained $\lambda _0$ value where condition $E(m_{\lambda _0})=E(m_{\lambda _0+\Delta \lambda })$ is imposed. Since it is not possible to know which is the exact value, the maximum and minimum values are chosen, namely $T_{\text {eff}}^{\text {max}}$ and $T_{\text {eff}}^{\text {min}}$, respectively. The $T_{\text {eff}}^{\text {const}}$ value is also reported. It can be noted that $T_{\text {eff}}^{\text {const}}$ belongs to the region of possible $T_{\text {eff}}$ values, illustrating the possible error associated with the assumption $E(m_{560 nm})=E(m_{700nm})$. It can be noticed that the obtained $T_{\text {eff}}^{\text {max}}$ and $T_{\text {eff}}^{\text {min}}$ depend on the reconstruction method retained (piecewise linear reconstruction or polynomial fitting). In addition, their values are expected to depend also on the size of the interval $\Delta \lambda$. The values $T_{\text {eff}}^{\text {max}}$ and $T_{\text {eff}}^{\text {min}}$ obtained with both reconstruction methods are illustrated as a function of $\Delta \lambda$ in Fig. 11b. While increasing the interval size $\Delta \lambda$, the impact of the assumption $E(m_{\lambda _0})=E(m_{\lambda _0+\Delta \lambda })$ is expected to increase. The results slowly converge toward the $T_{\text {eff}}^{\text {const}}$ value as expected. Even if the extreme values of possible effective temperature depend on the postprocessing procedure, the proposed methodology allows having an estimation of the effective temperature value and of the spectral evolution of $E(m_\lambda )$ with a quantification of the uncertainties due to theoretical assumptions and the signal postprocessing. In this work, results were obtained using the piecewise linear reconstruction for $\Delta \lambda =$5 nm. Uncertainties due to the postprocessing are not considered in this work and should be included in future works.

Fig. 10

Postprocessing procedure to estimate $T_{\text {eff}}$ from the LII spectral emission: a piecewise linear reconstruction with an interval window of 20 nm; b reconstruction using a fifth-order polynomial fitting

Fig. 11

a Spectral dependence of the effective particle temperature $T_{\text {eff}}^{\lambda _0}$ obtained using Eq. (2) for $\lambda _0 \in [560,700]$ nm. The obtained $T_{\text {eff}}^{\text {max}}$ and $T_{\text {eff}}^{\text {min}}$ depend on the reconstruction method retained. b The effect of $\Delta \lambda$ on the estimation of $T_{\text {eff}}^{\text {max}}$ and $T_{\text {eff}}^{\text {min}}$ effective temperatures for different reconstruction methods