Laser-based nanoparticle sizing is important in a growing number of applications. It is crucial for assessing the impact of soot on human health and climate change, for example, which both depend on nanoparticle size, and to study soot formation and growth within internal combustion engines. Laser-based nanoparticle metrology has also been proposed as a way to control the large-scale synthesis of engineered nanoparticles.

**Time-resolved laser-induced incandescence** uses a short laser pulse to heat aerosolized nanoparticles within a small probe volume to incandescent temperatures, and then measures the spectral incandescence of the nanoparticles as they return to the ambient gas temperature. Since larger nanoparticles cool more slowly than smaller ones, the particle size distribution can be determined from the observed spectral incandescence decay rate.

If the aerosol is monodisperse (i.e. all nanoparticles are the same size) the nanoparticle diameter is calculated directly from an energy balance,

$\rho {c_p}\dfrac{{\pi d_p^3}}{6}\dfrac{{d{T_p}}}{{dt}} = - {q_{cond}} - {q_{evap}} - {q_{rad}}$

where ρ and c_{p} are the density and specific heat of soot, and q_{cond}, q_{evap} and q_{rad} are the conduction, evaporation, and radiation heat transfer rates from the nanoparticle. (Radiation is usually negligible at near-ambient pressures.) In the free-molecular regime, evaporation heat transfer is found from

${q_{evap}} = \pi d_p^2\,\Delta {H_v}\,{N''_v}\left( {{P_v},{T_p}} \right)$

where ΔH_{v} and N"_{v} are the latent heat of formation and the number flux of sublimed species from the nanoparticle surface, which are both found using the Clausius-Clapeyron equation. Free molecular heat conduction is given by

${q_{cond}} = \pi d_p^2\,{N''_g} \langle{E_p}-{E_g}\rangle=\pi d_p^2\,{N''_g}\alpha \langle{E_{p,max}}-{E_g}\rangle$

where N"_{g} is the number flux of incident gas molecules, <E_{p}−E_{g}> is the average increase of a gas molecule that scatters from the laser-energized nanoparticle, and α is the thermal accommodation coefficient. If the nanoparticles are not small relative to the mean free molecular path within the gas, a transition regime model must be employed.

The above analysis is straightforward if the nanoparticles all have the same size, but in many scenarios the aerosols contain a range of nanoparticle sizes. In this instance, the incandescence signal observed at any instant is due to emission from all nanoparticle size classes, and recovering P(d_{p}) requires deconvolution of a Fredholm integral equation of the first kind,

where Q_{abs,λ} is the absorption efficiency of nanoparticles of diameter d_{p} at wavelength λ and I_{b,λ} is the blackbody spectral intensity at T_{p}.

This is an ill-posed inverse problem, since many candidate P(d_{p}) distributions exist that could explain the observed data^{1}. More often, the spectral incandescence decay is measured at multiple wavelengths, and these incandescenses are used to define a pyrometric effective temperature. The unknown distribution parameters for P(d_{p}) are then inferred by nonlinear regression of modelled temperatures to measured temperatures. Our group is currently developing techniques based on robust Bayesian analysis to quantify the nanoparticle size distributions and volume fractions, along with their uncertainties.

While TiRe-LII is most often applied to measure soot primary particles, our research is focused on using it to size non-carbonaceous nanoparticles. We recently collaborated with Prof. Christof Schulz's group at the Centre for Nanointegration Duisburg Essen (CENIDE) to size silicon nanoparticles. In this study^{2}, silicon nanoparticles were first synthesized in a plasma reactor, and then measured using two-color TiRe-LII.
The nanoparticle size distribution parameters were inferred from the TiRe-LII data, using a thermal accommodation coefficient derived from molecular dynamics, and uncertainties were quantified using robust Bayesian inference. The distribution parameters have large uncertainties due to the ill-posedness of this problem, but they correspond to a Sauter mean that matches one found through ex situ BET analysis.

We have also used TiRe-LII to size iron nanoparticles in various monatomic and polyatomic gases. Previous TiRe-LII studies have exclusively used gas-phase synthesis techniques to manufacture the aerosols; while this approach prevents agglomeration, the drawback is that the nanoparticle size depends on the nature of the carrier gas, which complicates any comparative analysis between gases.
In this work the nanoparticles are first formed in water by reducing ferrous iron (Fe^{2+}) with sodium borohydrate (NaBH_{4}), and are then capped with carboxymethyl cellulose (CMC) to prevent flocking. The nanoparticles are then aerosolized using a pneumatic atomizer. The droplet-laden gas stream flows through a diffusion dryer, and then enters the LII measurement chamber.

A robust Bayesian estimation procedure is used to infer the nanoparticle diameters and the thermal accommodation coefficients from the TiRe-LII data. (In contrast to silicon nanoparticles, the iron nanoparticles produced by this method can be approximated as monodisperse.) Iron nanoparticle sizes derived from TiRe-LII match those obtained through dynamic light scattering measurements carried out on the nanocolloid solution, and are consistent between gases. The experimentally-derived thermal accommodation coefficients are also consistent with MD-derived values.

- K. J. Daun, B. J. Stagg, F. Liu, G. J. Smallwood, D. R. Snelling, 2007, "Determining Aerosol Particle Size Distributions using Time-Resolved LII", Applied Physics B, 87, pp. 363.
- T. A. Sipkens, R. Mansmann, K. J. Daun, N. Petermann, J. T. Titantah, M. Karttunen, H. Wiggers, T. Dreier, C. Schulz, 2014, In situ nanoparticle size measurements of gas-borne silicon nanoparticles by time-resolved laser-induced incandescence," Applied Physics B, 116, pp. 623-636.