There are many applications, including time-resolved LII, in which it is crucial to model heat conduction accurately for nano-sized particles. Heat conduction from nanoparticles occurs in one of three regimes depending on the Knudsen number,

$Kn = \dfrac{{\lambda \left( {{T_g}} \right)}}{a}$

where λ(T_{g}) is the molecular mean free path in the gas and a is the particle radius. At large Kn, heat conduction occurs in the** free molecular regime**, in which gas molecules travel between the nanoparticle surface and the equilibrium gas without undergoing intermolecular collisions. The other limiting case is the **continuum regime** at small Kn, in which conduction is governed by the continuum equations and gas thermal conductivity.

Intermediate Knudsen numbers correspond to the **transition regime**, which bridges the free-molecular and continuum regimes. These conditions prevail in soot-laden aerosols at high pressures, such as in gas turbine combustors. Unlike the other two regimes, the equations governing heat conduction in the transition regime are analytically insoluble, so numerous schemes have been developed that interpolate between the results of the free-molecular and continuum equations for a particular Kn.

**1. Steady state heat conduction**

One of the most widely used interpolation schemes is Fuch’s boundary sphere method^{1,2}, in which the gas is divided into a free molecular regime (FMR) shell (Knudsen layer) surrounding the particle of thickness Δ, in turn surrounded by a continuum gas that extends to the equilibrium gas temperature, T_{g}. (A good biography of N. A. Fuchs is available here.) The FMR shell has a thickness approximately equal to the mean free path of the gas at the boundary, λ(T_{Δ}). The analysis proceeds by setting the FMR heat transfer between the particle surface at T_{p} and the boundary sphere at T_{Δ} equal to the continuum regime heat conduction between T_{Δ} and T_{g}, and solving for the unknown T_{Δ}.

While many implementations simply set Δ = λ(T_{Δ}), Filippov and Rosner^{3} use a simplified version of a statistical treatment proposed by Wright^{4},

$\Delta = \dfrac{{{a^3}}}{{\lambda _\Delta ^2}}\left({\frac{1}{5}\Lambda _1^5 - \frac{1}{3}{\Lambda _2}\Lambda _1^3 + \frac{2}{15}\Lambda _2^{\frac{5}{2}}}\right)-a, \, \Lambda _1 = \left[1 + \dfrac{{{\lambda _\Delta }}}{a}\right], \Lambda _2 = \left[1 + {\left( {\dfrac{{{\lambda _\Delta }}}{a}} \right)^2}\right]
$

which accounts for the curvature of the nanoparticle surface and the cosine distribution of diffusely scattered molecules. In our study^{5}, we explore the origin of this equation and demonstrate its affect on transition-regime conduction in TiRe-LII. We also use Direct Simulation Monte Carlo (DSMC) to visualize the molecular gas dynamics underlying transition regime conduction. In this approach, thousands of simulated gas molecules are tracked through collisions with molecules and other surfaces. While other studies^{6} have used DSMC to calculate heat transfer rates, we use this technique to calculate probability densities of the distances incident molecules have traveled from their last intermolecular collision, δ_{i}, the distance scattered molecules travel to their next intermolecular collision, δ_{o}, and the combined distance, l_{i+o}. These results highlight the fact that the collisionless Knudsen layer is an idealization, and the boundary sphere is not, in fact, collisionless. We also show how the influence of particle curvature on Knudsen layer changes with Knudsen number.

**2. Transient heat conduction**

While Fuchs's boundary sphere method provides an accurate estimation of steady state heat conduction, in TiRe-LII the nanoparticle temperature changes abruptly as it is heated with the laser pulse and then cools in the surrounding gas. There has been some speculation that transient gas dynamics phenomena could be responsible for some irregularities in observed nanoparticle cooling rates. Specifically:

- an enhanced cooling rate often observed immediately after the laser pulse, particularly pronounced in LII experiments carried out in ambient gas, could be due to a non-stationary gas layer surrounding the nanoparticle;
- at higher fluences, the gas and sublimed nanoclusters may expand supersonically, resulting in shockwaves;
- sublimed nanoclusters may recondensce on the nanoparticle surface, resulting in a reduced cooling rate.

To investigate these phenomena, Farzan Memarian pioneered the use of transient DSMC to model heat conduction in laser induced incandescence experiments. Transient DSMC works by carrying out a sequence of independent time-resolved DSMC simulations, and then averaging the ensemble results to reduce variance. Contrary to speculation in the literature, Farzan's simulations^{7} showed no shockwave formation, even at extremely high fluences. There is an enhanced cooling rate due to the nonstationary gas, however, and, a considerable amount of sublimed material will return to the nanoparticle surface due to collisions with gas molecules and other sublimed nanoclusters.

- N. A. Fuchs, 1934, "Über die Verdampfungsgechwindigkeit kleiner Tröpfchen in einer Gasatmosphäre (in German)," Physikalische Zeitschrift der Sowjetunion, 6, pp. 224-243
- N. A. Fuchs (as N. A. Fuks), 1958, "On the theory of the evaporation of small droplets," Soviet Physics Technical Physics, 3, pp. 140-144.
- A. V. Filippov, D. E. Rosner, 2000, "Energy transfer between an aerosol particle and gas at high temperature ratios in the Knudsen transition regime," International Journal of Heat and Mass Transfer, 43, pp. 127-138.
- P. Wright, 1960, "On the discontinuity involved in diffusion across an interface (the Δ of Fuchs)," Discussions of the Faraday Society, 30, pp. 100-112
- S. C. Huberman, K. J. Daun, 2012, "Influence of particle curvature on transition regime heat conduction from aerosolized nanoparticles", International Journal of Heat and Mass Transfer, 55, pp. 7668–7676.
- F. Liu, K. Daun, D. Snelling, G. Smallwood, 2006, "Heat conduction from a spherical nano-particle: status of modeling heat conduction in laser-induced incandescence, Applied Physics B: Lasers and Optics, 83 pp. 355-382.
- F. Memarian, K. J. Daun, 2013, "Gas dynamics of sublimed nanoclusters in high fluence time-resolved laser-induced incandescence,", Numerical Heat Transfer Part B: Fundamentals, 65, pp. 393-409