Investigating temporal variation in the apparent volume fraction measured by time-resolved laser-induced incandescence

Abstract

In many time-resolved laser-induced incandescence (TiRe-LII) experiments, it is common practice to relate the intensity emitted by laser-heated nanoparticles to the detected LII signal through a factor (here called the intensity scaling factor, ISF) that includes the particle volume fraction and other parameters that may not be the focus of the analysis. While, in the absence of evaporation or sublimation, the ISF should theoretically remain constant with respect to time, recent multi-wavelength measurements show that, in reality, it may vary with both time and fluence. We consider four candidate effects that contribute to this behavior: particle annealing; polydispersity in the nanoparticle-size distribution; background luminosity due to emission from nanoparticles in the line-of-sight before and behind the probe volume; and the temporal resolution of the detector. We demonstrate these effects by simulating TiRe-LII data for in-flame soot at atmospheric pressure, using new simplified heat transfer and annealing models. Analysis of experimental signals collected from flame-generated soot at atmospheric pressure reveals trends in the ISF similar to those predicted by simulations. These temporal variations provide important insights that can help to diagnose problems in TiRe-LII experiments and improve TiRe-LII models.

Appendices

Appendix A: heat transfer model

The properties for the heat transfer model in this work were chosen close to the median of the range of values given in the literature previously, most notably from Ref. [24]. This model is chosen to increase the information entropy contained in the model by preventing overtuning to specific laboratory settings, particularly given the disparity of soot properties given in the literature (cf. the range of trajectories for the nanoparticle temperature shown in Ref. [24]. While this model lends itself particularly well to statistical analyses (consideration of which are the topic of future work), point estimates produced by such a model are also beneficial from an information entropy perspective. The values used in this work are summarized in Table 3.

Table 3 Spectroscopic and heat transfer model parameter values used in generating simulated signals and, excluding the nanoparticle diameter, for interpreting TiRe-LII signals

Several notes are also made here on the function forms chosen for ΔH_v, p_v, and m_v. Above the melting point, we use the universal expression recently proposed by Román et al. [55]:

$$\Delta H_{\text{v}} = \Delta H_{\text{v,ref}} \exp \left[ {\left( {n - \beta_{\text{R}} } \right)\left( {\frac{{T_{\text{p}} - T_{\text{ref}} }}{{T_{\text{cr}} - T_{\text{ref}} }}} \right)} \right]\left( {\frac{{T_{\text{cr}} - T_{\text{p}} }}{{T_{\text{cr}} - T_{\text{ref}} }}} \right),$$

(35)

where ΔH_v,ref and T_ref are a reference latent heat of vaporization and temperature, n is the critical exponent, β_R = 0.37 [55], and T_cr is the critical temperature. Below the melting temperature, we use a linear expression that closely fits the data in Leider et al. [25]. Based on the correlations provided by Leider et al. [25], we separate the changes in m_v with temperature into three regimes: (i) low temperatures, where C₁ is the dominant evaporated species, though not in significant quantities; (ii) beyond 1900 K where C₃ becomes dominant; and (iii) finally, beyond 4500 K where the vapor becomes a mixture composed significantly of C₃, C₅, and C₇. Accordingly, we model the molecular mass of the vapor using a sigmoid function, so that the transitions between the different regimes are smooth. Finally, the vapor pressure is evaluated using the Kelvin equation, which accounts for the enhanced surface energy due to curvature:

$$p_{\text{v}} = p_{\text{v,o}} \exp \left( {\frac{{4\gamma_{\text{s}} }}{{d_{\text{p}} \rho R_{\text{s}} T_{\text{p}} }}} \right),$$

(36)

where γ_s = 0.18 J/cm² is the surface tension [56] and p_v,o is the nominal vapor pressure above a bulk surface, given by the Clausius–Clapeyron equation:

$$p_{\text{v,o}} = p_{\text{v,ref}} \exp \left[ {\frac{{ - \Delta H_{\text{v,ref}} }}{R}\left( {\frac{1}{{T_{\text{p}} }} - \frac{1}{{T_{\text{ref}} }}} \right)} \right],$$

(37)

where p_v,ref, T_ref, and ΔH_v,ref are reference values, in this case, different from those in Eq. (34). These quantities are generally consistent with the parameters of other TiRe-LII evaporation models presented by Michelsen et al. [24] and the original work of Leider et al. [25], although they predict slightly greater evaporation, and, consequently, steeper temperature decays, close to the laser pulse.

Appendix B: annealing model

The annealing model is based on the reducing sphere model of Butenko et al. [42] and assumes that graphitization initiates instantaneously over the primary particle surface. While this may not quantitatively represent the physics, it represents a simple model that accounts for the increased energy of the atoms at the surface, which makes it easier for the atoms to restructure. The transformation then progresses into the interior of the primary particle with a single rate, k. The transformed fraction of the individual primary particle X at any given time is described by the following [42]:

$$X = 1 - \left( {\frac{{d_{\text{p,t}} }}{{d_{\text{p}} }}} \right)^{3} ,$$

(38)

where d_p,t is the diameter of the untransformed material at any time. It can then be shown that the rate of change in X is given by the following:

$$\frac{{{\text{d}}X}}{{{\text{d}}t}} = \frac{\text{d}}{{{\text{d}}t}}\left( {1 - \frac{{d_{\text{p,t}}^{3} }}{{d_{\text{p}}^{3} }}} \right) = - \frac{{3d_{\text{p,t}}^{2} }}{{d_{\text{p}}^{3} }}\frac{{{\text{d}}d_{\text{p,t}} }}{{{\text{d}}t}} = \frac{{3\left( {1 - X} \right)^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} }}{{d_{\text{p}} }}\frac{{{\text{d}}d_{\text{p,t}} }}{{{\text{d}}t}}.$$

(39)

The progression of the transformed phase is governed by the Arrhenius equation:

$$\frac{{{\text{d}}d_{\text{p,t}} }}{{{\text{d}}t}} = 2k = 2A_{0} \exp \left( { - \frac{{E_{\text{A}} }}{RT}} \right),$$

(40)

where E_A is an activation energy and A₀ is a pre-exponential factor. The activation energy is taken as E_A = 400 kJ/mol, chosen on the lower end of the values reported in the literature (generally in the range of 400–1000 kJ/mol [43]), and the exponential prefactor is set to A₀ = 1.0 × 10⁵ s⁻¹. These quantities bring the model into agreement with Michelsen’s model at high temperatures, where it is considered reliable, while predicting slower annealing (on the order of milliseconds to hours) at flame temperatures and low fluences, consistent with Refs. [39, 59]. The final form of the rate of change in X is, thus:

$$\frac{{{\text{d}}X}}{{{\text{d}}t}} = \frac{{6A_{0} \left( {1 - X} \right)^{{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-0pt} 3}}} }}{{d_{\text{p}} }}\exp \left( { - \frac{{E_{\text{A}} }}{RT}} \right).$$

(41)

This form has the consequence that the rate of dX/dt is faster for smaller nanoparticles where there is less material to be transformed. The proposed model also allows for freezing of the transformation at values of X < 1 for moderate heating. We use X = 0 as an initial condition, as the annealing rate at flame conditions is assumed to be slow, and solve Eq. (40) using a fourth-order Runge–Kutta scheme.