Examination of the thermal accommodation coefficient used in the sizing of iron nanoparticles by time-resolved laser-induced incandescence

Abstract

While time-resolved laser-induced incandescence (TiRe-LII) shows promise as a diagnostic for sizing aerosolized iron nanoparticles, the spectroscopic and heat transfer models needed to interpret TiRe-LII measurements on iron nanoparticles remain uncertain. This paper focuses on three key aspects of the models: the thermal accommodation coefficient; the spectral absorption efficiency; and the evaporation sub-model. Based on a detailed literature review, spectroscopic and heat transfer models are defined and applied to analyze TiRe-LII measurements carried out on iron nanoparticles formed in water and then aerosolized into monatomic and polyatomic carrier gases. A comparative analysis of the results shows nanoparticle sizes that are consistent between carrier gases and thermal accommodation coefficients that follow the expected trends with bath gas molecular mass and structure.

Appendix: Uncertainty analysis through Bayesian estimation

TiRe-LII data are analyzed using Bayesian inference. In this technique, a posterior distribution, P(x|b), of the quantities-of-interest in x conditional on observed data in b is found using Bayes’ equation

$$P\left( {\left. {\mathbf{x}} \right|{\mathbf{b}}} \right) = \frac{{P\left( {\left. {\mathbf{b}} \right|{\mathbf{x}}} \right)P_{\text{pr}} \left( {\mathbf{x}} \right)}}{{P\left( {\mathbf{b}} \right)}}$$

(22)

where P(b|x) is the likelihood of the observed data in b occurring for a hypothetical x, P _pr(x) is a probability density that represents the state of knowledge of x prior to the measurement, and P(b) is the evidence

$$P\left( {\mathbf{b}} \right) = \int\limits_{{\mathbf{x}}} {P\left( {\left. {\mathbf{b}} \right|{\mathbf{x}}} \right)P_{\text{pr}} \left( {\mathbf{x}} \right){\text{d}}{\mathbf{x}}}$$

(23)

which scales P(x|b) so that the Law of Total Probability is satisfied. In this study, the quantities-of-interest are the nanoparticle size and the TAC, x = [d _p, α]^T, while b contains the expected values of the effective temperatures at various cooling times (i.e., the average of the effective temperatures obtained from individual shots, after outliers have been removed.) In contrast to Ref. [20], in which the effective temperatures in b are assumed to be independent, considerable covariance was observed in b, which likely arises from signal processing algorithms. We account for this by defining the variance–covariance matrix Γ_e, again based on the effective temperatures obtained from individual shots, and then defining the likelihood as

$$P\left({\mathbf{b}}|{\mathbf{x}} \right)\propto\exp \left\{\|({\mathbf{b}}^{\rm mod } - {\mathbf{b}}^{\rm exp}) \varGamma_{\rm b}^{-1}({\mathbf{b}}^{\rm mod} -{\mathbf{b}}^{\rm exp})\|_{2}^{2}\right\}$$

(24)

Uncertainties in the quantities-of-interest caused by uncertainties in the other “nuisance” model parameters, in this case Φ = [ρ, c _p, T _g, E(m)_r, P _g, T _cr, ΔH _v,b]^T, are incorporated into the analysis by treating them as additional stochastic variables to be inferred, so Bayes’ equation becomes

$$P\left( {{\mathbf{x}},{\varvec{\Phi}}|{\mathbf{b}}} \right) = \frac{{P\left( {{\mathbf{b}}|{\mathbf{x}},{\varvec{\Phi}}} \right)P_{\text{pr}} \left( {\mathbf{x}} \right)P_{\text{pr}} \left( {\varvec{\Phi}} \right)}}{{P\left( {\mathbf{b}} \right)}}$$

(25)

While an uninformative prior is used for x (i.e., P _pr(x) = 1), the analysis must incorporate prior probabilities that reflect the state of knowledge of the other model parameters, similar to the procedure followed by Crosland et al. [59]. In this work, the parameters in Φ are assumed to be normally distributed about their nominal values with a standard distribution of 10 %, which reflects the epistemic uncertainty associated with these parameters; for example, the range of values for E(m)_r corresponding to the various ellipsometry measurements on molten iron summarized in Table 2 are within ±10 % of the Drude theory prediction.

Finally, the nuisance parameters are “marginalized out” of the posterior density by integration,

$$P\left( {{\mathbf{x}}|{\mathbf{b}}} \right) = \int\limits_{{\varvec{\Phi}}} {P\left( {{\mathbf{x}},{\varvec{\Phi}}|{\mathbf{b}}} \right)\;} {\text{d}}{\varvec{\Phi}}$$

(26)

Instead of carrying out the integrations in Eqs. (23) and (26) explicitly, however, the marginalized posterior distribution P(x|b) is estimated using a Markov chain Monte Carlo (MCMC) procedure [60]. Nuisance parameters, Φ, are sampled directly from their prior distributions so as to keep the samples centered about their expected values. Error bounds correspond to one standard deviation of 75,000 MCMC samples.

Figure 12 shows a histogram estimating the density of MCMC samples for the Fe–CO data when considering conduction only cooling. Regions with a higher number of samples correspond to values of Fe–CO that are more likely. These values do not necessarily correspond to values where the posterior is maximum as there are multiple other dimensions that have been marginalized in producing this plot. The spread of the samples gives an indication of the uncertainty in the inferred parameter.

Fig. 12

Estimated posterior density for the Fe–CO data showing the density as a function of α/d_p during conduction only modeling. The regions of higher density correspond to values of α/d_p that are more likely