Abstract

For the ill-posed inverse problem of LII-based nanoparticle size measurement, recovered primary particle size distribution (PPSD) is sensitive to the uncertainty of LII model parameters. In the absence of reliable prior knowledge, the thermal accommodation coefficient (TAC) and fractal-dependent shielding factor are often required to be inferred simultaneously with the PPSD. In the simplified LII model for low fluence regime, TAC and fractal-dependent shielding factor are combined to define a new fractal-dependent TAC. The present study theoretically verified the feasibility of inferring PPSD and fractal-dependent TAC from the normalized LII signals. Moreover, the inversion is independent of prior knowledge of most full LII model parameters, which is attributed to low laser fluence, normalized signal, and fractal-dependent TAC.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Primary particle size distribution (PPSD) soot aggregate is an important property for human health and global climate modeling [1,2]. Laser-induced incandescence (LII) has developed as a powerful tool for estimating volume fraction, mean primary particle size, and PPSD of soot aggregate [3,4]. Compared with sampling-based ex-situ methods such as transmission electron microscope (TEM), LII has the advantage of non-invasive measurement and is applicable for fast online in-situ measurement [3].

To date, there have been many studies on the determination of PPSD with time-resolved laser-induced incandescence (TiRe-LII) signals. Roth et al. [5] presented an in situ method for inferring particle size distribution by fitting the intensity of TiRe-LII signals. Filippov et al. [6] applied this method to determine the PPSD of various types of aerosols. Kock et al. [7] demonstrated the capability of this method to obtain almost instantaneous soot PPSD in diesel engine combustion with rapidly varying conditions. Lehre et al. [8] extended this method to simultaneously measure PPSD and gas temperature. Lehre et al. [9,10] used the effective particle temperature derived from the ratio of TiRe-LII signals measured at two wavelengths to obtain PPSD, and proved that it can be used to infer PPSD and flame temperature simultaneously. Liu et al. [11] proposed a simple method to determine the soot PPSD by using the effective temperature in the non-sublimation regime. Danker et al. [12] retrieved the soot PPSD from two different characteristic time intervals of TiRe-LII curve.

The above LII-based PPSD recovery studies are hampered by the ill-posed nature of inverse problem, such that the recovered PPSD is sensitive to the uncertainties in the LII model parameters [13]. Under low fluence conditions, the accurate prior knowledge of the thermal accommodation coefficient (TAC or αT) is necessary for a reliable estimate of PPSD. Michelsen [14] and Daun [1517] used extrapolation from low temperature data using and excitation probability and molecular dynamics, respectively, to predict TAC prior considering LII data. Added factors, such as the influence of soot composition and maturity on the TAC values, may further complicate inference. Uncertainties are often lowered by assuming that the PPSD is log normal, reducing the PPSD inference problem to one determining the geometric mean primary particle diameter, dp, g, and the geometric standard deviation, σd, g. Kock et al. [18] demonstrated such a scheme, where ex-situ measurements were used to determine σd, g and the TAC and geometric mean diameter were inferred simultaneously from TiRe-LII signals. Sipkens et al. [13,19] later showed that inference of the TAC and particle size is restricted to the transition or high fluence regimes, as the structure of the problem (wherein the particle size and TAC appear as a product) make it nearly impossible to determine TAC and dp, g from a conduction-dominated temperature decay. On the other hand, the above studies are generally limited to cases where the thermal shielding effect caused by the fractal structure of aggregate is neglected. Bauer et al. [20] revealed that the fractal parameters highly correlated with the PPSD parameters amplifying the uncertainties in the recovered PPSD, As such, to determine the influence of fractal-dependent shielding factor, it is necessary to obtain prior knowledge of fractal structure and polydisperse distribution of aggregate size to infer the remaining particle properties. Unfortunately, fractal structure and aggregate size vary with the soot generation conditions or environmental conditions, and it is difficult to obtain their accurate values without additional sampling-based measurements.

In the absence of reliable prior knowledge, TAC and fractal-dependent shielding factor are required to be inferred with PPSD simultaneously, but it will not generally be possible to infer these quantities independently. Instead, we define a new fractal-dependent TAC, which combines the overall impact of TAC and fractal-dependent shielding factor on the simplified LII model for low fluence regime. The purpose of this paper is to theoretically evaluate the capability of inferring the PPSD and fractal-dependent TAC of soot aggregates from normalized TiRe-LII signals. In order to avoid the common inverse crime (see Section 3) in theoretical evaluation, this study uses a full LII model to synthesize LII data for evaluation, while uses a simplified model of the low-fluence regime to perform the inversion process. Since the dependence of LII model on aggregate structure, aggregate size distribution, and thermal accommodation coefficient is completely represented by the new parameter αT, f (see Section 3), the inference of PPSD and αT, f does not require the prior information of these properties. In addition, the inference of PPSD and αT, f does not require the prior information of absorption function at the measurement wavelength, the apparent fraction volume, and the calibration parameter when normalized LII signal is used for inversion analysis.

This paper is structured as follows. Section 2 describes the fractal-based full LII model used for LII data simulation, the simplified LII model in the low-fluence regime for inverse problem analysis, and the definition of fractal-dependent TAC. Section 3 explains the inversion process. Section 4 evaluates the feasibility of inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals, and analyzes log contour of the objective function. Section 5 summarizes the main conclusions of this paper.

2. Time-resolved laser-induced incandescence model

2.1 Full LII model

Laser-induced incandescence (LII) involves the effect of laser heating on the soot structure, please see Ref. [2126]. These works have provided extensive high-resolution TEM images of the soot aggregates and primary particles under a wide range of combustion conditions with the laser heating. These works provide important experimental support for the current modeling of LII signals. The full LII model (i.e. forward model) typically involves solving energy and mass balance equations, which include the effects of various submodels (cf. Fig. 1) [1]:

$${\dot{Q}_{\textrm{int}}} = {\dot{Q}_{\textrm{abs}}} + {\dot{Q}_{\textrm{cond}}} + {\dot{Q}_{\textrm{rad}}} + {\dot{Q}_{\textrm{sub}}} + {\dot{Q}_{\textrm{therm}}} + {\dot{Q}_{\textrm{ox}}} + {\dot{Q}_{\textrm{ann}}}$$
$$\dot{M} = {\dot{M}_{\textrm{sub}}} + {\dot{M}_{\textrm{ox}}}$$
where ${\dot{Q}_{\textrm{int}}}$, ${\dot{Q}_{\textrm{abs}}}$, ${\dot{Q}_{\textrm{cond}}}$, ${\dot{Q}_{\textrm{rad}}}$, ${\dot{Q}_{\textrm{sub}}}$, ${\dot{Q}_{\textrm{ox}}}$, and ${\dot{Q}_{\textrm{ann}}}$ present the change rate of particle energy caused by internal energy, absorption of laser energy, conduction, radiation, sublimation, thermionic emission, oxidation and annealing, respectively; $\dot{M}$ is the change rate of particle mass, ${\dot{M}_{\textrm{sub}}}$ and ${\dot{M}_{\textrm{ox}}}$ denote the mass change rate caused by sublimation and oxidation, respectively.

 

Fig. 1. Schematic of underlying heat and mass transfer processes involved in LII

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The internal energy change rate of any aggregate is assumed to result from a sum over the primary particles making up the aggregates, such that [1]:

$${\dot{Q}_{\textrm{int}}} = {N_\textrm{p}}{c_\textrm{s}}{\rho _\textrm{s}}\frac{\pi }{6}d_\textrm{p}^3\frac{{\textrm{d}T}}{{\textrm{d}t}}$$
where Np is the number of primary particle in a single soot aggregate; cs and ρs are respectively the specific heat and density of soot aggregate (taken as temperature-dependent in this study, cf. Reference [27]); dp and T are the primary particle diameter and mean temperature of a single soot aggregate, respectively; and t is time.

The laser energy absorption of a single soot aggregate is determined by its absorption cross-section Cabs [27]:

$${\dot{Q}_{\textrm{abs}}} = {N_\textrm{p}}{C_{\textrm{abs}}}E(t )= {N_\textrm{p}}{C_{\textrm{abs}}}Fq(t )$$
where E(t) is the temporal irradiance profile of the laser; F is the laser fluence; q(t) is the laser temporal variation function and is a Gaussian distribution with a standard deviation σLaser = 3.3 ns [20], and the spatial distribution of the laser beam is set as uniform top hat distribution; Cabs is the absorption cross-section of a single soot aggregate can be approximated by Rayleigh-Debye-Gans fractal aggregate (RDG-PFA) scattering theory [28]:
$${C_{\textrm{abs}}} = \frac{{{\pi ^2}d_\textrm{p}^3E(m )}}{{{\lambda _{\textrm{inc}}}}}$$
where λinc denotes the wavelength of the incident pulse laser; E (m) is the soot absorption function which is determined by spectral complex refractive index m. Here, λinc is 532 nm, and the corresponding E (m) is 0.3 [29].

The cooling terms in this model generally are prescribed in Ref. [27]. Conduction plays an important role in this model (and the simplified low-fluence model), such that it receives more attention here. A relatively accurate Fuchs boundary sphere model is applied to calculate heat conduction [30]:

$${\dot{Q}_{\textrm{cond}}} ={-} \pi {N_\textrm{p}}d_\textrm{p}^2{\alpha _\textrm{T}}\frac{{{P_\textrm{a}}}}{8}\sqrt {\frac{{8{R_\textrm{m}}{T_\delta }}}{{\pi {W_\textrm{a}}}}} \left( {\frac{{{\gamma^ \ast } + 1}}{{{\gamma^ \ast } - 1}}} \right)\left( {\frac{T}{{{T_\delta }}} - 1} \right)$$
where αT = 0.37 is the thermal accommodation coefficient [23]; Pa = 1 atm is the pressure of ambient air; Rm = 83.145 g·m3/(mol·K·s2) is the universal gas constant in effective mass units; subscript δ is the thickness of the boundary layer in the Fuchs approach; Tδ is the temperature in the limiting sphere; Wa = 28.74 g/mol is the molecular weight of air; γ* is the average value of the specific heat ratio. For a detailed description of this method, see Ref. [30].

The time-dependent particle temperature T(t) and the time-dependent primary particle diameter dp(t) can be obtained simultaneously by solving the system of differential equations. In this study, the coupled system of equations is solved by the fourth-order Runge-Kutta approach.

2.2 Fractal aggregate structure

As shown in Fig. 1(b), primary particles of soot aggregates are not isolated nanospheres but agglomerated in the form of fractal structure. For a soot aggregate containing Np primary particles, its fractal structure can be mathematically described by [31]:

$${N_\textrm{p}} = {k_\textrm{f}}{\left( {\frac{{2{R_\textrm{g}}}}{{{d_\textrm{p}}}}} \right)^{{D_\textrm{f}}}}$$
where Np is the number of primary particles in a single aggregate; kf is the fractal prefactor; Df is the fractal dimension; Rg is the gyration radius, dp is the diameter of the primary particles.

Filippov et al. [32] showed that the shielding effect can cause heat conduction rates from primary particles within a single aggregate to be an order of magnitude lower than that for isolated nanospheres, depending on the particle size, fractal structure, thermal accommodation coefficient, and the current flow regime [20]. To account for the shielding effect of soot aggregates, previous works [30,32] replaced the diameter in the conduction expression above with an effective diameter, Deff, of a single soot aggregate as follows [30]:

$${\dot{Q}_{\textrm{cond}}} ={-} \pi D_{\textrm{eff}}^2{\alpha _\textrm{T}}\frac{{{P_\textrm{a}}}}{8}\sqrt {\frac{{8{R_\textrm{m}}{T_\delta }}}{{\pi {W_\textrm{a}}}}} \left( {\frac{{{\gamma^ \ast } + 1}}{{{\gamma^ \ast } - 1}}} \right)\left( {\frac{T}{{{T_\delta }}} - 1} \right)$$

In their studies, Deff was assumed to be the projected-area equivalent diameter for fractal aggregates. Based on Eq. (7), Filippov et al. [32] and Liu et al. [30] both proposed a heuristic fractal-like relationship to parametrically express Deff. Their studies have not been further tested, and the physical connection between the effective diameter, and the fractal structure of the soot aggregate is complicated [20]. Given that, this study uses the shielding factor η to establish the implicit relationship between the shielding effect and specific geometric structure (including fractal structure and particle size) of the aggregate. The mathematical relationship between Deff and η is defined as follows [32]:

$$D_{\textrm{eff}}^2({d_\textrm{p}},\textrm{ }{N_\textrm{p}},\textrm{ }{k_\textrm{f}},\textrm{ }{D_\textrm{f}}) = d_\textrm{p}^2{N_\textrm{p}}\eta ({{N_\textrm{p}},\textrm{ }{k_\textrm{f}},\textrm{ }{D_\textrm{f}}} )$$

The value of η used to simulate LII signals is derived from the Eq. (9) and the heuristic fractal-like relationship in Ref. [30] and [32]. The two characteristic parameters kh and Dh in this relationship are determined by the quadratic function of αT in Ref. [30].

2.3 Simplified LII model and definition of fractal-dependent TAC

While simulation data will be evaluated using the full LII model above, analysis (i.e., the inverse problem) will use a simplified model for the low-fluence regime (this avoids inverse crime [33,34]). In this case, the energy and mass changes caused by the thermal radiation, sublimation, oxidation, and annealing are negligible, such that the system is reduced to

$${\dot{Q}_{\textrm{int}}} = {\dot{Q}_{\textrm{abs}}} + {\dot{Q}_{\textrm{cond}}}$$
and the mass changes are neglected.

For the simplified LII model, substituting Eqs. (3), (4), (8) and (9) into Eq. (11) yields dT/dt:

$$\frac{{\textrm{d}T}}{{\textrm{d}t}} = \frac{6}{{\pi {c_\textrm{s}}{\rho _\textrm{s}}d_\textrm{p}^3}}\left[ {\underbrace{{{C_{\textrm{abs}}}Fq(t )}}_{{\textrm{Absorption}}} - \underbrace{{\pi d_\textrm{p}^2\eta {\alpha_\textrm{T}}\frac{{{P_\textrm{a}}}}{8}\sqrt {\frac{{8{R_\textrm{m}}{T_\delta }}}{{\pi {W_\textrm{a}}}}} \left( {\frac{{{\gamma^ \ast } + 1}}{{{\gamma^ \ast } - 1}}} \right)\left( {\frac{T}{{{T_\delta }}} - 1} \right)}}_{{\textrm{heat conduction}}}} \right]$$

In Eq. (11), the TAC and fractal-dependent shielding factor appear as an isolated product. According to the findings of Ref. [19], it is generally impossible to simultaneously infer two parameters that appear as a product in an equation. Instead, these two quantities are combined to define a new fractal-dependent TAC as the product,

$${\alpha _{\textrm{T, f}}} \equiv \eta \cdot {\alpha _\textrm{T}}$$

The influence of aggregate structure on the simplified LII model can be fully quantified by η. After lumping η with αT, the influence of aggregate structure on the simplified LII model is completely represented by αT, f. Since η and αT is related to many factors [3], the prior value of αT, f tends to have large uncertainty. If the inversion of PPSD is carried out when αT, f is a prior model parameter, it is impossible obtain accurate estimate of PPSD because of the inherent deviation of prior value of αT, f. Therefore, αT, f and PPSD are simultaneously inferred in this study.

As more and more evidence shows that primary particle size is relatively constant within an aggregate and varies more significantly between aggregates [35,36], this study assumes that dp is constant in aggregates but obeys a narrow log-normal distribution between aggregates [37]:

$$p({{d_\textrm{p}}} )= \frac{1}{{{d_\textrm{p}}\sqrt {2\pi } \ln {\sigma _{\textrm{d, g }}}}}\textrm{exp} \left[ { - {{\left( {\frac{{\ln {d_\textrm{p}} - \ln {d_{\textrm{p, g}}}}}{{\sqrt 2 \ln {\sigma_{\textrm{d, g }}}}}} \right)}^2}} \right]$$
where dp, g and σd, g are the geometric mean primary particle diameter and the geometric standard deviation for the distribution of dp, respectively. The integration limits in Eq. (13) are 5 nm and 100 nm with enough refinement, so the integration error is negligible. It should be noted that this study is based on the assumption that primary particle size is basically the same within aggregate. The influence of the relatively large primary particle size distribution within aggregate on this study is beyond the scope of this paper and needs further study.

The aggregate size generally varies between aggregates and follow a log-normal distribution of Np:

$$p({{N_\textrm{p}}} )= \frac{1}{{{N_\textrm{p}}\sqrt {2\pi } \ln {\sigma _{\textrm{N, g }}}}}\textrm{exp} \left[ { - {{\left( {\frac{{\ln {N_\textrm{p}} - \ln {N_{\textrm{p, g}}}}}{{\sqrt 2 \ln {\sigma_{\textrm{N, g }}}}}} \right)}^2}} \right]$$
where Np, g and σN, g are the geometric mean primary particle number per aggregate and the geometric standard deviation for the distribution of Np, respectively. The integration limits in Eq. (14) are 1 and 600 with enough refinement, so the integration error is negligible.

The time-resolved laser-induced incandescence (TiRe-LII) signal of a single primary particle in any aggregate can be calculated as follows [29]:

$${S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )= \frac{{8{\pi ^3}h{c^2}E(m )d_\textrm{p}^3(t )}}{{\lambda _{\textrm{mea}}^6 \cdot \textrm{exp} [{hc/{\lambda_{\textrm{mea}}}{k_\textrm{B}}T({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )} ]- 1}}$$
where h = 6.626×10−34 J·s is the Plank’s constant; c = 2.998×108 m/s is the speed of light; kB = 1.381×10−23 J/K is the Boltzmann constant; λmea is the measurement wavelength; the mean particle temperature T is related to the fractal-dependent TAC.

For an aggregate with Np primary particles, the TiRe-LII signal can be calculated as follows [29]:

$${S_{\textrm{agg}}}({t,\textrm{ }{N_\textrm{p}},\textrm{ }{\alpha_{\textrm{T, f}}}} )= \int {{N_\textrm{p}}{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})\textrm{d}{d_\textrm{p}}}$$

The total TiRe-LII signal of N soot aggregates in the measurement volume can be calculated as follows [20]:

$${S_{\textrm{LII}}}({t,\textrm{ }{N_\textrm{p}},\textrm{ }{\alpha_{\textrm{T, f}}}} )= {C_{\textrm{exp} }}N\int\!\! \int{ {{N_\textrm{p}}{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} p({{N_\textrm{p}}} )\textrm{d}{d_\textrm{p}}\textrm{d}{N_\textrm{p}}}$$
where Cexp is a calibration parameter determined by experimental conditions, which is related to the detection geometry and collection efficiency of the detector.

In this study, the relative intensity of (normalized) TiRe-LII signal RSLII(t) is used as the input for the inverse problem, defined by the ratio of the absolute intensity and maximum value of the TiRe-LII signal [20]:

$$R{S_{\textrm{LII}}}({t,\textrm{ }{N_\textrm{p}},\textrm{ }{\alpha_{\textrm{T, f}}}} )= \frac{{\int {\int {{N_\textrm{p}}{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} p({{N_\textrm{p}}} )\textrm{d}{d_\textrm{p}}\textrm{d}{N_\textrm{p}}} }}{{\int {\int {{N_\textrm{p}}{S_\textrm{p}}({{t_{\max }},\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} p({{N_\textrm{p}}} )\textrm{d}{d_\textrm{p}}\textrm{d}{N_\textrm{p}}} }}$$
where tmax is the time when the SLII(t) signal reaches its maximum $S_{\textrm{LII}}^{\max }$.

It should be noted that even if Df, kf, and αT are constant for all aggregates, the fractal-dependent TAC, αT, f, may be constant within aggregates but varies with the polydispersity of Np between aggregates. So, we emphasized the dependence on αT, f in Eqs. (16)–(18) for different aggregates. According to Eqs. (10)–(12), if αT, f is inferred simultaneously with PPSD, temporal mean particle temperature T can be obtained in the simplified LII model without prior knowledge of fractal structure. However, according to Eqs. (15)–(18), prior information of fractal parameter Np is still required to simulate normalized TiRe-LII signals. In order to make the inversion process based on the simplified LII model independent of the prior information of the fractal structure, Eqs. (16)–(18) are restated below.

First, Sagg(t) in Eq. (16) is assumed to be the product of an equivalent primary particle number per aggregate, Np, eff and the integral of Sp:

$${S_{\textrm{agg}}}(t )\equiv {N_{\textrm{p, eff}}}\int {{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})\textrm{d}{d_\textrm{p}}}$$
where Np, eff is constant for different aggregates.

Considering the definition of αT, f in Eq. (12), αT, f varies for different aggregates. But in order to obtain a constant parameter as the inverse problem variable, the definition of αT, f must be extended for all aggregates within the measurement volume. According to the final definition of αT, f, it considers the overall impact of TAC and fractal-dependent shielding factor of all aggregates on the simplified LII model. The target value of αT, f can be obtained by minimizing the difference between the LII signals generated by the full LII model and the simplified LII model when the other two variables (dp, g and σd, g) are fixed at their target values. In this study, the target value of αT, f is 0.26.

Based on Eq. (19), Eq. (17) can be restated as:

$${S_{\textrm{LII}}}(t )= {C_{\textrm{exp} }}N{N_{\textrm{p, eff}}}\int {{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} \textrm{d}{d_\textrm{p}}$$

Then, Eq. (18) is restated as follows

$$R{S_{\textrm{LII}}}(t )= \frac{{{C_{\textrm{exp} }}N{N_{\textrm{p, eff}}}\int {{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} \textrm{d}{d_\textrm{p}}}}{{{C_{\textrm{exp} }}N{N_{\textrm{p, eff}}}\int {{S_\textrm{p}}({{t_{\max }},\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} \textrm{d}{d_\textrm{p}}}} = \frac{{\int {{S_\textrm{p}}({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} \textrm{d}{d_\textrm{p}}}}{{\int {{S_\textrm{p}}({{t_{\max }},\textrm{ }{\alpha_{\textrm{T, f}}}} )p({d_\textrm{p}})} \textrm{d}{d_\textrm{p}}}}$$

Finally, substituting Eq. (15) into Eq. (21) yields

$$R{S_{\textrm{LII}}}(t )= \frac{{\int {\frac{{d_\textrm{p}^3(t )p({d_\textrm{p}})\textrm{d}{d_\textrm{p}}}}{{\textrm{exp} [{hc/{\lambda_{\textrm{mea}}}{k_\textrm{B}}T({t,\textrm{ }{\alpha_{\textrm{T, f}}}} )} ]- 1}}} }}{{\int {\frac{{d_\textrm{p}^3({{t_{\max }}} )p({d_\textrm{p}})\textrm{d}{d_\textrm{p}}}}{{\textrm{exp} [{hc/{\lambda_{\textrm{mea}}}{k_\textrm{B}}T({{t_{\max }},\textrm{ }{\alpha_{\textrm{T, f}}}} )} ]- 1}}} }}$$

It can be seen from Eq. (22) that the inversion of αT, f and PPSD does not require the prior information of most full LII model parameters, especially aggregate structure, aggregate size distribution, and TAC, which greatly reduces the uncertainty from the model parameters.

3. Inversion process

In the present study, we consider simultaneous determination of PPSD parameters and fractal-dependent TAC, i.e., variables x = [dp, g, σd, g, αT, f], from the relative intensity of TiRe-LII signal using a weighted least-squares approach, with an objective function

$${f_{\textrm{obj}}} = \left\|{\frac{{{{\textbf {b}}_{\textrm{mea}}} - {{\textbf {b}}_{\textrm{est}}}}}{{{{\textbf {b}}_{\textrm{mea}}}}}} \right\|_2^2$$
where bmea and best are the measured and the estimated values of the measurement data, taken as the normalized TiRe-LII signal at multiple time points, i.e. b = [RSLII (t1), RSLII (t2), RSLII (t3), …]. In this study, the covariance matrix adaptive evolution strategy (CMA-ES) algorithm is used to minimize objective function value, with the details provided in Refs. [38,39].

The overall inverse method presented in this work is evaluated by simulating data instead of real measurements, using some target value of objective parameters xtar. The present work avoids inverse crime [33,34] (wherein identical forward and inverse models are used, resulting in a trivial procedure where the target parameters are returned exactly) in two ways.

First, noise is added to the signals. Two different noise models were considered. The first model follows from our previous studies [39,40], and only considering a signal-independent Gaussian error [41]:

$${{\textbf {b}}_{\textrm{mea}}} = {{\textbf {b}}_{\textrm{tar}}}(1 + \gamma ){{\textbf {n}}^\textrm{G}}$$
where btar is the target signal vector; γ is the scale factor in Gaussian error; nG is the standard normal random vector for Gaussian error. The second noise model is the general noise model of Sipkens et al. [41], which considers the Poisson-Gaussian error in single-shot signals and the shot-to-shot error,
$${{\textbf {b}}_{\textrm{mea}}} = {{\textbf {b}}_{\textrm{tar}}} + \underbrace{{\tau n{{\textbf {b}}_{\textrm{tar}}}}}_{{\textrm{Shot - to - shot error}}} + \underbrace{{{{[\theta (1 + \tau n){{\textbf {b}}_{\textrm{tar}}}]}^{1/2}} \circ {{\textbf {n}}^\textrm{P}}}}_{{\textrm{Poisson error}}} + \underbrace{{\gamma {{\textbf {n}}^\textrm{G}}}}_{{\textrm{Gaussian error}}}$$
where n is a standard normal random variable; τ, θ, and γ are the characteristic parameters of the general model, which are the scale factors in the shot-to-shot error, Poisson error and Gaussian error, respectively; nP and nG are respectively the standard normal random vector for Poisson error and Gaussian error.

Second, we employ a double-model approach (cf. Fig. 2), in which simulated data is generated using the full LII model, Eqs. (1) and (2), and then interpreted with simplified LII model of Section 2.3. Characteristics of the inverse procedure can be evaluated based on the degree to which it can return the target value, with appropriate consideration of uncertainties.

 

Fig. 2. Flowchart of double-model inversion process

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4. Results and discussion

4.1 Laser fluence selection

Since this study is based on a low fluence case where particle temperature and mass changes caused by thermal radiation, sublimation, oxidation, and thermionic are expected to be negligible, it is necessary to determine the appropriate laser fluence range. A simple method is provided in Ref. [42], in which the transition fluence (the point on the fluence curve where sublimation effects become significant) is given by

$${F_{\textrm{ref}}} = \frac{{{\lambda _{\textrm{inc}}}{\rho _\textrm{s}}{c_\textrm{s}}({{T_{\textrm{ref}}} - {T_\textrm{g}}} )}}{{6\pi E(m )}}$$
where Tref is the transition temperature, which depends on the sublimation model parameters; Tg = 300 K is the ambient gas temperature. This yields an estimated reference fluence of 0.126 J/cm2. Due to the more complex model, polydispersity, particle size variation, and Np effects considered in this work, we seek to validate this value and expand this discussion using simulated temperature traces.

In Fig. 3(a), as the laser fluence increases from 0.10 J/cm2 to 0.15 J/cm2, the peak portion of the primary particle temperature curve becomes more prominent, leading to higher peak temperatures and longer high-temperature duration, so in Fig. 3(b), the particle mass loss increases with the increase of laser fluence. When the laser fluence exceeds the 0.15 J/cm2, the diameter of the primary particles cannot be regarded as a constant because of the continuous mass loss. Figure 3 is obtained when Np = 1 and dp = 10 nm. For more Np and larger dp, the thermal shielding effect will be more significant and the specific surface area will be smaller, resulting in a higher peak temperature and a flatter cooling curve, and then more particle mass loss. Therefore, from the perspective of particle mass loss, a suitable fluence of laser should not exceed 0.15 J/cm2.

 

Fig. 3. Effects of different laser fluences on temperature and diameter decay rate of primary particles when Np = 1 and dp = 10 nm

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On the other hand, the simplified LII model only considers the energy change caused by laser absorption and heat conduction, so the laser fluence should be also selected according to the errors caused by the energy equation approximation. As shown in Fig. 4, when laser fluence exceeds 0.10 J/cm2, the ratio of the energy sum of laser absorption and heat conduction to the total energy is less than 90% at high temperature. In terms of the energy equation error in the simplified LII model, a suitable fluence of laser should be lower than 0.10 J/cm2. But lower laser energy density is not conducive to reducing the overall error, because lower laser energy density will make the signal-to-noise ratio of TiRe-LII signal worse [18]. Therefore, a laser fluence of 0.10 J/cm2 is selected in the present study.

 

Fig. 4. (a) when Np = 1 and dp = 10 nm and (b) when Np = 600 and dp = 150 nm, the effect of different laser fluence on the ratio of the energy sum of laser absorption and heat conduction to the total energy

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4.2 Simultaneously inferring PPSD parameters and fractal-dependent TAC

In this study, the simultaneous inference of PPSD parameters and fractal-dependent TAC are performed in different cases. Table 1 summarizes the basic settings of these cases. To investigate the influence of the measurement wavelength number and measurement noise level on the retrieval precision, two different measurement wavelength schemes were used under 4 different measurement noise levels (8 test cases in total) to evaluate the capability of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII. Figure 5 shows a set of noisy LII signal synthesized in this study.

 

Fig. 5. Noisy LII signals synthesized by (a) the general noise model and (b) Gaussian model

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Tables Icon

Table 1. Basic settings of the test cases

Table 2 shows the retrieval results of three objective parameters in all 8 test cases. For the 4 cases of the general noise model, the noisy signals used for one inversion are synthesized from an average of 100 shots, while for the 4 cases of the Gaussian noise model, the noisy signals used for one inversion are from a single shot. Given the stochastic nature of the noisy signal synthesis, 100 different sets of noisy signals are evaluated in each case. Each case is evaluated in the following aspects: (1) SR: The success rate of the all 100 independent inversions; (2) Avg: The average value of all successful inversions of one objective parameter; (3) Std: The standard deviation of all successful inversions of one objective parameter; (4) ɛrel: Relative error of Avg relative to the target value. Here, successful inversion means that the inversion process can eventually converge to a specific value.

Tables Icon

Table 2. Retrieval results of three objective parameters in 8 test cases

Regarding relative error of retrieval results, whether for Gaussian noise or general noise the ɛrel of αT, f, dp, g, and σd, g inferred at low noise level do not exceed 5%. For the cases of Gaussian noise, it is found that the ɛrel of αT, f and σd, g inferred at high Gaussian noise level exceeds 10%, while the relative error of dp, g is thoroughly below 5%. For the cases of general noise, the ɛrel of all three objective parameters basically increase as noise level becomes stronger. When using noisy signals measured at two wavelengths, the ɛrel of all objective parameters do not exceed 5% except for the ɛrel of dp, g and σd, g at high general noise. When using noisy signals at three wavelengths, the ɛrel of all objective parameters do not exceed 5% except for the ɛrel of σd, g at high general noise. The ɛrel of αT, f is thoroughly below 5% in all the cases of the general noise. In addition, the success rate of 100 inversions drops significantly at high noise level, and as the wavelength number of noisy signals increases, the success rate of 100 inversions increases. Table 2 indicates that the simultaneous inference of αT, f, dp, g, and σd, g can be accurately achieved at a relatively low noise level, but when the noise reaches a certain level, the resulting αT, f, dp, g, and σd, g may deviate seriously from their target values.

Regarding the standard deviation of retrieval results, the Std of inferred dp, g is close to its value magnitude, while for the inferred αT, f and σd, g, the Std is at least one order of magnitude smaller than its value. It indicates that the resulting dp, g of 100 evaluations are distributed in a wider range, i.e. the uncertainty of dp, g is greater than that of αT, f and σd, g. The magnitude of the uncertainties of dp, g and σd, g are consistent with the results of Bayesian analysis in Ref. [20]. The uncertainty of the inversion results of all objective parameters increases as the noise becomes stronger. Therefore, in order to obtain more accurate inversion results, it is necessary to average the results over multiple independent inversions, especially for dp, g.

According to our previous studies [39,40], the retrieval precision of objective parameters is typically improved when increasing the measurement wavelength number, since it brings more available information for inversion. In this study, for the cases under general noise, the retrieval results using three-wavelength noise signals are significantly better than those obtained from two-wavelength noise signals. It indicates that under general noise, increasing measurement wavelength number can effectively improve the inversion accuracy and reduce the uncertainty. But for all cases under Gaussian noise, the increase of measurement wavelength number has little effect on the relative error and standard deviation of three objective parameters.

In addition, it is not comprehensive to evaluate the inversion of the PPSD only from the retrieval precision of characteristic parameters dp, g and σd, g. For a more direct evaluation, Fig. 6 shows the PPSD profiles drawn from the target value of dp, g and σd, g together with PPSD profiles drawn from retrieval results of dp, g and σd, g in Table 2. It can be seen that the inferred PPSD is sensitive to the noise level. The greater the noise level, the wider the inferred PPSD profiles, and the farther the inferred PPSD profiles deviate from the target profile. For the PPSD profiles inferred under Gaussian noise in Fig. 6(a), there is only a slight difference between PPSD profiles inferred from two-wavelength (2λ) noisy signal and three-wavelength (3λ) noisy signal. However, for the PPSD profiles inferred under general noise in Fig. 6(b), the PPSD profiles inferred from three-wavelength (3λ) noise signals are obviously closer to target PPSD profiles than PPSD profiles inferred from two-wavelength (2λ) noise signals at high noise level. It is consistent with the trend of the relative error of dp, g and σd, g in Table 2, indicating that using noisy signals measured at more wavelengths can improve retrieval results, especially at high general noise level.

 

Fig. 6. Primary particle size distribution profiles derived from retrieval results of (a) Gaussian noise and (b) general noise cases in Table 2 together with the target value

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4.3 Log contour analysis of the objective function

Solving inverse problem of inferring three objective parameters is to find the point with the lowest objective function value from the three-dimensional (three-variable) domains. Therefore, the involved inverse problem can be analyzed from the perspective of the log contours of the objective function. Since it is computation-consuming to analyze all log-contours of the objective function from the three-dimensional (three variables) search domain with sufficient resolution, only three typical ones are selected from a large number of log contours of the objective function for analysis. These three log contours of the objective function are obtained by respectively fixing three variables to their target values, such as, the log contours of the objective function on the αT, f-dp, g plane is obtained when σd, g is fixed at 1.2. Figure 7 shows the log contours of the objective function on (a) σd, g-dp, g plane, (b) αT, f-dp, g plane, and αT, f-σd, g plane. The objective functions used in Fig. 7(a), (b) and (c) are based on a set of the normalized LII noise signal that can accurately infer αT, f, dp, g, and σd, g.

 

Fig. 7. Log contours of the objective function on (a) σd, g - dp, g plane, (b) αT, f - dp, g plane, and αT, f - σd, g plane

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Considering the analysis in Ref. [13], the objective function topography of Fig. 7(a) reveal a robust estimate of σd, g and dp, g. According to the objective function topography of Fig. 7(c), the estimate of αT, f and σd, g is likely to be robust. In Fig. 7(b), there is no minimum point, but a minimum valley. The objective function topography of Fig. 7(b) is consistent with that in Ref. [19] and [13], arising from the conduction-dominated temperature decay. Previous studies [13,19] concluded that it was impossible to infer αT and dp, g from conduction-dominated LII. But in this study, reasonable αT, f and dp, g can be inferred from the conduction-dominated LII. In order to further verify the possibility of inferring αT, f and dp, g from the LII data, Fig. 8 shows the logarithmic objective function of valley lines of Fig. 7 along the coordinate axes.

 

Fig. 8. Logarithmic objective function of valley lines of Fig. 6 along the coordinate axes

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Figure 8(a) and Fig. 8(d) are respectively the valley line of Fig. 7(a) along the dp, g-axis and σd, g-axis. In Fig. 8(a) and Fig. 8(d), the bottom of the valley line is U-shaped, with steep sides, from which the lowest point can be easily distinguished. The lowest points of dp, g and σd, g are respectively at 20 nm and 1.2, which are consistent with their target values. Due to the large slopes on both sides of the lowest point, the inversion process can easily converge to the lowest point, which indicates an accurate estimate of dp, g and σd, g when αT, f, is perfectly known. Figure 8(b) and Fig. 8(e) are respectively the valley line of Fig. 7(b) along the dp, g-axis and αT, f-axis. Although the bottom of the valley line is extremely flat, the insets of Fig. 8(b) and (e) prove the existence of the lowest points of dp, g and αT, f, at about 16.8 nm and 0.22, respectively. Due to the extremely small slopes on both sides of the lowest point, the inversion process is difficult to converge to the lowest point and may require a large number of iterations, which indicates that it is difficult but theoretically feasible to estimate dp, g and αT, f when σd, g is perfectly known. Figure 8(c) and Fig. 8(f) are respectively the valley line of Fig. 7(c) along the σd, g-axis and αT, f-axis. In Fig. 8(c) and (f), the lowest points of σd, g and αT, f are respectively at 1.2 and 0.26, which are consistent with their target values. But in Fig. 8(f), the valley line has two U-shaped bottoms with steep sides. The one on the right is the second lowest point of αT, f, which is about 0.36. Given that, it can be expected to accurately estimate σd, g and αT, f when dp, g is completely known, but it may happen that the retrieval process is trapped in the second lowest point where αT, f is about 0.36.

Figure 7 and Fig. 8 actually correspond to three relatively simple two-variable inversions. Table 3 shows the retrieval results of these three two-variable inversions using the same noisy signal as Fig. 7 and Fig. 8. Each inversion is repeated 20 times to calculate the average results and the standard deviation. The 20 inversions finally converged to the same result, so the standard deviation of all variables is 0. The average estimated value of all variables is consistent with the minimum value in Fig. 8. the minimum value of αT, f and dp, g can be obtained stably (SR = 100%) in the current log contour topography of objective function. Regarding the average iteration counter, inferring αT, f and dp, g takes almost twice as much time as other two two-variable inversions. It can be seen that inferring αT, f and dp, g is much more difficult. In addition, the estimate of αT, f and σd, g has a 70% probability of failure. According to the results, the failed inversions are all be trapped in the area where αT, f is about 0.36. All these results verify our analysis of Fig. 8.

Tables Icon

Table 3. Retrieval results of three two-variable inversions

5. Conclusion

This proof-of-concept study numerically evaluates the feasibility of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals. All the numerical evaluations are performed in the low-fluence case where the overall impact of TAC and fractal-dependent shielding factor on the simplified LII model can be defined as the fractal-dependent TAC. The full LII model is employed to synthesize normalized TiRe-LII signals for numerical evaluations, while the simplified model is used for the inverse problem analysis, which avoids the inverse crime. Based on normalized TiRe-LII signals and fractal-dependent TAC, the simplified LII model is independent of prior knowledge of most full LII model parameters, including fractal structure, the polydispersity of aggregate size, and absorption function at the measurement wavelength, the apparent fraction volume and calibration parameters.

The possibility of inferring PPSD parameters (dp, g and σd, g) and fractal-dependent TAC (αT, f) is numerically evaluated in 8 different cases under Gaussian noise or general noise. Regarding the relative error of retrieval results, the inference of dp, g, σd, g, and αT, f is accurate at low noise level but significantly biased at high noise level. For Gaussian noise, when more measurement wavelengths are used, the retrieval results are only slightly improved. For general noise, when more measurement wavelengths are used, especially for high noise level, the retrieval results are significantly improved. Regarding the uncertainty of retrieval results, to obtain a reliable estimate of dp, g, it is necessary to average multiple independent estimates. Regarding the log contour analysis of the objective function, although the inversion process is expected to be difficult due to the flat bottom line, the unique solution existing on the valley line theoretically proves the feasibility of inferring dp, g and αT, f.

In summary, all numerical results reveal the potential capability of simultaneously inferring PPSD and fractal-dependent TAC from normalized TiRe-LII signals. In this proof-of-concept, the inversion method is evaluated by simulation data instead of real measurements. Considering the essential difference between the simulation data and real measurements, the feasibility and predictive power of the method needs to be validated in controlled experiments with real measurement data. Further research will be conducted to prove the feasibility of this method experimentally. Furthermore, considering the large uncertainty of the inversion results in the current study, a full Bayesian inference will be employed in our future work to obtain reliable uncertainty estimates.

Funding

National Natural Science Foundation of China (51976044); National Science and Technology Planning Project (2017-V-0016-0069); Foundation for Heilongjiang Touyan Innovation Team Program.

Acknowledgements

The authors would like to thank Dr. Timothy Sipkens, Prof. Steven Rogak, and Miss Yilin Zhao for their direct input, insightful discussions, and editorial work on this manuscript. This paper also supported by the China Scholarship Council.

Disclosures

The authors declare no conflicts of interest.

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4. T. R. Meyer, B. R. Halls, N. Jiang, M. N. Slipchenko, S. Roy, and J. R. Gord, “High-speed, three-dimensional tomographic laser-induced incandescence imaging of soot volume fraction in turbulent flames,” Opt. Express 24(26), 29547–29555 (2016). [CrossRef]  

5. P. Roth and A. V. Filippov, “In situ ultrafine particle sizing by a combination of pulsed laser heatup and particle thermal emission,” J. Aerosol Sci. 27(1), 95–104 (1996). [CrossRef]  

6. A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999). [CrossRef]  

7. B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002). [CrossRef]  

8. T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003). [CrossRef]  

9. T. Lehre, B. Jungfleisch, R. Suntz, and H. Bockhorn, “Size distributions of nanoscaled particles and gas temperatures from time-resolved laser-induced-incandescence measurements,” Appl. Opt. 42(12), 2021–2030 (2003). [CrossRef]  

10. T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005). [CrossRef]  

11. F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006). [CrossRef]  

12. S. Dankers and A. Leipertz, “Determination of primary particle size distributions from time-resolved laser-induced incandescence measurements,” Appl. Opt. 43(18), 3726–3731 (2004). [CrossRef]  

13. T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017). [CrossRef]  

14. H. A. Michelsen, “Derivation of a temperature-dependent accommodation coefficient for use in modeling laser-induced incandescence of soot,” Appl. Phys. B 94(1), 103–117 (2009). [CrossRef]  

15. K. J. Daun, “Thermal accommodation coefficients between polyatomic gas molecules and soot in laser-induced incandescence experiments,” Int. J. Heat Mass Transfer 52(21-22), 5081–5089 (2009). [CrossRef]  

16. K. Daun, F. Liu, and G. Smallwood, “Molecular dynamics simulations of translational thermal accommodation coefficients for time-resolved LII,” in Heat Transfer Summer Conference, 2008), 333–342.

17. T. A. Sipkens and K. J. Daun, “Effect of Surface Interatomic Potential on Thermal Accommodation Coefficients Derived from Molecular Dynamics,” J. Phys. Chem. C 122(35), 20431–20443 (2018). [CrossRef]  

18. B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005). [CrossRef]  

19. T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013). [CrossRef]  

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23. B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019). [CrossRef]  

24. R. L. Vander Wal and M. Y. Choi, “Pulsed laser heating of soot: morphological changes,” Carbon 37(2), 231–239 (1999). [CrossRef]  

25. R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004). [CrossRef]  

26. K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011). [CrossRef]  

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41. T. A. Sipkens, P. J. Hadwin, S. J. Grauer, and K. J. Daun, “General error model for analysis of laser-induced incandescence signals,” Appl. Opt. 56(30), 8436–8445 (2017). [CrossRef]  

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References

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  1. H. A. Michelsen, C. Schulz, G. J. Smallwood, and S. Will, “Laser-induced incandescence: Particulate diagnostics for combustion, atmospheric, and industrial applications,” Prog. Energy Combust. Sci. 51, 2–48 (2015).
    [Crossref]
  2. V. Ramanathan and G. Carmichael, “Global and regional climate changes due to black carbon,” Nat. Geosci. 1(4), 221–227 (2008).
    [Crossref]
  3. H. A. Michelsen, “Probing soot formation, chemical and physical evolution, and oxidation: A review of in situ diagnostic techniques and needs,” Proc. Combust. Inst. 36(1), 717–735 (2017).
    [Crossref]
  4. T. R. Meyer, B. R. Halls, N. Jiang, M. N. Slipchenko, S. Roy, and J. R. Gord, “High-speed, three-dimensional tomographic laser-induced incandescence imaging of soot volume fraction in turbulent flames,” Opt. Express 24(26), 29547–29555 (2016).
    [Crossref]
  5. P. Roth and A. V. Filippov, “In situ ultrafine particle sizing by a combination of pulsed laser heatup and particle thermal emission,” J. Aerosol Sci. 27(1), 95–104 (1996).
    [Crossref]
  6. A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999).
    [Crossref]
  7. B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002).
    [Crossref]
  8. T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003).
    [Crossref]
  9. T. Lehre, B. Jungfleisch, R. Suntz, and H. Bockhorn, “Size distributions of nanoscaled particles and gas temperatures from time-resolved laser-induced-incandescence measurements,” Appl. Opt. 42(12), 2021–2030 (2003).
    [Crossref]
  10. T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005).
    [Crossref]
  11. F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
    [Crossref]
  12. S. Dankers and A. Leipertz, “Determination of primary particle size distributions from time-resolved laser-induced incandescence measurements,” Appl. Opt. 43(18), 3726–3731 (2004).
    [Crossref]
  13. T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017).
    [Crossref]
  14. H. A. Michelsen, “Derivation of a temperature-dependent accommodation coefficient for use in modeling laser-induced incandescence of soot,” Appl. Phys. B 94(1), 103–117 (2009).
    [Crossref]
  15. K. J. Daun, “Thermal accommodation coefficients between polyatomic gas molecules and soot in laser-induced incandescence experiments,” Int. J. Heat Mass Transfer 52(21-22), 5081–5089 (2009).
    [Crossref]
  16. K. Daun, F. Liu, and G. Smallwood, “Molecular dynamics simulations of translational thermal accommodation coefficients for time-resolved LII,” in Heat Transfer Summer Conference, 2008), 333–342.
  17. T. A. Sipkens and K. J. Daun, “Effect of Surface Interatomic Potential on Thermal Accommodation Coefficients Derived from Molecular Dynamics,” J. Phys. Chem. C 122(35), 20431–20443 (2018).
    [Crossref]
  18. B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005).
    [Crossref]
  19. T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
    [Crossref]
  20. F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
    [Crossref]
  21. J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
    [Crossref]
  22. B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
    [Crossref]
  23. B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
    [Crossref]
  24. R. L. Vander Wal and M. Y. Choi, “Pulsed laser heating of soot: morphological changes,” Carbon 37(2), 231–239 (1999).
    [Crossref]
  25. R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004).
    [Crossref]
  26. K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
    [Crossref]
  27. H. A. Michelsen, “Understanding and predicting the temporal response of laser-induced incandescence from carbonaceous particles,” J. Phys. Chem. C 118(15), 7012–7045 (2003).
    [Crossref]
  28. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).
  29. H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
    [Crossref]
  30. F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
    [Crossref]
  31. C. M. Sorensen, “Light Scattering by Fractal Aggregates: A Review,” Aerosol Sci. Technol. 35(2), 648–687 (2001).
    [Crossref]
  32. A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000).
    [Crossref]
  33. D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer Nature, 2019), Vol. 93.
  34. J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
    [Crossref]
  35. U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
    [Crossref]
  36. J. Olfert and S. Rogak, “Universal relations between soot effective density and primary particle size for common combustion sources,” Aerosol Sci. Technol. 53(5), 485–492 (2019).
    [Crossref]
  37. P. Desgroux, X. Mercier, and K. A. Thomson, “Study of the formation of soot and its precursors in flames using optical diagnostics,” Proc. Combust. Inst. 34(1), 1713–1738 (2013).
    [Crossref]
  38. J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
    [Crossref]
  39. J.-Y. Zhang, H. Qi, Y.-F. Wang, B.-H. Gao, and L.-M. Ruan, “Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering,” Opt. Express 27(2), 1613 (2019).
    [Crossref]
  40. J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
    [Crossref]
  41. T. A. Sipkens, P. J. Hadwin, S. J. Grauer, and K. J. Daun, “General error model for analysis of laser-induced incandescence signals,” Appl. Opt. 56(30), 8436–8445 (2017).
    [Crossref]
  42. T. A. Sipkens and K. J. Daun, “Defining regimes and analytical expressions for fluence curves in pulsed laser heating of aerosolized nanoparticles,” Opt. Express 25(5), 5684–5696 (2017).
    [Crossref]

2020 (2)

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
[Crossref]

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
[Crossref]

2019 (4)

J. Olfert and S. Rogak, “Universal relations between soot effective density and primary particle size for common combustion sources,” Aerosol Sci. Technol. 53(5), 485–492 (2019).
[Crossref]

J.-Y. Zhang, H. Qi, Y.-F. Wang, B.-H. Gao, and L.-M. Ruan, “Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering,” Opt. Express 27(2), 1613 (2019).
[Crossref]

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
[Crossref]

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

2018 (3)

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
[Crossref]

J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
[Crossref]

T. A. Sipkens and K. J. Daun, “Effect of Surface Interatomic Potential on Thermal Accommodation Coefficients Derived from Molecular Dynamics,” J. Phys. Chem. C 122(35), 20431–20443 (2018).
[Crossref]

2017 (5)

T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017).
[Crossref]

H. A. Michelsen, “Probing soot formation, chemical and physical evolution, and oxidation: A review of in situ diagnostic techniques and needs,” Proc. Combust. Inst. 36(1), 717–735 (2017).
[Crossref]

T. A. Sipkens, P. J. Hadwin, S. J. Grauer, and K. J. Daun, “General error model for analysis of laser-induced incandescence signals,” Appl. Opt. 56(30), 8436–8445 (2017).
[Crossref]

T. A. Sipkens and K. J. Daun, “Defining regimes and analytical expressions for fluence curves in pulsed laser heating of aerosolized nanoparticles,” Opt. Express 25(5), 5684–5696 (2017).
[Crossref]

J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
[Crossref]

2016 (1)

2015 (1)

H. A. Michelsen, C. Schulz, G. J. Smallwood, and S. Will, “Laser-induced incandescence: Particulate diagnostics for combustion, atmospheric, and industrial applications,” Prog. Energy Combust. Sci. 51, 2–48 (2015).
[Crossref]

2013 (2)

T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
[Crossref]

P. Desgroux, X. Mercier, and K. A. Thomson, “Study of the formation of soot and its precursors in flames using optical diagnostics,” Proc. Combust. Inst. 34(1), 1713–1738 (2013).
[Crossref]

2011 (1)

K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
[Crossref]

2009 (2)

H. A. Michelsen, “Derivation of a temperature-dependent accommodation coefficient for use in modeling laser-induced incandescence of soot,” Appl. Phys. B 94(1), 103–117 (2009).
[Crossref]

K. J. Daun, “Thermal accommodation coefficients between polyatomic gas molecules and soot in laser-induced incandescence experiments,” Int. J. Heat Mass Transfer 52(21-22), 5081–5089 (2009).
[Crossref]

2008 (1)

V. Ramanathan and G. Carmichael, “Global and regional climate changes due to black carbon,” Nat. Geosci. 1(4), 221–227 (2008).
[Crossref]

2007 (2)

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
[Crossref]

2006 (2)

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
[Crossref]

F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
[Crossref]

2005 (2)

T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005).
[Crossref]

B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005).
[Crossref]

2004 (2)

S. Dankers and A. Leipertz, “Determination of primary particle size distributions from time-resolved laser-induced incandescence measurements,” Appl. Opt. 43(18), 3726–3731 (2004).
[Crossref]

R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004).
[Crossref]

2003 (3)

H. A. Michelsen, “Understanding and predicting the temporal response of laser-induced incandescence from carbonaceous particles,” J. Phys. Chem. C 118(15), 7012–7045 (2003).
[Crossref]

T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003).
[Crossref]

T. Lehre, B. Jungfleisch, R. Suntz, and H. Bockhorn, “Size distributions of nanoscaled particles and gas temperatures from time-resolved laser-induced-incandescence measurements,” Appl. Opt. 42(12), 2021–2030 (2003).
[Crossref]

2002 (1)

B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002).
[Crossref]

2001 (1)

C. M. Sorensen, “Light Scattering by Fractal Aggregates: A Review,” Aerosol Sci. Technol. 35(2), 648–687 (2001).
[Crossref]

2000 (1)

A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000).
[Crossref]

1999 (2)

R. L. Vander Wal and M. Y. Choi, “Pulsed laser heating of soot: morphological changes,” Carbon 37(2), 231–239 (1999).
[Crossref]

A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999).
[Crossref]

1996 (1)

P. Roth and A. V. Filippov, “In situ ultrafine particle sizing by a combination of pulsed laser heatup and particle thermal emission,” J. Aerosol Sci. 27(1), 95–104 (1996).
[Crossref]

Abrahamson, J.

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
[Crossref]

Abrahamson, J. P.

J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
[Crossref]

Apicella, B.

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
[Crossref]

Baldelli, A.

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
[Crossref]

Bauer, F. J.

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
[Crossref]

Bengtsson, P.-E.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Bladh, H.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Bockhorn, H.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005).
[Crossref]

T. Lehre, B. Jungfleisch, R. Suntz, and H. Bockhorn, “Size distributions of nanoscaled particles and gas temperatures from time-resolved laser-induced-incandescence measurements,” Appl. Opt. 42(12), 2021–2030 (2003).
[Crossref]

T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003).
[Crossref]

Boehman, A. L.

K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

Boïarciuc, A.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Carmichael, G.

V. Ramanathan and G. Carmichael, “Global and regional climate changes due to black carbon,” Nat. Geosci. 1(4), 221–227 (2008).
[Crossref]

Charwath, M.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Choi, M. Y.

R. L. Vander Wal and M. Y. Choi, “Pulsed laser heating of soot: morphological changes,” Carbon 37(2), 231–239 (1999).
[Crossref]

Ciajolo, A.

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
[Crossref]

Colton, D.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer Nature, 2019), Vol. 93.

Conrad, B. M.

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
[Crossref]

Corbin, J. C.

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
[Crossref]

Dankers, S.

Daun, K.

K. Daun, F. Liu, and G. Smallwood, “Molecular dynamics simulations of translational thermal accommodation coefficients for time-resolved LII,” in Heat Transfer Summer Conference, 2008), 333–342.

Daun, K. J.

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
[Crossref]

T. A. Sipkens and K. J. Daun, “Effect of Surface Interatomic Potential on Thermal Accommodation Coefficients Derived from Molecular Dynamics,” J. Phys. Chem. C 122(35), 20431–20443 (2018).
[Crossref]

T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017).
[Crossref]

T. A. Sipkens, P. J. Hadwin, S. J. Grauer, and K. J. Daun, “General error model for analysis of laser-induced incandescence signals,” Appl. Opt. 56(30), 8436–8445 (2017).
[Crossref]

T. A. Sipkens and K. J. Daun, “Defining regimes and analytical expressions for fluence curves in pulsed laser heating of aerosolized nanoparticles,” Opt. Express 25(5), 5684–5696 (2017).
[Crossref]

T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
[Crossref]

K. J. Daun, “Thermal accommodation coefficients between polyatomic gas molecules and soot in laser-induced incandescence experiments,” Int. J. Heat Mass Transfer 52(21-22), 5081–5089 (2009).
[Crossref]

Desgroux, P.

P. Desgroux, X. Mercier, and K. A. Thomson, “Study of the formation of soot and its precursors in flames using optical diagnostics,” Proc. Combust. Inst. 34(1), 1713–1738 (2013).
[Crossref]

Dreier, T.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Eckhardt, T.

B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002).
[Crossref]

Filippov, A. V.

A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000).
[Crossref]

A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999).
[Crossref]

P. Roth and A. V. Filippov, “In situ ultrafine particle sizing by a combination of pulsed laser heatup and particle thermal emission,” J. Aerosol Sci. 27(1), 95–104 (1996).
[Crossref]

Foucher, F.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Gao, B.-H.

Geigle, K.-P.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Gord, J. R.

Grauer, S. J.

Hadef, R.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Hadwin, P. J.

Halls, B. R.

Hill, F. A.

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
[Crossref]

Hofmann, M.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Huber, F. J. T.

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

Jefferson, A. M.

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J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
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T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
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U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
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B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005).
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J.-Y. Zhang, H. Qi, Y.-F. Wang, B.-H. Gao, and L.-M. Ruan, “Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering,” Opt. Express 27(2), 1613 (2019).
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J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
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J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
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J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
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J. Olfert and S. Rogak, “Universal relations between soot effective density and primary particle size for common combustion sources,” Aerosol Sci. Technol. 53(5), 485–492 (2019).
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Rogak, S. N.

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
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B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005).
[Crossref]

B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002).
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A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999).
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Ruan, L.-M.

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
[Crossref]

J.-Y. Zhang, H. Qi, Y.-F. Wang, B.-H. Gao, and L.-M. Ruan, “Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering,” Opt. Express 27(2), 1613 (2019).
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J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
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Russo, C.

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
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B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
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H. A. Michelsen, C. Schulz, G. J. Smallwood, and S. Will, “Laser-induced incandescence: Particulate diagnostics for combustion, atmospheric, and industrial applications,” Prog. Energy Combust. Sci. 51, 2–48 (2015).
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H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Shi, J.-W.

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
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J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
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T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017).
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T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
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U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
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Smallwood, G.

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Smallwood, G. J.

H. A. Michelsen, C. Schulz, G. J. Smallwood, and S. Will, “Laser-induced incandescence: Particulate diagnostics for combustion, atmospheric, and industrial applications,” Prog. Energy Combust. Sci. 51, 2–48 (2015).
[Crossref]

F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
[Crossref]

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
[Crossref]

Snelling, D. R.

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
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F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
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J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
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C. M. Sorensen, “Light Scattering by Fractal Aggregates: A Review,” Aerosol Sci. Technol. 35(2), 648–687 (2001).
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F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
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H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Suntz, R.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005).
[Crossref]

T. Lehre, B. Jungfleisch, R. Suntz, and H. Bockhorn, “Size distributions of nanoscaled particles and gas temperatures from time-resolved laser-induced-incandescence measurements,” Appl. Opt. 42(12), 2021–2030 (2003).
[Crossref]

T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003).
[Crossref]

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P. Desgroux, X. Mercier, and K. A. Thomson, “Study of the formation of soot and its precursors in flames using optical diagnostics,” Proc. Combust. Inst. 34(1), 1713–1738 (2013).
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R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004).
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B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
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Tribalet, B.

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
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U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
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Vander Wal, R. L.

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
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J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
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K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
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R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004).
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B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
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Wang, Y.-F.

Will, S.

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
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[Crossref]

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

Yang, M.

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
[Crossref]

Yehliu, K.

K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
[Crossref]

Zhang, J.-Y.

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
[Crossref]

J.-Y. Zhang, H. Qi, Y.-F. Wang, B.-H. Gao, and L.-M. Ruan, “Retrieval of fractal dimension and size distribution of non-compact soot aggregates from relative intensities of multi-wavelength angular-resolved light scattering,” Opt. Express 27(2), 1613 (2019).
[Crossref]

J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
[Crossref]

Zhang, Y.

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
[Crossref]

Zurita, M.

A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000).
[Crossref]

Aerosol Sci. Technol. (2)

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J. Olfert and S. Rogak, “Universal relations between soot effective density and primary particle size for common combustion sources,” Aerosol Sci. Technol. 53(5), 485–492 (2019).
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Appl. Opt. (3)

Appl. Phys. B (5)

T. A. Sipkens, N. R. Singh, and K. J. Daun, “Time-resolved laser-induced incandescence characterization of metal nanoparticles,” Appl. Phys. B 123(1), 14 (2017).
[Crossref]

H. A. Michelsen, “Derivation of a temperature-dependent accommodation coefficient for use in modeling laser-induced incandescence of soot,” Appl. Phys. B 94(1), 103–117 (2009).
[Crossref]

H. A. Michelsen, F. Liu, B. F. Kock, H. Bladh, A. Boïarciuc, M. Charwath, T. Dreier, R. Hadef, M. Hofmann, J. Reimann, S. Will, P.-E. Bengtsson, H. Bockhorn, F. Foucher, K.-P. Geigle, C. Mounaïm-Rousselle, C. Schulz, R. Stirn, B. Tribalet, and R. Suntz, “Modeling laser-induced incandescence of soot: a summary and comparison of LII models,” Appl. Phys. B 87(3), 503–521 (2007).
[Crossref]

F. Liu, M. Yang, F. A. Hill, D. R. Snelling, and G. J. Smallwood, “Influence of polydisperse distributions of both primary particle and aggregate size on soot temperature in low-fluence LII,” Appl. Phys. B 83(3), 383–395 (2006).
[Crossref]

F. J. Bauer, K. J. Daun, F. J. T. Huber, and S. Will, “Can soot primary particle size distributions be determined using laser-induced incandescence?” Appl. Phys. B 125(6), 109 (2019).
[Crossref]

Carbon (2)

J. P. Abrahamson, M. Singh, J. P. Mathews, and R. L. Vander Wal, “Pulsed laser annealing of carbon black,” Carbon 124, 380–390 (2017).
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R. L. Vander Wal and M. Y. Choi, “Pulsed laser heating of soot: morphological changes,” Carbon 37(2), 231–239 (1999).
[Crossref]

Chemosphere (1)

T. Lehre, H. Bockhorn, B. Jungfleisch, and R. Suntz, “Development of a measuring technique for simultaneous in situ detection of nanoscaled particle size distributions and gas temperatures,” Chemosphere 51(10), 1055–1061 (2003).
[Crossref]

Combust. Flame (3)

R. L. Vander Wal and A. J. Tomasek, “Soot nanostructure: dependence upon synthesis conditions,” Combust. Flame 136(1-2), 129–140 (2004).
[Crossref]

K. Yehliu, R. L. Vander Wal, and A. L. Boehman, “Development of an HRTEM image analysis method to quantify carbon nanostructure,” Combust. Flame 158(9), 1837–1851 (2011).
[Crossref]

B. Apicella, P. Pré, J. N. Rouzaud, J. Abrahamson, R. L. V. Wal, A. Ciajolo, A. Tregrossi, and C. Russo, “Laser-induced structural modifications of differently aged soot investigated by HRTEM,” Combust. Flame 204, 13–22 (2019).
[Crossref]

Fuel (2)

B. Apicella, A. Ciajolo, A. Tregrossi, J. Abrahamson, R. L. Vander Wal, and C. Russo, “HRTEM and EELS investigations of flame-formed soot nanostructure,” Fuel 225, 218–224 (2018).
[Crossref]

U. Trivanovic, T. A. Sipkens, M. Kazemimanesh, A. Baldelli, A. M. Jefferson, B. M. Conrad, M. R. Johnson, J. C. Corbin, J. S. Olfert, and S. N. Rogak, “Morphology and size of soot from gas flares as a function of fuel and water addition,” Fuel 279, 118478 (2020).
[Crossref]

Int. J. Heat Mass Transfer (2)

F. Liu, B. J. Stagg, D. R. Snelling, and G. J. Smallwood, “Effects of primary soot particle size distribution on the temperature of soot particles heated by a nanosecond pulsed laser in an atmospheric laminar diffusion flame,” Int. J. Heat Mass Transfer 49(3-4), 777–788 (2006).
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K. J. Daun, “Thermal accommodation coefficients between polyatomic gas molecules and soot in laser-induced incandescence experiments,” Int. J. Heat Mass Transfer 52(21-22), 5081–5089 (2009).
[Crossref]

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P. Roth and A. V. Filippov, “In situ ultrafine particle sizing by a combination of pulsed laser heatup and particle thermal emission,” J. Aerosol Sci. 27(1), 95–104 (1996).
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A. V. Filippov, M. W. Markus, and P. Roth, “In-situ characterization of ultrafine particles by laser-induced incandescence: sizing and particle structure determination,” J. Aerosol Sci. 30(1), 71–87 (1999).
[Crossref]

J. Colloid Interface Sci. (1)

A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000).
[Crossref]

J. Comput. Appl. Math. (1)

J. Kaipio and E. Somersalo, “Statistical inverse problems: Discretization, model reduction and inverse crimes,” J. Comput. Appl. Math. 198(2), 493–504 (2007).
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J. Heat Transfer (1)

T. Sipkens, G. Joshi, K. J. Daun, and Y. Murakami, “Sizing of molybdenum nanoparticles using time-resolved laser-induced incandescence,” J. Heat Transfer 135, 549–557 (2013).
[Crossref]

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H. A. Michelsen, “Understanding and predicting the temporal response of laser-induced incandescence from carbonaceous particles,” J. Phys. Chem. C 118(15), 7012–7045 (2003).
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T. A. Sipkens and K. J. Daun, “Effect of Surface Interatomic Potential on Thermal Accommodation Coefficients Derived from Molecular Dynamics,” J. Phys. Chem. C 122(35), 20431–20443 (2018).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

J.-W. Shi, H. Qi, J.-Y. Zhang, Y.-T. Ren, L.-M. Ruan, and Y. Zhang, “Simultaneous measurement of flame temperature and species concentration distribution from nonlinear tomographic absorption spectroscopy,” J. Quant. Spectrosc. Radiat. Transfer 241, 106693 (2020).
[Crossref]

Nat. Geosci. (1)

V. Ramanathan and G. Carmichael, “Global and regional climate changes due to black carbon,” Nat. Geosci. 1(4), 221–227 (2008).
[Crossref]

Opt. Commun. (1)

J.-Y. Zhang, H. Qi, Y.-T. Ren, and L.-M. Ruan, “Simultaneous identification of optical constants and PSD of spherical particles by multi-wavelength scattering–transmittance measurement,” Opt. Commun. 413, 317–328 (2018).
[Crossref]

Opt. Express (3)

Proc. Combust. Inst. (5)

H. A. Michelsen, “Probing soot formation, chemical and physical evolution, and oxidation: A review of in situ diagnostic techniques and needs,” Proc. Combust. Inst. 36(1), 717–735 (2017).
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B. F. Kock, T. Eckhardt, and P. Roth, “In-cylinder sizing of diesel particles by time-resolved laser-induced incandescence (TR-LII),” Proc. Combust. Inst. 29(2), 2775–2782 (2002).
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T. Lehre, R. Suntz, and H. Bockhorn, “Time-resolved two-color LII: size distributions of nano-particles from gas-to-particle synthesis,” Proc. Combust. Inst. 30(2), 2585–2593 (2005).
[Crossref]

B. F. Kock, C. Kayan, J. Knipping, H. R. Orthner, and P. Roth, “Comparison of LII and TEM sizing during synthesis of iron particle chains,” Proc. Combust. Inst. 30(1), 1689–1697 (2005).
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P. Desgroux, X. Mercier, and K. A. Thomson, “Study of the formation of soot and its precursors in flames using optical diagnostics,” Proc. Combust. Inst. 34(1), 1713–1738 (2013).
[Crossref]

Prog. Energy Combust. Sci. (1)

H. A. Michelsen, C. Schulz, G. J. Smallwood, and S. Will, “Laser-induced incandescence: Particulate diagnostics for combustion, atmospheric, and industrial applications,” Prog. Energy Combust. Sci. 51, 2–48 (2015).
[Crossref]

Other (3)

K. Daun, F. Liu, and G. Smallwood, “Molecular dynamics simulations of translational thermal accommodation coefficients for time-resolved LII,” in Heat Transfer Summer Conference, 2008), 333–342.

D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory (Springer Nature, 2019), Vol. 93.

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Figures (8)

Fig. 1.
Fig. 1. Schematic of underlying heat and mass transfer processes involved in LII
Fig. 2.
Fig. 2. Flowchart of double-model inversion process
Fig. 3.
Fig. 3. Effects of different laser fluences on temperature and diameter decay rate of primary particles when Np = 1 and dp = 10 nm
Fig. 4.
Fig. 4. (a) when Np = 1 and dp = 10 nm and (b) when Np = 600 and dp = 150 nm, the effect of different laser fluence on the ratio of the energy sum of laser absorption and heat conduction to the total energy
Fig. 5.
Fig. 5. Noisy LII signals synthesized by (a) the general noise model and (b) Gaussian model
Fig. 6.
Fig. 6. Primary particle size distribution profiles derived from retrieval results of (a) Gaussian noise and (b) general noise cases in Table 2 together with the target value
Fig. 7.
Fig. 7. Log contours of the objective function on (a) σd, g - dp, g plane, (b) αT, f - dp, g plane, and αT, f - σd, g plane
Fig. 8.
Fig. 8. Logarithmic objective function of valley lines of Fig. 6 along the coordinate axes

Tables (3)

Tables Icon

Table 1. Basic settings of the test cases

Tables Icon

Table 2. Retrieval results of three objective parameters in 8 test cases

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Table 3. Retrieval results of three two-variable inversions

Equations (26)

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Q ˙ int = Q ˙ abs + Q ˙ cond + Q ˙ rad + Q ˙ sub + Q ˙ therm + Q ˙ ox + Q ˙ ann
M ˙ = M ˙ sub + M ˙ ox
Q ˙ int = N p c s ρ s π 6 d p 3 d T d t
Q ˙ abs = N p C abs E ( t ) = N p C abs F q ( t )
C abs = π 2 d p 3 E ( m ) λ inc
Q ˙ cond = π N p d p 2 α T P a 8 8 R m T δ π W a ( γ + 1 γ 1 ) ( T T δ 1 )
N p = k f ( 2 R g d p ) D f
Q ˙ cond = π D eff 2 α T P a 8 8 R m T δ π W a ( γ + 1 γ 1 ) ( T T δ 1 )
D eff 2 ( d p ,   N p ,   k f ,   D f ) = d p 2 N p η ( N p ,   k f ,   D f )
Q ˙ int = Q ˙ abs + Q ˙ cond
d T d t = 6 π c s ρ s d p 3 [ C abs F q ( t ) Absorption π d p 2 η α T P a 8 8 R m T δ π W a ( γ + 1 γ 1 ) ( T T δ 1 ) heat conduction ]
α T, f η α T
p ( d p ) = 1 d p 2 π ln σ d, g  exp [ ( ln d p ln d p, g 2 ln σ d, g  ) 2 ]
p ( N p ) = 1 N p 2 π ln σ N, g  exp [ ( ln N p ln N p, g 2 ln σ N, g  ) 2 ]
S p ( t ,   α T, f ) = 8 π 3 h c 2 E ( m ) d p 3 ( t ) λ mea 6 exp [ h c / λ mea k B T ( t ,   α T, f ) ] 1
S agg ( t ,   N p ,   α T, f ) = N p S p ( t ,   α T, f ) p ( d p ) d d p
S LII ( t ,   N p ,   α T, f ) = C exp N N p S p ( t ,   α T, f ) p ( d p ) p ( N p ) d d p d N p
R S LII ( t ,   N p ,   α T, f ) = N p S p ( t ,   α T, f ) p ( d p ) p ( N p ) d d p d N p N p S p ( t max ,   α T, f ) p ( d p ) p ( N p ) d d p d N p
S agg ( t ) N p, eff S p ( t ,   α T, f ) p ( d p ) d d p
S LII ( t ) = C exp N N p, eff S p ( t ,   α T, f ) p ( d p ) d d p
R S LII ( t ) = C exp N N p, eff S p ( t ,   α T, f ) p ( d p ) d d p C exp N N p, eff S p ( t max ,   α T, f ) p ( d p ) d d p = S p ( t ,   α T, f ) p ( d p ) d d p S p ( t max ,   α T, f ) p ( d p ) d d p
R S LII ( t ) = d p 3 ( t ) p ( d p ) d d p exp [ h c / λ mea k B T ( t ,   α T, f ) ] 1 d p 3 ( t max ) p ( d p ) d d p exp [ h c / λ mea k B T ( t max ,   α T, f ) ] 1
f obj = b mea b est b mea 2 2
b mea = b tar ( 1 + γ ) n G
b mea = b tar + τ n b tar Shot - to - shot error + [ θ ( 1 + τ n ) b tar ] 1 / 2 n P Poisson error + γ n G Gaussian error
F ref = λ inc ρ s c s ( T ref T g ) 6 π E ( m )

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