## Abstract

Fluence curves are a powerful tool for understanding the mechanisms underlying nanosecond pulse laser heating of aerosolized nanoparticles, which is relevant to laser-induced incandescence (LII). This paper presents analytical expressions encompassing the entirety of the fluence domain considered in LII and uses them to formally define fluence regimes. The derived expressions and non-dimensional parameters facilitate one of the first comparisons of published experimental fluence curves. This procedure provides physical insight into the laser-nanoparticle interaction and highlights inconsistencies in the application of LII models to analyze the data.

© 2017 Optical Society of America

## 1 Introduction

Laser heating of nanoparticles often arises in gas phase nanoparticle synthesis and characterization. A prominent example is laser-induced incandescence (LII), which is used to characterize soot-laden [1,2] and synthetic nanoaerosols [3–5]. The technique works by heating nanoparticles in a sample volume of aerosol to incandescent temperatures using a laser pulse. The resulting spectral incandescence is most often used to define a pyrometric temperature via a spectroscopic model. Parameters that include the particle volume fraction, nanoparticle diameter, and other properties, are then inferred from the temperature data.

Fluence studies, which examine the relationship between peak nanoparticle temperature and laser fluence, elucidate the physics underlying the laser heating and subsequent cooling processes [1]. Eckbreth [6] pioneered this analysis; he attributed trends in experimental fluence and peak temperature data to a balance between the laser energy input and nanoparticle evaporation. Subsequent experimental fluence studies [7–23] provide confirmation of the existence of a *saturation* or *plateau* regime identified by Eckbreth [6]. Some LII practitioners have also extended this technique to synthetic nanoparticles, including: iron [3,21]; molybdenum [3,24–26]; silver [3]; silicon [27]; and tungsten [28]. Fluence curves have also been used to understand laser-based nanoparticle synthesis of tungsten nanoparticles by photolysis [29–31] and share the same characteristics as those used in LII applications.

The variation of peak temperature with laser fluence has also been explored numerically. Snelling et al. [32] and Smallwood et al. [33] simulated how the temporal and spatial laser profiles, initial nanoparticle size, absorption efficiency, and detector gate width effect the fluence curve. Schraml et al. [34] compared peak temperature versus laser irradiance curves to infer information about the evaporation properties of their soot samples. Bladh and Bengtsson [35] simulated the effect of polydispersity and the spatial laser profile on numerical fluence curves. Bladh et al. [36], Delhay et al. [15], and Olofsson et al. [37] used numerical simulations to inform complimentary experiments. De Iuliis et al. [38] used fluence curves to infer an absorption efficiency at the laser wavelength. Michelsen et al. [39,40], López-Yglesias et al. [41], Goulay et al. [42], and Lemaire and Mobtil [43] have also compared experimental measurements to numerical fluence curves with some discussion of the underlying physics. Some analytical expressions have also been derived for microparticles [44], although the mechanisms governing laser absorption and evaporative cooling are fundamentally different than the nanoparticles considered in this study.

In this paper we present the first analytical expressions that describe the relationship between peak temperature and laser fluence over a wide range of fluences. We elucidate three distinct fluence regimes. The lower fluence regime is dominated by a linear increase in the peak temperature, which has been observed previously [38,45]. The higher or plateau fluence regime is dominated by evaporation or sublimation from the nanoparticle, similar to what was observed by Eckbreth [6]. A third transitionary regime exists between the low and high fluence limits. We also propose a set of non-dimensional parameters that can reduce most fluence curves obtained on different types of carbonaceous and non-carbonaceous nanoparticles onto a single curve. Finally, the predictive capabilities of this analytical treatment are evaluated by interpreting experimental LII data from published sources.

## 2 Heat transfer model

The nanoparticle temperature is described by an energy balance on the nanoparticle,

where*U*is the internal (sensible) energy of the nanoparticle,

_{int}*q*is the rate of energy absorption by the nanoparticle,

_{laser}*q*is the energy lost due to evaporating atoms,

_{evap}*q*is conduction between the nanoparticle and the gas, and

_{cond}*q*represents other terms, such as radiation and annealing, as summarized in [1]. The rate of sensible energy change is given by [1]where

_{oth}*M*is the mass of the nanoparticle and

*c*is the specific heat capacity of the nanoparticle material. Over the duration of the laser pulse, the sensible energy change, laser absorption, and evaporation usually dominate Eq. (1) and are several orders of magnitude larger than other terms.

_{p}If the nanoparticle diameter is smaller than the laser wavelength [4,32,46]

*f*

_{0}(

*t*) is the temporal energy profile of the laser,

*Q*= 4

_{abs,λ}*πxE*(

**m**

*) is the absorption efficiency,*

_{λ}*x*=

*πd*/

_{p}*λ*is the size parameter,

**m**

*is the complex index of refraction of the nanoparticle material at the laser wavelength, and*

_{λlaser}*f*(

_{0}*t*) is the temporal laser profile, in units of W/m

^{2}. (This model also holds for aggregates containing primary particles having a size parameter much smaller than unity, via Rayleigh-Debye-Gans Fractal Aggregate theory [47].) It is generally more useful to define the laser energy in terms of the laser fluence,

*F*,since this quantity can be readily measured with a power meter. Temporal laser profiles in numerical studies are generally modeled as Gaussian with heavy tails [32,44,47–49]. Koechner [50] showed that Q-switching an Nd:YAG laser can produce these kind of temporal laser profiles, which was confirmed by Bladh et al. [51]. The laser profiles are also often spatially non-uniform, which produces a spatial distribution of nanoparticle temperatures within the probe volume at any instant (e.g [32,35,52–54].).

_{0}The rate of evaporative cooling in the free molecular regime is given by [1,33,48]

where Δ*h*is the latent heat of vaporization [J/kg],

_{v}*m*is the mass of the vapor molecule, and

_{v}*N*is the vapor number flux. The rate of change of nanoparticle mass is defined by [1,4,48]

_{v}*A*=

_{c}*πd*

_{p}^{2}is the cross-sectional area of the nanoparticle,

*n*=

_{v}*p*/(

_{v}*k*) is the vapor number density,

_{B}T_{p}*k*is the Boltzmann constant,

_{B}*p*is the vapor pressure, and

_{v}*c*= [8

_{v}*k*/(

_{B}T_{p}*πm*)]

_{v}^{1/2}is the mean thermal speed of the vapor. The vapor pressure is given by the Clausius-Clapeyron equation [48]where

*A*is a material constant and

*R*is the specific gas constant. This treatment presumes thermal quasi-equilibrium across the phase boundary at any instant, which is usually reasonable over the nanoscale time duration typical of laser heating. Equations (1) and (6) are coupled by the evaporation rate and nanoparticle diameter (

_{s}*d*= [{6·

_{p}*M*(

_{p}*t*)/(

*π*·

*ρ*[

*T*(

_{p}*t*)]}

^{1/3}) and must be solved simultaneously to obtain the nanoparticle temperature and mass.

Although Eqs. (1) and (6) can be solved numerically to obtain the fluence curve, additional insights can be made if we impose the following assumptions:

- A1. The laser beam temporal profile is modeled as a step function, i.e. the laser energy rate is constant over the period from
*t*= 0 until the end of the laser pulse,*t*=*t*, and is given as_{lp}where

*H*(*x*) is the Heaviside delta. - A2. The specific heat capacity, density, heat of vaporization and other intensive thermodynamic properties are assumed to be constant over the entire temperature range considered.
- A3. Evaporation occurs in the free molecular regime, absorption occurs in the Rayleigh limit, and there is no enhanced evaporation associated with interface curvature (cf [4,21,55].).
- A4. Free-molecular evaporation is the only significant cooling mode and there is only one evaporated species. This means that the vapor pressure will be defined by a single Clausius-Clapeyron equation. (In the case of carbon, the only evaporated species is assumed to be C
_{3}, which has the lowest vaporization temperature [48].) - A5. Nanoparticles sizes in the sample aerosol are monodisperse.

The effect of these assumptions is discussed further in Sec. 5.1 below.

## 3 Simulated fluence curves

To identify fluence regimes and inform the derivation of analytical expressions, numerical solutions to the coupled ordinary differential equations, Eqs. (1) and (6), are solved using a Runge-Kutta scheme subject to the assumptions described above, for soot. Values of 8 ns, 1500 K, 30 nm, and 1064 nm are chosen for *t _{lp}*,

*T*,

_{g}*d*

_{p}_{,0}, and

*λ*respectively. Temperature-independent material properties for soot are taken from the Kock model [56] as defined in [48], with the exception of

_{l}*E*(

**m**

*) = 0.4 [45]. The resulting curves, shown in Fig. 1, reveal three distinct regimes in the temporal variation of temperature. The low fluence regime is distinguished by a linear increase in temperature with respect to time, with the peak occurring at the end of the laser pulse. Without a significant heat loss mechanism, the temperature remains constant after the laser pulse. At moderate fluences, evaporation induces a nonlinear variation in temperature with respect to time at the longest times and highest temperatures and causes the temperature to decline significantly following the pulse. At high fluences, the peak temperature occurs during the laser pulse as the evaporative cooling rate exceeds the laser heating rate at longer times.*

_{λl}Plotting the peak temperatures as a function of laser fluence further highlights these three regimes. In this scenario, the low fluence regime occurs for *F*_{0} < 0.1 J/cm^{2}, the transition of moderate fluence regime occurs for 0.1 J/cm^{2} < *F*_{0} < 0.2 J/cm^{2}, and the plateau or high fluence regime occurs for *F*_{0} > 0.2 J/cm^{2}.

## 4 Fluence regimes and analytical expressions

#### 4.1 Low fluence regime

At low fluences, two additional simplifications can be made. First, evaporation is negligible over the laser pulse duration so the nanoparticle mass remains constant. Because laser heating far exceeds the magnitude of the cooling modes, the peak temperature must occur the end of the laser pulse, *t* = *t _{lp}*. Consequently,

*f*

_{0}(

*t*) are independent of time, one can invoke Eq. (4) to obtainNote that, in this regime,

*T*is independent of the temporal profile of the laser pulse. Combining Eq. (9)-(11) gives the following expression for (

_{peak}*T*–

_{peak}*T*) as a function of fluence,

_{g}We also use this equation to define a dimensionless fluence, Φ,

While the value of *T _{ref}* is arbitrary, below we discuss a convenient value of

*T*that is useful in defining the regimes. This expression should hold regardless of the material being considered, irrespective of

_{ref}*E*(

**m**

*), the temporal width and shape of the laser pulse, and*

_{λlaser}*d*. It also has the added advantage of that a unit change in the dimensionless fluence results in an equal change in the dimensionless temperature on the left-hand side. This expression is plotted in Fig. 2, along with curves found by solving Eqs. (1) and (6).

_{p}#### 4.2 High or plateau fluence regime

Eckbreth [6] suggested that the temporal variation of soot temperature could be determined by equating the laser input and evaporation from the nanoparticle. Figure 1 affirms this observation and suggests the existence of a critical fluence at which the heating rate exactly balances evaporative cooling at the end of the laser pulse, coinciding with the peak temperature. At higher fluences, the peak temperature occurs during the laser pulse, corresponding to the instant when laser heating exactly balances evaporative cooling, *q _{laser}* =

*q*. Following assumption A2, this can be used to give

_{evap}*d*and therefore also depends on the particle mass. Assuming that the nanoparticle size is approximately constant with temperature, this expression can be rearranged into

_{p}*W*

_{-1}is the lower branch of Lambert W (or product log) relation. This can equivalently be stated in terms of the dimensionless fluence

*C*

_{1}, defined as

This expression can be used to define a useful reference temperature, *T _{ref}*, corresponding to the point where Eqs. (14) and (19) intersect, which is assigned the coordinate (1,1) on the dimensionless fluence curve. This gives

*t*, the nanoparticle size, and the gas temperature. The corresponding reference fluence can then be evaluated from

_{lp}*F*

_{0}≪

*F*corresponds to the low fluence regime, while

_{ref}*F*

_{0}≫

*F*, corresponds to the high fluence regime. Also, note that measurements carried out in the low and high fluence regimes contain different information about the unknown parameters. For instance, the high fluence regime contains information about the nanoparticle size,

_{ref}*d*, while the peak temperature in the low fluence regime is insensitive to this parameter. This analysis suggests that it may be able to compliment inference of nanoparticle sizes through time-resolved LII analyses by examining the fluence curve.

_{p}#### 4.3 Moderate or transitional fluence regime

At fluences beyond the reference fluence, Eq. (21), evaporation influences the peak temperature reached by the nanoparticles, but the evaporative cooling term remains lower in magnitude than laser heating. In this scenario, the energy rate balance gives

Because Eq. (22) is analytically-intractable, we instead propose a blending function [57] to model the smooth transition between two intersecting functions, *f*_{1} and *f*_{2}, with known asymptotic properties, *f*(*x*) = [*f*_{1}* ^{n}*(

*x*) +

*f*

_{2}

*(*

^{n}*x*)]

^{1/2}. Increasing |

*n*| sharpens the transition between the two functions, and

*n*= −20 gives good agreement to numerical simulations, as shown in Fig. 2.

## 5 Application

#### 5.1 Effect of assumption relaxation

We now examine how relaxing the simplifications used to derive the analytical expression influence the simulated peak temperatures.

The largest departure from true experimental conditions comes from modeling the temporal profile of the laser pulse with a step function (A1), while most often the time-average fluence profile is skewed normal [32,45,47–49]. To this end, we fit a lognormal distribution to the profile given in [48], yielding *μ _{g}* = 10.8 ns and

*σ*= 1.34. Using a different profile requires specification of an effective

_{g}*t*in the above equations, which we assign as the full-width half maximum (FWHM) of the distribution. Figure 3 shows the difference between the proposed expressions and simulation results obtained by relaxing A1. Changes to the low fluence regime are negligible as one would expect per Eqs. (9)-(12). Near the transition fluence, the tail effects of the smooth temporal fluence profile results in a lower nanoparticle temperature compared to the one obtained by assuming a step function. This can be somewhat accounted for by using

_{lp}*n*= −10, also shown in Fig. 3. The discrepancy disappears for Φ > 2.5.

Relaxing A2 allows the material properties to change with nanoparticle temperature. In this work, we consider temperature-dependent *ρ*, *c _{p}*, and Δ

*h*as defined by the Michelsen model in [47]. Reference values for these parameters are calculated at

_{v}*T*, (

_{g}*T*+

_{g}*T*)/2, and

_{b}*T*for

_{b}*ρ*,

*c*, and Δ

_{p}*h*respectively. The key effect is a heightened nanoparticle temperature in the low fluence regime due to dilation or

_{v}*ρ*(

*T*), wherein the absorption cross-section increases as the nanoparticle expands. This effect is not observed in the high fluence regime, likely because the nanoparticle temperature varies less during the duration of laser heating.

We relax A3 by considering the effect of multiple evaporated species, specifically C, C_{2}, C_{4}, and C_{5} as defined by the Michelsen model in [48]. (The introduction of conduction into the model alone has an indistinguishable effect of the results and is not further considered.) Including this effect lowers the nanoparticle temperatures at high fluences, since the additional evaporation modes lowers the temperature at which evaporation balances the laser input.

We relax A5 by assuming the nanoparticle diameters obey a lognormal distribution *p*(*d _{p}*) having a Sauter mean diameter,

*d*, corresponding to the monodisperse nanoparticle size [3,4,58] and

_{p,32}*σ*= 1.2. Spectral incandescence is then found by [4]

_{g}*I*is the blackbody spectral intensity. The spectral incandescence at two wavelengths is used to derive an effective pyrometric temperature. As the signal is dominated by larger nanoparticles, which cool more slowly, the peak temperatures are higher in the high fluence regime, where evaporative cooling is significant.

_{b,λ}Figure 3 also shows the combined effect of simultaneously relaxing A1, A2, A4, and A5. The net effect of these assumptions amounts to a maximum difference less than 10% from the more exact treatment, however, which is sufficient for the objectives of this technique.

#### 5.2 Experimental comparison

We now examine experimental LII fluence curves for soot reported in the literature, specifically [1,22,23,37–39,54], as well as unpublished data from an LII experiment carried out at the National Research Council of Canada (matching the experimental conditions given in Snelling et al. [45]). The laser fluences in these studies range from 0.083 to 0.189 J/cm^{2}. There is some discord in the literature regarding the radiative properties of soot, and moreover these are known to vary between experiments based on factors that include fuel type, local combustion conditions, and the “age” of the soot [37,59,60]. Accordingly, a value for *E*(**m*** _{λl}*) is estimated for each study based on low fluence regime data and is given in Table 1. The large range of estimated

*E*(

**m**

*) is characteristic of that found in the literature [61]. Note that the estimated*

_{λl}*E*(

**m**

*) for Maffi et al. [23] and Olofsson et al. [37], who collect data under similar experimental conditions, are the same, as one would expect. The gas temperature is inferred by extrapolating the peak temperature data back to zero fluence, which generally results in temperatures consistent with the range given in each study. The nanoparticle size and laser pulse width are taken from the respective works and summarized in Table 1. The remaining material properties are taken from the Kock model [49]. For comparative purposes, the temperature data is derived assuming a wavelength-independent*

_{λl}*E*(

**m**

*) as is prominent in the literature [61,62]. This requires a correction of the original temperatures reported by Maffi et al. [23] and De Iuliis et al. [38], which were calculated assuming a wavelength-dependent*

_{λ}*E*(

**m**

_{λ}) [63] that is distinct from other values reported in the literature.

Figure 4 shows that the analytical model reproduces experimentally-observed trends, and the non-dimensional parameters reduce experimental data onto a single curve. Moreover, the predicted reference fluences for each study, given in Table 1, coincide with the transition points in the experimental fluence curves. Remaining discrepancies between the analytical expression and experimental data suggest that the models do not capture all of the key physical processes that underlie the measurements, which may motivate further investigation. For example, in cases where the analytical expression overpredicts the experimentally-observed peak temperature, the discrepancy could be due to a laser pulse profile that differs from the one assumed to derive the analytical expression or because of enhanced evaporation of multiple species. Deviation of experimental data from the curves may also highlight issues with model implementation or data analysis. As a case in point: the adjustment made to the temperatures reported by Maffi et al. [23] and De Iuliis et al. [38], which is consistent with more recent experimental measurements of the *E*(**m**_{λ}) function [61,62], also brings this data in better alignment with the analytical expression and experimental fluence curves reported by other sources.

## 6 Conclusions

This paper presents the first comprehensive analytical model of the relationship between laser fluence and peak nanoparticle temperature during pulsed laser heating. We identify three fluence regimes: the low/linear fluence regime; the moderate/transition fluence regime; and the high/plateau fluence regime. Analytical expressions are derived for the low and high fluence regime, while an interpolating function is specified for the transition regime. Based on these expressions, a set of non-dimensional parameters is proposed that reduces most fluence curves to a single curve and allows for the formal definition of the fluence regimes about a non-dimensional fluence of unity. The expressions are capable of reproducing experimental trends, with the defined non-dimensional parameters reducing the curves to a single curve.

These results highlight the large amount of information contained in fluence curves. Important physical insights that can be obtained from examining the relationship between laser fluence and peak temperature, while deviations between the modeled trends and experimental results can be used to identify deficiencies in the spectroscopic and heat transfer models used to interpret the data.

## Funding

Natural Science and Engineering Research Council (NSERC) CGS D3 Fellowship; NSERC Discovery Grant (356267).

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