#### Abstract

The optical constants of a liquid hydrocarbon such as liquid n-octane are basic material properties that may be used to evaluate their thermal radiation transfer capabilities. In this study, the ellipsometry method was used to measure the optical constants of liquid n-octane in the midinfrared wavelength range of 2.0–16.0 *μ*m at temperatures of 20, 50, and 80°C. Experimental analyses indicate the significant effect of temperature on the refractive index, although it has little effect on the absorption index. With increasing temperature, the refractive index shows a linear decrease, and reduced density leads to weaker absorption intensities. The radiative properties of n-octane droplets, including the absorption and scattering efficiency factors of single droplets with droplet radii *r* = 10, 20, 50, and 100 *μ*m and the absorption and scattering coefficients in a droplets-air system of droplet volume fractions *f*_{v} = 2%, 3%, and 4%, were calculated using Mie theory. The numerical results indicate that, with increasing temperature, the absorption efficiency factor slightly decreases, and the variation trend of the scattering efficiency factor is more complicated. With increasing droplet radius, the absorption efficiency factor increases within the studied wavelength range, except for certain absorption peaks, but the scattering efficiency factor tends to decrease. While the absorption is greater, the scattering is weaker for a given droplet radius. With an increasing volume fraction of n-octane droplets, the absorption and scattering coefficients increase linearly within the studied wavelength range.

#### 1. Introduction

The thermal radiation transfer of liquid fuels is key to energy utilization, which plays an important role in the diagnosis and regulation of liquid-fuel ignition, liquid-droplet combustion, and fuel composition measurement and analysis in thermal devices such as automobile internal combustion engines and aerospace engines [1–10]. The optical constants (i.e., the complex refractive index , including the refractive index *n* and the absorption index *κ*) of liquid fuels and droplet radiative properties are important material properties and the basis for calculating radiation transfer, which is indispensable in researching the optical and radiative properties of liquid fuels. The optical constants of liquid fuel, which reflect the interaction with radiation, are vital to ensure accuracy in radiative heat transfer modeling of combustion engines and diagnosis during fuel combustion [11–15].

With the development of experimental conditions, numerous experimental approaches have been proposed to investigate the optical properties of liquids, such as the attenuated total reflection (ATR) method [16–20], the transmission method [21–24], the combined transmission-reflection method [25, 26], and the ellipsometry method [27–33]. The ATR method is suitable for the measurement of absorption indices in strong absorption regions, whereas the transmission method is especially suitable for measuring the absorption indices of liquids in weak absorption regions. Due to the use of the Kramers–Kronig relation, the reliability of the ATR method and the transmission method largely depends on the studied spectral ranges. Moreover, the ellipsometry method directly obtains the real and imaginary part of the complex refractive index by measuring changes in the intensity and phase of polarized light. Due to its high sensitivity to material optical constants and high measurement accuracy, the ellipsometry method is widely used in a variety of fields. Tiwald et al. [34] measured the optical constants of several lubricants by using the infrared ellipsometry and ATR cell. In their studies, the polarized light was vertically incident on the ATR prism. They ignored the multiple reflections and transmissions among the prism surfaces. Thus, this leads to inaccuracies in the measured data. Wang et al. [35, 36] proposed a modified ellipsometry-transmission method to solve this problem, ensuring that the polarized light was vertically incident on the prism. This approach provided improved accuracy, by considering the multiple reflections and transmissions among the prism surfaces.

The importance of the absorption and scattering properties of fuel droplets in the radiative heat transfer process has been widely discussed. Absorption and scattering of thermal radiation can be applied to various problems, including heating evaporation of droplets, autoignition of fuel vapor, attenuation of radiation by droplets and sprays, and optical diagnostics of sprays [37–39]. As the absorption efficiency factor *Q*_{abs} and scattering efficiency factor *Q*_{sca} of droplets depend on the optical constants of droplets and droplet size parameter , optical constants are crucial input parameters for the calculation of radiation characteristics. Experimental studies and computer simulations of temperature and droplet size distribution in combustion equipment have been reported by various researchers [40, 41]. However, much less attention has been paid to the temperature dependence of radiative properties of liquid-fuel droplets. Few studies have investigated the effect of droplet volume fraction on the radiative properties of the droplet-air system formed by a spray in a combustion chamber. Dombrovsky et al. [39, 42–44] calculated the radiative properties of fuel droplets in the wavelength range from 0.2 to 10 *μ*m using Mie theory. However, the optical constants used for their calculations were obtained at a single temperature, thereby neglecting the temperature dependence of optical constants. Therefore, it is necessary to study the temperature-dependent absorption and scattering efficiency factors and coefficients of fuel droplets, which are essential parameters for radiative heat transfer modeling in thermal devices.

Liquid n-octane, a component of industrial gasoline, is a colorless and transparent combustible organic compound fuel. At present, this liquid fuel is widely used in theoretical and experimental spray combustion research on the combustion chambers of internal combustion engines and aerospace engines [45–47]. It is, therefore, essential to determine the temperature-dependent optical constants and radiative properties of liquid n-octane and n-octane droplets.

In this study, the ellipsometry method is used to measure the temperature-dependent optical constants of liquid n-octane in the spectral wavelength range 2.0–16.0 *μ*m. The temperature effects on spectral optical constants are also analyzed further, and the effects of temperature and size parameters on the radiative properties of n-octane spherical droplets, including absorption and scattering efficiency factors calculated using Mie theory, are discussed. Furthermore, the effects of droplet volume fractions in a droplets-air system on absorption and scattering coefficients calculated using Mie theory are synchronously investigated.

#### 2. Methodology

##### 2.1. The Ellipsometry Method and Experimental Setup

In this study, the ellipsometry method was used to measure the optical constants of liquid n-octane using the IR-VASE ellipsometer (J.A. Woollam, Inc.). The light source of the IR-VASE ellipsometer is based on the Michelson Fourier transform infrared interferometer and the wavelength range spans from 1.5 to 30 *μ*m. Diagrams of ellipsometry measurements based on the “prism-liquid” system are shown in Figure 1. Zinc selenide prism glass consisting of an isosceles trapezoid with an angle of 45° was used for the optical windows in the “prism-liquid” system ellipsometry apparatus. The optical constant of the zinc selenide prism was measured using the double-thickness transmission method, and the data are given in Ref. [48]. The spectroscopic ellipsometer consisted of a polarized infrared light source and detector. The specific spectral requirements of wavelength and resolution could be set directly by using the computer.

The sample cell and optical windows were used for containing the liquid to be measured. Before performing the experiment, a protective shell made of PVC was placed on the test bench and filled with inert gas. In the measurements, the polarized light was vertically incident on one side of the ATR prism. The light, reflected by the ATR prism-liquid interface, was then emitted vertically from the other side of the ATR prism. The polarized light entered the detector via the rotary compensator and analyzer to measure changes in intensity and phase. The light from the infrared light source changed its polarization state, which was expressed by the amplitude and phase, when passing through the polarizer and being received by the detector. The ellipsometric parameters, which include the amplitude difference and phase difference are denoted as and . Considering the multiple reflections among the prism-air and prism-liquid interfaces, the reflection coefficient and the complex ratio at wavelength *λ* are provided by [35,36]where the subscripts 0, 1, and 2 denote the air, prism, and liquid. The superscript “*i*” denotes the imaginary unit. and are *p-*polarized and *s-*polarized reflection coefficients. denotes the amplitude reflection coefficient at the interface between layer *i* and *j*, which are functions of the optical constants (complex refractive index) , and denotes the phase thickness, where *l* denotes the path length of the light. *θ*_{i} and *θ*_{j} are the angles of incidence and refraction between layers *i* and *j* [49]. As the polarized light was vertically incident on the prism surface, and in equation (2) are equivalent to each other for measured media. The ZnSe-air system measurements may be conducted to obtain and , and then, the optical constants of the target liquids may be obtained by solving equation (2) for the ZnSe-liquid system [35, 36].

The measurement uncertainties are evaluated using the error propagation method provided in Ref. [50]. The specific calculation method is given in an earlier paper published by our research group [36]. Deviations of the refractive indices and absorption indices are plotted with error bars.

The ceramic heater was pasted onto the back side of the liquid cell using high-temperature resistant inorganic adhesive. The liquid was connected to a real-time pressure monitor, and the pressure was maintained at 1 atm by adjusting the manual feed on the liquid injector. The spectral resolution was set to 23.14 cm^{−1}, and the wavenumber range for the experimental measurements was 625 − 5000 cm^{−1}, providing a measured wavelength range of 2.0 – 16.0 *μ*m. The purity of the analytically pure n-octane provided by Shanghai Aladdin Bio-Chem Technology Co., LTD. was 99.9%. As liquid n-octane has a boiling point of approximately 125 − 127°C, in order to prevent the liquid from boiling during the heating measurement, which would lead to inaccurate results, the optical constants of liquid n-octane were measured at 20, 50, and 80°C, respectively.

##### 2.2. Radiative Property Calculations for Liquid Droplets

In the radiative heat transfer modeling of combustion engines and diagnosis during fuel combustion, liquid fuels are presented in the form of droplets. Considering the complexity of nonspherical particle calculations and similarity of fuel droplets to pure spheres, Mie theory is generally used to calculate the radiative properties of a single spherical droplet, including the absorption, scattering, and extinction efficiency factor [51, 52].

The radiative properties of a single droplet depend on two independent parameters: the size parameter and complex refractive index , where *r* represents the droplet radius, represents the wavelength of the irradiated light, and *n* and *κ* represent the refractive index and absorption index of liquid forming the droplet, respectively [53, 54].

Based on Mie theory, the extinction efficiency factor *Q*_{ext}, scattering efficiency factor *Q*_{sca}, and absorption efficiency factor *Q*_{abs} can be calculated as follows [55–57]:where the symbol Re denotes the real parts of plurality and *a*_{n} and *b*_{n} denote the Mie scattering coefficients, which are calculated by [55–57]where and are Ricatti–Bessel functions that satisfy the following recursive relationship [55–57]:where , ; , and .

In the case of a monodisperse spray, considering that fuel droplets in combustion chambers such as diesel engines are usually present with droplet radii of 10 − 100 *μ*m [43, 44], droplets with radii *r* = 10, 20, 50, and 100 *μ*m are calculated. For the studied wavelength range from 2.0 to 16.0 *μ*m and droplet radii from 10 to 100 *μ*m, the size parameter of the studied droplets is from 3.9 to 314.2. Thus, the droplets may be considered to be medium and large particles for most circumstances.

When a liquid fuel forms a spray using a nozzle, a dispersed droplet system is produced, which is composed of many droplets. In engineering applications, calculations involving polydisperse sprays are usually simplified using a monodisperse approximation, with droplet radii equal to an average radius of droplets in a polydisperse spray [43, 44]. To study the radiative transfer of the abovementioned droplet system, firstly, we consider the interaction between the droplets. Brewster and Tien [58] concluded the widely used criteria of independent scattering , where *c* is the interparticle clearance. In this study, the minimum particle radius *r* = 10 *μ*m, and the corresponding minimum size parameter *χ* = 3.93 (for *λ* = 16 *μ*m). Based on this criterion, the independent scattering condition is met along as *fv* > 0.39. Therefore, the cases (*fv* = 2%, 3% and 4%) in this study satisfy the independent scattering condition. The scattering and absorption effects of the droplet system can, then, be considered as a superposition of individual droplets. If this droplet system contains only one kind of droplet and the size distribution of the droplet is uniform, the spectral transport extinction coefficient *k*_{ext}, scattering coefficient *k*_{sca}, and absorption coefficient *k*_{abs} can be expressed as [55–57]where *D* denotes the diameter of the particle, *D* = 2*r*, *N*_{0} denotes the number density of droplets, and *f*_{v} denotes the volume fraction of the droplets.

#### 3. Results and Discussion

##### 3.1. Spectral Optical Constants of Liquid n-Octane at Different Temperatures

The IR-VASE ellipsometer was used to measure the ellipsometric parameters and of the “ATR prism-liquid” system, and in combining the specific mathematical models, the temperature-dependent optical constants of liquid n-octane were obtained. Figure 2 presents the optical constants of liquid n-octane measured by the ellipsometry method at 20, 50, and 80°C in the spectral wavelength range 2.0–16.0 *μ*m. As shown in Figure 2(a), temperature has a significant effect on the refractive index *n*, and the refractive index values are within the range 1.288–1.454 for the studied wavelength and temperatures. With increasing temperature, the refractive index shows an approximately linear decrease due to the decrease in liquid density. For instance, the refractive index *n* of liquid n-octane at a wavelength of 3.51 *μ*m decreases from 1.454 to 1.420 as the temperature increases from 20°C to 80°C, and the changing rate is about 5.67 × 10^{−4}/°C. The values of *n* at 7.32 *μ*m decrease from 1.395 to 1.367 as the temperature increases from 20°C to 80°C, and the changing rate is about 4.67 × 10^{−4}/°C. By contrast, Figure 2(b) shows the much smaller effect of temperature on the absorption index *κ*. The measured uncertainties of the absorption indices are larger for 13–16 *μ*m wavebands due to the weakening of the light source intensity and the enhanced absorption of the zinc selenide window, thus resulting in obvious differences of the absorption index at three different temperatures. The absorption peaks in the midinfrared spectral range are results of intermolecular interactions, intramolecular bending, and symmetric or asymmetric stretching vibrations of C–H groups. As shown, in the studied temperature region, temperature shows little effect on the absorption peaks representing C–H vibrations.

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##### 3.2. Effect of Temperature on Absorption and Scattering Efficiency Factors of a Single Droplet

We assume that droplets of liquid n-octane that are formed during spraying are spherical. Absorption and scattering of thermal radiation by droplets can be calculated using Mie theory [44, 59]. According to Mie theory, the absorption efficiency factor *Q*_{abs} and scattering efficiency factor *Q*_{sca} depend on the droplet size parameter and the complex refractive index of liquid n-octane. The studied spectral range 2∼16 *μ*m corresponds to a maximum radiative temperature range of 180∼1449 K, which is much broader than spray combustion.

Figure 3 presents the Mie-calculated absorption and scattering efficiency factors of n-octane droplets at different temperatures, with radii *r* = 10, 20 *μ*m. The absorption efficiency factor *Q*_{abs} represents the rate of the absorption cross section and the projected surface area of the spherical droplet. The scattering efficiency factor *Q*_{sca} represents the rate of scattering cross section and the projected surface area of the spherical droplet. As shown in Figures 3(a) and 3(c), the absorption efficiency factor *Q*_{abs} decreases with increasing temperature due to the slight decrease of the absorption index *κ* of liquid n-octane. The changing trends of *Q*_{abs} with increasing wavelength are similar to that of the measured absorption index *κ* of liquid n-octane, which is shown in Figure 2(b). Figure 3 indicates that stronger absorption generally leads to weaker scattering of the droplet. As shown in Figures 3(b) and 3(d), the scattering efficiency factor *Q*_{sca} of droplets shows regular damped oscillation characteristics with variations in the size parameter *χ*. However, the trends related to increasing temperature, which leads to a decrease of *n* and *κ*, on *Q*_{sca} differ for different spectral ranges. According to the Mie theory, the scattering efficiency factor is closely related to the complex refractive index and size parameter of the droplets. For different size droplets, due to the change of the complex refractive index and size parameter, the scattering efficiency factors show different tendency with the increase of incident wavelength. For instance, the *Q*_{sca} of a n-octane droplets with radius *r* = 10 *μ*m in the wavelength range 3.7–4.3 *μ*m and 6.3–10.9 *μ*m increase with increasing temperature, although *Q*_{sca} decreases in the wavelength ranges 4.4–5.7 *μ*m and 11.3–16.0 *μ*m.

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##### 3.3. Size Parameter Effects on Absorption and Scattering Efficiency Factors of a Single Droplet

Figure 4 presents the calculated absorption and scattering efficiency factors of n-octane droplets with different droplet radii at temperatures of 20°C. As shown in Figure 4, within the studied wavelength range except the absorption peaks around 3.5 *μ*m and 7 *μ*m, the absorption efficiency factor *Q*_{abs} increases with increasing radius of the spherical droplet. The changing trend of the scattering efficiency factor *Q*_{sca} is much more complicated as the droplet radius increases. *Q*_{sca} does not monotonically change with the increase of the droplet radius, but presents a vibrating trend around the value “2.0”, especially for large size droplets. With increasing droplet size parameter *χ*, the extinction efficiency factor (*Q*_{ext} = *Q*_{abs} + *Q*_{sca}) tends to its asymptotic value ‘2’ for a short-wave band. It can be seen from Figure 4 that scattering is weaker where absorption is greater for a given droplet radius. With increasing droplet size parameter *χ*, the extinction efficiency factor (*Q*_{ext} = *Q*_{abs} + *Q*_{sca}) tends to its asymptotic value ‘2’ for a short-wave band.

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##### 3.4. The Absorption and Scattering Coefficients of the Droplets System

The effects of temperature and size parameter on the absorption and scattering efficiency factors of an individual spherical droplet have been evaluated. The thermal radiation transfer properties of the droplet system in an air medium formed by a nozzle, approximated as a monodisperse system, were then, calculated using Mie theory. Figure 5 presents the Mie-calculated absorption and scattering coefficients of n-octane droplets-air systems with various volume fractions at temperatures of 20°C, with a droplet radius *r* = 10 *μ*m. The absorption coefficient *k*_{abs} represents the rate at which incident energy is absorbed by the droplet-air system. In addition, the scattering coefficient *k*_{sca} represents the rate at which incident energy is scattered in other directions by the system. When the temperature of the monodisperse system is equal, the radiation transfer properties of the droplet-air system are a superposition of the radiation properties of an individual droplet, as expressed in equation (14) and equation (15). Thus, as can be seen from Figure 5, the absorption and scattering coefficients of the droplet-air system increase linearly across the whole studied spectral range as the volume fraction increases.

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#### 4. Conclusions

In this study, the optical constants of liquid n-octane, the absorption and scattering efficiency factors of a single spherical droplet, and the absorption and scattering coefficients of droplets-air systems formed by sprays with various droplet radii and droplet volume fractions were investigated at 20, 50, and 80°C in the midinfrared wavelength range of 2.0–16.0 *μ*m. The absorption and refractive indices of liquid n-octane were obtained experimentally using the improved ellipsometry method. The absorption and scattering efficiency factors of n-octane droplets of various temperatures and radii, together with the absorption and scattering coefficients of droplet-air systems of various volume fractions, were then calculated using Mie theory. The following conclusions were drawn.

The refractive index of liquid n-octane presents an approximately linear relationship with temperature, while temperature has a much smaller effect on the absorption index. With increasing temperature, a reduction in density leads to a smaller refractive index and weaker absorption intensities of the n-octane, which leads to the decrease of the absorption efficiency factor of a single spherical n-octane droplet. According to the Mie theory, the scattering efficiency factor is closely related to the complex refractive index and size parameter of the droplets. For different size droplets, due to the change of the complex refractive index and size parameter, the scattering efficiency factors show different tendency with the increase of incident wavelength. With increasing droplet radius, the absorption efficiency factor increases within the studied wavelength range except the absorption peaks around 3.5 *μ*m and 7 *μ*m. The changing trend of the scattering efficiency factor is much more complicated as the droplet radius increases. With the increase in droplet size parameter , the extinction efficiency factor (*Q*_{ext} = *Q*_{abs} + *Q*_{sca}) oscillates around a value of 2 for a short-wave band.

#### Data Availability

All data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors have no conflicts of interest relevant to this manuscript.

#### Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (51576052).