# Error Sources and Distinctness of Materials Parameters Obtained by THz-Time Domain Spectroscopy Using an Example of Oxidized Engine Oil

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{−1}. While the average of measurements taken with different setup configurations did not yield significant differences for different TO times, a single, fixed setup would be able to discern all investigated oil species across the entire frequency range of 0.5–2.5 THz. The absorption coefficient measurement showed greater discernibility than the measurement of the refractive index.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Thermal Oxidation (TO) of Lubricating Oil

#### 2.2. THz-TDS Spectrometer

_{s}the refractive index of the sample, f the frequency, ${c}_{0}$ the speed of light in vacuum, ${d}_{c}$ the probe volume length, φ

_{S}(f) and φ

_{R}(f) the Fourier phases of the sample and reference measurements, respectively, ${n}_{w}=1.535$ is the refractive index of the window and $t\left(f\right)$ is the transmission ratio of the amplitude between sample and reference measurements at frequency f.

^{−1}for α ≈ 0.6 cm

^{−1}at 1 THz) for absorption coefficient. According to Equation (3) an error due to inaccuracy of the measurement of the window material, Δn

_{W}, has no influence on the accuracy of the refractive index because its influence is cancelled by the reference measurement of the empty cuvette. According to Equation (4) the absorption coefficient is weakly dependent on the window refractive index due to a modification of the reflection at the oil-window interface. Its influence is marginal, Δα = 0.16/cm·Δn

_{W}. An error Δn

_{W}~10

^{−3}results in an error of the absorption coefficient of 1.6 × 10

^{−4}cm

^{−1}. that is, one order of magnitude smaller than the influence of the error in sample thickness. Therefore, we neglected any error in n

_{W}. Another source of a systematic error is the speed of light, c

_{0}, in Equation (1). For the visible, the refractive index of air had been determined to n

_{Air}≈ 1.00027 [26], dependent on environmental parameters like temperature, humidity and pressure. For dry nitrogen, that was used to purge the setup during measurements, we assume a refractive index even closer to 1 due to absence of water vapor. As we have no means for measuring the refractive index of dry nitrogen at THz frequencies to this accuracy, we assumed n

_{N2}= 1, corresponding to a potential systematic offset of the cuvette length in the range of 4 µm at worst for the 15 mm cuvette. This is about 2.5 times smaller than the error due to disassembly and reassembly and would not influence the discriminability as it results in a systematic offset identical for all measurements.

## 3. Results

#### 3.1. Measurement Error Analysis

#### 3.1.1. Statistical Errors and Repeatability

_{c}= 15 mm. The general trend of the refractive index is a strictly monotonous decrease with increasing frequency for all TO times. The refractive index of different TO times differs only in the 4th digit. The total error due to cuvette positioning and due to statistical errors of the measurement technique caused fairly frequency-independent errors in the refractive index (e.g., Δn

_{rep}= 7 × 10

^{−5}(0.0048%) at 1.0 THz). On a first glance, the data suggest that the different TO times are well discriminable and that n increased with TO time. The refractive indices were 1.4666, 1.4667, 1.4670 and 1.4672 at 1.0 THz respectively for thermal oxidation times of 0 h, 48 h, 96 h and 144 h. However, the cuvette needed to be disassembled for cleaning between measurements of oils with different TO times, giving rise to additional systematic errors caused by preparation of the cuvette which are discussed in the next section. Figure 3 therefore does not allow for conclusions on discernibility of the TO times by THz measurements.

_{rep}= 1.4 × 10

^{−3}cm

^{−1}(0.22%) at 1.0 THz. Measurements of oils with different TO times seem well discernible. TO leads first to a reduction of absorption coefficient, indicating destruction of polar components in the lubricant, while it strongly increases again for TO times longer than 48 h.

^{th}refractive index measurement or i

^{th}absorption coefficient measurement and $\overline{x}$ is the mean value for the respective TO time. The term N-4 in the denominator takes into account that the four mean values are statistically dependent on the measurement values.

^{−1}, respectively for TO times of 0 h, 48 h, 96 h and 144 h. Highly significant differences (p < 0.0001) were found among thermal oxidation times across the 0.5–2.5 THz range. However, also these values are prone to systematic errors by dis- and reassembly and do not allow for judging on discernibility.

#### 3.1.2. Systematic Errors Due to Sample Preparation

_{N2}d

_{c}/c

_{0}, the length of the probe volume in the cuvette, d

_{c}, was determined, yielding a systematic error due to disassembly and reassembly of the 15 mm cuvette of σ = 10 µm ± 4 µm (CI: 19.6 µm) assuming n

_{N2}= 1. This is much more accurate than a mechanical length measurement with a caliper. The absolute error of the refractive index of the oils with different TO times due to disassembly and reassembly of the cuvette according to Equation (3) was Δn

_{Sys1}= 5.95 × 10

^{−4}(0.04%) at 1 THz and below 0.0405% over the whole range up to 2.3 THz, that is, about an order of magnitude larger than Δn

_{Rep}.

_{d}= 3.1 × 10

^{−3}cm

^{−1}at 1 THz according to Equation (4), yielding a total error of $\mathsf{\Delta}\alpha =\sqrt{\mathsf{\Delta}{\alpha}_{d}^{2}+\mathsf{\Delta}{\alpha}_{rep}^{2}}=3.4\times {10}^{-3}$ cm

^{−1}(0.56% for α = 0.6 cm

^{−1}at 1 THz). The measurements of the absorption coefficient remained discernible. The absorption index can therefore be considered as more robust for data analysis and sample comparison for samples with an absorption coefficient as low as about 0.5 cm

^{−1}.

#### 3.1.3. Systematic Errors Due to the Spectroscopy System

_{Sys2}= 1.9 × 10

^{−3}(0.13%), that is, 3 times larger than if the same setup (case II) was used and in excellent agreement with measurement errors obtained in [6] where standard deviations around 10

^{−3}(CI = 2 × 10

^{−3}) were reported. The CI of the absorption index was Δα

_{Sys2}= 0.052 cm

^{−1}at 1 THz (8.49%, averaged over all TO times), that is, 15 times larger than in case II causing a small overlap of the confidence intervals.

#### 3.2. Discussion of Error Sources

^{−1}as long as care is taken to properly align the samples.

_{c}, due to dis- and reassembly are of statistical nature. They can only be prevented if the same cuvette was used without dis- and reassembly for cleaning. Such errors would therefore not be present in a fixed, permanently filled cuvette for oil quality monitoring in a car engine. Such a fixed setup might not even experience repeatability errors from case I as long as other error sources, like thermal expansion of the probe volume can be prevented.

_{c}in Equations (3) and (4) decreases. Even if ratio Δd/d

_{c}remained constant, the error in absorption coefficient is dominated by the last term in Equation (4) for the samples investigated in this paper. This term scales as Δd/d

_{c}

^{2}, further reducing the error in absorption coefficient at large probe volume lengths.

_{R}. Since the sample features a refractive index n > 1 different from that of the reference measurement, that is, air (n ≈ 1), introduction of the sample increases the optical path length in the setup and, hence, modifies the Gaussian beam propagation and beam waist position as illustrated in Figure 6b. This leads to focusing errors, reducing the power transmitted from source to receiver and an overestimation of the absorption. Therefore, the sample needs to be thin vs. the Rayleigh length of the Gaussian beam. A particularly bad sample position for thick samples is in the vicinity of a sharp focal point, where the divergence angle of the beam is large, as for example, in ref. [8], which resulted in a fairly large standard deviation error of the refractive index Δn~10

^{−2}, being an order of magnitude worse than reported in this manuscript. For the samples studied here, we did not see a major influence of the sample thickness, proving that the probe volume length is much smaller than the Rayleigh length of the collimated beam and the system was well aligned.

## 4. Conclusions

_{rep}= 5 × 10

^{−5}for a refractive index around 1.467 and an error in absorption coefficient of <1 × 10

^{−3}cm

^{−1}for an absorption around 0.6 cm

^{−1}at 1 THz for a cuvette length of 15 mm. Sample preparation caused slight variations in the probe volume length. At 1 THz, the refractive index experienced an error of Δn

_{Sys1}= 3 × 10

^{−4}and the absorption coefficient experienced an error of Δα

_{Sys1}= 1.59 × 10

^{−3}cm

^{−1}. While the refractive index error caused an overlap of the 95% confidence intervals, the error in α is much smaller than the differences in absorption of all investigated TO times. Mean absorption values at 1.0 THz were 0.5989, 0.5657 0.6099 and 0.6330 cm

^{−1}, respectively, for TO times of 0 h, 48 h, 96 h and 144 h. Finally, variations in the setup configuration were examined by using cuvette lengths between 5 and 15 mm and variations in the alignment of the setup, averaging time of spectra and window size. Different measurement configurations caused confidence intervals in the refractive index in the third digit (Δn

_{Sys2}= 1.9 × 10

^{−3}= 0.13%) and in the second digit of the absorption coefficient (Δα

_{Sys2}= 0.052 cm

^{−1}= 8.49%) for all investigated samples. While this error source causes severe problems for comparing results from different laboratories, it usually causes a systematic offset rather than a statistical error and therefore does not affect the discernibility if a single, fixed setup is used, where the error is at least an order of magnitude smaller. We therefore conclude that THz-TDS demonstrated good potential for distinguishing differences in engine oil caused by thermal oxidation. Based on this study, continued exploration of THz-TDS for engine oil contaminants is warranted to determine the extent of the THz-TDS potential to distinguish other engine oil contaminants.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Liu, H.-B.; Chen, Y.; Bastiaans, G.J.; Zhang, X.-C. Detection and identification of explosive RDX by THz diffuse reflection spectroscopy. Opt. Express
**2006**, 14, 415–423. [Google Scholar] [CrossRef] [PubMed] - Abdul-Munaim, A.M.; Méndez Aller, M.; Preu, S.; Watson, D.G. Discriminating gasoline fuel contamination in engine oil by terahertz time-domain spectroscopy. Tribol. Int.
**2018**, 119, 123–130. [Google Scholar] [CrossRef] - Jeon, T.-I.; Grischkowsky, D. Characterization of optically dense, doped semiconductors by reflection THz time domain spectroscopy. Appl. Phys. Lett.
**1998**, 72, 3032. [Google Scholar] [CrossRef] - Piesiewicz, R.; Jansen, C.; Wietzke, S.; Mittleman, D.; Koch, M.; Kürner, T. Properties of Building and Plastic Materials in the THz Range. Int. J. Infrared Millim. Terahertz Waves
**2007**, 28, 363–371. [Google Scholar] [CrossRef] - Abdulmunem, O.M.; Born, N.; Mikulics, M.; Balzer, J.C.; Koch, M.; Preu, S. High accuracy terahertz time-domain system for reliable characterization of photoconducting antennas. Microw. Opt. Technol. Lett.
**2016**, 59, 468–472. [Google Scholar] [CrossRef] - Withayachumnankul, W.; Fischer, B.M.; Lin, H.; Abbott, D. Uncertainty in terahertz time-domain spectroscopy measurement. J. Opt. Soc. Am. B
**2008**, 25, 1059–1072. [Google Scholar] [CrossRef] - Withayachumnankul, W.; Naftaly, M. Fundamentals of Measurement in Terahertz Time-Domain Spectroscopy. J. Infrared Millim. Terahertz Waves
**2014**, 35, 610–637. [Google Scholar] [CrossRef] - Yang, F.; Liu, L.; Song, M.; Han, F.; Shen, L.; Hu, P.; Zhang, F. Uncertainty in Terahertz Time-Domain Spectroscopy Measurement of Liquids. J. Infrared Millim. Terahertz Waves
**2017**, 38, 229–247. [Google Scholar] [CrossRef] - Ornik, J.; Watson, D.G.; Balzer, J.C.; Koch, M. Experimental characterization of dielectric parameter extraction uncertainty for low absorbing liquids using THz TDS. In Proceedings of the 42nd International Conference on Infrared, Millimeter and Terahertz Waves (IRMMW-THz), Cancun, Mexico, 27 August–1 September 2017; pp. 1–2. [Google Scholar]
- Lansdown, A.R. Lubrication and Lubricant Selection a Practical Guide, 3rd ed.; The American Society of Mechanical Engineers: New York, NY, USA, 2004. [Google Scholar]
- Soleimani, M.; Sophocleous, M.; Glanc, M.; Atkinson, J.; Wang, L.; Wood, K.; Taylor, I. Engine oil acidity detection using solid state ion selective electrodes. Tribol. Int.
**2013**, 65, 48–56. [Google Scholar] [CrossRef][Green Version] - Mortier, R.M.; Fox, M.F.; Orszulik, S.T. Chemistry and Technology of Lubricants, 3rd ed.; Springer: New York, NY, USA, 2010. [Google Scholar]
- Macian, V.; Tormos, B.; Gomez, Y.A.; Salavert, J.M. Proposal of an FTIR Methodology to Monitor Oxidation Level in Used Engine Oils: Effects of Thermal Degradation and Fuel Dilution. Tribol. Trans.
**2012**, 55, 872–882. [Google Scholar] [CrossRef][Green Version] - Faure, D.; Hipeaux, J.C.; Guevellou, Y.; Legros, A. Oxidation stability of gasoline engine lubricants: Effect of base-oil chemistry in laboratory and engine tests. Lubr. Sci.
**1999**, 5, 337–360. [Google Scholar] [CrossRef] - Zuidema, H.H. The Performance of Lubricating Oils; Reinhold Publishing Corporation: New York, NY, USA, 1959. [Google Scholar]
- Ikeda, T.; Matsushita, A.; Tatsuno, M.; Minami, Y.; Yamaguchi, M.; Yamamoto, K.; Tani, M.; Hangyo, M. Investigation of inflammable liquids by terahertz spectroscopy. Appl. Phys. Lett.
**2005**, 87, 034105. [Google Scholar] [CrossRef] - Al-Douseri, F.M.; Chen, Y.; Zhang, X.-C. THz wave sensing for petroleum industrial products. Int. J. Infrared Millim. Terahertz Waves
**2006**, 27, 481–503. [Google Scholar] [CrossRef] - Naftaly, M.; Miles, R.E. Terahertz Time-Domain Spectroscopy for Material Characterization. Proc. IEEE
**2007**, 95, 1658–1665. [Google Scholar] [CrossRef] - Tian, L.; Zhou, Q.; Jin, B.; Zhou, K.; Zhao, S.; Shi, Y.; Zhang, C. Optical property and spectroscopy studies on the selected lubricating oil in the terahertz range. Sci. China Ser. G
**2009**, 52, 1938–1943. [Google Scholar] [CrossRef] - Tian, L.; Zhao, K.; Zhou, Q.-L.; Shi, Y.-L.; Zhang, C.-L. Quantitative Analysis for Monitoring Formulation of Lubricating Oil Using Terahertz Time-Domain Transmission Spectroscopy. Chin. Phys. Lett.
**2012**, 29, 043901-1–043901-3. [Google Scholar] [CrossRef] - Abdul-Munaim, A.M.; Reuter, M.; Koch, M.; Watson, D.G. Distinguishing Gasoline Engine Oils of Different Viscosities using Terahertz Time-Domain Spectroscopy. Int. J. Infrared Millim. Terahertz Waves
**2015**, 36, 687–696. [Google Scholar] [CrossRef] - Abdul-Munaim, A.M.; Reuter, M.; Abdulmunem, O.M.; Balzer, J.C.; Koch, M.; Watson, D.G. Using terahertz time-domain spectroscopy to discriminate among water contamination levels in diesel engine oil. Trans ASABE
**2016**, 59, 795–801. [Google Scholar] [CrossRef] - Ofunne, G.C. Studies on the Ageing Characteristics of Automotive Crankcase oils. Tribol. Int.
**1989**, 22, 401–404. [Google Scholar] [CrossRef] - Egharevba, F.; Maduako, A.U. Assessment of oxidation in automotive crankcase lube oil: Effects of metal and water activity. Ind. Eng. Chem. Res.
**2002**, 41, 3473–3481. [Google Scholar] [CrossRef] - George, D.K.; Markelz, A.G. Terahertz Spectroscopy of Liquids and Biomolecules. In Terahertz Spectroscopy and Imaging; Peiponen, K.-E., Zeitler, A., Kuwata-Gonokami, M., Eds.; Springer: Berlin, Germany, 2013; pp. 229–250. [Google Scholar]
- Ciddor, P.E. Refractive index of air: New equations for the visible and near infrared. Appl. Optics
**1996**, 35, 1566–1573. [Google Scholar] [CrossRef] [PubMed] - Hurlbert, S.H. Pseudoreplication and the design of ecological field experiments. Ecol. Monogr.
**1984**, 54, 187–211. [Google Scholar] [CrossRef] - SAS. SAS Enterprise Guide 7.1; SAS Institute: Cary, NC, USA, 2016. [Google Scholar]
- Krüger, M.; Funkner, S.; Bründermann, E.; Havenith, M. Uncertainty and Ambiguity in Terahertz Parameter Extraction and Data Analysis. J. Infrared Millim. Terahertz Waves
**2011**, 32, 699–715. [Google Scholar] [CrossRef] - Pupeza, I.; Wilk, R.; Koch, M. Highly accurate optical material parameter determination with THz time-domain spectroscopy. Opt. Express
**2007**, 15, 4335–4350. [Google Scholar] [CrossRef] [PubMed]

**Figure 3.**(

**a**) Mean refractive index (left axis) and mean absorption coefficient (right axis) of gasoline engine oil (5W20) oxidized over four different times from three measurements of the 15 mm cuvette of gasoline engine oil with 95% confidence interval bars; (

**b**) Relative values for the CI of the refractive index (left axis) and absorption coefficient (right axis).

**Figure 4.**(

**a**) Mean refractive index and absorption coefficient of gasoline engine oil (5W20) oxidized over four different times from three measurements of the 15 mm cuvette; (

**b**) 95% confidence intervals for n and α, taking the CI of the probe volume thickness into account.

**Figure 5.**(

**a**) Refractive index and absorption coefficient with 95% confidence intervals from 5 different measurement setups; (

**b**) 95% confidence interval with N = 5 different setups.

**Figure 6.**Examples of beam propagation errors caused by a sample. Black: beam propagation without sample. Red: deviation of the beam propagation with sample inserted. (

**a**) beam walk off due to a small inclination angle of the sample and (

**b**) focusing error due to imperfect beam collimation or by a very long sample with a thickness longer than the Rayleigh length, d

_{c}> z

_{R}, causing a shift of the Gaussian beam waist in the measurement path (indicated by f) and a focal shift at the receiver. Both cases not only cause pointing or focusing errors, they also alter the beam profile, leading to reduced transmission form source to receiver. Deflection by the silicon lenses is not shown here.

Repeatability Error (Case I) | Total Error Incl. Sample Preparation (Case II) | Inter-System Comparability Error (Case III) | |
---|---|---|---|

Δn at 1 THz | 0.0048% | 0.04% | 0.13% |

Δα at 1 THz | 0.22% | 0.56% | 8.49% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Méndez Aller, M.; Abdul-Munaim, A.M.; Watson, D.G.; Preu, S.
Error Sources and Distinctness of Materials Parameters Obtained by THz-Time Domain Spectroscopy Using an Example of Oxidized Engine Oil. *Sensors* **2018**, *18*, 2087.
https://doi.org/10.3390/s18072087

**AMA Style**

Méndez Aller M, Abdul-Munaim AM, Watson DG, Preu S.
Error Sources and Distinctness of Materials Parameters Obtained by THz-Time Domain Spectroscopy Using an Example of Oxidized Engine Oil. *Sensors*. 2018; 18(7):2087.
https://doi.org/10.3390/s18072087

**Chicago/Turabian Style**

Méndez Aller, Mario, Ali Mazin Abdul-Munaim, Dennis G. Watson, and Sascha Preu.
2018. "Error Sources and Distinctness of Materials Parameters Obtained by THz-Time Domain Spectroscopy Using an Example of Oxidized Engine Oil" *Sensors* 18, no. 7: 2087.
https://doi.org/10.3390/s18072087