Interpreting the spectral reflectance of advanced high strength steels using the Davies’ model

Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

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Interpreting the spectral reflectance of advanced high strength steels using the Davies’ model K. Lin∗, K.J. Daun Department of Mechanical and Mechatronics Engineering, University of Waterloo, 200 University Ave W, Waterloo, Ontario N2L 3G1, Canada

a r t i c l e

i n f o

Article history: Received 2 August 2019 Revised 6 December 2019 Accepted 7 December 2019 Available online 10 December 2019 Keywords: Advance high strength steel Spectral emissivity Rough surface Kirchhoff-Helmholtz integral Wavelet filtering Pyrometry

a b s t r a c t This study investigates the relationship between surface roughness and spectral reflectivity of DP980, an advanced high strength steel (AHSS) alloy. Five surface states were considered: as-received, polished, roughened, and two coupons annealed in a reducing atmosphere. Radiative properties were measured using a UV–Vis-NIR spectrophotometer and two FTIR spectrometers for directional-hemispherical and specular reflectance measurements, while surface roughness is determined using an optical profilometer. These measurements were interpreted in the context of a theoretical relationship derived from Davies’ model for specular reflectance. The reflectance of the polished coupon most closely matches this theory, and the remaining coupons become more aligned with this relationship after large-scale roughness has been removed from the profilograms using a wavelet filter. The results show that directionalhemispherical reflectance can be predicted using Davies’ model with local scale roughness as opposed to macroscopic artifacts, although this model is intended to capture the variation of specular reflectance with respect to wavelength. © 2019 Elsevier Ltd. All rights reserved.

λB , and ε λB /ε λA is the ratio of the spectral emissivities at these

1. Introduction Vehicle weight reduction is one of the most effective means for improving fuel economy and reducing energy consumption. This has motivated development of advanced high strength steels (AHSS), which have superior mechanical properties compared to traditional high strength low alloy (HSLA) steels and thereby permit the use of thinner and lighter parts that provide equivalent crash-performance. Achieving these properties requires a carefully-controlled intercritical annealing process that transforms the microstructure of the steel, thereby increasing both its tensile strength and ductility. However, thermal excursions during annealing, mainly attributed to pyrometry errors caused by wavelengthdependent variations in spectral emissivity, sometimes result in substandard steel mechanical properties [1–3]. The steel industry most often uses two-wavelength pyrometers to control AHSS strip temperature during intercritical annealing, which, after making Wien’s approximation [4], is estimated by

  5  1  1 JλA λA ελB T = C2 − /ln λB λA JλB λB ελA

(1)

where C2 = hc0 /kB = 1.439 μm·K, JλA and JλB are the spectral incandescences measured at the two detection wavelengths, λA and ∗

Corresponding author. E-mail address: [email protected] (K. Lin).

https://doi.org/10.1016/j.jqsrt.2019.106796 0022-4073/© 2019 Elsevier Ltd. All rights reserved.

wavelengths, sometimes called the “e-slope”, which must be specified. Thus, understanding how ε λ varies with respect to λ as the steel is processed is critical for accurate pyrometry. The evolution of spectral emissivity (and thus the e-slope) with thermal processing can often be categorized into two scenarios. If the steel is heated in an oxidizing atmosphere, a near-uniform oxide layer may form, causing a phase shift between incident and reflected waves. Constructive and destructive interference between these waves leads to coherent wavelike patterns in ε λ with respect to λ that shift with annealing time as the oxide layer grows [5]. This phenomenon had been observed in studies carried out on steels heated within an oxidizing atmosphere [5–7]. In contrast, when the steel is heated within a reducing atmosphere (e.g. 95% N2 /5% H2 ), as is done as part of the galvanizing process, the surface topography is often dominated by macroscopic rolling artifacts, and microscopic divots and tears, as well as isolated submicron oxide nodules [8]. The nature and extent of oxidation depends on the alloy content of the steel, the thermal processing history, and the composition of the atmosphere. For example, the moisture content as specified by the dew point determines the effectiveness of the reduction reaction MOy + yH2  M+yH2 O, where M is a metal atom, and thus the propensity for the steel surface to oxidize during annealing [9]. For transformation-induced plasticity aided (TRIP-aided) steel, for example, surface oxides tend to form a thin film under low dew point atmosphere, while under high dew

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K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

point atmosphere the oxide appeared as discontinuous separate islands [10]. In the case of a slightly rough surface, the effect of surface topography on radiative properties may be captured by a model proposed by Davies [11],

   4π σ 2 ρλ = ρλ,s exp − λ

(2)

where σ is the root-mean-square (RMS) roughness, and ρ λ and ρ λ,s are the specular reflectances of a slightly rough and a hypothetical perfectly-smooth surface, respectively. Temperaturedependent properties associated with changing electron mobility in the metal can be accommodated by modeling ρ λ,s using the Hagen-Rubens relation [12]

 r T 0 . 5  r T ( ) e( ) ρλ,s (T ) = 1 − 36.5 e − 464 λ λ

(3)

where

re = r0 + α × r0 (T − T0 )

(4)

is the temperature-dependent resistivity, ro denotes resistivity at a reference temperature T0 , and α is the temperature coefficient of resistivity. Pyrometry requires the spectral, directional emissivity, which is related to the directional-hemispherical reflectance through Kirchhoff’s law,

ελ (θ ) = 1 − ρλ,d−h (θ )

(5)

which is often required at normal or near-normal incidence for pyrometers measuring AHSS strip temperatures. In the case of an optically-smooth surface, the directional-hemispherical reflectance equals the specular reflectance obtained from Eq. (2). Eq. (2) is derived assuming that the surface roughness obeys a Gaussian distribution as well as Kirchhoff’s approximation, which requires the surface to be locally-smooth (i.e. the radius of curvature at each point of surface is large relative to the wavelength). The application of Eq. (2) to metallic surfaces is limited to σ /λ  1, under which conditions the surface can be modeled as opticallysmooth and therefore a specular reflector [13]. A major challenge in implementing this theory concerns how to characterize σ . In their study of aluminum alloys, Wen and Mudawar [14] measured σ using a contact profilometer. They applied Eq. (2) when σ /λ < 0.2, satisfying the so-called “Fraunhofer criterion” [15], while for rougher surfaces or shorter wavelengths they propose the theory of Agababov [16], which models the surface as a series of cavities between asperities, and radiant enclosure theory is used to predict how these cavities enhance ε λ . In the context of advanced high strength steel, Somveille et al. [17] examined DP780 and DP980 coupons heated in a reducing atmosphere of 95%/5% N2 /H2 and a dew point of −30 °C, which were then quenched at intermediate heating times using process gas. They found that ε λ evolved with heating time even though the RMS roughness found from contact profilometry remained constant. Ham et al. [8,18] obtained similar results for dual phase and TRIP AHSS processed under similar conditions. They hypothesized that the surface roughness measured by profilometry is governed by near-macroscopic surface artifacts imparted by the rolling/forming process, while radiative properties are sensitive to submicron features associated with the formation and growth of oxide nodules. They supported their hypothesis by demonstrating an empirical correlation between measured spectral emissivity and the roughnesses obtained through wavelet filtering of optical profilograms measured on ex situ samples. This study further explores the treatment by Ham et al. [8], and how it relates to the underlying theory by reviewing

the theoretical underpinnings of Davies’ model and comparing the predicted and measured diffuse and specular refectance of dual-phase AHSS (DP980) coupons via Davies’ model with filtered and measured roughness. Five different surface states were investigated: as-received, polished, roughened, as well as two coupons annealed within a reducing atmosphere using a galvanizing simulator [19]. The surface profiles of the coupons were characterized by optical profilometry, as well as optical, electron, and atomic force microscopy. Ex-situ directional-hemispherical reflectance measurements were done using a UV–Vis-NIR spectrophotometer (0.25–2.5 μm) and an FTIR infrared spectrometer (2–25 μm), both equipped with integrating spheres. Additional specular reflectance measurements were done using a Nicolet 6700 FTIR (2.5–15 μm) at an incident angle of 30° These measurements are interpreted in the context of the optical profilograms and the Kirchhoff-Helmholtz diffraction theory [11] that underlies Eq. (2). The results show that reflectances calculated using Davies’ model with wavelet-filtered roughnesses closely match directionalhemispherical reflectances measured using an integrating sphere at wavelengths longer than 2 μm for all samples, even though the model is supposed to represent specular reflectance. However, the modeled specular component is larger than the measured values for all samples except for the polished sample, indicating that the wavelet-filtered roughness corresponds to a hypothetical surface having a larger specular component and smaller diffuse component compared to the true surface. Nevertheless, Davies’ model with wavelet-filtered roughness could prove useful for parameterizing the wavelength-dependence of near-normal emissivity for multiwavelength pyrometry, which is the primary objective of this analysis. 2. Scattering of an electromagnetic wave from a rough surface When an electromagnetic wave impinges a randomly rough surface, as shown in Fig. 1, the waves are reflected off of each point source and interfere constructively and destructively in the farfield. The surface profile, ζ (x,y), determines the manner in which the waves interfere, and thus the far-field spectral reflectivity of the surface. In particular, the scattered far-field wave E2 at any point P at a distance r far away from the surface is given by the Helmholtz integral [20]

E2 ( P ) =

1 4π





E A

  eikr ∂ E ∂ eikr − dA ∂n r r ∂n

(6)

where both time variation and polarization of the wave is ignored. This analysis hinges on modeling the local reflection of the wave at every point on the surface. If the surface is locally flat relative to the incident wave, the integrand can be derived by invoking the Kirchhoff boundary approximation

Es = (1 + ψ )E1 ,

∂ Es = (1 − ψ )E 1 k1 · n ∂n

(7)

where Es is the field intensity at the surface, k1 is the incident wave vector, n is the local surface normal vector, and ψ is found by locally-applying Fresnel’s equation. For a rectangular surface having domains x ∈ [-X, X] and y ∈ [-Y, Y], after some manipulation, it can be shown that the specular reflectance is given by

 2  E  ρλ cos θ1 =  2 2  E 20

=

2

F A2

X X Y Y





eivx (x1 −x2 )+ivy (y1 −y2 ) eivz (ζ1 −ζ2 ) dx1 dx2 dy1 dy2

−X −X −Y −Y

(8)

K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

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Fig. 1. Schematic of E-M wave scattering from a rough surface. Points A and B show cases where the radius of curvature is large and small relative to the wavelength of interest.

where E20 is the electric field scattered from a smooth surface, v = k1 −k2 is the difference between the incident and scattered wave vectors, ζ 1 = ζ (x1 ,y1 ), ζ 2 = = ζ (x2 ,y2 ), and F is a coefficient that depends on θ 1 and θ 2 . If ζ is normally-distributed and isotropic, which often occurs when the surface roughness arises from a random process, and assuming σ /λ1, the integral in Eq. (8) can be carried out to yield the specular reflectance,

 

ρλ = ρλ,s exp −

4π σ cos θ1

2 

λ

(9)

which, in the case of near-normal incidence, simplifies to Eq. (2). The above result relies on the validity of Kirchhoff’s approximation, which requires the local radius-of-curvature, R, of the surface to be larger than the projection of the incoming wavelength onto the surface [21]. This is locally-enforced by [22]

2k1 |R|sin

3

φg > 1

(10)

where R is the local radius of curvature, k1 = 2π /λ is the modulus of the incident wave vector, and φ g is local grazing angle of incident field. This criterion can be globally enforced by 3

2k1 RRMS sin

φ>1

(11)

where RRMS denotes the root-mean-square radius of curvature and φ = π /2−θ 1 . If ζ is normally-distributed,



RRMS =

τ2



1+

√ 12σ

2σ 2

τ2

3/2

2  1 / 2 = ζ  (x )

√ 2σ

τx

nearly constant. Ham et al. [8] suggest that this roughness measurement is dominated by artifacts imparted by the rolling process that do not influence the spectral emissivity in the infrared. They employed wavelet filtering of optical profilograms to show the existence of a subscale roughness, ~ O(0.1 μm), that correlated with increases in the measured spectral emissivity as annealing progresses. The present study adapts this procedure as described below.

(12)

where τ is the correlation length, defined as the distance at which the autocorrelation between two points reaches 1/e = 0.3678. Finally, for a randomly rough surface, the correlation length can be related to the RMS surface slope by [23]



Fig. 2. Heating cycles for the annealed coupons [17].

(13)

where ζ ’(x) is the derivative of surface height in the x direction, which can be calculated from the surface profile data. The same procedure is applied to obtain the correlation length in the y direction, and the equivalent correlation length τ is taken as the average value of τ x and τ y . The above analysis presumes that the surface profile can be adequately defined by a single set of parameters, σ , and τ , while, in reality, it is more accurately represented by the superposition of different roughness scales that affect the spectral emissivity in different ways. Both Somveille et al. [17] and Ham et al. [8] found that spectral, directional-hemispherical reflectances of steel coupons evolved as they were annealed in a reducing atmosphere, even though the profilometry-derived roughness, σ ~ O(1 μm), remained

3. Coupon preparation and measurement techniques Coupons were cut from a coil of cold-rolled DP980, having an elemental composition that conforms to ASTM A1079-17 as indicated in Table 1 and an Mn/Si ratio of 25.7 (mass). One coupon is left in the as-received state, while a second surface was polished using a series of five polishing wheels with progressively finer grit and particle size (240 grit SiC, 320 grit SiC, 400 grit SiC, 600 grit SiC, diamond compound, Gamma alumina), resulting in a mirrorlike finish. The roughened coupon was made by abrading the surface with a 30 μm sand paper. Finally, two as-received (cold-rolled) coupons were annealed within a 95%/5% N2 /H2 atmosphere at a dew point of −30 °C according to the schedule shown in Fig. 2, using the galvanizing simulator described in Ref. [24]. The optical micrographs of the surfaces are shown for polished (Fig. 3(a), (b)), as-received (Fig. 3(c), (d)) roughened (Fig. 3(e), (f)), annealed “A” (Fig. 3(g), (h)) and annealed “Full” (Fig. 3(i), (j)); rolling striations are clearly visible in the as-received coupon, which have been removed by polishing or are obscured by roughening or oxide growth in the remaining coupons. Surface topography was measured using an optical surface profiler (WYKO NT1100) with a ver-

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K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796 Table 1 Chemical composition of tested coupon, DP980 [25]. % mass

C

P

S

Mn+Al+S

Cu

Ni

Cr+Mo

V+Nb+Ti

DP980

0.23

0.080

0.015

6.00

0.20

0.50

1.40

0.35

nealing. The EDS results indicate that the oxide nodules consist of manganese and silicon oxides, which is consistent with predictions obtained from an Ellingham diagram under annealing conditions. The statistical data obtained from the optical profilogram for the annealed “A” surface is compared to the topography obtained from an atomic force microscope (AFM, Pacific Nanotechnology NanoRTM ), with a lateral resolution of 0.02 μm and scanning area of 10μm × 10μm, to confirm the physical relevance of surface statistics. It should be noted, however, the AFM can only be used to characterize a very small sample area, which may not be representative of the overall surface. Consequently, the surface profiles of the tested coupons in this study are acquired from the optical profiler measurements. All coupons were cleaned in isopropanol, and then rinsed with acetone and distilled water to remove any additional particles and contaminants before the optical measurement. The near-normal directional-hemispherical spectral reflectance of each coupon was measured using a UV–Vis-NIR spectrophotometer (0.25–2.5 μm) and a Bruker Invenio-R spectrometer (2.5–25 μm), both equipped with integrating spheres, while the specular reflectances an incident angle of 30° were obtained using a Nicolet 6700 FTIR (2.5–15 μm) equipped with a VeeMAX III variable angle specular reflection apparatus. 4. Results and discussion

Fig. 3. Optical microscopy (left) and optical profilograms (right) of the polished (a, b), as-received (c, d), roughened (e, f), annealed “A” (g, h) and “Full” (i,j) coupons.

tical measurement resolution of 0.1 nm and horizontal resolution of 2 μm, which provided both the RMS roughness, σ , and the RMS surface slope in Eq. (13). Optical profilograms of these coupons are also shown in Fig. 3. Fig. 4 shows an SEM image of the DP980 “A” coupon, highlighting the oxide nodules that form during an-

Eqs. (11)–(13) are first employed to investigate the validity of the Kirchhoff approximation for the tested coupons, assuming a directional-hemispherical grazing angle of 80° (incident angle of 10°) since the incident light on the coupons is near-normal. The RMS radius of curvature is calculated through Eq. (12). The profilometer-inferred σ values are shown in Table 2. Fig. 5 verifies that this condition is satisfied for all coupons, although it is most in line with the polished coupon. The measured spectral, directional-hemispherical, and specular reflectances for the five coupons are plotted in Fig. 6(a), which qualitatively confirm the trends predicted by Eq. (2). There is a larger variation in the specular reflectances between the surfaces compared to the hemispherical reflectances. It is also observed that the measured specular reflectances of all coupons are generally lower than the hemispherical ones except for the polished surface, indicating the fact that hemispherical reflectance would approximate the specular reflectance when the surface is optically-smooth. For both types of measurements on the un-annealed coupons, the polished coupon has the highest reflectance, followed by the asreceived coupon and roughened coupon. The spectral reflectances

Fig. 4. Scanning electron micrographs of the annealed “A” coupon, highlighting the oxide nodules formed during annealing. The image on the right corresponds to the white boxed region in the image on the left.

K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

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Table 2 RMS roughness inferred by linear regression of measured ρ d-h and ρ sp via Eq. (14), along with surface RMS roughness inferred from the profilograms before and after filtering.

Polished As-received Roughened Annealed, A Annealed, Full

σ profiler [μm]

σ optical,

0.08 0.61 1.55 0.37 0.34

0.091 0.092 0.094 0.131 0.129

Fig. 5. Verification of the applicability of the Kirchhoff approximation using Eq. (11) using profilometry-derived surface parameters.

of the two annealed coupons, “A” and “Full”, are nearly indistinguishable, and stand out from the others; they have a lower reflectance at short wavelengths, and approach the reflectance of the polished coupon at longer wavelengths. Further insights into these trends is found by taking the logarithm of Eq. (9) and rearranging to get



ln

 4π σ 2 ρλ =− cos θ ρλ,s (λ ) λ

(14)

where the spectral reflectance from smooth surface, ρ λ,s , is found from Eq. (3)-(4) with re, 0 = 1.43×10−7 ·m (carbon electrical steel at room temperature [26]). The incident angles, θ , are set

d -h

[μm]

σ filtering [μm]

σ optical, sp [μm]

– 0.106 0.103 0.138 0.118

0.084 0.228 1.061 0.304 0.301

equal to 0° and 30° for ρ λ,d-h and ρ λ,sp , respectively. Replotting the reflectance data accordingly in Fig. 6(b) shows nearly linear trends for the non-annealed coupons, while the data obtained from the annealed coupon is more curved. In contrast, replotting the reflectance data accordingly in Fig. 7(b) shows nearly linear trends for only polished coupon and the data obtained from the other coupons are more curved, particularly for the roughened coupon. The σ values obtained from linear regression of ln(ρ λ /ρ λ,s ) vs 1/λ2 are shown in Table 2, and are denoted “optical roughness”. For hemispherical measurements, the RMS roughness measured with the optical profilometer, σ profiler , closely matches the one found from Eq. (14) for the polished coupon, while for the other coupons the profilometer-derived roughnesses, σ optical,d-h , are much larger. For specular measurements, the optically-derived roughnesses, σ opical,sp are generally close to the values derived from profilometer measurements. Next, we use 2-D wavelet filtering to remove the low-frequency micrometer-scale roughness artifacts, following Ham et al. [8]. In this work the Daub2 wavelet transform technique was adopted wherein the surface is decomposed into four levels having different frequencies [27,28]; the 2-D surface profile is then reconstructed using the highest frequency wavelets (first level wavelets) as illustrated in Fig. 8 for the as-received surface. The RMS roughness values obtained by reconstructing the surface with different levels of wavelets is shown in Fig. 9. The RMS roughness of the surface decreases as the level of reconstructing wavelets decreases, and the surface composed of first level wavelet approaches the opticallyderived values. The surface topography of the roughened coupon before and after wavelet filtering is shown in Fig. 10. As indicated in Table 2, the filtered RMS roughness values, σ , are much closer to the ones obtained via Eq. (14) compared to the unfiltered values. In order to test the robustness of surface data measured by optical profilometry and the profile derived from wavelet filtering, the statistical distribution of surface height is compared with that measured by the AFM for the annealed “A” coupon, as shown

Fig. 6. Measured spectral, directional-hemispherical reflectance obtained using spectrometers equipped with integrating spheres. (a) Plot of ρ λ vs λ and (b) plot of ln(ρ λ /ρ λ,s ) vs 1/λ2 . The trends found through linear regression via Eq. (14) are shown as dashed lines. The reflectance of the “A” and “full” coupons are nearly indistinguishable.

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K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

Fig. 7. Measured spectral, specular reflectance obtained using the specular reflectance apparatus. (a) Plot of ρ λ vs λ and (b) plot of ln(ρ λ /ρ λ,s ) vs 1/λ2 . Since the incident angle is 30° instead of normal incidence, Eq. (9) is used and the trends found are shown as dashed lines. There are larger differences between the specular reflectances between each coupon compared to the hemispherical reflectances. The reflectance of the “A” and “full” coupons are nearly indistinguishable.

Fig. 8. Surface profiles obtained using wavelet filtering for the as-received coupon.

Fig. 9. Comparison of surface RMS roughness composed of different levels of wavelet for the roughened coupon. The RMS roughness decreases as the level of reconstructing wavelet decreases, and the value approaches that of optically-derived one when the surface is composed of the 1st level wavelet.

in Fig. 11. The AFM was set to contact mode, with a sampling length of 10 μm, and has a lateral and vertical resolution of 0.02 μm and 0.1 nm, respectively. It is therefore capable of resolving the oxide nodules and other submicron features that affect the ra-

diative properties. The roughness distributions found through the AFM and filtered optical profilogram are very similar, while the RMS roughnesses are nearly identical. These results suggest that the surface statistics found from the filtered optical profilograms may be physically-relevant. The histograms for the unfiltered surface profile, shown in Fig. 12, are distinguished by how they were processed: the roughness of the polished coupon is caused by random, cumulative manufacturing processes and most closely approximates a normal distribution, while surface artifacts imparted on the as-received (rolling) and roughened (abraded) surfaces exhibit anisotropicallydistributed surface heights, and are much larger than the opticallyinferred RMS roughness. The surface roughness of the annealed coupon follows a near-normal distribution, and exhibits less skewness than the as-received and roughened coupons. After removing the directional-hemispherical roughness ζ becomes normallydistributed for the as-received, roughened, and annealed coupons, suggesting that the surface features at the micrometer length scale originate through random processes, and, in this respect, are more in line with the assumptions that underlie Eq. (2). It is not clear, however, whether the assumption of “local flatness” is satisfied by the filtered images. Scanning electron microscopy of the annealed specimen, shown in Fig. 4, reveal that the oxide nodules that partly constitute the fine surface roughness

K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

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Fig. 10. Surface topography of the roughened coupon before (left) and after (right) the removal of directional-hemispherical texture.

Fig. 11. Comparison of statistical heights between the filtered surface profile (left) with that of measured by atomic force microscopy (right) for the annealed “A” coupon. Although the optical profiler cannot capture the detained surface data at lateral resolutions below 2μm, similar distributions are observed from both profiles that result in almost same surface height deviation, σ . This demonstrates the robustness of using optical profiler and filtering techniques to obtain the overall surface parameters.

Fig. 12. Probability density of surface roughness from filtered and unfiltered profilograms.

have diameters O(100 nm), which would appear to violate this criterion. The measured spectral, directional-hemispherical and specular reflectances, along with the modeled values found using Eq. (2), are plotted in Fig. 13(a)–(e). For the directional-hemispherical measurement, a normal incident angle (θ =0° ) is assumed in the model, which is close to the near-normal angle of incidence used in the spectrometers, and is often the case for pyrometers used

to measure steel strip temperature during continuous galvanizing. For the specular measurement, an incident angle of 30° is used. Fig. 13(a) compares the measured reflectances of the polished surface to the modeled values obtained from Eq. (2) using the profilometer-derived RMS roughness. In this case the modeled specular ρ λ matches the measured directional-hemispherical reflectance for wavelengths that satisfy the Fraunhofer criterion, σ /λ < 0.03, which, for an RMS roughness of 0.10 μm, is λ > 2 μm; over

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K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

Fig. 13. Comparison of measured and estimated reflectance for (a) polished, (b) as-received, (c) roughened, (d) annealed “A” and (e) annealed “Full” coupons. The solid red lines represent the measured directional-hemispherical reflectance at near normal incident. The long dash line in (a) denotes the estimation with constant ρ λ,s . The solid black line in each figure represent the measured specular reflectance at 30° of incident angle. Modeled curves in (b)–(e) were achieved by applying Eq. (2), using Hagen– Rubens derived ρ λ,s with profiler measured σ (dotted line) and filtered σ (dashed line), respectively. The Fraunhofer criterion based on the optical roughness is shown as a vertical dashed line for all cases.

K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

this range the surface is sufficiently optically-smooth to model the interaction as specular [29]. The measured specular reflectance also approaches the hemispherical reflectance, indicating that the surface is optically-smooth since the diffuse component of reflectance is small compared to the specular component. At shorter wavelengths the modeled values underestimate the measured values because the diffuse component of reflectance, which is captured by the integrating sphere but not accounted for by the Davies’ model, becomes important. Similar results have been reported elsewhere [30,31]. Fig. 13(a) also shows the fit one would obtain using the opticalderived roughness σ and a wavelength-independent value of ρ λ,s that is left as a fitting parameter, instead of the Hagen-Rubens relation. Note that the Hagen-Rubens relation applies only to clean metallic surfaces, but can be used here because the scanning electron micrographs reveal only oxide nodules as opposed to a fullyformed oxide layer (Fig. 4). While a thin passivating oxide layer must exist on the surface, it is nearly transparent, and the dominant effect of the oxide is topographical. Accordingly, the Hagen– Rubens equation can be applied. A nonlinear least-squares regression between the modeled spectral reflectances and the portion of the measured spectral reflectances that satisfy the Fraunhofer criterion indicates ρ λ, s = 0.96 for the polished surface, but this inferred reflectance does not match the ρ λ,s obtained from the Hagen–Rubens relation, highlighting the importance of accounting for the wavelength-dependence of ρ λ,s . For the other coupons, Fig. 13(b)–(e) show that the measured directional-hemispherical reflectances differs significantly from modeled reflectances calculated using the profilometerderived roughness. Instead of measured directional-hemispherical reflectances, the modeled reflectances using Eq. (9) with profilometer-derived roughness are closer to the measured specular reflectances due to the significant diffuse reflectance components on the optically-rough surfaces. These results also reflect the findings in Table 2, where the profilometer-derived roughness are more consistent with the optically-derived values from specular measurements. This affirms the finding of Ham et al. [8] that, for optically-rough surfaces, the profiler-measured roughness is not directly relevant to spectral emissivity, which is related to the directional-hemispherical reflectance. Much better agreement between the measured directional-hemispherical reflectances and modeled reflectances is obtained when the filtered roughness is used instead of profilometer-derived values. This is a surprising result, since Davies’ model is intended to predict specular reflectance, and not directional-hemispherical reflectance. The predictive ability of Davies’ model for directional-hemispherical relflectance could be due to a superposition effect where the overall reflectance is due to the superposition of specular reflectance of each local surface point, although this hypothesis requires further analysis. As the surfaces becomes optically rougher, based on their optical and filtered RMS roughness relative to the wavelength, a larger fraction of the incident energy is reflected outside of the specular direction, which gives rise to increased deviation between the prediction and measurement. The measured directional-hemispherical reflectances of the two annealed surfaces, Fig. 13(d) and (e), stand out from the other coupons; the reflectances of these surfaces at shorter wavelengths are significantly lower than that of the other surfaces, likely due to scattering by the oxide nodules shown in Fig. 4. This is also reflected by the larger values of σ shown in Table 2, indicating that these surfaces are optically rougher at the local scale compared to the un-annealed surfaces. To further elucidate the link between roughness scale and radiative properties, the measured directional-hemispherical and modeled specular reflectances of the roughened surface using different levels (1st, 2nd, 3rd, 4th) of decomposed wavelets is shown

9

Fig. 14. Estimated reflectances via Eq. (2) using several different levels of filtered roughness scales. It can be seen that the 1st level roughness is most in line with the measured ρ d-h compared to the cases of the other roughness scales.

Fig. 15. Comparison of estimated spectral reflectance given different temperatures with the same surface roughness, σ =0.134, found from -(Eq. (2)–(4). This result highlights how the temperature-dependent electrical resistivity can affect the predictions and motivates the development of strategies for measuring the spectral reflectance of AHSS coupons at high temperatures during annealing.

in Fig. 14. It can be seen that the modeled reflectance using the first wavelet level (filtered surface) best predicts the directionalhemispherical reflectance, but the modeled values deviate from the measurements as the level of wavelet increases, suggesting that it is the basic wavelet comprising the secondary roughness (local roughness) of a rough surface that can be used to predict the directional-hemispherical reflectance via Eq. (2), possibly due to the “local applicability” of Davies’ model. This phenomenon also reflects those depicted in Fig. 9, where the surface roughness composed of first level wavelet is consistent with the optically-derived value from directional-hemispherical measurement. Nevertheless, more analysis and supporting measurements are required to understand this relationship and the mechanism behind “local applicability”. Finally, we consider how the spectral reflectance may change with temperature. Fig. 13(a) showed that it is necessary to account for the spectral dependence of ρ s,λ according to the Hagen– Rubens relation in order to capture the spectral variation of ρ λ for wavelengths that satisfy the Fraunhofer criterion. Given the wide range of temperatures encountered during intercritical annealing, this result suggests that the temperature dependence of the radiative properties via the electrical resistivity may also be important to consider. Fig. 15 plots the estimated spectral reflectance using Eqs. (2)-(4) at different temperatures (25 °C, 400 °C, 800 °C), given the same surface roughness (σ =0.134) and assuming a value

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K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796

of α = 0.005 K−1 , which is typical for steel [32]. It can be seen that the temperature-dependent electrical resistivity of steel significantly affects the modeled spectral reflectance, even excluding the additional variation one would expect as the surface phase evolves with processing. This result motivates the development of strategies for measuring the spectral reflectance of AHSS coupons at high temperatures, in order to derive results that are directly applicable to industrial conditions, and highlights the importance of including temperature-dependence in the pyrometry measurement equations.

5. Conclusions and future work This study examines the relationship between surface topography and the radiative properties of advanced high strength steel alloys having different surface states. Both the measured specular and directional-hemispherical reflectance follow the trends expected from Davies’ model for specular reflectance, and, in the case of the polished coupon, the optical roughness inferred from these trends match the value measured with an optical profilometer. The results indicate that the profilometerderived roughnesses can be used to approximate the specular reflectances of all the surfaces via Davie’s model, but fail to predict the directional-hemispherical reflectances for opticallyrough surfaces. It is found that the Davies’ model reproduces the experimentally-measured hemispherical-directional reflectance when the filtered roughness is used. This finding implies that, for the wavelengths important to steel pyrometry, the spectral emissivity of the steel is dominated by how the E-M wave interacts with local (sub-micrometer) roughness scale artifacts, although the exact relationship between the predicted specular reflectance and the measured directional-hemispherical reflectance requires further analysis, including directionally-resolved reflectance measurements. These results also show that it is important to account for the spectral dependence of ρ s,λ in Davies’ model using the Hagen– Rubens relation; while the focus of this study is on how the evolving surface oxide may affect emissivity, this result also highlights that the temperature-dependent electrical properties of the substrate steel are also important. This finding motivates the development of strategies for measuring the spectral emissivity of AHSS coupons at high temperatures in the context of industrial conditions. High-pass wavelet filtering provides a roughness scale that explains the spectral reflectance of the steel and shows that the rough surface composed of 1st level wavelets has the roughness scale that can be used to well predict the directional-hemispherical reflectance via Davies’ model. The vertical resolution of the optical profilometer can capture length scales in the tens of nanometers, but the horizontal resolution is much lower, making it impossible to resolve the oxide nodules that are suspected to cause the spectral emissivity to evolve with processing history. High fidelity techniques, like atomic force microscopy, can provide high resolution local surface profiles, but are impractical to carry out over the large surface areas needed to derive a statistically-robust estimate of the RMS roughness. Future work will use Fourier transform filtering of the profilograms to explicate the relationship between length scale and optical roughness, including the mechanism behind the superposition effect of local roughness on specular reflectance with regard to its approximation to the measured directional-hemispherical values. A broader range of annealed specimens that have been quenched at intermediate heating times will also be examined to connect the evolving surface state with variations in spectral reflectivity. This information can be used directly to improve the accuracy of pyrometrically-inferred

temperature measurements, thereby improving the efficiency of the steel-making process. Authorship statement All persons who meet authorship criteria are listed as authors, and all authors certify that they have participated sufficiently in the work to take public responsibility for the content, including participation in the concept, design, analysis, writing, or revision of the manuscript. Furthermore, each author certifies that his material or similar material has not been and will not be submitted to or published in any other publication before its appearance in the Journal of Quantitative Spectroscopy and Radiative Transfer. Declaration of competing interest None Acknowledgements This research is sponsored by the International Zinc Association through the Galvanized Autobody Partnership and the Natural Science and Engineering Research Council of Canada (NSERC CRDPJ 521291-17). The authors are especially grateful for helpful discussions with, and encouragement from, Dr. Frank Goodwin (IZA), Mr. Michel Dubois (John Cockerill Industry) and Professor Myriam Brochu (Ecole Polytechnique du Montreal.) Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.jqsrt.2019.106796. References [1] Chrzanowski K. Problems of determination of effective emissivity of some materials in MIR range. Infrared Phys Technol 1995;36(3):679–84. [2] Thiessen RG, Bocharova E, Mattissen D, Sebald R. Temperature measurement deviation during annealing of multiphase steels. Metallur Mater Trans B 2010;41(4):857–63. [3] Mosser P, Somveille Q, Daun KJ, Brochu M. Effect of temperature deviation during intercritical annealing of HSLA and DP980 steels. In: Proceedings of the 11th international conference on zinc and zinc alloy coated steel sheet, Tokyo, Japan. Tokyo; 2017. [4] Tanaka F, Dewitt DP. Theory of a new radiation thermometry method and an experimental study using galvannealed steel specimens. Trans Soc Instrum Control Eng 1989;25(10):1031–7. [5] del Campo L, Pérez-Sáez RB, Tello MJ. Iron Oxidation Kinetics Study by Using Infrared Spectral Emissivity Measurements Below 570°C. Corros Sci 2008;50(1):194–9. [6] Iuchi T, Furukawa T, Wada S. Emissivity modeling of metals during the growth of oxide film and comparison of the model with experimental results. Appl Opt 2003;42(13):2317–26. [7] Kobayashi M, Otsuki M, Sakate H, Sakuma F, Ono A. System for measuring the spectral distribution of normal emissivity of metals with direct current heating. Int J Thermophys 1999;20(1):289–98. [8] Ham SH, Carteret C, Angulo J, Fricout G. Relation between emissivity evolution during annealing and selective oxidation of TRIP steel. Corros Sci 2018;132:185–93. [9] Khondker R, Mertens A, McDermid JR. Effect of annealing atmosphere on the galvanizing behavior of a dual-phase steel. Mater Sci Eng A 2007;463(1–2):157–65. [10] Liu H, He Y, Swaminathan S, Rohwerder M, Li L. Effect of dew point on the surface selective oxidation and subsurface microstructure of TRIP-aided steel. Surface and Coatings Technology 2011;206(6):1237–43. [11] Davies H. The reflection of electromagnetic waves from a rough surface. Proc IEE-Part IV: Inst Monogr 1954;101(7):209–14. [12] Howell JR, Menguc MP, Siegel R. Thermal radiation heat transfer. CRC press; 2015. [13] Bennett H, Porteus J. Relation between surface roughness and specular reflectance at normal incidence. J Opt Soc Am 1961;51(2):123–9. [14] Wen C-D, Mudawar I. Modeling the effects of surface roughness on the emissivity of Aluminum alloys. Int J Heat Mass Transf 2006;49(23–24):4279–89. [15] Pinel N, Boulier C. Electromagnetic wave scattering from random rough surfaces. John Wiley & Sons; 2013.

K. Lin and K.J. Daun / Journal of Quantitative Spectroscopy & Radiative Transfer 242 (2020) 106796 [16] Agababov S. Effect of secondary roughness on the emissivity properties of solid bodies. Teplofizika Vysokikh Temperature 1970;8(1):220–2. [17] Somveille Q, Mosser P, Brochu M, Daun KJ. Effect of oxidation on emissivity for DP780 and DP980 steels. In: Proceedings of the 11th international conference on zinc and zinc alloy coated steel sheet, Tokyo, Japan. Galvatech; 2017. [18] Ham S, Ferté M, Fricout G, Depalo L, Carteret C. In-situ spectral emissivity measurement of alloy steels during annealing in controlled atmosphere. In: Proceedings of the 13th quantitative infrared thermography conference, ´ Gdansk, Poland; 2016. [19] Bellhouse EM, Mertens A, McDermid JR. Development of the surface structure of TRIP steels prior to hot-dip galvanizing. Mater Sci Eng A 2007;463(1–2):147–56. [20] Gordon W. Far-field approximations to the Kirchhoff-Helmholtz representations of scattered fields. IEEE Trans Anten Propag 1975;23(4):590–2. [21] Brekhovskikh L. The diffraction of waves by a rough surface, part I and II. Zh. Eksper. I Teor. Fiz 1952;23:275–89. [22] Thorsos EI. The validity of the Kirchhoff approximation for rough surface scattering using a Gaussian roughness spectrum. J Acoust Soc Am 1988;83(1):78–92.

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[23] Tang K, Buckius RO. A statistical model of wave scattering from random rough surfaces. Int J Heat Mass Transf 2001;44(21):4059–73. [24] K. Lin, K.J. Daun, Impact of roughness length scale on spectral emissivity during intercritical annealing of advanced high strength steels. Material and Science Technology Symposium, Columbus, Ohio, 2018. [25] Standard ASTM A1079-17, (2017). [26] Davis J, Mills K, Lampman S. Metals handbook. vol. 1. Properties and selection: irons, steels, and high-performance alloys. ASM International; 1990. [27] Graps A. An Introduction to wavelets. IEEE Comput Sci Eng 1995;2(2):50–61. [28] Meyer Y. Wavelets and applications. Springer-Verlag; 1992. [29] Ticconi F, Pulvirenti L, Pierdicca N. Models for scattering from rough surfaces. InTech; 2011. Electromagnetic WavesISBN: 978-953-307-304-0. [30] Torrance KE, Sparrow EM. Off-specular peaks in the directional distribution of reflected thermal radiation. J Heat Transf 1966;88(2):223–30. [31] Torrance KE, Sparrow EM. Theory for off-specular reflection from roughened surfaces. J Opt Soc Am 1967;57(9):1105–14. [32] Haynes WM. Handbook of chemistry and physics. CRC Press; 2014.