Theoretical Modeling and Analysis of Directional Spectrum Emissivity and Its Pattern for Random Rough Surfaces with a Matrix Method

: The emissivity is an important surface property parameter in many ﬁelds, including infrared temperature measurement. In this research, a symmetry theoretical model of directional spectral emissivity prediction is proposed based on Gaussian random rough surface theory. A numerical solution based on a matrix method is determined based on its symmetrical characteristics. Inﬂuences of the index of refraction n and the root mean square (RMS) roughness σ rms on the directional spectrum emissivity ε are analyzed and discussed. The results indicate that surfaces with higher n and lower σ rms tend to have a peak in high viewing angles. On the contrary, surfaces with lower n and higher σ rms tend to have a peak in low viewing angles. Experimental veriﬁcations based on infrared (IR) temperature measurement of Inconel 718 sandblasted surfaces were carried out. This model would contribute to understand random rough surfaces and their emitting properties in ﬁelds including machining, process controlling, remote sensing, etc.


Introduction
As a surface property parameter, the precise determination of emissivity is important in many fields, including infrared temperature measurement. Experimental studies have shown that the main factors affecting the emissivity of a given rough surface are the chemical composition [1], temperature [2][3][4], wavelength [2,3] and surface morphology [4,5], as shown in Figure 1. Closely related to the speed of light in the material, the index of refraction n depends mainly on the material, and is affected by temperature, wavelength and other factors. Surface morphology has an effect on reflection and shadowing, influencing the emissivity ε. In this paper, if not specifically indicated, the term emissivity refers to directional spectrum emissivity. The indirect influences of temperature, wavelength and other factors on emissivity can be experimentally measured and are relatively clear. Studies on Pt-10Rh and SiC indicate that the emissivity of rough surface increases with the increase of temperature and the decrease of wavelength [6]. Similar phenomena are found in difficult-to-process materials including Inconel 718 and Ti6Al4V [3,[7][8][9]. Other factors including chemical composition [1], heat treatment [2,3] and oxidation [10,11] may also have an effect on emissivity. The influence of chemical composition and heat treatment are combined in the influence of the index of refraction n. The oxide layers are neglectable if not specifically oxidized.
The influence of surface morphology is complicated and not yet well understood. It is found that coarsely sandblasted samples, compared with finely sandblasted samples, have higher total hemispherical emissivity at low temperature, but lower emissivity in high temperature [12]. It is also reported that differently processed surfaces with close surface roughness Ra may have very different emissivities [13]. Further information is needed to understand this ambiguity.
Theoretical modelling studies have been carried out to predict the emissivity of rough surfaces. This was predicted considering the shadowing effect [14]. The surface was however assumed as a grey body, which limits its application. The average probability that infrared radiation emitted hits the surface directly was calculated, and the influence of reflection on emissivity of rough surfaces was estimated by Wen and Mudawar [15]. Reflectivity of the rough surface is also affected by viewing angle. Thus, errors were introduced in applying the average probability. Emissivity model of rough surfaces considering shadowing effect and reflection was established [16], while the reflection was assumed no more than once. Emissivity model of sea surfaces have been carried out by Wu and Smith [17] and Li et al. [18][19][20][21] respectively. However, sea surfaces have certain patterns (perpendicular to the wind), and are composed merely by sea water. Their models lack in generality.
Despite this valuable work, due to the complexity of the emission processes, the influences of the index of refraction n and surface morphology on the directional emissivity ε are to be further studied, especially the influence of the index of refraction n. In this research, influences of the index of refraction n and root mean square (RMS) roughness σ rms on emissivity ε are modeled, as shown in Figure 1. Geometric optics are assumed and a directional spectral emissivity prediction model based on the Gaussian random rough surface is briefly analyzed, as similar models have already been proposed and derived in other fields [18,22]. Then, a novel matrix method was introduced, and sectional Romberg integral is applied, to numerically solve the proposed model. Influences of the index of refraction n and the root mean square (RMS) roughness σ rms on the directional emissivity ε are analyzed and discussed. It is found that emissivity of surfaces with lower σ rms and higher n tends to have a peak in high viewing angles. On the contrary, the emissivity of surfaces with higher σ rms and lower n tends to have a peak in low viewing angles. Finally, experimental verification based on IR temperature measurement of an Inconel 718 machined surface is carried out.

Methodology
In the model part, the surface is assumed to be Gaussian, and limited to the area of geometrical optics (GO). According to the number of reflecting processes involved, the emissivity is divided into components. To contribute to one component, an infrared ray must experience processes including emission, shadowing and reflection. As the surface is random, each process can only be discussed statistically by a possibility density function, and each component calculated by integration. The expression of emissivity is derived by summarizing the components. In this part, it is noticed that the emission, shadowing and reflection processes have no inner connection, and can be expressed separately. Similar processes of different components share the same properties, and can be depicted by one formula.
The expression of emissivity is too complex to be solved analytically. In the numerical solution part, considering the discoveries above, a matrix method is designed. Emission, shadowing and reflection processes are represented and computed by matrixes. Nested loops are avoided and computational time is shortened.
Experimental verification based on IR temperature measurement of an Inconel 718 machined surface is carried out. Temperature and emissive power are measured simultaneously to calculate the emissivity. Other experimental studies are also cited to further verify the model. The framework of this paper is displayed in Figure 2.

Analysis of Emissivity of Rough Surfaces
To predict the emissivity of rough surfaces, three-dimensional Gaussian random rough surface is assumed, as shown in Figure 3. The three-dimensional Gaussian random rough surface is generated by a stationary stochastic process, as expressed by Equation (1) [22]: where r is the position vector, and x, y, z are the coordinates of r. We have: where p, q are the slopes of the surface in x, y direction, respectively. t N (p, q, σ rms ) is the probability density function of p, q. The Gaussian random rough surfaces have the following characteristics: • The surface is isotropic (and statistically symmetry); • p and q are randomly distributed according to a Gaussian distribution; • No correlation between p and q. In this study, emissivity is defined as the emissive power of the rough surface compared with a black body, and geometric optics is assumed. According to Fresnel's law and the reflection law: where ε F is emissivity, n is the index of refraction, θ S is the viewing angle.
The main phenomena in the emission process of rough surfaces are Fresnel's law and reflection enhancement. Figure 3b shows that the smooth surface consists of many horizontal microelements, while the rough surface consists of elements with varied directions, each having different emissivity in the same direction. Reflection enhancement refers to the phenomenon whereby reflection induces higher emissivity near the normal direction in rough surfaces, as depicted in Figure 3c. Reflection enhancement may also cause lower emissivity in high viewing angles. Due to their combined effect, the influence of surface morphology is complicated and not yet well understood.

Modelling of Emissivity of Rough Surfaces
Directional spectral emissivity refers to the ability of measured surface to emit in a certain wavelength and direction compared with a black body. A statistical model is established to calculate the emissivity of rough surfaces.
Infrared radiation emitted by an element may leave the surface directly (zeroth-order), reflect once (first-order), twice (second-order), and so on. Considering that the correlation between emissivity and the emitted power is linear, the emissivity could be divided into different order components, and the components can be computed separately. Components of second order and above are called high-order components collectively, as presented in Equation (5): Zeroth-order component refers to emissivity related to infrared radiation that leaves the surface directly. To contribute to zeroth-order component, an infrared ray must be emitted, and not be blocked by another part of the surface. Different elements have different directions, and thus have different local viewing angle θ l , as illustrated in Figure 4b. According to Equation (4), their local emissivities ε l are not equal. The emitted radiation ray may also be blocked by another element, and the possibility is a function of its slope. In Figure 4a: where η is the normal direction of the element. As shown in Figure 4b, projection of the element distorts depending on η and the viewing direction Ω S . Thus, area projection coefficient A p is: where η yoz is the component of η in plane yoz.
where t (θ s , p, q, σ rms ) is the possibility that the slope of the emitting element is p, q.
Smith's shadowing function S(θ s , σ rms ) [ 24,25] is the possibility that infrared radiation is not blocked. It has been reported [ 26,27] that correlation between elements is not significant and can be ignored. Smith's shadowing function is estimated as Equation (9): The zeroth-order component is, In Equation (13), we note that S(θ s , σ rms ) is not a function of p, q. The integration part depicts the emission process, while S(θ s , σ rms ) stands for not being blocked.
The first-order component refers to emissivity related to infrared radiation that reflects once before leaving the surface. To contribute to a zeroth-order component, an infrared ray must be emitted, hit another part of the surface, be reflected, and not hit the surface again. The process could be divided into two parts: emission and blocking, reflecion and exiting. Figure 5 shows that the infrared radiation emitted by the first element hits the second element in direction Ω 1 , and the reflected beam leaves the surface in direction Ω S . By applying the local coordinate system x'y'z', the emission process is similar to the zeroth-order component. According to the reflection law [28]: which is: By substituting Equations (11) and (12) into Equations (7) and (8), t(θ 1 , p, q, σ rms ) and ε l (n, p, q, θ s ) could be expressed as t(θ s , θ 1 , φ 1 , σ rms ) and ε l (n, θ 1 , φ 1 , θ s ). According to Kirchhoff's law we have [28]: where α, ρ are the absorptivity and reflectivity, respectively. Noting that Gaussian random rough surface is isotropic and statistically symmetry, we have: where Equation (18) refers to the reflection process, and Equation (19) refers to the emitting process. Noting that S(θ s , σ rms ) is not a function of p, q. Equation (13b) is mainly about emission and blocking, while Equation (18) is about reflection and exiting. In Equation (19), the integration part depicts the emission process, while [1 − S(θ s , σ rms )] stands for blocking, which is similar to Equation (13). In Equation (19), S(θ s , σ rms ) stands for exiting, and the rest for reflection. As shown in Figure 6, second-order component refers to emissivity related to infrared radiation that reflects twice before leaving the surface. By applying a local coordinate system to the first element and the second element, we have: Third order component and above are modeled similarly. We note that emissivity is not a function of ϕ and the model is symmetrical.

Numerical Solution Based on a Matrix Method
The directional spectral emissivity of the rough surfaces is too complex to be solved analytically. Generally, multiple integrals are computed by nested loops, and the operation period of nested loops increases exponentially with multiplicity. In this section, a novel matrix method is introduced, and sectional Romberg integral is applied to obtain numerical solutions quickly and precisely.
As shown in Section 3.2, high-order components are related to two basic process, emission and refraction. As mentioned in Section 3.1, a Gaussian random rough surface is isotropic (and symmetrical), and no correlation between p and q is assumed. From another point of view, the optics processes involved are emission, reflection and shadowing. Thus, these processes can be represented by irrelevant matrixes. The numerical integration could be obtained by multiplying matrixes, which is much quicker than nested loops. We define: (23) where Der(n) is the n-th order emissivity in the viewing angle θ s k . Emi is the emission matrix. S is the shadowing matrix. Rbg 2 is the Romberg matrix. Definitions of Rbg 2 and theoretical verification of the matrix method are presented in Appendix A. rft is the reflection matrix. This gives: where θ s k are, respectively.
From the viewpoint of optics, as sketched in Figure 7, Emi, (1 − S), Rbg 2 , rft are defined, representing the emission, shadowing, reflection and leaving processes, respectively. emi k is the average emissivity of a random element in direction θ K . Elements of (1 − S) are the possibility that a ray in direction θ K hits the surface again. Rbg 2 is a constant matrix introduced by Romberg integration. Rft lk is the average energy reflected to direction θ K when a ray from direction θ l hits a random element. Thus, the infrared rays are traced statistically by a matrix method. Multiple integrations are generally computed by nested loops, in which t (θ S , p, q, σ rms ) and ε F (n, p, q) must be calculated repeatedly in every step of every integration. Thus, computational time increases exponentially with the order n. In matrix method, they are computed only in obtaining the matrixes. Therefore, the computational time hardly increases when n > 0. The amount of calculation is presented in Table 1. Table 2 illustrates that the estimated error of the matrix method is acceptable. Estimation of the error is summarized in Appendix A.

Results and Discussion
The effects of the index of refraction n and the RMS roughness σ rms on the emissivity ε are discussed by setting different parameters. The numerical integral was carried out with the MATLAB R2018a program, using a personal computer running with an Intel Core I7-4720HQ, 3.6 GHz. CPU. In industry, n is mostly between 2 and 16, σ rms is mostly under 1. The values of n and σ rms is chosen inside this range by proportional sequences, to cover the majority of rough surfaces encountered in industry.

Pattern of Different Components
The patterns of different components are discussed. Figure 8 displays that higher order components are considerably smaller in scale. The proportion of higher order components increases with increasing σ rms . The components of first order and above are almost 0 when the surface is smooth, and consists of only about 20% when the surface is relatively rough.

Effects of the Index of Refraction n on Emissivity ε
The index of refraction n depends on the chemical composition of the surface and varies with wavelength and temperature [28].
When the RMS roughness σ rms is relatively high, ε would be dominated by σ rms , which is to be discussed in Section 5.3. When σ rms is relatively low, p, q converge to zero, and the surface would be too smooth for an infrared ray emitted to hit the surface again. Thus, ε depends mainly on ε 0 , and ε l ≈ ε F . Equation (6) could be transformed into the following form: The patterns shown in Figure 9b are reported in experimental studies [14,29]. It is shown in Figure 9 that ε F and ε have similar patterns. Obviously, n affects ε l (ε l ≈ ε F ), and thus have an effect on ε. In another word, n influences ε l by influencing the emission characteristics of the elements. emi (θ S , n, σ rms ) is the same function that stands for the emission process mentioned in Section 4. In Equation (4), it can be analytically demonstrated that the extreme point is the polarizing angle, which is really interesting. Such a phenomenon could be valuable for further research.

Effects of RMS Roughness σ rms on Emissivity ε
Referring to the root mean square of the slope of the surface elements, RMS roughness σ rms is a surface integrity parameter. According to Equations (29) and (30), σ rms affects both S (θ S , σ rms ) and emi (θ S , n, σ rms ). That is, σ rms effects both of emission and shadowing. ε decreases with increasing σ rms in high viewing angle, increases with increasing σ rms in middle viewing angle, and first increases then decreases with increasing σ rms in low viewing angle (near normal direction). Surfaces with very high σ rms may have a peak in low viewing angle. It should be noted that high, middle and low here are a relative concept. As higher order components are relatively small, reflection is not discussed in details here.
As shown in Figure 10, on the one hand, the shadowing effect becomes more notable with increasing σ rms . Figure 10a shows that S (θ S , σ rms ) decreases with increasing σ rms in high viewing angle. Thus, ε decreases with increasing σ rms in high viewing angle. On the other hand, these blocked infrared rays are redistributed in all directions, including middle viewing angles. Thus, ε increases with increasing σ rms in middle viewing angle. On the other hand, emi (θ S , n, σ rms ) varies with increasing σ rms . Local emissivity of the elements ε F (θ S , n) are not a function of σ rms , but the distribution of the elements can be influenced by σ rms . Thus, σ rms have an effect on the emissivity ε. As shown in Figure 11, emi (θ S , n, σ rms ) could be considered as a convolution of ε F (θ S , n). Thus, the curve is flattened and the emissivity decreases with increasing σ rms in high viewing angles, and increases with increasing σ rms in middle viewing angles. The emissivity ε is dominated by σ rms , and n have little effect on ε, as mentioned in Section 5.2. Similar patterns are reported experimentally [13,20]. Figure 11. Effect of RMS roughness σ rms on emitting process and emissivity ε.
Near normal direction, the effect of σ rms in middle viewing angle still works, and ε increase with increasing σ rms firstly. While σ rms is very high, 2·σ rms may proceed the polarizing angle, where ε F starts to decrease with increasing θ S . Thus, ε near normal direction may decrease with increasing σ rms . Due to this phenomenon near normal direction, surfaces with very high σ rms tend to have a peak in low viewing angles.

Combined Effect of n and σ rms on ε
Influences of the index of refraction n and root mean square (RMS) roughness σ rms on emissivity ε are discussed based on Sections 5.2 and 5.3. n have an impact on emi (θ S , n, σ rms ) by ε F , and thus influences ε. σ rms influences both emi (θ S , n, σ rms ) and S (θ S , σ rms ). In optics point of view, n influences primarily emission process, while σ rms influences both emission and shadowing, as shown in Figure 12. When σ rms is relatively high, ε is dominated by σ rms . It is found that ε decreases with increasing σ rms in high viewing angle, increases with increasing σ rms in middle viewing angle, and first increases then decreases with increasing σ rms near normal direction. Therefore, surfaces with very high σ rms may have a peak in low viewing angle. When σ rms is relatively low, ε tend to decrease with increasing n, and have a steeper peak in high viewing angles. The extreme point is the Brewster's angle, which increases with increasing n.
Summarizing the conclusions above, surfaces can be divided into three areas depending on index of refraction n and RMS roughness σ rms . Surfaces with lower σ rms and higher n tends to have a peak in high viewing angles [30][31][32]. On the contrary, surfaces with higher σ rms and lower n tends to have a peak in low viewing angles. There is an intermediate region between the two areas [29,32], as shown in Figure 13.

Verification by Simulated Experiment
According to Plunk's law: where C 1 ≈ 3.7418·10 −16 W·m 2 , C 2 ≈ 1.4388·10 −2 m·K. Thus, the emissivity of the surface could be calculated by measuring the temperature T and the emissive power E (λ, T). Figure 14a depicts the principle of the proposed verification experiment. The temperature T of the sample is measured by thermocouples. The emissive powe E (λ, T) is measured by an IR thermometer from different directions. All data is imported to a personal computer, by which the emissivity ε is calculated according to Equation (31). All samples are heated to one same temperature (900 K), in order to minimize the error. As shown in Figure 14b, the samples introduced are polished Inconel 718 columns, φ 27 × 18 mm, with a blind hole drilled to place the thermocouple. Inconel 718 is a difficultto-process material. The surface roughness of the samples were Ra 0.17 ± 0.009 µm, and the RMS roughness were 1.16 ± 0.05. State and properties of the sample were referred in [33]. As shown in Figure 14c, the temperature values of the samples were measured by a DAQ-2401 thermometer (OMEGA, city, state abbrev if USA, country) equipped with a type-K thermocouple.
The emissive power E (λ, T) was measured by the IR thermometer. The wavelength of the IR thermometer is 1.7 µm. Further information was reported by Zhao [34] et al. The upper surface of the sample was heated to 900 K to minimum the environmental interference. Five points in one sample are chosen for each group in the experimental study. As shown in Figure 15, the experimental results match well with the model. Sandblasted surfaces have similar pattern with Gaussian random rough surface. Emissivity of Inconel 718 sandblasted surfaces were reported by Campo [35] et al. Figure 16 shows that the experimental results match well with the theoretical results. Deviations at higher viewing angles might due to diffraction.

Conclusions
In this research, a theoretical model considering the index of refraction n and the root mean square (RMS) roughness σ rms is established out to predict the emissivity ε of rough surfaces. The directional spectral emissivity prediction model based on the Gaussian random rough surface is proposed. Then, the sectional Romberg integration based on a novel matrix method is applied to numerically solve the proposed model. Influences of the index of refraction n and the RMS roughness σ rms on the emissivity ε are analyzed and discussed. Experimental verification is carried out with Inconel 718 samples. In this research, diffraction is neglected, and the differences between real [36] and Gaussian rough surface are also neglected. Their potential effects on emissivity are to be further researched. The main conclusions drawn from this work are as follows: (1) A directional spectral emissivity prediction model of rough surfaces is established considering the shadowing effect and the reflection enhancement. Sectional Romberg integration based on a novel matrix method is applied to numerically solve the proposed model and is theoretically verified. The computational time of the matrix method is compared with that of the traditional method. Errors induced in the matrix method are estimated. (2) The influence of the index of refraction n and RMS roughness σ rms on the emissivity ε is discussed according to the model. It is found that higher order components are smaller in scale and their energy is more concentrated near the normal direction.
(3) n influences ε by influencing ε F (or from an optics point of view, the emission process). When σ rms is relatively low, ε tends to decrease with increasing n, and have a steeper peak in high viewing angles. The extreme point is the Brewster's angle, which increases with increasing n. When the RMS roughness σ rms is relatively high, ε would be dominated by σ rms , and influence of n would be quenched. (4) σ rms influences both S (θ S , σ rms ) and emi (θ S , n, σ rms ) (both emitting and shadowing).
That is, σ rms effects both of emitting and shadowing. ε decreases with increasing σ rms in high viewing angle, increases with increasing σ rms in middle viewing angle, and first increases then decreases with increasing σ rms near normal direction. Surfaces with very high σ rms may have a peak in low viewing angle. (5) Surface with lower σ rms and higher n, due to Fresnel's law, tends to have a peak in high viewing angles. On the contrary, surface with higher σ rms and lower n, mainly due to reflection enhancement, tends to have a peak in low viewing angles. In the intermediate region, due to their combined effect, the emissivity has no recognizable peak and decreases with increasing viewing angle.

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Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.
High-order components could be verified similarly. The error was estimated according to Equation (A10).