Single-frequency apparent eddy current conductivity assessment of metallic coating thicknesses over nonmagnetic metals

1. Introduction

With the use of nickel and titanium alloys, weld overlay and metal spraying technologies deliver attractive solutions to the industry as they exhibit excellent corrosion resistance behavior and high strength characteristics while operating at high temperatures [1, 2, 3]. The implementation of these dissimilar metal-joining processes has been gaining rapid popularity at design and manufacturing levels in Aviation and Oil and Gas industries as they deliver cost-effective solutions with the rising cost of these alloys. The application of these technologies has also been extended to the service sector of these industries in support of repairing their damaged components. This mandates the use of reliable and practical nondestructive evaluation techniques to assess metallic coating thicknesses over nonmagnetic metals.

In compliance with the International Organization for Standardization (ISO), the assessment of a metallic or nonmetallic coating thickness over a metallic or magnetic substrate is controlled with different standards made available to the industry using magnetic induction and eddy current technologies. Phase-sensitive eddy current technique is typically implemented to assess metallic coating thicknesses over nonmagnetic metals per ISO 21968 [4]. It calls for calibrating the eddy current measurement system over nonmagnetic metal coating/substrate calibration blocks of different coating thicknesses to establish a probe characteristic curve. However, coating and substrate conductivities delivered in manufacturing processes slightly vary from those used in the calibration blocks. This can adversely affect the accuracy of estimating coating thicknesses as eddy current measurements over coated structures deviate from the probe characteristic curve. Additionally, accuracy in coating thickness estimation requires placing the probe over a coated structure at a lift-off distance consistent with the one used in establishing the probe characteristic curve. This can be accounted for with the implementation of a lift-off compensation procedure, which calls for a new probe characteristic curve measured at a controlled lift-off distance consistent with that used over the coated structure. This is typically arranged by placing a plastic shim of a known thickness, which is equivalent to the lift-off distance of interest, between the probe and the coated calibration blocks. In all cases, lift-off over the calibration blocks and coated structure shall be consistent to deliver the accuracy needed in estimating coating thicknesses.

Assuming a nonmagnetic-layered structure, Dodd and Deeds exact solutions of forward eddy current problem [5, 6] can be used for coating thickness estimation [7] assuming pure inductive coupling. In the forward model, analytical solution of the frequency-dependent coil impedance can be determined with the input of numerous electromagnetic constraints such as conductivities of the coated/substrate structure, coil parameters, lift-off distance, lead wire lengths, etc. [5, 6, 7, 8]. Alternatively, commercially available tools such as COMSOL, Vic-3D, etc. can be used to estimate the complex impedance coil spectrum numerically [8, 9]. These analytical and numerical forward models lend themselves for impedance-based inversion algorithm solutions to assess metallic coating thicknesses from the frequency-dependent coil impedance Z ⁢ ( f ) measured over coated structures [7]. It further requires measuring the complex coil impedance spectrum over the nonmagnetic substrate Z 0 ⁢ ( f ) to estimate the impedance change Δ ⁢ Z ⁢ ( f ) = Z ⁢ ( f ) - Z 0 ⁢ ( f ) needed to assess nonmagnetic coating thicknesses [7]. Other than the need to measure the impedance change in a relatively broad frequency range, these measurements are sensitive to conductivity deviations of substrates in coated structures relative to the conductivity of the uncoated substrate used as a reference block. Furthermore, the dependency of the impedance-based model on lift-off makes the technique less practical as it requires matching the lift-off over the coated structure with that used over the reference block to deliver 10% uncertainty in coating thickness estimation. Based on numerical simulations of impedance-based forward and inverse models, a ± 25.4 μ m lift-off deviation from the reference block delivers 30% uncertainty in coating thicknesses estimation [10]. Alternatively, hyperbolic regression taking the lift-off effect into consideration can be adopted for a more flexible calibration method to evaluate the relationship between the coil impedance and coating thicknesses with higher accuracy [11]. To overcome time limitations set with the use of eddy current coil impedance spectroscopy, time-domain analysis using pulsed eddy current (PEC) technology presents a promising solution in assessing coating thicknesses with speed in such nonmagnetic structures. Unfortunately, PEC delivers 13% uncertainty in estimating metallic coating thicknesses over nonmagnetic substrates mainly due to the dependency of the forward model on lift-off [12]. Emerging technologies such as PEC and electromagnetic optical imaging techniques need further investigation in the fields of coating thickness assessments [13].

Recent developments in eddy current capabilities allow capturing apparent eddy current conductivity (AECC) spectrums with relatively high accuracy and precision [14]. Implementing an optimized coil design, together with a high-order calibration algorithm, minimizes AECC sensitivity to sample conductivity and lift-off deviations from those used over the calibration blocks. It allows the measurement system to operate up to 80–100 MHz with less than 0.1% uncertainty in AECC measurements within ± 5% conductivity and ± 25 μ m lift-off ranges [15, 16, 17]. The later represents a practical lift-off variation just from placing the probe alone over the coated sample. More importantly, the accuracy in estimating the AECC spectrum does not depend on the conductivity of the substrate used as a reference block following the impedance-based forward model. Instead, different uncoated conductivity calibration blocks are used to bracket the apparent conductivity of interest. As a forward model, this makes AECC-based spectroscopy a suitable replacement to overcome the impedance-based measurement limitations. Until recently, AECC-based inversion models require continuous and smooth depth-dependent conductivity profiles [9, 18], which do not apply to coated structures of rectangular conductivity profiles. Recent development in AECC-based inversion models demonstrated the potential capabilities of using the forward AECC spectroscopy to assess metallic coating thicknesses over nonmagnetic metals [10]. It can deliver one order of magnitude improvement in coating thickness assessment over existing impedance-based inversion models in a lift-off range of ± 25.4 μ m. Unfortunately, the technique requires capturing the AECC spectrum over a broad frequency range using a relatively large coil design ( D = 50 mm) to meet the plane-wave approximation [10]. This makes the technique relatively unpractical and difficult to compete with potential solutions offered using PEC technology [12] or frequency scanning using eddy current grid technique [19, 20], which uses a significantly smaller coil diameter. However, AECC-based models not only present a new way of estimating coating thicknesses of rectangular conductivity profiles but also decreases the measurement sensitivity to conductivity and lift-off deviations from those used over the calibration blocks.

This study simplifies the previously introduced AECC spectroscopy measurement technique by assessing coating thickness at a single frequency with equivalent robustness to lift-off variations. Following the new approach not only makes it a practical technique, but also brings it a step closer for industrial applications. For a given coating/substrate conductivity combination, the inspection frequency is optimized based on AECC sensitivity to frequency change, and the coil design is reduced to 15 mm diameter to meet the plane-wave approximation at the required inspection frequency. Moreover, with the use of a systematic correction factor, the coil diameter can be further reduced to as low as 3 mm with the required robustness to lift-off and accuracy in estimating coating thicknesses. This effort is validated in close comparison between the plane-wave approximation and COMSOL simulations at the optimized inspection frequency and selected coil designs. More importantly, a new AECC-based inversion algorithm is developed and validated for single-frequency AECC measurements over different coating thicknesses relevant to the industry. Since the measurement is required at a single frequency, commercially available eddy-current-based conductivity meters can be easily leveraged for this application. This study not only offers a new way of estimating coating thicknesses of rectangular conductivity profiles but also broadens the range of AECC measurement applications and competes with the speed and accuracy of potential of PEC solutions.

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Metallic coatings over nonmagnetic metals represent nonmagnetic-layered structures with depth-dependent inhomogeneous conductivity profiles. At a given inspection frequency, the previously mentioned AECC is defined as the conductivity of an equivalent homogeneous medium producing similar complex coil impedance over a depth-dependent inhomogeneous conductivity profile [21]. Assuming an infinitely large coil diameter, the AECC spectrum can be directly estimated over a known depth-dependent conductivity profile using the plane-wave approximation [9]. From a practical prospective, AECC spectroscopy can be indirectly estimated using two separate steps. The first step requires establishing a methodology to estimate the complex coil impedance spectrum Z ⁢ ( f ) at a given lift-off distance. For a given coil design, this step can be accomplished theoretically [5, 6, 7], numerically [22, 23, 24, 25, 26, 27, 28, 29] and experimentally [7, 17, 30]. In this study, the previously demonstrated capabilities of COMSOL simulations is used to execute this step numerically [10]. The second step requires a system calibration to evaluate AECC from a coil complex impedance measured or simulated over an inhomogeneous sample at a given frequency. In this step, coil complex impedances measured or simulated over two homogeneous calibration blocks with ( l = s ) and without ( l = 0) lift-offs are needed to bracket the coil complex impedance measured or simulated over the inhomogeneous sample at an unknown lift-off distance within the calibration range. Implementing a simple four-point linear interpolation algorithm can be used to estimate the AECC at a given frequency.

This is a significant deviation from phase-sensitive or impedance-based calibration techniques, which call for coating/substrate calibration blocks with different coating thicknesses or an uncoated reference substrate, respectively. In the latter two cases, any deviation in the conductivity of the nonmagnetic-layered structure from the conductivity of the calibration block(s) can adversely affect the accuracy of coating thickness estimation. This is not the case in assessing the AECC of a nonmagnetic-layered structure as the measurement itself captures a physical property of the sample rather than the coil complex impedance alone [21]. Recent development in coil designs and calibration algorithms used in high-frequency AECC spectroscopy reduced the measurement sensitivity to as low as 0.1% in a lift-off range of ± 25.4 μ m [17]. The capabilities offered through AECC technology makes it a suitable technique to be leveraged for coating thickness estimation.

The robustness of using AECC spectroscopy has been previously established over nonmagnetic metals assessing depth-dependent conductivity variations up to 3% due to surface enhancement methods such as shot peening and low plasticity burnishing [9, 18], and the loss of AECC due to surface roughness introduced from shot peening [31]. The capability of AECC spectroscopy to cover a broader conductivity variation ( ≈ 56%) has been recently demonstrated to assess metallic coating thicknesses over nonmagnetic metals [10]. The accuracy in covering this range of AECC variations can be demonstrated for Ti-6Al-4V ( σ c = 1.05%IACS) 0.5-mm coating thickness over SS304 ( σ s = 2.40%IACS) semi-infinite substrate shown in Fig. 1a. This was validated directly using the plane-wave approximation (solid line) in close comparison to the indirectly simulated AECC spectrum using COMSOL (markers) as shown in Fig. 1b. In these COMSOL simulations, a relatively large coil diameter ( D = 50 mm) was used [10]. In comparison to the plane-wave approximation (solid lines), Fig. 2 further demonstrates the potential capabilities of AECC spectroscopy using COMSOL simulations (markers) to cover similar coating/substrate conductivity combination of various coating thicknesses [10].

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A schematic representation of AECC COMSOL simulations using 2D axisymmetric model of a coil over metallic nonmagnetic coating/substrate material is shown in Fig. 3. This electromagnetic simulation was built using COMSOL’s AC/DC module with 1 V excitation to copper coils of different diameters ( D ). A Ti-6Al-4V coating of σ c = 1.05%IACS is simulated over a semi-infinite SS304 substrate of σ s = 2.40%IACS. The coating thickness ( t c ) ranges between 0.148 and 1.688 mm as previously illustrated in Fig. 2, and it is significantly smaller than the substrate thickness ( t s ) to meet the plane-wave approximation at low frequencies. Both coating and substrate alloys are nonmagnetic, so their relative magnetic permeability ( μ r ) equals to unity. To simulate a realistic eddy current probe, a pull-back distance of 100 μ m was set between the lower side of the coil and the lower side of the probe body. The lift-off distance ( l ), i.e., the separation distance between the lower side of the probe and the coating’s top surface, will be simulated in a ± 25.4 μ m lift-off range to illustrate the robustness of the proposed technique. Both coated samples and coil designs are surrounded with air.

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At its present state, applying AECC spectroscopy for metallic coating thickness estimation over nonmagnetic substrate presents some interconnected challenges that can render the technique unpractical if not addressed at an early technology level. One of the clear challenges in AECC spectroscopy is the need for a relatively large coil design to meet the plane-wave approximation and cover the broad frequency range illustrated in Fig. 2 [10]. This is certainly unpractical even in comparison to the 6- to 15-mm-diameter coil designs commercially used in eddy-current-based conductivity measurement systems. Moreover, relatively large coil diameters lack the spatial resolution needed for industrial applications. Operating the AECC-based technique at broad frequency range certainly presents another set of challenges that are comparable to the impedance-based method. First, a single spot measurement is rather time consuming to cover the entire frequency range to deliver 0.1% accuracy in AECC spectroscopy, which makes it difficult to compete with potential PEC solutions. Certainly, high-precision impedance analyzers can sweep through the frequency range in a matter of seconds [19, 20] but it affects the accuracy in measuring the coil impedance [14, 15]. Second, the capacitive effect at large frequencies can present a major deviation from the previously introduced inductive-based plane-wave approximation and COMSOL simulations. For similar reasons, the use of relatively large coil designs can adversely affect the measurement sensitivity to lift-off and it would typically call for multiple coil designs to cover the frequency range of interest [16]. Finally, operating at relatively low frequencies can limit the technique capabilities as it assumes dealing with semi-infinite substrates, which is not the case in coated structures. Other than the challenges set with the use of large coil diameters and the need to cover AECC measurements at a broad frequency range, covering AECC variations ( ≈ 56%) mandate the use of numerous calibration blocks to deliver the accuracy needed for metallic coating thickness estimation over nonmagnetic substrate. In the following subsections, these challenges are addressed to bring this technology a step closer to industrial applications.

3. Single-frequency AECC-based inverse model

A recent study introduced a new AECC-based inverse model [10], which extended the capabilities of AECC spectroscopy to estimate metallic coating thicknesses over nonmagnetic metals. However, it mandates measuring the AECC spectrum over a broad frequency range, which makes the measurement technique rather unpractical. With the use of the proposed single-frequency AECC coating thickness assessment of depth-dependent rectangular conductivity profiles, a new AECC-based inverse model is needed. To build the new inverse model, the rectangular conductivity profile shown in Fig. 1 is used to simulate Ti-6Al-4V ( σ c = 1.05%IACS) coating over SS304 ( σ s = 2.40%IACS) semi-infinite substrate where the coating thickness is relatively small in comparison to the sample’s overall thickness. The coating thickness t c used here is 0.5 mm and the selected numerical interpolation method is the bisection method since a single AECC measurement is used.

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The proposed flow diagram for the new AECC-based inverse algorithm is illustrated in Fig. 8. To start the process, the measured AECC change Δ ⁢ Γ m at the selected frequency f 0 along with the initial guesses of coating thicknesses t 0 and t 1 are defined by the upper and lower control limits to cover the thickness range of interest. Then a tolerance level ∈ is specified as the accepted criteria for the iterations to pass through. The conductivities of coating and substrate materials are assumed to be known or measured separately, which is consistent with existing impedance-based and AECC-based inversion models [4, 7]. Starting with the first iteration ( n = 1), Δ ⁢ Γ n - 1 and Δ ⁢ Γ n at the selected frequency f 0 are estimated for coating thicknesses t n - 1 and t n using the plane-wave approximation. As previously illustrated, the 15-mm coil design delivers AECC measurements in agreement with the plane-wave approximation at the selected frequency. This further exploits the plane-wave approximation for the proposed AECC-based inversion model. A new estimated thickness t n + 1 is introduced following the bisection method as follows:

t n + 1 = 10 log 10 ⁡ t n - 1 + log 10 ⁡ t n 2 ,(1)

due to the logarithmic nature of the problem. Taking this new thickness value, the corresponding AECC change Δ ⁢ Γ n + 1 is estimated at f 0 again using the plane-wave approximation. The difference between the measured AECC change ( Δ ⁢ Γ m ) with both Δ ⁢ Γ n - 1 and Δ ⁢ Γ n + 1 are taken and multiplied with each other. If the product yields a negative value, then t n + 1 and Δ ⁢ Γ n + 1 replace t n and Δ ⁢ Γ n respectively to proceed further with the algorithm. If the product yields a positive value, then t n + 1 and Δ ⁢ Γ n + 1 replace t n - 1 and Δ ⁢ Γ n - 1 respectively. This process is repeated until the absolute difference between the measured AECC change Δ ⁢ Γ m and the simulated AECC change Δ ⁢ Γ n + 1 using the plane-wave approximation reaches acceptance tolerance criteria. The next section discusses the accuracy of the proposed algorithm as well as the further reduction of the coil diameter with the use of a systematic correction factor. Moreover, it also compares thickness estimates using the impedance-based method against the proposed single-frequency AECC-based model with the reduced coil diameter.

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This section demonstrates the convergence of the proposed single-frequency AECC-based inverse model using the coating/substrate combination illustrated in Fig. 1a, and will further be extended for different coating thicknesses using different coil diameters. Moreover, the performance of the proposed AECC-based model is compared with the existing impedance-based one for a relatively small coil diameter. Using a 15-mm diameter coil, which meets the plane-wave approximation at the selected frequency of f 0 = 0.32 MHz, delivers a “measured” or simulated AECC change of Δ ⁢ Γ m = - 26.90% over 0.5-mm coating thickness as shown in Fig. 7a. To start the proposed inverse model, initial thickness values of t 0 = 0.1 mm and t 1 = 1.1 mm were selected as well as a tolerance limit of ∈ = 1 × 10 - 5 was specified. Figure 9a shows the convergence of the estimated AECC change using the 15-mm diameter coil at a frequency of 0.32 MHz. It takes about 10 iterations for the AECC change Δ ⁢ Γ n to converge to the measured one Δ ⁢ Γ m = - 26.90%. The corresponding convergence of coating thicknesses t n to the estimated coating thickness t e ⁢ s ⁢ t = 0.505 mm is demonstrated in Fig. 9b. It slightly over estimates the actual coating thickness ( ≈ 1% over estimation) due to the reduced coil size. Similar analysis is presented in Fig. 9c using a 3-mm diameter coil where AECC change Δ ⁢ Γ n certainly converges to the measured one Δ ⁢ Γ m = - 29.51% taken from Fig. 7(b). However, it delivers an estimated coating thickness of t e ⁢ s ⁢ t = 0.543 mm since the measured AECC change corresponds to an equivalent coating thickness using the plane-wave approximation. This systematic over estimation of ≈ 8.6% is mainly due to the use of a very small coil diameter.

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In comparison to actual coating thicknesses, Fig. 10a shows the corresponding coating thickness estimates measured at 0.32 MHz with 25.4 μ m lift-off using different coil diameters. Analyzing this data clearly indicates the systematic nature of this error as a function of coil diameter. It simply shows that decreasing the coil diameter deviates AECC measurements from the plane-wave approximation. This systematic error can be easily corrected for using best-fit analysis as follows:

t e ⁢ s ⁢ t = a ⁢ t e ⁢ s ⁢ t ⁣ * b ,(2)

where t e ⁢ s ⁢ t is the estimated coating thickness from the proposed single-frequency AECC-based model and t e ⁢ s ⁢ t ⁣ * is the corrected coating thickness estimate. The constants a and b are estimated once and for all using the best-fit analysis for a given coil design and coating/substrate conductivity combination. Accordingly, t e ⁢ s ⁢ t ⁣ * can be evaluated by rearranging the above equation as follows

t e ⁢ s ⁢ t ⁣ * = ( t e ⁢ s ⁢ t a ) 1 b(3)

Using the latter equation, corrected coating thickness estimates show an agreement with actual coating thicknesses as illustrated in Fig. 10b regardless of the coil diameter. Applying the correction factors, which capture to systematic nature of measurement deviations, allow reducing the coil diameter to as low as 3 mm. If the coil diameter meets the plane-wave approximation at the optimized frequency for a coating/substrate combination, i.e., D ⩾ 15 mm in this study, then there is no need to apply any corrections. The use of smaller coil diameters requires applying these corrections. Since the measurement is required at a single frequency, commercially available eddy-current-based conductivity meters can be easily leveraged for this application. Tabulating the measurement correction constants a and b associated with a given coil diameter to be used over a specific coating/substrate combination to cover a predefined coating thickness range at an optimized inspection frequency allows inverting the measured AECC to the corrected estimated thickness in Eq. (3).

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Figure 11 further illustrates the robustness of the proposed model, where coating thickness estimation is simulated following both multi-frequency impedance-based model [7] and the proposed single-frequency AECC-based model with a 3 mm coil in a ± 25.4 μ m lift-off range. Detailed analysis of the multi-frequency impedance-based model can also be found in a recent study demonstrating its capabilities against the multi-frequency AECC-based model [10]. As expected, using the plane wave approximation delivers negligible deviation ( ≈ 0.1%) in estimating coating thicknesses. Using COMSOL, the corrected thickness estimates using a 3-mm diameter coil delivers an average of 3% variations in a ± 25.4 μ m lift-off range. This shows an excellent agreement with the previously introduced multi-frequency AECC model which used a much larger coil diameter ( D = 50 mm) [10]. Following the impedance model using the 3-mm diameter coil in a ± 25.4 μ m lift-off range, the impedance-based model delivers 30% uncertainty in coating thickness estimation. For coating thicknesses higher than 0.5 mm, the impedance-based model loses its sensitivity since the coil diameter is rather small to meet the standard eddy current penetration depth. This further validates the robustness of the proposed single-frequency AECC-based measurement technique to lift-off variations and deems the AECC-based model a more efficient technique when assessing metallic coating thicknesses over nonmagnetic materials.

The previously introduced AECC-based model demonstrated the potential capabilities of AECC spectroscopy to assess metallic coating thickness over nonmagnetic metals. However, it mandates covering the AECC spectrum over a broad frequency range and the use of a relatively large coil diameter to meet the plane-wave approximation in that frequency range. This study introduced a new AECC-based technique to assess these coating thicknesses using a single frequency, which makes the technique more practical for the use in industrial applications. Optimizing coil design and inspection frequency along with the implementation of a new AECC-based inversion model helped in making this possible. The technique allowed reducing the coil diameter to 15 mm to deliver the 3% uncertainty in coating thickness estimation, which is in agreement with the more complicated multi-frequency AECC-based model. With the implementation of correction factors, which need to be done once and for all for a given coil design and coating/substrate conductivity combination, the coil diameter can be reduced to as low as 3 mm with the same level of uncertainty in coating thickness estimation in a ± 25.4 μ m lift-off range. This is at least one-order of magnitude improvement over the multi-frequency impedance-based model using similar coil design over the lift-off range of interest, and consistent with the previously introduced multi-frequency AECC-based model.