2021 Volume 61 Issue 4 Pages 1064-1070
The electrochemical hydrogen penetration measurement technique, to which a sinusoidal perturbation method was applied, was modified using a signal containing multiple frequency components. The technique was successfully applied to measurement of the hydrogen diffusion coefficient in a ferric sheet specimen. A series of numerical calculations for the technique, in which the constituent frequencies of the signal were selected from the measurement result, also provided the same diffusion coefficient and verified the validity of the technique. The use of this technique enables rapid determination of the hydrogen diffusion coefficient in a specimen.
Steel materials suffer from the so-called hydrogen embrittlement caused by hydrogen that has penetrated into the materials.1) In order to establish effective measures against hydrogen embrittlement, an understanding of the behavior of hydrogen in steel materials is necessary. In general, hydrogen that has penetrated into metal diffuses according to Fick’s law, but hydrogen is trapped by crystal defects such as dislocations and the apparent value of hydrogen diffusion coefficient is reduced. Reversibly trapped hydrogen is known to contribute to hydrogen embrittlement,2) and the hydrogen diffusion coefficient is therefore a useful indicator for analyzing the hydrogen embrittlement sensitivity of metal.
The technique using a dual electrochemical cell, called a Devanathan-Stachurski (DS) cell as schematically illustrated in Fig. 1, has been widely used to estimate the hydrogen diffusion coefficient in metal.3,4) In a DS cell, a metal sheet specimen is sandwiched between two electrochemical cells, and one side of the specimen is cathodically polarized while the other side is anodically polarized. On the cathodically polarized side (hereinafter referred to as the entry side), hydrogen evolution reaction occurs and a part of the adsorbed hydrogen (Hads) generated during the reaction is absorbed into the specimen. The absorbed hydrogen (Habs) diffuses through the specimen and pops out from the anodically polarized surface (hereinafter referred to as the exit side) as adsorbed hydrogen. The adsorbed hydrogen on the exit side surface is then removed from the surface by combining with hydroxide ion and becomes water. The amount of hydrogen that has penetrated through the specimen can be quantified from integration of the electrical current measured in the exit side cell. The response of the exit side current is delayed from that of the entry side due to the transport of the hydrogen through the specimen. The hydrogen diffusion coefficient in the specimen is usually determined from the time lag between the initiation of the hydrogen ingress and the rise of the exit side current.4) There is another technique for measuring the hydrogen diffusion coefficient utilizing a sinusoidal signal. In this technique, a sinusoidal perturbation is superimposed on the hydrogen entry process and the delay is measured as the phase shift.5) The relationship between the phase shift and the value of hydrogen diffusion coefficient was reported by Sekine and is described in the following section.
Schematic illustration of a Devanathan-Stachurski (DS) cell.
The utilization of sinusoidal perturbation provides the advantagesof elimination of background noise and realization of temporal measurement, as reported previously.5,6) Importantly, input signals for the entry process are not limited to a single sine wave. When an arbitrary waveform containing multiple frequency components is superimposed on the hydrogen entry process, the response will be the sum of the responses for all frequency components. This allows simultaneous measurements of the responses to multiple sine waves with different frequencies. Although such a technique has already been used in electrochemical impedance spectroscopy,7,8) it is expected that the technique can also be applicable to hydrogen penetration measurement using sinusoidal perturbation and that the use of the technique would enable the measurement period to be shortened without sacrificing accuracy.
In this study, the multi-sine technique was used for the electrochemical hydrogen penetration measurement. An overview of the technique, the results of a series of electrochemical hydrogen penetration tests, and the results of numerical calculations to confirm the validity of the technique are presented in this paper.
The behavior of hydrogen migration in a metal due to the concentration difference can be described by Fick’s law of diffusion. When the thickness L of the metal is sufficiently small, hydrogen diffuses from the entry side to the exit side in a one-dimensional direction according to the following 1st and 2nd laws.
| (1) |
| (2) |
| (3) |
| (4) |
| (5) |
| (6) |
| (7) |
When multiple sinusoidal signals with different frequencies (f1, f2, …, fk, ..., fn) are simultaneously applied as a perturbation of the electrode potential on the entry side, the boundary conditions on the entry side can be expressed as follows:
| (8) |
The data processing flow to obtain the value of D in actual measurement is schematically illustrated in Fig. 2. Firstly, multiple sinusoidal frequencies are determined as mentioned afterward and the sinusoidal waves with the frequencies are synthesized. The synthesized wave is applied to the entry side potential Eentry, and current Ientry flowing through the entry side is measured as well as current Iexit flowing through the exit side. The phase shift θk is obtained by Fourier transform of the waveforms of Ientry and Iexit and comparison of the phase values for the kth frequency component. The value of αk is obtained by inverse interpolation of the value of θk using the α–θ curve of Eq. (6), as plotted beforehand. The value of D is obtained by substituting the values of αk, fk and L into the following equation obtained by modifying Eq. (7).
| (9) |
| (10) |
Data processing flow diagram for the determination of hydrogen diffusion coefficient D from electrochemical hydrogen penetration with potential perturbation by a synthesized wave signal. (Online version in color.)
There are several things to consider in selecting frequencies used for a synthesized signal applied to the entry side. If the frequency used is too low, the phase shift becomes almost zero, making quantification of the D value difficult. If the frequency used is too high, on the other hand, the phase shift exceeds 360°, which complicates the quantification. According to our previous study,6) it is desirable to select a frequency to achieve 0.5 < α < 2.
Even if the signal applied to the entry side is completely sinusoidal, the current waveform detected on the exit side is sometimes distorted.6) The distorted waveform frequently contains harmonic components of frequency of the applied signal. If a harmonic component overlaps with a constituent frequency of the applied signal, the phase value for Iexit at that frequency might be affected. Selection of constituent sine waves for which frequencies are co-prime to each other is effective for avoiding an overlap. For example, a combination of sinusoidal waves with frequencies of a prime number sequence is recommended. The sampling rate for data acquisition should also be selected appropriately. In order to record an accurate waveform, it is necessary to set the sampling frequency higher than twice the highest constituent frequency of the synthesized wave. The number of samples for Fourier transform should also be taken into account. To obtain an accurate phase, it is necessary to set the number of data samples to contain an exact integer number of the sine wave of interest in the sample range.
A pure iron sheet (99.99% Fe, 800 μm in thickness) purchased from Nilaco was used as a specimen. The sheet was cut into a 12 mm × 15 mm rectangular shape and heated at 1173 K for 1 h. The sheet was then furnace-cooled to reduce internal stresses. The annealed sheet was electro-polished at a current density of 60 mA cm−2 for 600 s in Jacquet solution (18.5 vol% perchloric acid (purity 70%) + 76.5 vol% anhydrous acetic acid + water) at 273 K.9) The exit side of the sheet was electroplated with nickel at a current density of 4 mA cm−2 for 600 s in a Watt bath (1 mol dm−3 nickel sulfate + 0.2 mol dm−3 nickel chloride + 0.6 mol dm−3 boric acid) at room temperature10) to ensure smooth hydrogen withdrawal. The thickness of the plated nickel layer was calculated to be ~0.3 μm with a reported Faradaic efficiency of ~40% for the deposition.
3.2. Electrochemical MeasurementHydrogen penetration measurement was conducted using a DS cell with dual flow channels reported previously.6) A schematic illustration of the measurement system is shown in Fig. 3. Each cell has a three-electrode configuration: an iron sheet specimen as a common working electrode (WE), a platinum wire as counter electrode (CE) and a silver-silver chloride electrode (SSE) with saturated potassium chloride solution as a reference electrode (RE). The surface area of the WE was restricted to 0.50 cm2 by an O-ring with an inner diameter of 8.0 mm. The entry-side cell of the DS cell was filled with 0.3 mol dm−3 borate buffer solution with pH of 8.4, while the exit-side cell was filled with 0.2 mol dm−3 sodium hydroxide solution. During the measurement, electrolyte solutions in both sides of the DS cell were flowed by bimorph pumps (BPS-215i, Nitto Kohki) at a constant rate of 0.625 cm3 s–1. The temperature of both solutions was kept constant at 25°C throughout the measurement by a thermostat (NCB-1200, EYELA).
Equipment configuration for the electrochemical hydrogen penetration measurement with a flow-type DS cell.
Electrochemistry of the hydrogen penetration measurement was conducted using a bi-potentiostat (SDDP-212m, Syrinx) controlled by a LabVIEW (National Instruments) program. The entry side potential was modulated by a multiple signal from five sinusoidal waves as follows:
| (11) |
Under the initial condition of Eq. (3) and the boundary conditions of Eqs. (4) and (8), the one-dimensional diffusion equations of Eqs. (1) and (2) were solved by calculations using a finite element method (FEM). The following constants were used in the calculations according to experimental conditions: L = 800 μm, D = 5 × 10−10 m2 s−1, n = 5, f1 = 1 mHz, f2 = (5/3) mHz, f3 = (7/3) mHz, f4 = (11/3) mHz, f5 = (13/3) mHz, and b = 3.93546. The values of c0 and Δc were set to 10−3 mol dm−3 and 10−5 mol dm−3, respectively, for the calculations, although these values were not obtained from measurements.
All of the FEM calculations for which results are presented in the manuscript were performed using a numerical calculation software program, COMSOL Multiphysics® 5.2.
The results of the electrochemical hydrogen penetration measurement using the synthesized signal are shown in Fig. 4. The potential perturbation signal Eentry applied to the entry side induced the same perturbation in current Ientry of the entry side. A similar perturbation in current Iexit of the exit side was clearly detected. This is due to the transmission of the variation in hydrogen concentration in the specimen. Although some spike-like noises are observed in Iexit, they are not intrinsic to the experiment but seem to be due to a minor instrumental error. The waveform of Iexit is slightly delayed relative to that of Ientry as shown in the close view of Fig. 4(b). In order to quantify the delay as phase shifts, Fourier transforms were performed for a couple of portions of Ientry and Iexit. Figure 5 is an example of power spectra obtained by the Fourier transforms for 9 ks in Fig. 4(a). For both Ientry and Iexit, five distinct peaks are clearly identified at the input elemental frequencies of 1, 1.6667, 2.3333, 3.6667, and 4.3333 mHz. It is thought that the frequencies used in the measurements are well within the appropriate range. The phase shifts from a comparison of the phase values of the entry and exit sides at these frequencies. The phase shifts between Ientry and Iexit for five constituent frequencies derived by Fourier transforms of the samples of 4.5 ks before and after each time are shown at the bottom of Fig. 4. It is apparent that the phase shift increases with increase in frequency, although the dependency was obtained from Eqs. (6) and (7). In Fig. 4(a), the values of phase shifts were almost constant despite small variations, while Ientry gradually decreased and Iexit gradually increased. This trend might be caused by the alteration of the entry side surface over time as reported previously.6,11)
Temporal changes in currents Ientry and Iexit of the entry and exit sides and the phase shifts between them during electrochemical hydrogen penetration measurement with potential perturbation by a synthesized wave signal (a) from 0 to 40 ks and (b) from 0 to 6 ks. Phase shifts at frequencies of (red dense-dotted line) 1 mHz, (yellow sparse-dotted line) 1.6667 mHz, (olive dense-broken line) 2.3333 mHz, (green broken line) 3.6667 mHz, and (blue sparse-broken line) 4.3333 mHz. (Online version in color.)
Power spectra obtained by Fourier transforms of the waveforms of (upper) Ientry and (lower) Iexit.
Figure 6(a) shows diffusion coefficients derived from the measured phase shifts using Eqs. (6) and (9). The values of D for the constituent frequencies show a rather large variation of 4.5 − 6.2 × 10–9 m2 s–1. The averaged value of D and its standard deviation σD are shown in Table 1. The values of D were in the range of 5.0 − 5.5 × 10–9 m2 s–1 independent of frequency, whereas the values of σD tended to be smaller at higher frequencies. Although the reason will be discussed afterwards, the diffusion coefficient obtained from this method is in good agreement with the previous value obtained by using a single sinusoidal wave signal under a similar condition.6) Figure 6(b) shows the average values of α2 plotted against frequency. Eq. (6) was used to find α from the phase shift of each constituent frequency. It is clear that α2 is in proportion to f with a slope of 386.46 s. Since the proportional constant corresponds to πL2/D in Eq. (10), the values of D was calculated to be 5.2 × 10−9 m2 s−1. It is also in good accordance with the previously reported value6) and is an intermediate value of the D values obtained from Fig. 6(a). There are two possible reasons for the relatively large σD for the low frequency region shown in Table 1. One possible reason is the number of waves included in the sample range used for the Fourier transform. In the calculation, the number of data samples used for Fourier transform was the same for all frequencies, although the sampling at a high frequency included many waves in the sample range, resulting in averaged-out values. The other possible reason is the dependence of α–θ. Figures 7(a) and 7(b) show the α–θ curve of Eq. (6) and its differential form α-dθ/dα, respectively. The value of θ has a linear relationship with α when α is large. On the other hand, it exhibits a quadratic behavior when α is small. Therefore, the slope dθ/dα varies with the value of α (Fig. 7(b)). The larger the value of dθ/dα is, the smaller is the variation of α against the fluctuation inθ caused by measurement error. In short, a larger value of dθ/dα is more favorable for obtaining D from θ. The value of dθ/dα has a maximum value at α = 0.94. The value of α = 0.96 obtained at 2.3333 mHz was the most similar to this value and thus the most suitable for this experimental condition. These may have led to the variation in the standard deviation.
(a) Hydrogen diffusion coefficient determined from the data shown in Fig. 4(a) at frequencies of (red dense-dotted line) 1 mHz, (yellow sparse-dotted line) 1.6667 mHz, (olive dense-broken line) 2.3333 mHz, (green broken line) 3.6667 mHz, and (blue sparse-broken line) 4.3333 mHz. (b) Relation between f and α2. (Online version in color.)
| f/mHz | D/10−9 m2 s−1 | σD/10−9 m2 s−1 |
|---|---|---|
| 1 | 5.5 | 0.3 |
| 1.6667 | 5.2 | 0.4 |
| 2.3333 | 5.0 | 0.1 |
| 3.6667 | 5.2 | 0.2 |
| 4.3333 | 5.2 | 0.0 |
(a) Relationship between θ and α expressed by Eq. (6) and (b) its differential form. The value of α experimentally obtained for each constituent frequency is indicated by dotted lines.
Since the synthesized wave signal comprising five constituent sine waves was used in this measurement, the measurement data can be acquired five-times faster than that by the conventional one-by-one method. Since the number of waves that can be applied is not limited to five, a larger number may be used to increase the speed of data acquisition if desired.
Figure 8 shows temporal changes in fluxes of hydrogen at the entry and exit sides when the fluxes were numerically calculated for application of the synthesized waveform signal to the entry side. Both fluxes show almost the same behaviors as those of currents of the entry and exit sides as shown in Fig. 4(b), indicating that the numerical calculation successfully simulates the diffusion of hydrogen in a steel sheet during polarization with a synthesized wave form at the entry side since the current is in proportion to the flux. Temporal changes in the phase shift between fluxes are also shown at the bottom of Fig. 8.
Temporal changes in numerically calculated fluxes Jentry and Jexit of the entry and exit sides and the phase shifts between them for electrochemical hydrogen penetration measurement with potential perturbation by a synthesized wave signal. Phase shifts at frequencies of (red dense-dotted line) 1 mHz, (yellow sparse-dotted line) 1.6667 mHz, (olive dense-broken line) 2.3333 mHz, (green broken line) 3.6667 mHz, and (blue sparse-broken line) 4.3333 mHz. (Online version in color.)
Figure 9 depicts derivation of the D value in the same manner as that in Fig. 6. It was confirmed from Fig. 9(a) that the diffusion coefficients derived from the phase shifts for five constituent frequencies are all 5.00 × 10−9 m2 s−1, which completely agrees with the set value for the calculation. The slope of the f − α2 plot in Fig. 9(b) was 401.76 s, from which D = 5.00 × 10–9 m2 s–1 is derived according to Eq. (10). The result verified that the phase shift can successfully be quantified even if several sinusoidal waves are simultaneously applied as the perturbation signal to the entry side process. In other words, it was shown that the response to the synthesized wave is the sum of the responses to independent frequency components. This is consistent with the experimental results.
(a) Hydrogen diffusion coefficient determined from the results of calculations shown in Fig. 8 at frequencies of (red dense-dotted line) 1 mHz, (yellow sparse-dotted line) (5/3) mHz, (olive dense-broken line) (7/3) mHz, (green broken line) (11/3) mHz, and (blue sparse-broken line) (13/3) mHz. These five plots are completely overlapped. (b) Relation between f and α2. (Online version in color.)
As a consequence, the feasibility of the electrochemical hydrogen permeation measurement method with multi-sine wave perturbation was confirmed both experimentally and computationally. During analyses of the results, it was found that the frequency to achieve α = 0.94 is theoretically the most effective for reducing measurement error. The multi-sine method is expected to be effective especially in industrial applications such as the evaluation of complicated materials whose hydrogen diffusion coefficients have not been investigated. In the measurement of hydrogen diffusion coefficient of such materials with the conventional single-sine method, it takes a long time to change the measurement frequency stepwise since the appropriate frequency is unknown. However, the multi-sine method enables us to measure the response from one synthesized waveform signal for multiple frequencies and to select the most suitable frequency, at which α = 0.94, i.e., θ–45°, is accomplished, for the evaluation of diffusion coefficient afterward.
This method can be utilized not only for speeding up data acquisition but also for measurement of electrochemical hydrogen penetration of a steel sample, for which the characteristics of the surface would frequently vary because of the corrosion process. Since the process for determination of D is independent of the intensity of Ientry and Iexit, it is expected that surficial and internal changes in the hydrogen penetration process can be investigated separately. The multi-sine wave technique enhances the usefulness of electrochemical hydrogen penetration measurement.
The multi-sine wave technique was used for the electrochemical hydrogen penetration measurement in order to realize fast data acquisition. The technique was applied to measure the hydrogen diffusion coefficient D in a ferric sheet sample. Experimental results and results of calculations showed the following.
• The obtained value of D is in good agreement with that obtained by the single sine technique.
• The validity of this technique was verified by numerical calculations.
• The frequency of a constituent sine wave affects the deviation of the obtained value of D: The frequency achieving α = 0.94 is theoretically the best to minimize the deviation.
