Numerical Interpretation of Hydrogen Thermal Desorption Spectra for Iron with Hydrogen-Enhanced Strain-Induced Vacancies

We used pure iron; Table I shows its chemical composition. It was annealed for 1800 seconds at 1173 K to reduce defects. Figure 2 schematically shows the shape of the specimens for the tensile test and L-TDS measurement.

The procedure for the experiment for TDS measurement is illustrated in Figure 3.

To introduce vacancies, after pre-charging with H atoms to the specimen for 1800 seconds, we applied the tensile strain to the specimen (Figure 2(a)) at \({8.3\ \times \ 10^{-6}\, \hbox {s}^{-1}}\) strain rate until the total tensile strain became 0.25 and was charged with H atoms by the cathodic electrolysis method concurrently. In the cathodic electrolysis charging, we applied \({50\,\hbox {A}\,\hbox {m}^{-2}}\) in the electric current to the specimen during immersing in a 0.1 N \({\hbox {NaOH}+\hbox {NH}_4\hbox {SCN}}\) 5 g \({\hbox {L}^{-1}}\) solution at 303 K. After cutting out a square plate specimen from the tensile test specimen, we removed H atoms from the cut-out specimen by holding it at 303 K for 3600 s. In addition, we mechanically and chemically polished the square plate specimen at room temperature to obtain the specimen shown in Figure 2(b). It took about 5400 s for those processes. We labeled this specimen as the H + strain specimen. For comparison, we also prepared another tensile test specimen to which strain was applied without H pre-charging and H charging and then from that cut out the square specimen, and the specimen shown in Figure 2(b) was obtained by the same processes as those for the H + strain specimen. Although this specimen was not H charged, the annealing process at 303 K for 3600 s was also applied for comparing with the case of the H + strain specimen. The polishing process was the same as for the H + strain specimen. We labeled this specimen as the strain specimen. By immersing both specimens in a \({0.1\,\hbox {N},\hbox {NaOH}+\hbox {NH}_4\hbox {SCN}}\) 20 g \({\hbox {L}^{-1}}\) solution at 303 K for 3600 s, the specimens were H charged as a tracer for L-TDS measurement. After the specimens were held in the liquid nitrogen, we measured the thermal desorption spectrum for each specimen using the L-TDS apparatus mainly with \({60 \,\hbox {K}\,\hbox {h}^{-1}}\) as the heating rate.

We employed a model based on the McNabb-Foster theory, which is represented by the following reaction-diffusion equation:Here, \({C^{H}_{L}}\) represents the concentration of H atoms in the normal lattice site (the interstitial site), and \({C^{H}_{V}}\) and \({C^{H}_{1}}\) represent the concentration of H atoms trapped by vacancies and vacancy clusters and by defects except for them, respectively. The LHS is their time evolution term, and the RHS is the diffusion term. The diffusion coefficient is given by \({D_H=D_{0H}\exp (-Q_H/RT)}\), where \({D_{0H}}\) is the coefficient independent of temperature, \({Q_H}\) is the barrier energy of H diffusion, T is temperature, and R is the gas constant. We used \({D_{0H}=4.2\times 10^{-8}\, \hbox {m}^{2}}\), \({Q_H=3.9\, \hbox {kJ} \,\hbox {mol}^{-1}}\) [21], and \({R=8.314\,\hbox {J}\,\hbox {mol}^{-1}\, \,\hbox {K}^{-1}}\). The one-dimensional form on the RHS of Eq. [1] is considered a valid approximation because the specimens used for our L-TDS measurement were thin. The time evolution of \({C^H_{1}}\) is given bywhere the first and second terms in the last line mean the trapping and detrapping processes, respectively, and \({K(X,T)=\exp (-X/RT)}\). In these terms, \({E_1}\), \({N_1}\), and \({\theta _1}\) are the H trapping energy, trap site concentration, and H occupancy of the trap sites, respectively, for defects except the vacancies and vacancy clusters. The pre-exponential factors for K(X, T) in the two terms are represented by \({k_{01}}\) and \({p_{01}}\), respectively.

While Eq. [2] is the usual form used so far, we employed the following equation for \({C^H_{V}};\)where i is the size of vacancy clusters, namely the number of vacancies composing one vacancy cluster. Vacancies are represented by \({i=1}.\) The meaning of \({E_{V_i}},\) \({N_{V_i}},\) \({k_{0V_i}},\) and \({p_{0V_i}}\) is the same as in Eq. [2]. Since the time evolution of vacancies and vacancy clusters brings about the change of \({N_{V_i}}\), we incorporated the effect to the model by adding the term \({\sum _i \frac{\partial N_{V_i}}{\partial t}\theta _{V_i}}\) to Eq. [3]. Based on the reports[28,29], we employed the following equations for estimating \({\frac{\partial N_{V_i}}{\partial t}};\)The equations represent the time evolution of the concentration \({C_{V_i}}\) of vacancies and vacancy clusters whose maximum size is n. The first and the second terms on the RHS of these equations represent the diffusion process and the annihilation process at sink sites such as dislocations, respectively. The terms with the coefficients \({b^{+}_{V_1+V_{i}}}\) and \({b^{-}_{V_{i}}}\) represent the agglomeration and dissociation processes, respectively. The diffusion coefficient is given by \({D_{V_i} = D_{0V_i} \exp (-Q_{V_i}/RT)},\) \({S_{V_i}}\) is the sink strength, and \({C_{V_i}^{eq}}\) represents the thermal equilibrium concentration. Table II shows \({D_{0V_i}}\) and \({Q_{V_i}}\) until \({i=6},\) and we assumed that vacancy clusters > \({i=6}\) are immobile. We used \({C_{V_1}^{eq}=\exp (-2.045\times 10^{5}\ \mathrm{J mol}^{-1}/RT)}\)[29] and assumed \({C_{V_i}^{eq}=0}\) for \({i>1}.\) The reaction constants are given by \({b^{+}_{V_1+V_j} =4\pi (r_{V_1}+r_{V_j})(D_{V_1}+D_{V_j})}\) and \({b^{-}_{V_j} = 4\pi (r_{V_1}+r_{V_j})(D_{V_1}+D_{V_j}) N_s \exp (-E^b_{V_i}/RT)},\) respectively, where \({r_{V_i}=(3i\varOmega /4\pi )^{1/3}+r_0}\) [28]. \({N_s}\) is the number of available sites, \({\varOmega }\) is the atomic volume, \({r_0=3.3}\)Å, and \({E^b_{V_i}}\) is the binding energy of the vacancy cluster \({V_i}\) to a vacancy. In the model, the growth and dissociation of vacancy clusters are represented by the absorption and emission of a single vacancy, respectively. According to the report[30], the value of \({E^b_{V_i}}\) is evaluated up to \({V_4}\) by first-principles calculation, and the value for clusters larger than \({V_4}\) is given by extrapolation. Table II shows \({E^b_{V_i}}\) until \({i=6},\) and for \({V_5}\) and \({V_6},\) the value obtained by extrapolation is shown. We used \({N_s=N_L(i-1)/i}\) for \({i>1}\) where \({N_L=5.1\ \times \ 10^{29}\, \hbox {m}^{-3}}\) is the normal lattice site concentration. As for \({S_{V_i}},\) we employed \({2.5\ \times \ 10^{14}\, \hbox {m}^{-2}}\) for all i. The trap site concentration is given by \({N_{V_i}=m_{V_i}C_{V_i}}\) where \({m_{V_i}}\) is the number of trap sites of an i-size vacancy cluster. In other words, \({m_{V_i}}\) is the maximum number of H atoms that one \({V_i}\) can trap.

According to the report[32], when a vacancy traps H atoms, its barrier energy of diffusion increases, so that the jump frequency of the vacancy is several orders of magnitude smaller than the case without trapping H atoms. The similar situation may occur for vacancy clusters. Thus, we assumed that vacancies and vacancy clusters with \({\theta _{V_i} \ge 1}\) are immobile. In addition, since \({\theta _{V_i}}\) takes a continuous value in the model, the number of H atoms trapped by vacancies and vacancy clusters, \({\theta _{V_i}m_{V_i}}\) can become a positive real number < 1. In such a case, we regarded the value of \({\theta _{V_i}m_{V_i}}\) as the ratio of vacancies or vacancy clusters that trap a H atom to all vacancies or vacancy clusters. Hence, we assumed that vacancies or vacancy clusters of \({(1-\theta _{V_i}m_{V_i})C_{V_i}}\) can migrate in the case of \({0 \le \theta _{V_i}m_{V_i} < 1}\).

We evaluated the H trapping energy \({E_{V_i}}\) for vacancy clusters by the molecular static simulation with the EAM potential proposed by Wen et al.[33]. Table III summarizes the value of \({E_{V_i}}\) until \({V_9}\). We ignored trap sites whose H trap energy is < \({10.0\,\hbox {kJ}\, \hbox {mol}^{-1}}\). According to the table, we can see that \({m_{V_i}}\) increases as the cluster size increases and \({E_{V_i}}\) depends on the number of trapped H atoms. As previously described, the number of trapped H atoms, \({\theta _{V_i}m_{V_i}}\) becomes a positive real number. Thus, we interpolated linearly between two H trapping energies, \({E_{V_i}(s+1)}\) and \({E_{V_i}(s)}\), as follows:where s is a positive integer for the number of trapped H atoms.

We determined the other parameters such as \({E_1},\) \({N_1},\) \({N_{V_i}},\) \({k_{01}},\) \({p_{01}},\) \({k_{0V_i}},\) and \({p_{0V_i}}\) from the experimental thermal desorption spectra shown in Section IV using the procedure described in Section V.

We solved Eqs. [1] though [4] simultaneously using the finite difference method in a one-dimensional calculation region. Since the specimen shape is a thin square plate, most of the H atoms are desorbed from the widest surface after diffusing in the thickness direction in which the length is shortest. Thus, the one-dimensional calculation region is a valid approximation. This calculation region is adapted by several reports[34‐36]. Due to symmetry, we used the half-thickness of the specimen for the calculation region. Also, we chose the initial and boundary conditions reflecting the experimental condition.

First, we applied the model in the previous section to the simulation of the degassing process for the H + strain specimen or the annealing process for the strain specimen, and then we applied it to the simulation of behavior of H atoms, vacancies, and vacancy clusters in the polishing process for both specimens. In the degassing process, as its initial state, we supposed that H atoms and vacancies distribute homogeneously and there is no vacancy cluster. Thus, in the simulation, we set the initial values for the H concentration and vacancies in the one-dimensional calculation region of \({2.5\ \times \ 10^{-4}\,\hbox { m}}\). Furthermore, we assumed the equilibrium state of H atoms between the normal lattice site and trap sites. In the simulation for the annealing process, we set the initial value of H atoms to zero. We performed the simulation isothermally at 303 K for 3600 s under the boundary condition that the concentration of H atoms, vacancies, and vacancy clusters becomes zero on the surface. From the degassing or the annealing simulation, we obtained the distributions of H atoms, vacancies, and vacancy clusters and used them as the initial state of the simulation for the polishing process. Strictly speaking, in the polishing process, the specimen thickness decreases; however, we employed the same one-dimensional calculation region with the degassing or the annealing simulation because the influence of the thickness decrease is minute. We simulated the process at 298 K for 5400 s under the same boundary condition as the case of the degassing or the annealing simulation. Next, we simulated the process for charging the specimen of \({3.0\ \times \ 10^{-4}\,\hbox { m}}\) with tracer H atoms. In the simulation, we set a flux J at the surface boundary of the calculation region with length \({1.5\ \times \ 10^{-4}\,\hbox { m}}\), and we held the temperature at 303 K. Then, using the states of H atoms, vacancies, and vacancy clusters, which were obtained by the H charging simulation, we simulated the heating process of the L-TDS measurement at the heating rate of \({60 \,\hbox {K} \,\hbox {h}^{-1}}\). During the simulation, we counted the amount of desorbed H at a constant interval.

Section V describes the determination of the initial values of vacancies in the degassing simulation and of vacancies and H atoms in the annealing simulation and the flux J in the charging simulation.

First, we determined the simulation parameters that are not already set in Section III from the experimental results shown in the previous section. As the H trapping energy of Peak 1, we used \({E_1=26.6\,\hbox {kJ}\,\hbox {mol}^{-1}}\), which is obtained from the difference between the evaluated detrapping activation energy of Peak 1 of the H + strain specimen and the barrier energy of H diffusion, \({Q_H}\). We chose \({p_{01}=4.6\times 10^{2}\,\hbox {s}^{-1}}\) so that the peak temperature of Peak 1 becomes that of the experimental spectrum and \({k_{01}=9.0\times 10^{-28}\,\hbox {m}^3\,\hbox {s}^{-1}}\) from the relation \({k_{01}=p_{01}/N_L}.\) As for \({N_1}\), according to our several preliminary experiments, the amount of H desorption does not almost change even if the H charging was severer than the current condition. Thus, we can suppose that the trap site of Peak 1 with H trapping energy of \({E_1}\) is almost in saturation after charging with tracer H atoms. Hence, from the desorbed H atoms of Peak 1 in Table IV, we assumed that \({\theta _1=0.95}\) after the charging of tracer H atoms and estimated \({N_1}\) to be \({1.1\times 10^{24}\,\hbox {m}^{-3}}\) and \({1.2\times 10^{24}\,\hbox {m}^{-3}}\) for the strain and H + strain specimens. In addition, we chose the flux J for the H charging simulation so that \({\theta _1=0.95}\) is satisfied under the experimental condition of charging the specimens with tracer H atoms.

Meanwhile, we determined \({p_{0V_i}}\) and \({k_{0V_i}}\) as follows. Although the existence of vacancy clusters can be supposed after charging with tracer H atoms, we considered only vacancies and assumed that \({\theta _{V_1}=0.3}\) and then calculated the value of \({C_L^{H, a.c.}\exp (E_{V_1}(\theta _{V_1})/RT)(1-\theta _{V_1})/\theta _{V_i}}\), where \({C_L^{H, a.c.}}\) is the H concentration in the normal lattice site after charging. The calculated value corresponds to \({p_{0V_1}/k_{0V_1}}\). Even though \({p_{0V_1}/k_{0V_1}}\) becomes \({N_L}\) when the H trap energy is a constant, we can suppose that it does not have to be \({N_L}\) since \({E_{V_1}}\) depends on \({\theta _{V_1}}\) in the case of vacancies. Thus, we chose \({p_{0V_1}=5.7\ \times \ 10^{5}}\) s\({^{-1}}\) so that the peak temperature of Peak 2 approaches the experimental result and then \({k_{0V_1}=2.7\ \times \ 10^{-30}\,\hbox { m}^3 \,\hbox {s}^{-1}}\) is obtained using the calculated value above. We also used the same values with \({p_{0V_1}}\) and \({k_{0V_1}}\) for all the other vacancy clusters. As for the initial value of the concentration of vacancies \({C_{V_1}^{init}}\) for the degassing simulation, we chose the value so that the total amount of H atoms trapped by vacancies and vacancy clusters before the heating simulation becomes the experimental value of Peak 2 in Table IV. This determination is possible because the amount of H atoms trapped by defects depends on the amount of the defects. In addition, the size distribution of vacancy clusters before the L-TDS measurement depends on the migration behavior of vacancies in the pre-measurement processes such as degassing, polishing, and charging, and the migration of vacancies is suppressed by their trapped H atoms. Therefore, the initial concentration of H atoms trapped by vacancies in the degassing process influences the thermal desorption spectra. In the degassing simulation, by varying the total H concentration in the calculation region in equilibrium calculation, we determined the concentration of H atoms trapped by vacancies that makes the shape of simulated desorption spectra close to the experimental spectra as the initial value \({C_{V_1}^{H,init}}\). We also determined the initial concentration of H atoms trapped by defects for Peak 1 \({C_{1}^{H,init}}\) from the equilibrium calculation simultaneously.

Herein, we show the simulation results for the pre-measurement process. Figure 7 shows the change of the amount of H atoms trapped by each kind of trap site and of the concentration of vacancies and vacancy clusters in the strain specimen during the annealing, polishing, and charging processes. Figure 8 shows the result for the H + strain specimen where the degassing process was simulated instead of the annealing process.

The initial values of the simulation for the annealing process or the degassing process are as follows: \({C_{1}^{H, init}=0}\), \({C_{V_1}^{H, init}=0}\), and \({C_{V_1}^{init}=2.8\ \times \ 10^{-7}}\) at% for the strain specimen, and \({C_{1}^{H, init}=0.25}\) ppm (0.97), \({C_{V_1}^{H, init}=0.043}\) ppm (0.34), and \({C_{V_1}^{init}=1.2\ \times \ 10^{-6}}\) at% for the H + strain specimen where the value in the parentheses means the occupancy of H atoms for the corresponding trap site. These values were chosen so that the simulated spectra could reproduce the experimental spectra in Figure 4. As for the strain specimen, since no H atoms suppressing vacancy migration are contained before the charging process, as seen in Figure 7(a), vacancy clusters are generated from the beginning of the annealing process because of vacancy migration, as seen in Figure 7(b). However, it is seen that the generation of vacancy clusters is not so large that the decrease in vacancy (\({V_1}\)) concentration is small. In the charging process, the concentration of vacancies and vacancy clusters is almost constant because their migration is suppressed by H atoms. In the case of the H + specimen shown in Figure 8, vacancy clusters are not generated from the beginning to about 0.8 h in the degassing process because of the suppression of vacancy migration by H atoms. After that, since vacancies start to migrate until the end of the polishing process because of the decrease of H atoms, the \({V_1}\) concentration decreases slightly, and vacancy clusters are generated rapidly. Compared with the case of the strain specimen, since the initial \({V_1}\) concentration is larger, the generation of vacancy clusters is also larger. The tendency in the charging process is similar to that of the strain specimen.

Figure 9 shows the amount of H atoms trapped by vacancies, vacancy clusters, and trap sites of Peak 1 and the concentration of vacancies and vacancy clusters. These values are ones before the heating process of TDS. According to the figure, H atoms trapped by \({V_1}\) are the largest after Peak 1, and \({V_5}\) traps the largest amount of H atoms among vacancy clusters. The tendency of trapped H atoms reflects the tendency of the concentration of vacancies and vacancy clusters. Thus, it means that the amount of trapped H atoms depends strongly on the concentration of vacancies and vacancy clusters rather than the trapping energy. It is also found that the H atoms trapped by \({V_2}\) and \({V_3}\) are smaller than those for \({V_4}\) and \({V_5}\) in the strain specimen and those for \({V_4}\) to \({V_6}\) in the H + strain specimen. This is because the concentration of the former is smaller than that of the latter. The reason why the concentration of \({V_2}\) and \({V_3}\) is smaller is that their diffusion is faster than that of the vacancies and other clusters. The tendency is almost the same between both specimens though the trapped H atoms and the concentration of vacancies and vacancy clusters of the strain specimen are smaller than those of the H strain specimen. The total amount of H atoms trapped by vacancies and vacancy clusters is 0.006 and 0.037 ppm for the strain and H + strain specimens, respectively, and the amount of trapped H atoms for Peak 1 is 0.22 and 0.24 ppm for the strain and H + strain specimens, respectively. Note that these values are the same as in the experimental value in Table IV, as intended.

Figure 10 shows the thermal desorption spectra simulated in the heating process starting from the distribution of trapped H atoms, vacancies, and vacancy clusters shown in Figure 9. Comparing the simulated spectra to the experimental spectra in Figure 4, we found that the shape of Peaks 1 and 2 is mostly reproduced. However, we can see a spike at about 343 K, and the plateau part and the tail of the high-temperature side of Peak 2 are shorter than those of the experimental spectra. The change of trapped H atoms and of the concentration of vacancies and vacancy clusters in the heating process is shown in Figures 11 and 12. According to the figures, although the amount of trapped H atoms and the concentration of vacancies and vacancy clusters of the H + strain specimen are larger than those of the strain specimen before the heating process, their tendency to change during the heating process is similar. The trapped H atoms start to decrease before 273 K for Peak 1 and just after 273 K for vacancies and vacancy clusters, and by 398 K they become < \({10^{-8}}\) ppm for all kinds of trap sites. Accompanying the decrease of H atoms trapped by \({V_1},\) \({V_1}\) starts to migrate and vacancy clusters are additionally generated. The generation of \({V_2}\) and \({V_3}\) is especially more remarkable than that of the others and the decrease of \({V_2}\) and \({V_3}\) is also so rapid. The rapid decrease causes the rapid release of H atoms trapped by \({V_2}\) and \({V_3}\) so that the rapid release forms the spike in the simulated spectra at about 343 K in Figure 10. According to Figures 11(a) and 12(a), we observed that the desorption spectra in the temperature range > 343 K are formed by H atoms released from vacancy clusters > \({V_3}\). Those vacancy clusters remain slight even beyond the temperature of 373 K and \({V_1}\) is also remaining.

Figure 13 shows the thermal desorption spectra obtained by simulating the case that the degassing process of the H + strain specimen is replaced by the annealing process.

The figure corresponds to Figure 6. Comparing both figures, we revealed that the decrease of the desorption rate of Peak 2 against the aging temperature was almost reproduced. Thus, the model can properly represent the decrease of vacancies and vacancy clusters for temperature. However, the change of Peak 1 was not simulated in Figure 13. As for Peak 1, the peak temperature in Figure 4 is close to that of the corresponding peak in the spectra of the tempered martensitic steel[27], and the H trapping energy evaluated in Section IV was close to that of screw dislocations. Thus, it can be considered that Peak 1 is formed by H atoms released mainly from dislocations as reported[27]. Figure 7 indicates that by the aging process at ≥ 323 K, the H trap sites decrease and the H trapping energy changes from the case of 303 K. If it is supposed that the defect for Peak 1 is dislocations, although its decrease by the aging process of high temperature is well known, the increase of its H trapping energy is not so obvious. It might relate to that dislocations absorb vacancies and vacancy clusters as sink sites. However, the mechanism of the change of dislocations is not so clear that it can be incorporated into the model. In addition, since dislocation trap sites might not change at a temperature of about 281 K, which is the peak temperature of Peak 1, it is considered that their change does not almost influence the result of the TDS measurement. Therefore, we did not consider the change of defects for Peak 1 in the model, so that the change of Peak 1 was not simulated in Figure 13.