Influence of Specimen Thickness on Thermal Desorption Spectrum of Hydrogen in High Strength SCM435 Steel

Abstract

Thermal desorption analysis (TDA) of hydrogen was carried out in high strength SCM435 steel in which the strain field of dislocation presumably is a major hydrogen trap site. Cylindrical specimens of radius 0.5–5 mm were cathodically charged until saturated and were heated at a rate ranging from 25 to 200°C/hr, aiming to vary the condition of hydrogen desorption from detrapping- to diffusion-control. For specimens of radius 0.5 mm the trap energy of hydrogen evaluated from the Choo-Lee (C-L) plot was as high as 33.8 kJ/mol, while for thicker specimens it was significantly smaller, i.e. 25.6–27.5 kJ/mol. The possible causes for the dependence of the trap energy on specimen thickness are discussed in terms of the influence of initial hydrogen distribution on the peak temperature and the deviation from local equilibrium of hydrogen during desorption. If pre-exposure is carried out for a sufficiently long time prior to TDA, the C-L plot seems to give a correct detrap energy even in the mixed-control desorption.

1. Introduction

Thermal desorption analysis (TDA) is widely used to identify the trap site and the state of presence of hydrogen in steel.¹⁾ In a previous study we developed a finite difference computer program which simulates the spectrum of thermal desorption analysis (TDA) of hydrogen in steel.²⁾ The simulation model was based upon the McNabb-Foster model³⁾ with the hydrogen trap sites of discrete binding energies. The key parameters, e.g. trap energy and detrapping coefficient, were determined by fitting with the spectra of cylindrical specimens of 2.5 mm in radius by trial and error. From the value of the detrapping coefficient p₀=25–30 s⁻¹,²⁾ it was thought that hydrogen desorption seemed to occur almost in the detrapping-controlled condition because the interval of detrapping events, i.e. inverse of p₀, was calculated to be greater than the diffusion time to the surface of the specimen.

The TDA spectrum is highly dependent on the specimen thickness. In a thin specimen lattice diffusion of hydrogen does not play a significant role in desorption because detrapped hydrogen almost instantly diffuses to the surface of the specimen. On the other hand, in a thick specimen lattice diffusion has a major influence on the spectrum shape because hydrogen atoms can escape from the trap site to such an extent that the hydrogen concentration in the lattice maintains equilibrium with that in the trap site. This is called diffusion-controlled desporption. If peak temperatures are measured at several ramp rates, the detrap energy of hydrogen can be determined from the Choo-Lee (C-L) plot in detrapping-controlled condition.⁴⁾ The C-L plot is also applicable to diffusion-controlled desorption if a sufficient pre-exposure is carried out prior to TDA.^5,6) Except these two extreme cases much less attention has been paid to desoption in which lattice diffusion has a substantial influence and equilibrium of hydrogen is achieved only partially between the lattice and the trap site. This is called mixed-control desorption henceforth. In this report, TDA of hydrogen was carried out using SCM435 steel specimens of varying thickness with the aim to see if C-L plot yields a correct energy of hydrogen escape in mixed-control desorption.

2. Mixed-control Desorption

The previously developed finite difference code²⁾ is used for simulation. The McNabb-Foster (M-F) theory consists of two equations. One is the diffusion equation with source (or sink) term expressed by,   

$$\frac{\partial c}{\partial t}+N_t\frac{\partial {\theta }_t}{\partial t}=D_L\left(\frac{{\partial }^{2}c}{\partial r^2}+\frac{1}{r}\frac{\partial c}{\partial r}\right)$$(1)

where c is hydrogen concentration in the lattice, N_t is the density of trap sites, θ_t is occupancy fraction at the trap site, D_L (=5.8×10⁻⁸ exp(−4.5×10³/RT) m²/s,⁷⁾) is the lattice diffusivity of hydrogen. r is the radial coordinate and t is time. The other equation describes the rate of change of the occupancy fraction and is given by,   

$$\frac{\partial {\theta }_t}{\partial t}=kc(1-{\theta }_t)-p{\theta }_t$$(2)

Here,   

$$k=k_{0}exp\left(-\frac{E_D}{RT}\right)$$(3)

and,   

$$p=p_{0}exp\left(-\frac{E_B+E_D}{RT}\right)$$(4)

are, respectively, trapping and detrapping coefficients, E_D (=4.5 kJ/mol) is the activation energy of lattice diffusion and E_B is the binding energy of hydrogen with the trap site. R is the gas constant and T is temperature. The pre-exponential factors k₀ and p₀ are related to each other as k₀=p₀/N_L⁸⁾ where N_L (=5.08×10²⁹ m⁻³) is the number of available sites of hydrogen in the lattice of bcc Fe. N_t was evaluated from the relationship N_t=νρ/a, where ν is the number of hydrogen occupation sites per Fe atom plane, a is the atom spacing (=2.49×10⁻¹⁰ m), and ρ is the dislocation density in the martensite lath, which was assumed to be 5×10¹⁵ m⁻² in this study.⁹⁾ Moreover, ν was assumed to be 10, which implies that the effective radius of dislocation as a trap site is ~1 nm.^2,7) At the start of simulation the concentration of hydrogen is uniform and in equilibrium between the lattice and the rap site, which often makes the initial evolution rate very high.

When the specimen size is so small that both re-trapping and diffusion of hydrogen can be ignored, the desorption rate is equal to the rate of escape from the trap site. It is well known that in this condition the detrap energy, denoted E_d, can be evaluated from the slope of ln (ϕ/T_p²) vs 1/T_p plot proposed by Choo-Lee (C-L),⁴⁾ where T_p is peak temperature and ϕ is ramp rate. On the other hand, when the specimen size is so large that the diffusion time is longer than the time interval of detrapping, the desorption is controlled by diffusion of hydrogen, and hydrogen desorption is described by the diffusion equation without a source or sink term using the so-called effective diffusivity.¹¹⁾ The C-L plot is also applicable to diffusion-controlled desorption if pre-exposure is carried out for a sufficiently long time,^5,6) whereas, the ln (ϕ/T_p) vs 1/T_p plot, another formula of Kissinger-type desorption proposed by Lee and Lee,¹²⁾ does not yield a correct value of detrap energy.

In practice a C-L plot is often carried out without paying due attention to the condition of desorption. To the best of the author’s knowledge there has been no discussion as to the validity of C-L plot in desorption between these two extremes. One may be able to examine this issue experimentally by performing TDA using specimens of varying thickness. SCM435 steel seems to be appropriate for this purpose because the TDA spectrum of hydrogen is relatively simple and the proposed trap energies do not scatter widely in low alloy martensitic steels.¹³⁾ Moreover, the M-F model can deal with the diffusion and trapping of hydrogen in any condition between detrapping- and diffusion-control without relying upon the local equilibrium assumption.¹⁴⁾ In the model one can calculate the parameter defined by,   

$$R_{LE}=\frac{{\theta }_L}{{\theta }_{LE}}$$(5)

to observe the degree of deviation from local equilibrium during desorption, where θ_L =c/N_L is the occupancy fraction of hydrogen in the lattice and θ_LE is the value at equilibrium with the trap site given by,¹¹⁾   

$${\theta }_{LE}=\frac{{\theta }_t}{1-{\theta }_t}\cdot exp\left(-\frac{E_B}{RT}\right).$$(6)

R_LE is close to zero in the detrapping-controlled condition in which hydrogen atoms that escaped from the trap site are released almost instantly from the specimen surface. On the contrary, R_LE is equal to unity in the diffusion-controlled desorption in which local equilibrium of hydrogen is maintained, i.e. θ_L ~θ_LE, and thus, it increases from zero to unity with specimen thickness. It should also be noted that θ_L, θ_t, θ_LE and R_LE are all functions of the radial coordinate r. The amount of deviation from local equilibrium is thus dependent upon the position within the specimen.

3. Experimental Procedure

SCM435 is a high-strength medium carbon Cr–Mo steel. The alloy contained 0.35 mass%C, 0.22 mass%Si, 0.78 mass%Mn, 1.16 mass%Cr and 0.17 mass%Mo, as has been presented elsewhere.²⁾ The alloy was machined into a cylinder of 0.5(R1), 1.25(R2), 2.5(R3), and 5 mm (R4) in radius (r₀) and 100 mm in length, see Table 1. Specimens were austenitized at 950°C for 1 h under Ar atmosphere and quenched into an oil bath maintained at 60°C. They were then tempered at 300°C for 1 hr under Ar and quenched into water. The microstructure was tempered lath martensite presumably with a small amount of retained austenite.¹⁵⁾

Table 1. Condition of hydrogen charge and detrap energy deduced from Choo-Lee plot.

Specimen	radius, mm	Charging time, h	Amount of hydrogen, ppm	Detrap energy Ed, kJ/mol	Detrapping coefficient p0, s−1
R1	0.5	48	1.72	38.3	4000
R2	1.25	96	3.84	31.9	50
R3	2.5	96	4.06	30.1	10
R4	5.0	96	4.39	32.0	5

Specimens were cathodically charged with hydrogen at a current density of 0.2 mA/cm² in a 3%NaCl aqueous solution with the addition of NH₄SCN (3 g per litre) as a catalyst at ambient temperature. They were then kept in liquid nitrogen. TDA was carried out using gas-chromatograph at constant heating rates from 25 to 200°C/hr. Since the heating rate employed in TDA is slow, it is designated ramp rate hereafter. The amount of released hydrogen was measured at the interval of 5 min using Ar as a carrier gas, and the total amount of charged hydrogen was obtained from the accumulated hydrogen flux at the end of the analysis, assuming that charged hydrogen was all diffusible. The procedure of TDA has also been described elsewhere.²⁾

4. Results

4.1. Cathodic Charge of Hydrogen

Figure 1(a) displays the TDA spectra of specimen R4 at various charging times, and in Fig. 1(b) the amount of charged hydrogen was plotted against charging time for all specimens. It seems that specimens of radius 1.25 mm or larger were saturated with hydrogen by charging for 96 h. Only one charging time (48 hr) was employed for R1. The amount of charged hydrogen is considerably less than those of the other specimens presumably because several specimens were charged simultaneously in an electrolysis cell. Table 1 shows the charging time and the concentrations of charged hydrogen in the specimens used for TDA.

Fig. 1.

a) TDA spectra of specimens of radius 5 mm cathodically hydrogen-charged for varying times. b) The amount of charged hydrogen plotted against charging time.

4.2. Variation of TDA Spectrum with Ramp Rate and Choo-Lee Plot

Figures 2(a) through 2(d) display TDA spectra of all specimens at a ramp rate of ϕ=100°C/hr. At the lowest ramp rate (ϕ=25°C/hr) the hydrogen evolution rate continued to decrease in all specimens except specimen R4. The highest peak temperature (T_p) was 189°C of R4 at ϕ=200°C/hr. The peaks became flatter and broader with specimen thickness. Since the amount of hydrogen released from the specimen was measured at a time interval of 5 minutes, the temperature interval increased significantly at a faster ramp rate, which may give rise to an error of ±8.3°C in T_p at ϕ=200°C/hr. In order to remedy this situation cubic spline interpolation was adopted for the determination of T_p for thick specimens.

Fig. 2.

TDA spectra of specimens of varying thickness at a ramp rate of a) 25°C/h, b) 50°C/h, c) 100°C/h and d) 200°C/h for specimens R1(+), R2(□), R3(△) and R4(○). Solid curves were calculated with E_B=35 kJ/mol, and p₀=5×10³ p⁻¹, see section 5.1.

Figures 3(a) through 3(d) present the C-L plots for all specimens. As shown in Table 1, E_d values determined from the slope are in good agreement with that assumed previously (E_B=27 kJ/mol),²⁾ while that of R1 is significantly greater. The p₀ values were evaluated by fitting with the observed T_p’s for each specimen. They are also similar to or even smaller than the value previously determined (p₀~30 s⁻¹) except R1.

Fig. 3.

Choo-Lee plot of experimental peaks (open squares), calculated peaks assuming initial homogeneous hydrogen distribution (solid circles), and calculated peaks of pre-exposed specimens (open circles) for specimens a) R1, b) R2, c) R3, and d) R4.

5. Discussion

5.1. Re-evaluation of E_d and p₀

Assuming that the binding energy of hydrogen trap site is identical in all specimens, the peak temperatures are calculated in all specimens using the E_d and p₀ values of specimen R3. Figure 4(a) shows clearly that they do not reproduce T_p’s of the other specimens. A small p₀ value implies that desorption occurred under detrapping-control and thus, T_p depends weakly on specimen thickness. On the other hand, the E_d and p₀ of R1 fit much better with T_p’s of the other specimens. By trial and error one can get an overall fit at E_B=35 kJ/mol and p₀=5×10³ s⁻¹ as displayed in Fig. 4(b). Although the agreement was worse, the best fit E_d and p₀ values were similar for ρ=10¹⁶ m⁻². These E_B values are somewhat smaller than the trap energy of hydrogen at an edge dislocation in bcc Fe,^16,17) presumably because the core of dislocation in lath martensite that formed upon quenching are occupied by carbon^10,18) and thus the sites of a higher binding energy of dislocation are not available for hydrogen.

Fig. 4.

Fitting with experimental peaks of all specimens with a single set of E_B and p₀ values. a) E_B=25.6 kJ/mol, and p₀=10 s⁻¹, and b) E_B=35 kJ/mol, and p₀=5×10³ s⁻¹. Open and solid symbols indicate peak temperatures of experimental and simulated peaks, respectively.

Figure 5 presents the Arrhenius plot of the Oriani’s effective diffusivity,¹¹⁾   

$$D_{eff}=\frac{D_L}{\text{1}+\frac{N_t}{N_L}exp\left(\frac{E_B}{RT}\right)}$$(7)

calculated with E_B=35 kJ/mol for three ρ values. The intersection of two straight lines extrapolated from the low and high temperature parts, denoted T_eff, varies considerably with N_t/N_L and thus, dislocation density. D_eff begins to deviate from the straight line at a temperature significantly below T_eff.

Fig. 5.

Arrhenius plot of effective diffusivity of hydrogen with varying dislocation density of lath martensite. D_L is the diffusion coefficient of hydrogen in αFe.

Figures 2(a) through 2(d) include TDA spectra simulated using E_B=35 kJ/mol and p₀=5×10³ s⁻¹ for comparison with experimental spectra. The variations of R_LE near the center (r/r₀=0.05) and the edge of a specimen (r/r₀=0.95) are presented in Figs. 6(a) and 6(b), respectively. The arrows indicate the temperature at which the hydrogen evolution rate falls to one hundredth of the peak height, i.e. the right edge of the peak. For reference R_LE of a thin specimen of r₀=0.1 mm is included (dotted curves); the value of R_LE is quite small throughout the peak in the center (~0.15 at the arrowed temperature, Fig. 6(a)) and near the edge (~0.07, Fig. 6(b)) and thus, desorption may occur almost under detrapping-control. In specimens R3 and R4 R_LE is close to unity both in the center and near the edge, indicating that the desorption occurred in an almost diffusion-controlled condition. On the other hand, desorption is likely to have occurred under mixed-control in R1 and R2. In these specimens, after the peak is over, R_LE increased to unity presumably because only diffusion of a minute amount of hydrogen is taking place in the specimen.

Fig. 6.

Variation of R_LE parameter with temperature during TDA a) in the center, and b) near the edge of the specimen. Arrows indicate the temperature at which the hydrogen evolution rate fell to one hundredth of the peak evolution rate, i.e. the end temperature of the peak. Dotted lines indicate R_LE of specimens of radius 0.1 mm.

5.2. Comparison of Measured and Calculated Detrap Energies

In Fig. 7 E_d’s obtained from experimental and simulated peaks with E_B=33 (grey dashed line) and 35 kJ/mol (black dashed line) are plotted against specimen radius and are compared with the sum of E_B and E_D, here denoted E_d° (horizontal dotted line). Simulations were carried out for specimens of r₀=0.1–10 mm. E_d calculated for thin specimens (r₀ ≤ 0.2 mm) is equal to E_d° and decreased with specimen radius, in good agreement with experiment for specimens R1 and R2. In these specimens both experimental and simulated E_d’s are smaller than E_d°. This is probably caused by the uniform distribution of hydrogen at the start of TDA. The calculated E_d then increases with specimen radius and as a result, it becomes greater than experimental E_d and even E_d° in specimen R4. The increase in calculated E_d above E_d° is probably because peaks become close to T_eff, in other words they protrude from the temperature region in which the low temperature form of D_eff is applicable. The small experimental E_d could be attributed to the extensive carbon segregation to dislocation caused by the slower cooling rates in a thicker specimen during quenching. With E_B=33 kJ/mol, however, calculated E_d seems to fit with experiment up to r₀~2.5 mm (R3), indicating that calculated E_d depends quite sensitively on the E_B and ρ values as well.

Fig. 7.

Variation with specimen thickness of E_d from experimental and calculated peaks. Calculations were carried out assuming E_B=33 and 35 kJ/mol. Only E_d° (=39.5 kJ/mol) with E_B= 35 kJ/mol is shown (horizontal dotted line). Open circles indicate E_d of pre-exposed peaks.

5.3. Simulation of TDA of Pre-exposed Specimens

It was mentioned that Kissinger-type desorption occurs in purely detrapping-controlled and purely diffusion-controlled desorption. In the later case, however, a sufficiently long pre-exposure is required. During the exposure at a low temperature hydrogen escapes slowly from the specimen surface. In purely diffusion-controlled desorption the concentration distribution in the cylindrical specimen can be described in terms of D_eff by the equation,   

$$c\left(\text{r,}\tau \right)=\frac{2c_0}{r_0}\sum _{n=1}^{\infty }\mathrm{\hspace{0.17em} }\frac{J_0({\alpha }_{n}r)}{{\alpha }_n{J}_1({\alpha }_n{r}_0)}exp(-{\alpha }_n{\hspace{0.17em}}^2{r}_0^2\tau )$$(8)

where   

$$\tau =\frac{\underset{0}{\overset{t}{\int }}{D}_{eff}dt}{r_0^2}$$(9)

is the dimensionless time, c₀ is the initial concentration of hydrogen, J₀ and J₁ are the Bessel function of the first kind of order zero and the first order, respectively, and α_n’s are the roots of J₀(α_nr₀)=0. Kissinger-type desorption is achieved where the 2^nd and the higher order terms in the r.h.s. of Eq. (8) become negligibly small compared to the first term.⁵⁾ The critical time for achieving it is τ=0.119 at which the fraction of hydrogen remaining in the specimen is 0.35 for a cylindrical specimen, see Table 1 of Ref. 6) Under this circumstance it is interesting to see if the Kissinger-type desorption can be achieved in the mixed-control condition.

Figures 8(a) and 8(b) display the concentration distribution of hydrogen in the lattice and the trap site, respectively, calculated for specimen R2 at various times of pre-exposure at 20°C. The concentration of trapped hydrogen on the specimen edge is not zero at an early stage because hydrogen atoms can be captured in the trap site from the inside of the specimen. The escape of hydrogen in the trap site is delayed compared to hydrogen in the lattice and this makes R_LE small near the edge at an early stage of pre-exposure. Figure 8(c) shows TDA spectra of the corresponding pre-exposed specimens. It is seen that the peak moves slowly to a higher temperature with exposure time. In experiment the shift of peak in pre-exposed specimens is often attributed to the continuum or multi-energy levels of trap sites in the strain field of dislocation. However, it can happen due to the change of initial hydrogen distribution during exposure even with the trap site of a single energy level. In order to achieve Kissinger-type desorption the time for exposure was taken to be t=3×10⁴ s for specimen R2. Figure 8(d) indicates that with this exposure time R_LE in the center and near the edge becomes close to each other from the beginning, albeit it does not approach unity. As seen in Eq. (9), the pre-exposure time required for achieving Kissinger-type desorption is proportional to the square of specimen diameter. It was thus taken to be 5×10³, 1×10⁵ and 3×10⁵ s for specimens R1, R3 and R4, respectively. E_d evaluated from the C-L plot of pre-exposed specimens is shown in Fig. 7 (open circles). It is almost equal to E_d° irrespective of specimen thickness, in other words from detrapping- to diffusion-controlled desorption. Particularly, it is noteworthy that E_d = E_d° even if desorption occurs in a mixed mode or with R_LE significantly below unity.

Fig. 8.

a) Diffusion profiles of hydrogen in the lattice and b) concentration distribution of trapped hydrogen at various exposure times in R2 specimen. c) TDA spectra of pre-exposed specimens for various times at room temperature. d) Comparison of R_LE of specimens (r₀= 1.25 mm) with and without pre-exposure. The pre-exposure time was 3×10⁴ s.

Choo-Lee plots are often carried out without paying attention to the condition in which desorption occurs. The above results indicate that the slope of the plot or detrap energy differs significantly from the true value (E_d°) due to the influence of initial hydrogen distribution and/or the condition of desorption. It is emphasized that pre-exposure can remove or mitigate very efficiently the influence of these factors. The critical exposure time of a cylindrical specimen is the time at which approximately two thirds of hydrogen escapes from the specimen. Hence, the specimen should not be too thick to avoid an excessively long exposure to achieve Kissinger-type desorption.

6. Summary

Thermal desorption analysis (TDA) of hydrogen was carried out in high strength SCM435 steel using cylindrical specimens of radius from 0.5 to 5 mm, aiming to vary the desorption condition from detrapping- to diffusion-control. The specimens were cathodically charged at ambient temperature until saturated and were heated at a ramp rate from 25 to 200°C/h. All specimens exhibited a single peak except thin specimens in which the evolution rate of hydrogen decreased monotonically at the smallest ramp rate. The trap energy of hydrogen was evaluated from the Choo-Lee (C-L) plot for specimens of each radius. The trap energy obtained from the thinnest specimens (r₀=0.5 mm) gave a much better account for peak temperatures of all specimens. A significantly lower E_B value for specimens of radius 1.25–5 mm can be attributed to the substantial involvement of lattice diffusion, more specifically the hydrogen distribution prior to TDA. The influence of initial hydrogen distribution can be removed if pre-exposure is carried out for a sufficiently long time even in mixed-control desorption. To evaluate the trap energy of hydrogen thin specimens are preferred because not only the influence of diffusion is reduced, but also the pre-exposure time can be shortened and peak temperatures do not enter the temperature region in which the Arrhenius plot of D_eff deviates from the straight line.

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