Corrigendum

This article refers to:

Mobile effect of hydrogen on intergranular decohesion of iron: first-principles calculations

An error occurred in Philosophical Magazine, Volume 92, Issue 11, 11 April 2012, pp. 1349–1368, in the article ‘Mobile effect of hydrogen on intergranular decohesion of iron: first-principles calculations’, by M. Yamaguchi (MY), et al.

For the calculation of a reduction in the cohesive energy (2γ_int) at a bcc FeΣ3(111) symmetrical tilt grain boundary (GB) by mobile and immobile hydrogen segregation on the GB and its fracture surfaces (FS's), the internal segregation energy terms () have been only considered in Equation (12) in the MY paper, as a first approximation,

where γ_s/gb is the FS/GB energy not affected by hydrogen segregation, A is the unit-cell area. Γ is defined by the number ratio of segregated- or dissolved-hydrogen atoms (n _gb/s/b) to its sites over the unit-cell area (N _gb/s/b). The subscripts of gb, s, and b represents the grain boundary, fracture surface, and bulk, respectively. The Γ _i term is explained later.

This approach was found to be incorrect. Instead, 2γ_int should be calculated as a free energy difference before and after segregation on GB/FS () that are represented by the internal segregation energy and the temperature-dependent configurational entropy of hydrogen. Therefore, its correct formula can be re-written by

The term is derived in the following way. The free energy of the system with GB/FS segregation defined over the unit-cell area of A () can be described by

where configurational entropy per a single atom (S _c(Γ)) is given by −k _B[Γ ln Γ + (1 − Γ)ln(1 − Γ)]. Considering that the total number of hydrogen atoms remains the same before and after segregation in the system, the free energy difference of the system () is indicated by

where δΓ _b = n _gb/s/N _b. It is reasonable to postulate that the number of segregated atoms (n _gb/s) is much smaller than that of dissolved hydrogen atoms in the bulk (n _b). Thus, the difference between S _c(Γ _b) and S _c(Γ _b + δΓ _b) associated with and without segregation, respectively, can be approximated by

where μ_b is the chemical potential of a dissolved hydrogen atom in the bulk given by k _B T ln[Γ _b/(1 − Γ _b)]. Using Equations (c1), (c3), and (c4), 2γ_int for the conditions of constant composition (fast fracture); Γ _i = n _gb/(2N _s), n _i = n _gb/2, and chemical potential (slow fracture); Γ _i = Γ _s, n _i = n _s, has a valid form of Equation (c5).

In Equation (c5), the last two terms are added to Equation (12) in the MY paper. The configurational entropy of segregated hydrogen on the GB and two FS's, i.e., −T [2S _c(Γ _i)N _s − S _c(Γ _gb)N _gb]/A, is very small (about 0.1 − 0.2 J/m²) at ambient temperatures. The last term {−(2n _i − n _gb)μ_b/A} has a significant contribution under the condition of constant chemical potential, as the dissolved hydrogen content in the bulk (C _H = 6Γ _b) is reduced, leading to a very high negative value of μ_b. On the other hand, the last term disappears under the condition of constant composition with n _i = n _gb/2.

According to Equation (c5), the value of 2γ _int for slow and fast fracture decreases monotonically and parallel with increasing log₁₀ C _H as shown in Figure 8 (corrected) and Figure 10 (corrected). The onset of slow decohesion occurs above C _H = 10⁻¹³ atomic fraction (300 K). In a realistic range of C _H around 10⁻⁵–10⁻⁸ atomic fraction in iron and ferritic steels, the re-calculated 2γ _int at 300 K decreases about 30–40% at most under the condition of constant chemical potential (Figure 8 (corrected)). Although the re-calculated 2γ _int is not largely reduced for slow fracture especially in a low region of C _H, the combined effects of mobile and immobile hydrogen still bring about more prominent GB decohesion of iron, compared with a sole effect of immobile hydrogen. Finally, it should be pointed out that the segregation behavior of hydrogen, influencing the diffusion time, is not changed by the modified analysis of 2γ _int, as illustrated in Figure 10 (corrected).

Figure 8 (corrected). Comparison of reduction in the cohesive energy (2γ_int) of the bcc Fe Σ3(111) GB caused by only immobile effect of hydrogen under the condition of constant composition (fast fracture) and combined effects of mobile and immobile hydrogen under the condition of constant chemical potential (slow fracture) with increasing bulk hydrogen content (C _H) at 300 K (a) and 170 K (b).

Figure 10 (corrected). Comparison of variations of diffusion time for fast (constant composition) and slow (constant chemical potential) fracture, relevant to the change in 2γ_int at 300 K (b) and 170 K (c).