4.1. Macroscopic Characterization of Hydrogen Trap
According to Doyle´s theory [
18], nickel grain boundaries are short circuit diffusion paths, resulting in apparent hydrogen diffusivities being higher than the effective hydrogen diffusivity. However, the results found herein showed the opposite trend, as shown in
Table 2. Thus, the apparent hydrogen diffusivity was generally lower than the effective hydrogen diffusivity, indicating that the grain boundaries of 2.25Cr-1Mo-0.25V steel are not short circuit paths. On the contrary, the grain boundaries of 2.25Cr-1Mo-0.25V steel can be regarded as a kind of lattice defect, and it may hinder hydrogen diffusion. The grain boundaries are often denoted as hydrogen traps. The hydrogen trap density
(m
−3) can be estimated as follows [
20,
21,
22]:
where
is the lattice diffusion coefficient of hydrogen (m
2·s
−1,
m
2·s
−1 in ferrite [
22,
23] and
is Avogadro’s number (
mol
−1). The hydrogen trap density values calculated with this expression are listed in
Table 3. The values for the first permeation (
) were higher than those of the second permeation (
). This difference implies that the hydrogen traps were not completely uniform. According to previous studies [
24,
25], hydrogen traps can be divided into reversible and irreversible
and
, respectively). Both the reversible and the irreversible hydrogen traps were empty before the first permeation. During the permeation process, hydrogen atoms occupy all types of hydrogen traps. After the permeation and once the discharge process has started, hydrogen atoms can be easily released from the reversible traps. Conversely, the irreversible traps favored hydrogen to remain adsorbed. Thus, the hydrogen trap values for the first permeation involved both reversible and irreversible traps, while the second permeation values only involved reversible hydrogen traps, as follows:
Table 3 shows the reversible and irreversible hydrogen trap densities for RS and AS. As shown in
Table 3, more than 90% of the hydrogen was irreversibly trapped (RS 95.3%, higher than that of AS). The reversible hydrogen trap density of RS was ca. one third of that of AS. Nevertheless, the irreversible hydrogen trap density of RS was ca. 1.42 times higher than that of AS.
Figure 7 summarizes the effective hydrogen diffusivity of RS and AS for the first and second permeation tests.
The effective hydrogen diffusivities of RS and AS increased by 237 and 240%, respectively, after the second permeation process. In addition, the hydrogen subsurface concentration of RS and AS decreased by 84% and 63%, respectively, after the second permeation process. These results were consistent with the hydrogen trap theory. During the first permeation process, the diffusion process was hindered because of the resistance of the irreversible hydrogen traps, resulting in lower diffusivities. Thus, a considerable amount of hydrogen atoms remained at the subsurface of the specimens in the charging cell, resulting in higher hydrogen subsurface concentrations. Conversely, during the second permeation process, higher diffusivity values and lower hydrogen subsurface concentrations were found.
The large differences between RS and AS can be explained based on the metallography results. The chemical composition and microstructure of RS were similar to those of AS. Nevertheless, both samples differed in their grain size and their ferrite microstructure. The average grain size of RS was ca. half that of RS (12.5 and 24.1 , respectively), which resulted in grain boundaries ca. 8 times larger than that of AS in a three-dimensional crystal structure. As for the microstructure of ferrite, RS contained larger amounts of lath ferrite than AS. Each path of lath ferrite was formed by independent nucleation via austenite transformation. The joining of two or three units of lath ferrite can be defined as a sub-boundary. Similar to the grain boundaries, massive carbon atoms can remain in the sub-boundaries region (i.e., carbon segregation). During cooling, cementite precipitates from the supersaturated ferrite, which is close to the grain boundaries and sub-boundaries regions. Thus, the different hydrogen diffusion values of RS and AS can be essentially explained in terms of their cementite content. Cementite acts as a type of hydrogen trap, hindering its diffusion. RS showed higher cementite contents, and this resulted in higher hydrogen trap density (especially the irreversible hydrogen trap density) and lower hydrogen diffusivity. Conversely, AS had a lower cementite content, resulting in lower hydrogen trap density (especially the irreversible hydrogen trap density) and higher hydrogen diffusivity.
4.2. Microscopic Mechanism for Hydrogen Trap
Cementite is distributed at the grain boundaries and at the interfaces of the lath ferrite, acting as important hydrogen traps. Thus, the hydrogen trap mechanism of cementite is relevant for this work. This mechanism can be explained by the following two steps. First, the diffusion thermodynamics theory was used to explain the two kinds of diffusion, namely, down-hill diffusion and up-hill diffusion (i.e., along and against the concentration gradient, respectively). Second, the Lennard–Jones potential theory was used to explain the up-hill diffusion of hydrogen from ferrite to cementite, which resulted in cementite trapping hydrogen atoms.
According to Fick’s first law applied to hydrogen diffusion:
the diffusion is carried out along the hydrogen concentration gradient. When the hydrogen concentration gradient tends to zero, the flux tends to zero, and the diffusion stops. However, according to the thermodynamics, hydrogen diffusion is driven by a chemical potential gradient along a direction from regions of high chemical potential to regions of low chemical potential. In most cases, the concentration gradient follows the same trend as the chemical potential gradient, although this is not an absolute rule. In contrast, the diffusion process, described by thermodynamics, is more universal. At constant temperature and pressure, the essential of a diffusion process is that the change of the Gibbs free energy in a solid solution is less than zero,
. Thus, in the hydrogen diffusion process, the driving force for the diffusion of hydrogen atoms
along the
-direction can be determined by:
where
is the chemical potential of hydrogen. The diffusion rate
is proportional to
:
where the proportional coefficient
is defined as the mobility, which is related to the moving resistance. Thus, the hydrogen flux
is:
where
is the hydrogen concentration. According to Equations (8)–(11):
In a non-ideal solid solution, the chemical potential of hydrogen can be determined by:
where
is defined as the chemical potential of pure hydrogen,
is the gas constant, and
is the activity coefficient, which is used to correct the deviation from ideality, as predicted by the Raoult’s law. If the activity coefficient is less than 1, deviation from the Raoult’s law is negative, which implies that the components present attractive forces. On the contrary, if the activity coefficient is higher than 1, deviation from the Raoult’s law is positive, and the components present repellent forces [
26].
is the mole fraction of hydrogen:
where
is a constant. Then, the flux can be simply calculated as:
As shown in Equation (8), compared with Fick’s first law, the diffusivity of hydrogen can be calculated as:
Therefore, the sign of the diffusivity depends on the factor . If , diffusion takes place along the concentration gradient, from a region of high concentration to a region of low concentration. This is called down-hill diffusion. If , diffusion takes place along the fixed direction, regardless the concentration gradient. Thus, the concentration of the region with low chemical potential increases at the expense of those of other regions. This is called up-hill diffusion.
Since hydrogen is significantly smaller than iron, solute hydrogen atoms are located at the interstitial sites of the ferrite crystal. As shown in
Figure 8, there are two types of interstitial sites, namely, octahedral sites and tetrahedral sites. Hydrogen atoms placed at the same type of interstitial sites are equal. The arrows show hydrogen diffusion direction through the (001), (100), and (110) crystallographic planes in an octahedral sites solid solution and through the (111) and (100) crystallographic planes in a tetrahedral sites solid solution.
Figure 9a,b show the projections of a
supercell of cementite in the (001) and (010) crystallographic planes. As shown in
Figure 9b, cementite has a layered structure, with each layer consisting of identical and continuous trigonal prisms (highlighted with a black solid line for the first layer and a red dashed line for the second layer). Each trigonal prism consists of 6 Fe atoms and 1 C atom at the corners and center, respectively, as shown in
Figure 9c.
Figure 9d shows an orthorhombic (space group Pnma) crystal cell of cementite. However, the position of hydrogen in cementite remains unclear. Thus, it can be assumed that hydrogen atoms are located at the regions having the largest interstices.
Since the iron crystal remains unchanged upon diffusion of hydrogen atoms, the multi-compound system formed by hydrogen and steel can be considered a solid solution. Thus, it can be assumed that there are no chemical bonds between hydrogen and other atoms in steel, such that they exist independently. The hydrogen atoms and iron atoms are nonpolar such that the Lennard–Jones potential [
14] can be used to determine the interatomic forces. The Lennard–Jones potential theory explains the interaction between a pair of neutral atoms or molecules. In the crystal, which is formed by inert gases atoms, the interatomic van der Waals force is successfully described by the Lennard–Jones potential theory, and the calculated values of lattice constant extremely approximate to the experimental values. The most common expression of the Lennard–Jones potential is:
where
is the depth of the potential well,
is the finite distance at which the interatomic potential is zero, and
is the distance between the atoms. The force between two atoms is:
Regarding a crystal with
atoms, the total potential can be expressed as:
Considering
as the distance between the two closest atoms, then
. Equation (19) can be simplified to:
where,
The balanced interatomic distance
, which is also the lattice constant, can be determined as follows:
then
can be calculated as:
and
can be determined with the bulk modulus
:
which can be simplified to:
The bulk modulus in Equation (25) can be determined experimentally or by theoretical calculations. The bulk modulus of ferrite uses the experimental results of J.A. Rayne [
27], and is determined as:
where
is the Young’s modulus and
is the Poisson ratio. The bulk modulus of cementite has not been determined experimentally yet.
Table 4 shows the elastic moduli calculated from first principle calculations by different researchers [
28,
29,
30,
31].
The bulk modulus can be determined by the Voigt–Reuss–Hill method [
32]:
The
C11 and
C12 values calculated by different researchers were similar (
Table 4). Thus, it is reasonable to select arbitrary theoretically calculated values as true elastic moduli values of cementite. In this study, we selected the values from Jiang [
28] to calculate the Lennard–Jones potential. The lattice constant and bulk modulus of ferrite and cementite are summarized in
Table 5.
A
supercell was used as a model for calculating the Lennard–Jones potential and force. The hydrogen was located in the center cell. The force between one hydrogen atom and another atom can be calculated by Equation (18). The vector sum of the force can be qualitatively considered the force of the hydrogen atom in ferrite or cementite.
Table 6 shows the results of these calculations, and
Figure 10 shows the force of a hydrogen atom on random layers of the ferrite and cementite crystal cells. It is obvious that the force of a hydrogen atom is significantly larger in cementite than that in ferrite. Thus, cementite has a stronger attractive force towards hydrogen atoms and therefore the activity coefficient
of cementite is lower than that of ferrite. We considered that the direction of hydrogen diffusion from ferrite to cementite is positive. According to Equation (16), if the concentration of hydrogen in cementite is lower, the diffusivity is greater than zero,
. Also, hydrogen atoms diffuse along the concentration gradient. If the concentration of hydrogen in cementite is higher, the diffusivity is less than zero,
. The direction of diffusion is opposite to the hydrogen concentration gradient (i.e., up-hill diffusion). Then, the concentration of hydrogen in cementite increased continuously. Furthermore, hydrogen atoms trapped by cementite are hardly released (i.e., a high activation energy must be overcome). This is the hydrogen trap mechanism of cementite.