Long-range hydrogen-binding effects of carbide interfaces in iron

A micromechanics model was developed to evaluate the elastic binding energy between carbide precipitates and hydrogen interstitials using Eshelby's equivalent inclusion method. Density functional theory (DFT) simulations were performed to obtain the material-specific quantities, e.g., lattice constants and the elastic constants, for the continuum model. Using this model, we find that for coherent carbide precipitates, hydrogen atoms are more likely to bind on the broad surfaces of the disk-like precipitates, which is consistent with experimental observations. For semicoherent and incoherent precipitates, our model suggests that it is possible for semicoherent precipitates to have significant hydrogen binding capability while there is no hydrogen-binding capability of incoherent precipitates, which also agrees with experimental findings. In addition, several factors that influence the binding energies between hydrogen atoms and carbide precipitates were quantitatively analyzed, including the precipitate size, morphology, orientation, and interface. These collective results include both the position and the value of the strongest hydrogen-binding interaction for a wide range of carbide stoichiometries, which contributes to our understanding of hydrogen trapping in steel-based materials.

Figure 1

(a) A schematic of absorbed H in steels. The blue dots represent H atoms and the orange ellipses represent the carbide particles. (b) A schematic illustrating our simplified model: The blue sphere represents the point defect (a H atom) and the orange ellipsoid represents the inclusion (a carbide particle). The global coordinates ${\widehat{x}}_i$ are set along the semiaxes of the ellipsoidal inclusion. The local coordinates ${\tilde{x}}_i^D$ and ${\tilde{x}}_i^I$ represent the orientation of the point defect and the inclusion, respectively.

Figure 2

(a) The components of the computed elastic dipole tensor for a H atom in $\gamma$-Fe as a function of the supercell size. The blue squares denote $P_{11}$ for H at the octahedral interstices, and red circles denote $P_{11}$ for H at the tetrahedral interstices. Due to the symmetry of $\gamma$-Fe, the elastic dipole tensor is isotropic $P_{11}=P_{22}=P_{33}$. (b) The energy correction term $\mathrm{\Delta }{E}^{\mathrm{corr}}$ versus the supercell size. $\mathrm{\Delta }{E}^{\mathrm{corr}}$ represents the energy correction to DFT calculations for the periodic images of the point defect. The blue squares represent $\mathrm{\Delta }{E}^{\mathrm{corr}}$ for H at the octahedral interstices and red circles represent $\mathrm{\Delta }{E}^{\mathrm{corr}}$ for H at the tetrahedral interstices.

Figure 3

(a) The octahedral interstices (orange spheres) in a conventional unit cell of $\alpha$-Fe with the defined local coordinates ${\tilde{x}}_i^D$. The three different interstitial positions are labeled as 1, 2, and 3, respectively. (b) The elastic dipole components of H at the octahedral P3 position, marked by the red sphere in (a), where $P_{33}>P_{11}=P_{22}$. (c) The tetrahedral interstices (orange spheres) in a conventional unit cell of $\alpha$-Fe with the defined local coordinates ${\tilde{x}}_i^D$. The three different interstitial positions are labeled as 1, 2, and 3, respectively. (d) The elastic dipole components of H at the tetrahedral P3 position, marked by the red spheres in (c), where $P_{33}<P_{11}=P_{22}$.

Figure 4

A schematic of a coherent TiC precipitate in $\alpha$-Fe adopting the Baker-Nutting OR [7, 16]. (a) A TiC platelet is modeled as an oblate spheroidal inclusion with the semiaxes $a_1=a_2>a_3$ where the global coordinates ${\widehat{x}}_i$ are set along the semiaxes. (b) A schematic of the atomic configuration on the plane II in the subplot (a). Coherent interfaces (light-blue region) form on the broad surfaces of the oblate spheroid, which lead to the induced strain from the lattice mismatch. Near the edge of the oblate spheroid, incoherent interfaces (light-pink region) form and no significant strain is induced along the direction ${\widehat{x}}_3\parallel {\left[100\right]}_{\alpha }\parallel {\left[100\right]}_{\mathrm{TiC}}$.

Figure 5

The elastic interaction energy ($E_{\mathrm{int}}$) between the TiC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the octahedral interstice of $\alpha$-Fe. The Baker-Nutting OR is chosen where the plane normal of the coherent interface is ${\widehat{x}}_3\parallel {\left[100\right]}_{\alpha }\parallel {\left[100\right]}_{\mathrm{TiC}}$. (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the octahedral P1 position. (c) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (d) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the octahedral P2 and P3 positions. In the subplots (b) and (d), the circled numbers show the site at the inclusion interface associated with the largest hydrogen-binding energy.

Figure 6

The elastic interaction energy $E_{\mathrm{int}}$ between the TiC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the tetrahedral interstice of $\alpha$-Fe. The Baker-Nutting OR is chosen where the plane normal of the coherent interface is ${\widehat{x}}_3\parallel {\left[100\right]}_{\alpha }\parallel {\left[100\right]}_{\mathrm{TiC}}$. (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the tetrahedral P1 position. (c) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (d) $E_{\mathrm{int}}$ on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the tetrahedral P2 and P3 positions. In the subplots (b) and (d), the circled numbers represent the point at the inclusion interface associated with the largest hydrogen-binding energy.

Figure 7

The elastic interaction energy $E_{\mathrm{int}}$ between the TiC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the octahedral interstice of $\gamma$-Fe. (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane. The orange-triangle marks the site at the inclusion interface associated with the largest hydrogen-binding energy.

Figure 8

The minimum elastic interaction energy $E_{\mathrm{int}}$ on the surface of an oblate spheroidal inclusion ($a_1:a_3=5$) for the carbides with a FCC metal sublattice plotted as a function of the carbon concentration. (a) H at the octahedral interstices inside $\alpha$-Fe; (b) H at the tetrahedral interstices inside $\alpha$-Fe; (c) H at the tetrahedral interstices inside $\gamma$-Fe; (d) H at the octahedral interstices inside $\gamma$-Fe. In the subplot (a), the right-vertical axis represents the elastic-strain interaction energy referenced to the ground-state energy of the tetrahedral-interstice H atom, $E_{\mathrm{int}}\left(\min \right)+\mathrm{\Delta }{E}_{ot}$. In the subplot (c), the right-vertical axis represents the elastic-strain interaction energy referenced to the ground-state energy of the octahedral-interstice H atom, $E_{\mathrm{int}}\left(\min \right)-\mathrm{\Delta }{E}_{ot}$.

Figure 9

The elastic interaction energy $E_{\mathrm{int}}$ between the WC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the octahedral interstice of $\alpha$-Fe. The Pitsch-Shrader OR is chosen where the plane normal of the coherent interface is ${\widehat{x}}_3\parallel {\left[001\right]}_{\alpha }\parallel {\left[0001\right]}_{\mathrm{WC}}$ (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the octahedral P1 position. (c) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ for H at the at the octahedral P2 and P3 positions; (d) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the octahedral P2 position. In the subplots (b) and (d), the circled numbers show the site at the inclusion interface associated with the largest hydrogen-binding energy.

Figure 10

The elastic interaction energy $E_{\mathrm{int}}$ between the WC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the tetrahedral interstice of $\alpha$-Fe. The Pitsch-Shrader OR is chosen where the plane normal of the coherent interface is ${\widehat{x}}_3\parallel {\left[001\right]}_{\alpha }\parallel {\left[0001\right]}_{\mathrm{WC}}$. (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the tetrahedral P1 position. (c) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ for H at the tetrahedral P2 and P3 positions and (d) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane for H at the tetrahedral P2 position. In the subplots (b) and (d), the circled numbers represent the point at the inclusion interface associated with the largest hydrogen binding energy.

Figure 11

The elastic interaction energy $E_{\mathrm{int}}$ between the WC oblate spheroidal inclusion ($a_1:a_3=5$) and a H atom in the octahedral interstice of $\gamma$-Fe. (a) $E_{\mathrm{int}}$ plotted along the semiaxes of the oblate spheroidal inclusion ${\widehat{x}}_i$ and (b) $E_{\mathrm{int}}$ plotted on the ${\widehat{x}}_1{\widehat{x}}_3$ plane. The orange triangle marks the site at the inclusion interface associated with the largest long-range hydrogen binding energy.

Figure 12

The minimum elastic interaction energy $E_{\mathrm{int}}$ on the surface of an oblate spheroidal inclusion ($a_1:a_3=5$) for the carbides with a hexagonal metal sublattice plotted as a function of carbon concentration. (a) H at the octahedral interstices inside $\alpha$-Fe; (b) H at the tetrahedral interstices inside $\alpha$-Fe; (c) H at the tetrahedral interstices inside $\gamma$-Fe; and (d) H at the octahedral interstices inside $\gamma$-Fe. In the subplot (a), the right-vertical axis represents the elastic-strain interaction energy referenced to the ground-state energy of the tetrahedral-interstice H atom, $E_{\mathrm{int}}\left(\min \right)+\mathrm{\Delta }{E}_{ot}$. In the subplot (c), the right-vertical axis represents the elastic-strain interaction energy referenced to the ground-state energy of the octahedral-interstice H atom, $E_{\mathrm{int}}\left(\min \right)-\mathrm{\Delta }{E}_{ot}$.

Figure 13

The values of $E_{\mathrm{int}}\left(\min \right)$ for oblate spheroidal inclusions with $a_1/a_3$ ranging from 1.5 to 5. (a) The TiC coherent precipitate in $\alpha$-Fe; (b) the TiC coherent precipitate in $\gamma$-Fe; (c) the WC coherent precipitate in $\alpha$-Fe; (d) the WC coherent precipitate in $\gamma$-Fe.

Figure 14

The calculated lattice constant and heat capacity using the Debye-Grüneisen model compared with the experimental data. Experimental data of the lattice constant for (a) $\alpha$-Fe and (c) $\gamma$-Fe: Feng [82], Basinski et al. [83]. Experimental data of the heat capacity of (b) $\alpha$-Fe and (d) $\gamma$-Fe: Chase [84].

Figure 15

(a) The computed average elastic long-range interaction between H atoms inside $\gamma$-Fe via DFT simulations; (b) The computed average elastic long-range interaction between H atoms inside $\gamma$-Fe via DFT simulations. The light green region in the subplot (b) represents the threshold concentration of hydrogen in ferritic/martensitic steels with its ultimate tensile stress (UTS) is 1 GPa.

Figure 16

(a) The constraint strain ${\varepsilon }_{ij}^c$ along the ${\widehat{x}}_3$ axis of the cylindrical inclusion and the ellipsoidal inclusion. The shape is chosen as $a_1:a_2:a_3=5:5:1$ and the eigenstrain is set isotropic on the ${\widehat{x}}_1{\widehat{x}}_2$ plane as ${\varepsilon }_{11}^{*}={\varepsilon }_{22}^{*}={\varepsilon }^{*}$. The constraint strain ${\varepsilon }_{ij}^c$ is scaled with respect to ${\varepsilon }^{*}$ (b) The relative difference in the constraint strain ${\varepsilon }^c$ at $(0,0,a_3)$ between the cylindrical and ellipsoidal inclusions.

Figure 17

(a) The oblate spheroidal TiC particle inside $\alpha$-Fe, where $a_1:a_2:a_3=5:5:1$ and the Baker-Nutting orientation relation is adopted. (b) The magneto-elastic energy $\mathrm{\Delta }{E}_{\sigma }\left(\vec{\alpha }\right)$ caused by the stress field around the TiC particle with respect to the magnetization orientation $\vec{\alpha }(\theta ,\varphi )$.