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\begin{frontmatter}
\title{Effect of hydrogen on the surface energy of ferrite and austenite}

\author[gift]{Eun Ju Song}
\author[gift,cam]{H. K. D. H. Bhadeshia}
\author[gift]{Dong-Woo Suh}

\address[gift]{Graduate Institute of Ferrous Technology,
POSTECH, Republic of Korea}
\address[cam]{Department of Materials Science and Metallurgy,
University of Cambridge, U.K.}

\date{\today}

\begin{abstract}
Calculations indicate that the introduction of hydrogen into the
body--centred cubic and face-centred cubic allotropes of iron in
both cases reduces the $\{100\}$ surface energy. The reduction is
rather small in magnitude so this mechanism cannot present the
major cause of the well--known hydrogen embrittlement phenomenon. Consistent with the theory of grain boundary embrittlement in iron, {carbon is confirmed to increase the surface energy, thereby rendering cleavage fracture less likely assuming that other factors governing fracture are maintained constant.}

\end{abstract}

\begin{keyword}
hydrogen embrittlement \sep decohesion theory \sep first principles calculation \sep surface energy
\end{keyword}
\end{frontmatter}

\section{Motivation}

The embrittlement that occurs when small concentrations of
hydrogen are introduced into body-centred cubic iron was first
noted in 1875 \cite{Johnson:1875}. The mechanisms postulated to
explain this phenomenon include the internal pressure theory
\cite{Zapffe:1941} where diffusible hydrogen accumulates at
critical defects, combines to form molecular hydrogen and the
resulting pressure helps initiate fracture. The second may involve
a reduction in  the cohesive energy of iron, thus making it easier
to cleave \cite{Troiano:1960}. In the localised plasticity model,
hydrogen promotes dislocation mobility and if this happens in a
localised region then it can result in a shear instability which
in turn stimulates further failure modes such as cleavage
\cite{Beachem:1972}.

Recently, Takano \cite{Takano:2010} investigated the cohesive energy
in cleavage fracture on the $\{100\}$ (cleavage) planes of single
crystal $\alpha$-iron with hydrogen occupying octahedral
interstices. The fracture in the method was simulated by monitoring
the energy change as the displacement of a $\{100\}$ surface
relative to the underlying plane is increased. Decohesion is said to
occur when the energy no longer changes with displacement. However,
those calculations did not consider the lattice expansion due to the
hydrogen addition into the octahedral interstices of ferrite; this
can be seen from the fact that there was no increase in energy as
the separation of $\{100\}$ planes increase from  0 to 0.06\,nm. So
in this case, the energy reduction due to the hydrogen is
overestimated. It was found that hydrogen reduced the cohesive
energy, with the conclusion that it therefore assists crack
initiation.

It is important to assess this conclusion again, taking account of
the issues highlighted above, and in addition, accounting for the
fact that hydrogen can occupy both the tetrahedral and octahedral
interstices. The calculations have been conducted for austenite in
addition to ferrite in order to provide a basis for comparison, and
the role of carbon in influencing the surface energy is also
examined. {In the present work, the
calculations were carried out for the case of hydrogen atoms; molecular hydrogen is not addressed, and neither is intergranular failure, which has been the subject of previous investigations \cite[for example]{Zhong:2000}.}

\section{Calculation procedure}

Density functional theory (DFT) \cite{Hohenberg:1964,Kohn:1965}
calculations were carried out with the Vienna ab initio simulation
package commonly referred to as VASP
\cite{Kresse:1993,Kresse:1996,Kresse:1996b} using the projector
augmented wave basis set \cite{Blochl:1994} and the generalized
gradient approximation (GGA) of the Perdew, Burke, and Ernzerhof
(PBE)  form for electron exchange and correlation
\cite{Perdew:1996}. An energy cutoff of 450\,eV was used for all
calculations.

The ferrite was simulated using a $2\times2\times4$ supercell of
conventional body-centred cubic structure with $6\times6\times3$
$k$--point sampling. For the austenite the supercell was
$2\times2\times4$  of the conventional face-centred cubic
structure with $8\times8\times4$  $k$--point sampling. Spin-polarised calculations were
performed for ferrite and non-spin-polarised calculations in the
case of austenite. Both cell shape and atomic positions were
allowed to relax for bulk calculations. Lattice parameters of 2.84
and 3.45\,\AA were obtained with ferrite and austenite,
respectively. The reference hydrogen was assumed to be an H$_2$
molecule and the total energy of the hydrogen molecule was
calculated by putting H$_2$ in a cubic box with 10\,\AA\,sides and
carrying out a $\Gamma$--point calculation. A bond length of
0.75\,\AA\,was obtained for H$_2$.

For modelling the surface, the lattice parameters obtained from
relaxed bulk Fe$_n$ and Fe$_n$H were fixed and then the thickness of
0.2-10\,\AA\ vacuum layer was inserted. Only the atoms located
in the top 5 layers were allowed to relax; the bottom three
layers were fixed at their bulk positions in order to maintain the
vacuum between the layers undergoing separation during the simulation of
cleavage. Same $k$--point samplings with the bulk metal were used
for systems with vacuum thickness less than 10.0\,\AA, while
$6\times6\times1$ and $8\times8\times1$  $k$--point samplings were
used for ferrite and austenite respectively, for the systems with
10\,\AA\ thickness vacuum.

\section{Results and discussion}

 \subsection{H atoms in ferrite and austenite}

Both octahedral and tetrahedral sites (Fig.~\ref{fig:1}) were
investigated to assess the dissolution of hydrogen in the bulk
matrix. Then the dissolution energy of hydrogen atoms in bulk Fe was
obtained as follows, with the reference state of pure metal and
H$_2$ molecule:
\begin{equation}
\Delta E_1=E({\rm Fe}_n{\rm H})-E({\rm Fe}_n)-\frac{1}{2}E({\rm
H_2})\label{eqn:1}
\end{equation}
Here, \emph{n} is the number of Fe atoms. Table~\ref{tab:1} shows
the dissolution energies (eV) of hydrogen in ferrite and austenite
from the present study and together with data from the literature
\cite{Manchester:2000,Jiang:2004,Counts:2010}. Our calculations are
in reasonable agreement with reported data, and the discrepancy with
reported experimental values comes from the zero-point energy
\cite{Jiang:2004,Counts:2010}. \footnote{\scriptsize The zero point
energy correction is usually negligible with heavy elements but for
Fe-H system, large corrections  have been reported
\cite{Jiang:2004,Counts:2010}. Consistent with the present work, the
correction is reported to be negligible for the Fe-C system
\cite{Jiang:2003}.  The large difference in the case of the light
atom hydrogen might be caused by the fact that the zero point energy
has kinetic origins.} In ferrite, hydrogen is favoured at
tetrahedral sites unlike the other interstitial atoms such as carbon
and nitrogen. The results are in a good agreement with the
literature \cite{Silva:1976,Norskov:1982,Juan:1999}.   For
austenite, hydrogen is favoured at octahedral sites and its
dissolution energy is smaller than in ferrite which explains the
larger solubility of hydrogen in austenite than in ferrite. From
equation 1, a positive $\Delta E_1$ implies an increase in energy
when hydrogen is dissolved. Note that a smaller $\Delta E_1$ favours
greater solubility.

The H-H interaction was investigated next, with the same supercell
but including two hydrogen atoms. The interaction energy  was
studied as a function of the number of intermediate layers
separating the two hydrogen atoms. For example, the ferrite atomic
structures with two atoms are in Fig.~\ref{fig:2}. By analogy with equation~\ref{eqn:1}, the
dissolution energy $\Delta E$ was calculated as follows:
\begin{equation}
\Delta E_2=\frac{E({\rm Fe}_n{\rm H}_m)-E({\rm
Fe}_n)-m\frac{1}{2}E({\rm H_2})}{m}
\end{equation}
Here, $m$ is the number of H atoms.
 It is evident from Table~\ref{tab:2} that the dissolution energy
increases as the distance between hydrogen atoms decreases, implying
that the interaction between the two hydrogen atoms in ferrite and
austenite lattice is repulsive.  This result is somewhat surprising
given that hydrogen is often said to combine within the steel to
form voids, but such combinations are postulated to occur at defect
sites where the thermodynamics must differ due to the existence of
free surfaces or strain fields. There is unrelated evidence in
chemisorption experiments on iron surfaces of specific repulsive
interactions between hydrogen atoms \cite{Bozso:1977} and from first
principles calculations of adsorption on particular iron surfaces
\cite{Jiang:2003b}. By studying the variation in dissolution energy
as a function of hydrogen concentration ($\Delta E_1$ are 0.2\,eV,
0.16\,eV, and 0.19\,eV in ${\rm Fe_2H}, {\rm Fe_{16}H}$ and ${\rm
Fe_{56}H}$, respectively.) in ferrite, which is assumed to relate to
a mean separation distance, Jiang and Carter \cite{Jiang:2004}
concluded that the variation was within numerical error (0.02\,eV)
and hence a H-H repulsion is non-existent. The numerical error in
the present work is $\pm0.0048$ and $\pm0.0070$\,eV, for ferrite and
austenite respectively, given that $k$--point and cut off energy
were chosen such that the energy difference is less than $0.00015$
and $0.00011$\,eV per atom.

 \subsection{Effect of H on the $\{100\}$ surface of ferrite and austenite}

The vacuum thickness  between the top (100) plane and the bulk, can
be considered as a displacement, thus the increase in total energy
is proportional to the force needed to implement this deformation.
The atomic structures for modelling surface in pure metal are in
Fig.~\ref{fig:3}. To see the effect of hydrogen, a hydrogen atom was
added at tetrahedral and octahedral site for ferrite and austenite,
respectively. Fig.~\ref{fig:4}. {As
an example, Fig.~\ref{fig:5} shows the energy difference as a
function of the vacuum thickness in the case of pure iron and when H
atom is added at the tetrahedral site and octahedral site for
ferrite and austenite, respectively.} The results shows hydrogen
reduces the force needed to deform. It is assumed that there is no
interaction across the 10\,\AA\ thickness of vacuum. {Similarly,
calculations for H at octahedral and tetrahedral sites for ferrite
and austenite were conducted and the dissolution energy $\Delta E_3$
was calculated as follows:
\begin{equation}
 \Delta E_3=E({\rm Fe}_n{\rm H}+{\rm vacuum})-E({\rm Fe}_n+{\rm vacuum})-\frac{1}{2}E({\rm H_2})
\end{equation}
The results are shown in Table 3.}
{The dissolution energy hydrogen
dramatically decreased for both ferrite and austenite. Also, for the
ferrite, H atom prefers octahedral sites at the (100) surface unlike
the bulk material \cite{Jiang:2004,Ramasubramanian:2009}. From those results, the surface energy can be
obtained as follows:}
\begin{equation}
\Delta E_s=\frac{E({\rm Fe}_n{\rm H}+{\rm vacuum})-E({\rm Fe}_n{\rm
H}_m)}{2\,A}
\end{equation}
Here, $n$ and $m$ are the number of Fe and H atoms, respectively and
$A$ is the surface area. {Note that the term $E({\rm
Fe}_n{\rm H}+{\rm vacuum})-E({\rm Fe}_n{\rm H}_m)$ represents the
energy difference between two supercells containing the same number
of atoms, with $E({\rm Fe}_n{\rm H}_m)$ representing the bulk and
$E({\rm Fe}_n{\rm H}+{\rm vacuum})$ the surface.} The calculated
surface energies for pure metal and hydrogen containing system for
ferrite and austenite are indicated in Fig.~\ref{fig:6}. Both for
ferrite and austenite, surface energy was reduced by adding a
hydrogen. The stable positions of H at (100) surface are octahedral
and tetrahedral sites for ferrite and austenite, respectively.
{The result is of the same order of
magnitude as the reported binding energy of hydrogen to a hollow
site at the (100) surface, which ranges from 0.78
\cite{Ramasubramanian:2009} to 0.82 \cite{krueger:1976}.  In this
study, the value was 0.99, and the difference with reported data
comes from the different H fractions at the surface and is discussed
in detail in the next paragraph.} The lowering of the surface energy
by hydrogen decreases force needed to form new surfaces. This may
explain a part of the reason why hydrogen embrittles iron, but it is
noteworthy that the reduction in surface energy is quite small, just
{7\,\%} in the case of ferrite and
9\,\% for austenite. This is not sufficient to explain the
well-established dramatic effect that hydrogen has on the toughness
of ferritic steel. The fact that hydrogen similarly affects the
surface energy of (100) austenite but does not exhibit pronounced
embrittlement is not surprising because its flow stress is much less
sensitive to temperature, and hence plastic flow occurs at a lower
stress than that required to cause cleavage.

{However, the extent of energy
reduction is sensitive to the fraction of the surface covered by
hydrogen atoms. There are $12$ stable octahedral positions of H at
the $2\times2$ supercell ferrite (100) surface. The ``covering
fraction" at the surface, $\mathit{\Theta}$, can be defined as:
\begin{equation}
\mathit{\Theta}=\frac{\hbox{number of occupied sites}}{\hbox{total
number of stable positions}}
\end{equation}
It follows that $\mathit{\Theta}$ for ferrite is 0.083 since one H
atom was placed at the surface. The surface energy of ferrite as
function of $\mathit{\Theta}$ was calculated. The lattice parameter
was optimised with H covered bulk and then this lattice parameter
was used in the calculations associated with the H covered surface
system. The lattice parameter change as a function of H
covering-fraction is shown in Fig.~\ref{fig:7}. These results were
obtained by comparing the energy of localised H bulk and that of the
H covered surface are shown in Fig.~\ref{fig:8}. In the absence of
the surface, the stable H concentration is very low, about
$2.68\times10^{-6}$ and $5.81\times10^{-6}$ atomic \% for ferrite
and austenite, respectively \cite{Manchester:2000}. Jiang and Carter
\cite{jiang:2004b} investigated the H effect on (110) surface of
ferrite. They calculated \{surface energy of pure iron / surface
energy of H covered system\}, $\gamma(\rm H)/\gamma(0)$, as a
function of H covering fractions. When one H atom was put at
$2\times2$ supercell ferrite, $\gamma(\rm H)/\gamma(0)$ was 0.90
while it is calculated as 0.96 in this study. They are of same order
but the difference indicates that H effect is likely to be larger in
(110) system. It should be noted that they calculated the surface
with the case that one H atom was placed at $2\times2$ supercell
ferrite, and obtained the curve for various H covering fractions by
decreasing the surface area. This led to an almost linear reduction
of surface energy due to the increase of H covering fraction.}

{Even though the H solubility in pure
iron is very small, we conducted surface energy calculations for
large $\mathit{\Theta}$ because it is possible that during
deformation, the surface could in principle attract hydrogen atoms
from the bulk. However, when there is no surface, H atoms are very
unlikely to segregate to a particular  (100) plane inside the
crystal. It follows therefore that in
 Eq. 4, the term $E({\rm Fe}_n{\rm H}_m)$ can
be susbstituted by $E^*({\rm Fe}_n{\rm H}_m)$ so that the stable
dissolution energy of hydrogen can remain 0.19\,eV.}
\begin{equation}
E^*({\rm Fe}_n{\rm H}_m)=0.19\,m+E({\rm Fe}_n)+m\frac{1}{2}E({\rm
H_2})
\end{equation}
{Here, all energies are in eV.}
{The surface energy is then obtained
by comparing the energy of unlocalized H in the bulk, and that at
the covered surface (red line, Fig.~\ref{fig:8}). H reduces the
surface energy and the amount of reduction increases with
$\mathit{\Theta}$.}

In addition, the effect of carbon on the same surface energy was
investigated. The obtained dissolution energy is shown in
Table~\ref{tab:4}. Then the surface energy of the system  with
carbon in octahedral site was investigated by using same method. The
results are shown in Fig.~\ref{fig:9}. Unlike the hydrogen, the
carbon increases the surface energy. This result is consistent with
grain boundary  theory, where embrittlement is construed by
comparing the energy of solutes at grain boundaries and free
surfaces. Carbon is the only solute of many segregating species
which enhances boundary cohesion \cite{Anderson:1987,Wu:1996}  some
of which is also based on first-principles calculations. This is
because the increase in  surface energy due to carbon outweighs the
change in grain boundary energy. It therefore becomes unfavourable
to expose boundaries. The influence of hydrogen is on the other hand
detrimental \cite{Anderson:1987}. It is speculated that when soluble
carbon is present in iron, the influence of hydrogen on reducing the
surface energy will be mitigated, especially if the presence of
carbon does not have other detrimental effects such as through
excessive strengthening.

In summary, the work presented here suggests that although hydrogen
reduces the energy of the classical cleavage plane in ferrite, the
effect is {dependent on the fraction
of the surface covered by hydrogen}, so it may not be safe to
attribute its embrittlement effect simply to changes in cohesive
energy.

{\scriptsize {Acknowledgments: {\scriptsize The authors are grateful
for support from the POSCO Steel Innovation Programme, and to the
World Class University Programme of the National Research Foundation
of Korea, Ministry of Education, Science and Technology, project
number R32--2008--000--10147--0.}}


%\bibliographystyle{unsrt}
%\bibliography{all}

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\newpage
\begin{figure}[htbp]
 \centering
      \subfigure(a){\includegraphics[height=5cm]{bccOct}}
      \subfigure(b){\includegraphics[height=5cm]{bccTet}}
      \subfigure(c){\includegraphics[height=5cm]{fccOct}}
      \subfigure(d){\includegraphics[height=5cm]{fccTet}}
    \caption{Atomic structure for dissolution of hydrogen in (a) ferrite, octahedral (b) ferrite, tetrahedral (c) austenite, octahedral and (d) austenite, tetrahedral site, respectively. }\label{fig:1}
\end{figure}


\begin{figure}[htbp]
 \centering
      \subfigure(a){\includegraphics[height=6cm]{bccFeH}}
      \subfigure(b){\includegraphics[height=6cm]{bccFeH1_2}}
      \subfigure(c){\includegraphics[height=4cm]{bccFeH1_2(2)}}
      \subfigure(d){\includegraphics[height=6cm]{bccFeH2}}
      \subfigure(e){\includegraphics[height=6cm]{bccFeH3}}
     \caption{Atomic structures for H--H interaction with different intermediate layers in ferrite (a) an atom, (b) two atoms with less than 1 intermediate layer (indicated as 0 intermediate layer in Table 2), (c) hydrogen configuration on (100) in the system (b), (d) 1 intermediate layer and (e)  3 intermediate layers}\label{fig:2}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure(a){\includegraphics[height=8cm]{VbccFe}}
      \subfigure(b){\includegraphics[height=8cm]{VfccFe}}
     \caption{Atomic structure for modelling the surface with $d$, vacuum distance of (a) ferrite (b) austenite}\label{fig:3}
\end{figure}

\newpage
\begin{figure}[htbp]
 \centering
      \subfigure(a){\includegraphics[height=8cm]{VbccFeH}}
      \subfigure(b){\includegraphics[height=8cm]{VfccFeH}}\\
     \caption{Atomic structure for modelling the surface with hydrogen in (a) ferrite (b) austenite }\label{fig:4}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure(a){\includegraphics[height=8cm]{bcc}}
      \subfigure(b){\includegraphics[height=8cm]{fcc}}
     \caption{The energy difference with $d$, vacumm distance of (a) ferrite (b) austenite for the pure iron and when H is added at a tetrahedral site and octahedral site, for ferrite and austenite, respectively}\label{fig:5}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure{\includegraphics[height=8cm]{surf}}
     \caption{The calculated surface energy of ferrite and austenite for pure metal and hydrogen containing system. }\label{fig:6}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure{\includegraphics[height=8cm]{lattice}}
     \caption{The obtained lattice parameter as a function of H covering fractions }\label{fig:7}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure{\includegraphics[height=8cm]{surf_fraction}}
     \caption{The calculated surface energy of ferrite as a function of H covering fractions }\label{fig:8}
\end{figure}

\begin{figure}[htbp]
 \centering
      \subfigure{\includegraphics[height=8cm]{surf2}}
     \caption{The calculated surface energy of ferrite and austenite for pure metal and hydrogen, carbon containing system. }\label{fig:9}
\end{figure}


\newpage
\begin{table}
\caption{Dissolution energy of hydrogen (eV) in ferromagnetic
ferrite and non-magnetic austenite. The values in parenthesis are
zero point energy corrected values.}\label{tab:1}
\resizebox{0.99\textwidth}{!}{\begin{tabular}{llccc}
\noalign{\smallskip}\midrule
 Phase & Sites & Present work & Published  calculated data & Measured data  \\
\midrule
ferrite & octahedral & 0.34  & 0.32 \cite{Jiang:2004}, 0.34 \cite{Counts:2010} & -- \\
& tetrahedral & 0.19  & 0.19 (0.30) \cite{Jiang:2004}, 0.21 (0.30) \cite{Counts:2010} & 0.30 \\
\midrule
austenite & octahedral & 0.07  & -- & 0.28\\
& tetrahedral & 0.51 & -- & -- \\
\midrule
\end{tabular}}
\end{table}

\begin{table}
\caption{Dissolution energy (eV) of hydrogen in ferrite and austenite as a function of the separation of two H atoms in units of the number of separating $(100)$ planes}\label{tab:2}
\begin{tabular}{ccccc}
\noalign{\smallskip}\midrule
 phase & single H atom & 0 layer & 1 layer & 3 layers  \\
\midrule
ferrite & 0.19 & 0.24 & 0.20 & 0.19 \\
austenite & 0.07 & 0.11 & 0.09 & 0.07 \\
\midrule
\end{tabular}
\end{table}

\begin{table}
\caption{Dissolution energy of hydrogen (eV) in ferromagnetic
ferrite and non-magnetic austenite (100) surface.}\label{tab:3}
{\begin{tabular}{llc} \noalign{\smallskip}\midrule
 Phase & Sites at bulk & Present work   \\
\midrule
ferrite & octahedral & -0.31   \\
& tetrahedral & -0.29   \\
\midrule
austenite & octahedral & -0.38  \\
& tetrahedral & -0.81  \\
\midrule
\end{tabular}}
\end{table}

\newpage
\begin{table}
\caption{Dissolution energy of carbon (eV) in ferromagnetic ferrite
and non-magnetic austenite.}\label{tab:4}
\begin{tabular}{llc}
\noalign{\smallskip}\midrule
 Phase & Sites & Present work   \\
\midrule
ferrite & octahedral & 0.66   \\
& tetrahedral & 1.44  \\
\midrule
austenite & octahedral & -0.13  \\
& tetrahedral & 2.72 \\
\midrule
\end{tabular}
\end{table}


\end{document}
