Atomistic simulations of hydrogen embrittlement

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 5 7 6 – 9 5 8 4

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Atomistic simulations of hydrogen embrittlement

Ryosuke Matsumoto a,b, Shinya Taketomi a,b, Sohei Matsumoto a, Noriyuki Miyazaki a,b,*a Department of Mechanical Engineering and Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto 606-8501, Japanb National Institute of Advaned Industrial Science and Technology (AIST), 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

a r t i c l e i n f o

Article history:

Received 29 May 2009

Received in revised form

12 September 2009

Accepted 19 September 2009

Available online 16 October 2009

Keywords:

Hydrogen embrittlement

Atomistic simulation

Molecular dynamics method

Molecular statics method

Crack propagation

Dislocation

* Corresponding author at: Department of Me606-8501, Japan. Tel.: þ81 75 753 5213; fax: þ

E-mail address: [email protected]/$ – see front matter ª 2009 Profesdoi:10.1016/j.ijhydene.2009.09.052

a b s t r a c t

It is well known that hydrogen weakens strengths of metals, and this phenomenon is

called hydrogen embrittlement. Despite the extensive investigation concerning hydrogen

related fractures, the mechanism has not been enough clarified yet. In this study, we

applied the molecular dynamics method to the mode I crack growth in a-Fe single crystals

with and without hydrogen, and analyzed the hydrogen effects from atomistic viewpoints.

We estimated the hydrogen trap energy in the vicinity of an edge dislocation in order to

clarify the distribution of hydrogen atoms, using the molecular statics method. We also

evaluated the energy barrier for dislocation motion under a low hydrogen concentration.

Based on these results, we propose a mechanism for hydrogen embrittlement of a-Fe under

monotonic loading.

ª 2009 Professor T. Nejat Veziroglu. Published by Elsevier Ltd. All rights reserved.

1. Introduction low to weaken the bonds. In the HELP mechanism, a plastic

Metals absorbing hydrogen show the reduction of ductility

due to hydrogen [1] and the acceleration of fatigue crack

growth [2]. This phenomenon is known as hydrogen embrit-

tlement. Much attention has been paid to hydrogen as a clean

energy source to solve environmental problems and to cope

with the global warming problem. Increase in hydrogen use

would result in increase of failure accidents related with

hydrogen embrittlement.

Various mechanisms for hydrogen embrittlement have been

proposed so far. Among them, the hydrogen enhanced deco-

hesion (HEDE) [3,4] and the hydrogen enhanced localized plas-

ticity (HELP) [5] are typical ones. In the HEDE mechanism, the

bonds of metal atoms are weakened by hydrogen atoms. It has

been, however, supposed that a hydrogen concentration is too

chanical Engineering an81 75 753 5719.ac.jp (N. Miyazaki).sor T. Nejat Veziroglu. Pu

behavior of a material is affected by hydrogen atoms. This

mechanism is supported by experimental results. For example,

increase in dislocation mobility is observed in in situ TEM

observations [6–8]. It is also observed that slip bands are local-

ized in the vicinity of a crack tip in fatigue tests using hydrogen-

chargedtest specimens [2].Thefracture phenomenon causedby

hydrogen embrittlement cannot be explained only by the HELP

mechanism.

Despite a lot of experimental works concerning hydrogen

embrittlement, for example Refs. [9] and [10], the mechanism of

hydrogen embrittlement is not fully understood. Hydrogen has

a high diffusivity and a low concentration in metal. It is, there-

fore, difficult to perform direct observation of hydrogen and to

answer the question from experimental results what is a correct

mechanism for hydrogen embrittlement. Atomistic simulations

d Science, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kyoto

blished by Elsevier Ltd. All rights reserved.

mailto:[email protected]

http://www.elsevier.com/locate/he

Nomenclature

Roman symbols

b Magnitude of Burger’s vector, m

KI Mode I stress intensity factor, MPaffiffiffiffiffimp

R Gas constant, J/mol K

t Time, s

T Absolute temperature, K

xeq Hydrogen concentration expressed by

the atomic ratio -

x0 Hydrogen concentration expressed by the

atomic ratio without hydrostatic stress, -

Greek symbols

dUH Partial molar volume of hydrogen, m3/mol

3 Ratio of number of hydrogen atoms to the

number of iron atoms, -

shyd Hydrostatic stress, MPa

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y 3 4 ( 2 0 0 9 ) 9 5 7 6 – 9 5 8 4 9577

such as the molecular dynamics method, the molecular statics

method and soonare powerful tools tostudythe mechanism for

hydrogen embrittlement.

In the present study, after choosing an adequate inter-

atomic potential for a-Fe and hydrogen system (hereafter

abbreviated as aFe-H system), we perform several kinds of

atomistic simulations for such a system, that is, the molecular

dynamics analyses of crack propagation and the molecular

statics analyses of the interaction between dislocation and

hydrogen atoms, and we propose a mechanism for hydrogen

embrittlement of a-Fe under monotonic loading, based on the

results of the atomistic simulations.

2. Interatomic potential for aFe–H system

Adequate selection of an interatomic potential is of crucial

importance for molecular dynamics and molecular statics

calculations. Only three kinds of interatomic potential have

been proposed so far for the aFe–H system. They are the

embedded-atom-method (EAM) potential by Ruda et al.

(abbreviated as EAM-R) [11], the Morse type potential by Hu

et al. (abbreviated as Morse) [12] and the EAM potential by Wen

et al. (abbreviated as EAM-W) [13]. EAM-W was formulated by

improving EAM-R so as to reproduce more properties of the

aFe–H system accurately. Thus, EAM-W is superior to EAM-R.

In comparison with Morse, EAM-W provides accurate results

for elastic constants of a-Fe and the properties of hydrogen in

a-Fe, as shown in Tables 1 and 2. The elastic constants of a-Fe

calculated from EAM-W and Morse are shown in Table 1,

compared with the experimental results [14]. The

Table 1 – Elastic constants of a-iron.

Exp. [14] EAM-W Morse

C11 (GPa) 243.1 230.2 244.3

C12 (GPa) 138.3 135.8 80.7

C44 (GPa) 138.1 116.7 80.7

experimental results agree well with the calculated results

using EAM-W. The heat of solution and migration energy of

hydrogen in a-Fe are shown in Table 2. They are also

compared with experimental results [15,16]. Again EAM-W

provides better results than Morse in comparison with the

experimental results. Morse neglects H–H interactions.

Moreover it takes account of the long-range (about 1 nm)

interaction between iron and hydrogen atoms, so that it

requires a large amount of computational time for a molecular

dynamics calculation despite a pair potential. Because of the

above reasons, we selected EAM-W as the best interatomic

potential for the aFe–H system and employed it in the subse-

quent atomistic simulations.

3. Molecular dynamics analyses of crackpropagation

We applied molecular dynamics simulations to the mode I

crack propagation in a-Fe single crystal with and without

hydrogen for several analysis conditions, i.e. two kinds of

crystal orientation and two levels of temperature.

3.1. Analysis model

Fig. 1 shows an analysis model whose shape is a circular disk

with 9.7 nm in radius, 2.8 nm in thickness and 0.7 nm in

thickness of the boundary region. The displacements corre-

sponding to KI of 0.9 MPaffiffiffiffiffimp

are prescribed to all atoms to

introduce an initial crack. The origin is set at the crack tip and

the x-axis and z-axis correspond to the forward direction of

the initial crack and the thickness direction, respectively. This

analysis model is a quasi three-dimensional model consisting

of about 71,000 atoms, on which a periodic boundary condi-

tion in the z-direction is imposed.

According to Ref. [17], stable locations of hydrogen atoms

are tetrahedral sites (T-sites), and a hydrogen distribution

depends on the hydrostatic stress. Thus, hydrogen atoms are

introduced at the T-sites in iron atoms in accordance with the

following equations [18], using random numbers:

xeq ¼(

x0exp�

shydðXÞdUHRT

�ðjXj � jr0jÞ

x0exp�

shydðr0ÞdUHRT

�ðjXj < jr0jÞ

(1)

where xeq is the hydrogen concentration expressed by the

atomic ratio at a location X, x0 the hydrogen concentration

without hydrostatic stress shyd, dUH the partial molar volume

of hydrogen in a-Fe, R the gas constant and T is the absolute

Table 2 – Heat of solution and migration energy ofhydrogen atom.

Exp. EAM-W Morse

heat of solution (eV) 0.30 [15] 0.28 –

migration energy (eV) 0.035 [16] 0.037a 0.05a

a potential energy difference between hydrogen atoms at a tetra-

hedral site and an octahedral site.

y [110]

{112} slip plain

a

x

y

z

9.7nm

2.8nm

0.7nm

boundary region

Fig. 1 – Analysis model.


temperature. We used dUH of 1.2�10�6m3/mol [18]. We

selected jr0j ¼ 5:0� 10�10 m to avoid an infinite quantity of the

hydrogen concentration at a crack tip. In the present study, we

changed the hydrogen concentration x0 from 0 (no hydrogen

atom) to 5.0�10�4 as an analysis parameter. The initial

distribution of hydrogen atoms is shown in Fig. 2 for

x0¼3.0�10�4 (z 5.4 mass ppm) and 400 K, where red points

and blue points denote iron atoms and hydrogen atoms,

respectively. It is found from the figure that the hydrogen

concentration is higher around the crack tip than in the

periphery of the analysis model because of stress concentra-

tion around the crack tip.

x [110]

crack surface

54.7°

54.7°

crack plane

3.2. Analysis conditions

Crack propagation analyses were performed for two kinds of

crystal orientation, crystal orientations (A) and (B), as shown in

Fig. 2 – Initial distribution of hydrogen atoms

(x0 [ 3.0 3 10L4).

Fig. 3, in which the gray planes indicate the {112} slip planes of

a-Fe. In the crystal orientation (A), the crack plane is the (112)

plane, and the forward direction of the initial crack is the ½110�direction. In this case, there is no slip plane in the xy-plane, and

we can expect no dislocation emission from the crack tip. In the

crystal orientation (B), the crack plane is the (110) plane, and

the forward direction of the initial crack is the [001] direction. In

this case, there exists {112} slip planes in the xy-plane, and we

can expect the emissions of dislocations from the crack tip.

Although the ductile-brittle transition temperature (DBTT)

for a-Fe is less than 100 K, the DBTT calculated using EAM-W is

between 200 K and 300 K. In the crystal orientation (B) where

dislocation emissions are expected, we performed the crack

propagation analyses at the initial temperature of 400 K, which

is above the DBTT, and that of 100 K, which is below the DBTT.

Dislocation emissions are expected at 400 K, and no dislocation

emission is expected at 100 K. In the crystal orientation (A), no

dislocation emission is expected at any temperature, so that

we performed the crack propagation analyses only at the initial

temperatures of 400 K. We can discuss the effect of dislocations

on the crack growth behavior by comparing the result of the

crystal orientation (A) at 400 K with that of the crystal orien-

tation (B) at 400 K. In the present analyses, we did not control

the temperature of the analysis system to keep the tempera-

ture at a constant value during crack propagation.

The crack propagation analyses were performed by

imposing the displacement rate corresponding to the rate of

z [001]

crack tip

x [001]

y [110]

z [110]

{112} slip plane

crack plane

54.7°

54.7°

crack tipcrack surface

b

Fig. 3 – Arrangement of [112] slip planes. (a) Crystal

orientation (A) (b) Crystal orientation (B).

Fig. 4 – Snapshots of crack propagation behavior for the crystal orientation (A) at 400 K (x0 [ 0).

Fig. 5 – Time evolution of crack growth length in the crystal

orientation (A) at 400k.


stress intensity factor dKI=dt ¼ 5:0� 109MPaffiffiffiffiffimp

=s on the

atoms in the boundary region of the analysis model. This

analysis condition is equivalent to the crack opening velocity

of 0.4 m/s, which is very slow for a molecular dynamics

analysis. In the present analyses, we performed several

numbers of analyses with different initial locations of

hydrogen atoms and different initial velocities of iron and

hydrogen atoms for the respective analysis conditions by

changing random numbers in order to avoid the effect of

initial conditions.

3.3. Results and discussion

3.3.1. The cases without dislocation emissionIn the crystal orientation (A), crack propagation analyses were

performed for four cases of the initial hydrogen concentra-

tion, x0¼ 0 (no hydrogen atom), 1.0�10�4, 3.0�10�4 and

5.0�10�4 at the initial temperature of 400 K. A crack propa-

gation behavior is shown for the case without hydrogen

(x0¼ 0) in Fig. 4, where a green part indicates the bcc crystal

structure and a black one other crystal structures. In this case,

a crack propagates straight without dislocation emission.

Although the figures are omitted here, similar crack propa-

gation behaviors were also obtained for the cases with

hydrogen atoms (x0¼ 1.0� 10�4, 3.0�10�4 and 5.0�10�4). Fig. 5

shows the time evolution of crack growth length in the crystal

orientation (A) at 400 K. Significant differences in the initiation

time for crack growth and the crack growth velocity are not

observed among the respective hydrogen concentrations.

In the crystal orientation (B), crack propagation analyses

were performed for x0 ¼ 0 (no hydrogen atom), 1.0� 10�4 and

2.0� 10�4 at the initial temperature of 100 K. Fig. 6 shows

a crack propagation behavior for x0¼ 1.0� 10�4. As expected,

no dislocation emission is observed because of a lower

temperature than the DBTT, and a crack propagates nearly

straight. No meaningful difference is observed between the

cases with and without hydrogen atom.

3.3.2. The cases with dislocation emissionsIn the crystal orientation (B), crack propagation analyses were

performed for five cases of the initial hydrogen concentration,

x0¼ 0 (no hydrogen atom), 0.5� 10�4, 1.0� 10�4, 3.0� 10�4 and

5.0� 10�4 at the initial temperature of 400 K. The distributions

of hydrogen atoms around dislocation cores are shown for

x0 ¼ 3:0� 10�4 in Fig. 7, where a grey circle and a black one

denote an iron atom and a hydrogen atom, respectively. It is

found from the figure that hydrogen atoms are trapped

around dislocation cores within 100ps. Detailed discussion on

the interaction between dislocations and hydrogen atoms will

be given in the following section.

Figs. 8(a), (b) and (c) respectively represent the crack prop-

agation behaviors for three cases of hydrogen concentration,

that is, (a) x0 ¼ 1:0� 10�4, (b) x0 ¼ 3:0� 10�4 and (c) x0 ¼ 0 (no

hydrogen atom). As shown in Figs. 8(a) and (b), not only dislo-

cation emissions from the crack tip but also crack propagation

along the {112} slip planes are observed in high frequency in the

cases with hydrogen atoms. On the other hand, as shown in

Fig. 8(c), only crack tip blunting caused by dislocation emis-

sions tends to be observed without crack propagation in the

case where no hydrogen atom is included. The fracture in the

slip plane is observed in only one case in the case without

hydrogen atom and in three cases in the cases with hydrogen

atoms out of four cases with different initial conditions.

Figs. 9(a) and (b) show the enlarged figures near the crack

tip during slip plane fracture. It is found that hydrogen atoms

gather on the {112} slip planes, from which slip plane fracture

occurs. The hydrogen atoms trapped near dislocation cores

seem to promote the crack propagation along the slip plane.

In the molecular dynamic analysis, we can deal with

atomic motion for only a very short period of pico-second

order. In actual, a lot of hydrogen atoms would be trapped for

a long period of time, and hydrogen atoms trapped with high

Fig. 6 – Snapshots of crack propagation behavior for the crystal orientation (A) at 100 K (x0 [ 1.0 3 10L4).


density at the dislocations emitted from a crack tip would

induce the fracture in the slip plane easier than expected in

the molecular dynamics simulations.

4. Molecular statics analyses of interactionbetween dislocation and hydrogen

In section 3, we presented the results of the molecular

dynamics analyses showing that hydrogen atoms gather

around the cores of edge dislocations on the {112} slip planes.

Here we will deal with this phenomenon in detail using the

molecular statics method. We will also show how hydrogen

atom affects dislocation motion. We have already published

the paper concerning these phenomena [19]. So we will show

the results briefly.

4.1. Hydrogen occupation sites around dislocation core

We consider the interaction between an edge dislocation on

the {112} slip planes and hydrogen atoms. Fig. 10 shows an

analysis model, in which the (112)[111] edge dislocation is

introduced on the xz-plane by removing an atomic plane and

relaxing the atomic structure using the conjugate gradient

(CG) method. The analysis model contains 8054 iron atoms,

and the dimensions of the unit cell are 11.05 nm in the

x-direction, 4.91 nm in the y-direction and 2.02 nm in the

z-direction. A periodic boundary condition was imposed on

Fig. 7 – Hydrogen distributions during cleavage in the slip plan

the x- and z-directions. The dislocation density in this system

is approximately 0.018nm�2. We used EAM-W as the inter-

atomic potential for the aFe–H system. A hydrogen atom was

allocated either at the T-site or the O-site near the dislocation

core, and then the positions of iron and hydrogen atoms were

relaxed to minimize the total potential energy using the CG

method. The hydrogen trap energy at each occupation site is

shown in Fig. 11. We can observe three regions with strong

hydrogen trap energy. The hydrogen trap energy is strongest

around the dislocation core. It is also relatively strong around

a high hydrostatic stress region (region A in Fig. 11) and along

a slip plane (region B in Fig. 11). It should be noted that the

region B has strong hydrogen trap energy. This could not be

predicted by the theory of elasticity that hydrogen and dislo-

cation interact mechanically as a result of lattice dilatation

caused by hydrostatic stress. Furthermore, the result suggests

that a lot of hydrogen atoms accumulate on the slip plane

around the dislocation core.

4.2. Effect of hydrogen on dislocation mobility

In situ TEM observation performed by Robertson et al. [6–8]

revealed that the distance between dislocations decreases

when hydrogen gas is introduced during TEM observation. This

fact indicates increase in dislocation mobility by hydrogen

atoms. The elastic analysis performed by Sofronis and Birn-

baum [20] showed that the shear stress acting on dislocation

decreases with increase in hydrogen concentration. This

e for the crystal orientation (B) at 400 K. (x0 [ 3.0 3 10L4).

Fig. 8 – Snapshots of crack propagation behavior for the crystal orientation (B) at 400 K. (a) x0 [ 1.0 3 10L4 (b) x0 [ 3.0 3 10L4

(c) x0 [ 0.


phenomenon is called the hydrogen-induced shielding effect.

They concluded that the hydrogen-induced shielding effect

causes increase in dislocation mobility. Their conclusion needs

to be examined, because they did not consider the effect of

Fig. 9 – Enlarged views near the crack tip during cleavage in th

(a) x0 [ 1.0 3 10L4 (b) x0 [ 3.0 3 10L4.

hydrogen atoms at a dislocation core and dealt with extremely

high hydrogen concentrations such as 3 (the ratio of the

number of hydrogen atoms to the number of iron atoms) of 0.1

and 0.01. It is not confirmed whether the hydrogen-induced

e slip plane for the crystal orientation (B) at 400 K.

Fig. 10 – Analysis model for molecular statics analyses of

interaction between dislocation and hydrogen.

Fig. 12 – Variations of the energy barrier for dislocation

motion.


shielding effect holds for a low hydrogen concentration.

Therefore we study the effect of hydrogen from the viewpoint

of energy barrier for dislocation motion.

According to the hydrogen trap energy obtained in 4.1, it is

the highest at a dislocation core. Thus the probability of

hydrogen occupation is the highest at the dislocation core.

From this reason, we placed hydrogen atoms at the disloca-

tion core. We used the same analysis model shown in Fig. 10.

We evaluated the energy barrier for the edge dislocation

motion of 1b (b: the magnitude of Burger’s vector) with and

without hydrogen, using the nudged elastic band (NEB)

method [21]. We obtained the energy barrier for the following

three cases; (a) without hydrogen atom, (b) with a hydrogen

atom at the dislocation core in the initial state and the dislo-

cation moving forward by 1b, and (c) with a hydrogen atom 1b

ahead of the initial dislocation and the dislocation moving to

the hydrogen atom. The hydrogen concentration of this

system is 2.24 mass ppm, and the number of hydrogen atoms

per unit length of a dislocation line is 0.49nm�1. The variations

of energy barrier with dislocation motion are shown in Fig. 12

for the case without a hydrogen atom and two cases with

a hydrogen atom. As shown in Fig. 12, the energy barrier for

dislocation motion is 2:65� 10�20J for the case without

a hydrogen atom, while it decreases to 2:35� 10�20J for the

Fig. 11 – Distribution of hydrogen trap energy at each site of

hydrogen.

case (b) with a hydrogen atom and 1:18� 10�20J for the case (c)

with a hydrogen atom. It is concluded that the energy barrier

for dislocation motion decreases due to hydrogen atoms.

Next we performed atomistic analyses in order to examine

whether the hydrogen-induced shielding effect holds under

a low hydrogen concentration. We obtained the stress field

around an edge dislocation based on atomistic model shown

in Fig. 10. The stress field around the dislocation is calculated

using the molecular statics method both for the case without

a hydrogen atom and for the case with hydrogen atoms. Fig. 13

shows the shear stress distributions along the slip plane near

a dislocation core. It is found from the figure that the shear

stress distribution is not affected by hydrogen atoms. Even if

the number of hydrogen atoms per unit length of a dislocation

line is increased up to 7.35nm�1, no significant difference in

the stress distribution is observed. So the hydrogen-induced

shielding effect is not observed under low hydrogen concen-

tration conditions.

Fig. 13 – Shear stress distributions along the slip plane near

a dislocation core.

Fig. 14 – Effect of hydrogen atoms on the surface energy.


It is shown that hydrogen at a dislocation core reduces the

energy barrier for dislocation motion. It is also shown that the

hydrogen-induced shielding effect is very small. It is therefore

suggested that one reason for increase in dislocation mobility

under low hydrogen concentration conditions is not the

hydrogen shielding effect but the reduction of the energy

barrier for dislocation motion due to hydrogen.

5. Mechanism for hydrogen embrittlement

The molecular statics analysis using EAM-W provides the

result that hydrogen atoms existing on a slip plane promote

the separation of the slip plane because of decrease in its

surface energy caused by hydrogen atoms. Fig. 14 shows the

effect of hydrogen atoms on the surface energy of a-Fe for

{100}, {110} and {112} surfaces. Considering this fact and the

results shown in sections 3 and 4, we can propose the

following mechanism for hydrogen embrittlement of a-Fe

under monotonic loading, as follows:

(1) Dislocations are emitted from a crack tip and they exist

along a slip plane.

(2) A lot of hydrogen atoms are trapped at dislocation cores

and along a slip plane in the vicinity of a dislocation core.

(3) The hydrogen atoms at a dislocation core reduce the

energy barrier for dislocation motion and increases dislo-

cation mobility. Thus the distance between dislocations is

reduced.

(4) Separation of a slip plane is caused due to the hydrogen

atoms trapped by a dislocation, and such separation is

connected among pile-up dislocations.

Our proposed mechanism is a hybrid of the HELP and the

HEDE. The fracture is associated with the HELP mechanism in

that plastic deformation with dislocations is needed prior to

the fracture. On the other hand, the fracture is associated with

the HEDE mechanism in that the fracture results from the

separation of a slip plane.

Our proposed mechanism for the hydrogen embrittlement

agrees well with several experimental observations [22,23]

showing that the fracture of a hydrogen-charged test spec-

imen occurs at {110} or {112} slip planes.

6. Concluding remarks

We chose EAM-W as the best interatomic potential for the

aFe-H system. We performed the molecular dynamics anal-

yses of crack propagation in a-Fe including hydrogen atoms

under monotonic loading. We estimated the hydrogen trap

energy in the vicinity of a (112)[111] edge dislocation in order

to clarify the distribution of hydrogen atoms. We also evalu-

ated the energy barrier for dislocation motion under a low

hydrogen concentration. Based on the above results, we have

proposed a mechanism for hydrogen embrittlement of a-Fe

under monotonic loading. Our proposed mechanism agrees

well with several experimental observations.

Acknowledgements

This research was performed as a part of the Fundamental

Research Project on Advanced Hydrogen Science funded by

the New Energy and Industrial Technology Development

Organization (NEDO).

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