A computational simulation for H-dislocated BCC Fe interaction

ELSEVIER

A computational

MaterialsChemistry rindPhysics55 ( 1998) 61-67

simulation for H-dislocated BCC Fe interaction Carolina Pistonesi ‘, Alfred0

Juan b-*

Abstract A model was performed to simulate hydrogen diffusion and trapping on an Fe35cluster with a ( 100) dislocation over three different paths, to obtain the relative amount of hydrogen atoms in different sites along each path. The results indicate that hydrogen locates preferentially on the surface zone and on the void produced by the dislocation. For hydrogen energies greater than certain values that depend on the path, the hydrogen concentration predominates on the void region. 0 1998 Elsevier Science S.A. All rights reserved. k’qwords:

Hydrogendiffusion: DislocatedFe cluster: Computersimulation:MonteCarlo

1. Introduction The interaction of hydrogen with transition metals and particularly with iron is of considerable technological relevance in relation to the passivated storage of hydrogen, to hydrides properties and, in particular, to the role played in the embrittlement susceptibility. The physicochemical phenomena involved in the dissolution of hydrogen in bulk iron, such as the chemisorption of atomic hydrogen, its surface and bulk diffusion and the trapping at interstices or defects, have been the subject of intensive experimental and theoretical work in recent years. The solubility of hydrogen in perfect crystals of pure iron is exceedingly small. A fraction of the diffused hydrogen is probably captured by lattice traps. Such behaviour can be explained assuming that the activation energy for diffusion within the traps is smaller than in the normal lattice. Myers et al. have analysed data from a series of implantation experiments in Fe, Ni and alloys [ I]. Besenbacher et al. have taken into consideration the predictions of effectivemedium theory along with other experimental information on the properties of hydrogen and defects in metals [ 21. It has been reported that relatively strong binding of the hydrogen occurs at monovacancies and small vacancy clusters, and as a result these traps dominate the behaviour of the system [X4]. It has been reported that whenever the electron density decreases below the optimum value at the interstices in met* Correspondingauthor.Fax: -t I 6072535524;E-mail:ajZ,c)@corncll.edu 0354.OS84/98/$19.000 199s ElsevierScienceS.A. All rights resrrved. P/lSO251-0584~98~00091-1

als, the associated defect will act as a trap for hydrogen and vice versa. There are many kinds of open defects in metals and it is widely accepted that the more open they are, the more strongly will they trap the hydrogen atom [ 51. Dislocations give smaller trapping energies than vacancies and voids. The theoretical study of the interaction between hydrogen with a transition metal, either when the hydrogen atom is an impurity in the bulk or in the case of hydrides, has been examined mainly by using the cluster approximation. Irigoyen et al. have used the semiempirical atom superposition and electron delocalization molecular orbital ( ASED-MO) formalism to study the H-Fe interaction. Calculations were carried out using clustersof the type Fe,, simulating both the adsorption and the absorptionof a hydrogen atom on/in the ( 100) face and in the ( 100) dislocation [ 61. Their results suggestthat when the hydrogen atomsresideinside the void (introduced by the dislocation), the total energy decreases. Moreover, there is a tendency for hydrogen atoms to accumulate in the void region. In this situation it is also possible that the hydrogen atomsform H-H pairs. The aim of the presentwork is to obtain the relative hydrogen concentration along the Fe,, cluster with a ( 100) dislocation. For this, we simulatethe diffusion of hydrogen atoms by meansof theMonte Carlo method, usingthe energyresults obtainedby Irigoyen et al. [ 61. 2. The diffusion model The relative oscillations of a local group of atoms in a crystalline solid can be suchthat one of the atomsflies out of

C. P&ones/, A. Jrran /Materiuis

Chemimy and Physics 55 (1998) 6147

-

Fe layer)

( 1% Fe

layer)

/ 0

2

I

I

distance view of the H paths; on the right side, a two-dimensional

4

Fig. I. The FeA5cluster model of dislocated Fe ( 100). On the left side, a tridimensional

its original site and comes to rest at a new adjoining site (or interstitial). The fraction of atom oscillations that leads to such a jump is extremely small, even at temperatures approaching the melting point. This indicates that diffusion is an activated process, that is, it involves some expenditure of energy and the elementary diffusive event occurs overcoming a potential barrier. Diffusion is a property sensible to the structure. Atom velocities in the crystal depend on its structure and the atomic disposition. Atoms change sites most frequently through the aid of defects, such as vacancies or dislocations, which always exist in some small concentration. So the presence of defects generally favours the diffusion process. In a previous work, Irigoyen et al. have calculated some energy curves for selected trajectories of hydrogen atoms inside an FeJ5 cluster with a (100) dislocation. They used the semiempirical atom superposition and electron delocalization molecular orbitai ( ASED-MO) formabsm to study the H-Fe interaction. This theory is a semiempirical method, which makes a reasonable prediction of molecular and electronic structures employing atomic valence orbital exponents and experimental ionisation potentials as input parameters. In Fig. 1 it can be seen the Fe,, cluster and its two-dimensional view in which the fifth layer is taken as the origin for the x coordinate axis. The position of the surface is 5.72 A from it. Three pathb for the hydrogen atom movements can also be seen. Irigoyen et al. suggested that H-H interaction in the void is a favourable process due to a strong indirect interaction with the Fe matrix and it is very difficult for the H-H pairs to diffuse from the void to the surface owing to an extremely high potential energy barrier [ 61. The aim of the present work is to create a mode1 for hydrogen diffusion along the paths indicated in Fig. 1 for the FesS cluster. For each path the interaction between the hydrogen atom and the dislocated lattice is given by the dashed curves shown in Figs. 2-4. To study the behaviour of hydrogen atoms, we have assumed the atoms to locate near the relative minima of the energy curve. For this, in each diffusive jump the atom is considered to pass from one minimum region to

/

path I

6

8

view,

Energy, (ev) “I

o

12

3

4

5

Distance,

Fig. 2. Approximated

6

7

89

(A)

potential energy curve for path I.

an adjacent one. The following are considered,

simplifications

to the probiem

2.1. The energy cases are replaced by selleral potential barriers These barriers are constructed from the original energy curve, using an equal area criteria. The zero potential energy point is chosen as corresponding to the absolute minimum. Figs. 2-4 show the approximated energy curves for each path. 2.2. The hydrogen atoms are considered to locate in the regions bettlaeen the barriers Thus, there are four sites for each trajectory that can be occupied by the hydrogen atoms. The location and spatial extension of each region can be observed from Figs. 2-4. The first site corresponds to the region beyond the surface; the second to the surface region; and the third and fourth sites are inside the cluster (bulk). The last site corresponds to the void region produced by the dislocation. The barrier and site characteristics are listed in Tables 1-3.

C. f’istonesi,

A. Juan/Materials

Chemist?

and Physics

5.5 (1998)

Table 2 Path II potential barriers and sites characteristics

Energy CeV)

Potential energy

64 5 4

63

61-67

Site 1 Barrier 1 Site 2 Barrier 2 Site 3 Barrier 3 Site 4

1

3t

j

11,. ,,--J 0

12

3

4

5

Distance

Fig. 3. Approximated

6

7

(eV)

Width (A,

0.123

0.750

0.423

2.750

0.000

0.906

2.043

0.578

0.983 1.213 0.941

0.562 0.406

Table 3 Path III potential barriers and sites characteristics 8

9

(A)

potential energy curve for path II. Site 1 Barrier 1 Site 2 Barrier 2 Site 3 Barrier 3 Sile 4

Energy, (ev) 6

Potential energy (ev)

Width (A,

0.329

0.750 2.750 0.906 0.578 0.562

0.629

0.176 2.249 1.189 1.419 0.000

:F

0

12

3

4

5

Distance,

Fig. 4. Approximated

6

7

8

9

(A)

potential energy curve for path III.

Table 1 Path I potential barriers and sites characteristics

Site 1 Barrier 1 Site 2 Barrier 2 Site 3 Barrier 3 Site 4

Potential energy (ev)

Width (A,

0.123 0.423

0.750 2.750

0.000

~0.906

2.043 0.983

0.578 0.562 0.406 1.312

1.213 0.823

2.125

2.3, The probabiliry of passing from one site to another is given by the truwnissiotz coejjicierlt of the barrier betwleen the TN’Osites Let an atom with energy E be in a given site characterised by a value of potential energy V,. We want to calculate the probability of passing to one of the two adjacent sites char-

0 Fig. 5. Potential calculations.

0.875

2.125

+

a energy

barrier

x used for

‘,(+

transmission

;:+

coefflcienti

acterised with V,. The potential barrier between these twc sites has the value V, and width n. This probability is approximated by the transmission coefficient of the potential bartie] shown in Fig. 5. For this potential we consider an incident particle coming from x= - ~0 (i.e. the region with potential energy V,, car. responding to the initial site). The transmission coefficien of the barrier gives the probability for the particle arriving from x = - m to pass the potential barrier and reach the regior on the other side of the barrier. In order to find the transmission coefficient of the barrier we consider two situations: E > V, and E < V,, being in botk cases E > V, and E > V,. In the second case the particle car pass the barrier owing to the tunnel effect. First, we have to determine the stationary states in the thret regions where the potential V(x) is constant [ 71. This i: shown in Table 4, where m is the particle (hydrogen atom: mass. Then requiring the continuity of q(x) and of d9ld.y at tht points where V(x) is discontinuous (x= 0 and x= n) ant defining the transmission coefficient T of the barrier as

1.0

In this equation v is the attempt frequency (roughly the vibrational frequency of the atoms), 1is the true jump length to the next adjacent site, cy is the dimensionality of space (a!= 1 for channelled systems) and II is the number ofjump directions available to the atoms. Ed is the energy barrier and T the temperature. Implicitly, in this equation we have assumed the transition state theory, i.e. that no recrossing or multiple jumps events occur so the H atom makes randomly oriented single jumps. At high temperatures multiple jumps and anharmonic effects can be important while classical mechanics break down at very low temperatures. The diffusion parameters are thus considered according to

* 2 .g

08-

E ;

0.6

-

-

iq,01

:.._,, :,

.

jj

.: :

E

2

‘5

E

s

5

3.-

F I-

0.4 -

II

-T

I,2 T 23 T

.__._CQ-

3,4

0.0 ’ I,

:, 1

,:,

,

2

3

,

Energy

,

(

,

4

5

6

, 7

W

Fig. 6. Transmission coefficients for path ILL

where J is the jump frequency. The attempt frequency I, was computed from z,=-

1 Ii 377L 112

where k is the force constant of a parabola fitted to the energy curve near the minima at the site regions and f?l is the mass of the H atom. In the analysis of diffusion of adatoms on FCC metallic surfaces Liu et al. [ 121 and Stumpf and Scheffler [ 131 have employed similar approximations for their calculations. The reader interested in the calculation of crossover between classical and quantum regimens is referred to the paper by Gillian [ 141 and for a more generalised analysis of H trapping to the paper by Iino [ 151

0.0

1

*’

2.2

’

’

2.3

’

’

’

’

2.4

25

2.6

2.7

Energy

2.8

W4

Fig. 7. Transmission coeflicients for path III (2.2-2.5

eV energy range).

Relative H concentraGon Site 2 0.8 I E=2.12

3. Results The transmission coefficients from Table 5 are displayed in Fig. 6 for energies above the corresponding site barrier in the path III. As can be seen the transmission coefficient from site 1 to site 2 CT,.?) has the most oscillatory behaviour of all studied situations. T,,? and T,,d become unit rapidly but T2,3 seems to be decisive and controls the jumps. Fig. 7 clearly shows, in a zoom for the region 2.2-2.S eV, the main contribution of T,,, to the H interaction with the energy potential curve. A variable site occupancy with H energy is shown in Figs. 8 and 9. The energy value for the maximum occupation of H at site 4 matches the maximum for the coefficient T2,3 (see also Fig. 11). After jumping from site 2 to site 3 there is approximately 70% probability ofjumping to site 4, as can be seen in Fig. 7. These results are completely analogous to those corresponding to the other two paths. The computed attempt and jump frequencies and oscillation amplitudes are shown in Table 6. The values for attempt

eV

0.6c 0.4 i

0.2

Site 1

i

Site 4 0.0 ,-. 0

1

2

3

Site 3 .-4 5 Distance

Fig. 8. Concentration

Ii7

8

Ji! 9

(A)

along path III for hydrogen atoms with 2.12 eV.

frequencies have the same order of magnitude as those mentioned in the literature for similar systetns [ 111. The results obtained by the execution of the program are the relative amount of hydrogen atoms at each site of the selected path for a given value of hydrogen atom energy. There exists a great change in concentration when hydrogen atom energy is greater than approximately 2.3 eV for

C. Pisronesf, A. Junn /Mrttrrinls

66

Table 6 Caiculated H-Fe vibrational

Site

frequencies (cm- ‘), jump site frequency and oscillation amplitude in a for path III Jump frequency (s- ’ )

Frequency (cm-‘) ’

Oscillation amplitude ’ (A)

kT= 2 eV i+i+

Site 1 Site 2 Site 3

Chanistty and Physics 55 (1998) 61-67

kT=3

1

i-+i-i

i-tifl

1.9 x 10” 2.6 x 10’” 7.5x 10”

1.2x 4.0x

eV

kT= 2 eV i-i-

Site length (A,

kT= 3 eV

1 0.750

812.6 1452.7 505.5

Site 4

8.6 x 1o1*

3.9x 10”

lo’? 10”

2.1 x 10’” 3.1 x 10’”

6.6 2.75

1.1 x IO’”

13.4

9.8

0.906 0.562 2.125

4. I 20.1

a To obtain frequency value ins-‘, each value has to be multiplied by c (speed of light). h Computed in equilibrium with a thermal bath [7]. Relative

H concentration

“07-

Relative

--

4

E=2.92

--

H concentration

~

eV

0.6

0.2

-‘...I

0.0 0

1

0.0’ 2.0

2.4

2.2

6

2 Distance

Fig. 9. Concentration

Site 1 , d

(A)

along path III for hydrogen atoms with 2.92 eV.

path III (or 2.1 eV for paths I and II). For lower energies almost all atoms reside at the second site which corresponds to the surface zone, and a very small fraction at the first site. We show the results in Fig. 8 only for path III because it is the minimum energy path. These results are completely analogous to the results corresponding to the other two paths. In this figure, the widths of the bars correspond to the spatial width of the site shown in Table 4. For higher energies most of the atoms reside at the last site which corresponds to the void due to the dislocation and most of the remaining atoms locate at the second site, being only a small fraction of atoms distributed between the first and third site. This is shown in Fig. 9. Making several executions of the program with different values of the hydrogen atom energies, we can also represent site concentration versus energy. Figs. 10 and 11 show an increment of concentration on site four while there is a decrease of concentration on site two. In bothcases saturation values are reached.

4. Discussion When the hydrogen atom energy is below the height of the higher battier, which is, for the three paths the one that separates sites two and three, the atom can pass through the

2.6 Enwa,

2.6

3.0

3.2

3.4

(ev)

Fig. 10. Hydrogen concentration variation with energy of sites two and four (pathI).

Relative

H concentration

me 4

0.0

0.6

0.4

0.2 IL

2.0

~

2.2

2.4

2.6 Energy,

2.8

3.0

3.2

3.4

W

Fig. 11. Hydrogen concentration variation with energy of sites two and four (path III).

barrier owing to the tunnef effect. The lower the energy values are from this value, the lower the transmission coefficient of the barrier. So the probability of passing the barrier takes negligible values. This obstructs the atoms passage from site two to site three. This is why the concentration of sites three and four are extremely small for low energies. This occurs for hydrogen atom energies below approximalely 2.1 or 2.3 eV (for paths I and II or path III, respectively). For higher-