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Progress in Nuclear Energy 50 (2008) 616e620www.elsevier.com/locate/pnucene

Simulation study of steels corrosion phenomenon in liquid leadebismuthcooled reactors using molecular dynamics methods

Alan Maulana a,c,*, Zaki Su’ud a, K.D. Hermawan b, Khairurrijal a

a Laboratory of Nuclear Reactor, Department of Physics, Bandung Institute of Technology, Jl. Ganesha 10, Bandung 40132, Indonesiab Department of Engineering Physics, Bandung Institute of Technology, Jl. Ganesha 10, Bandung 40132, Indonesiac Neutron Scattering Laboratory PTBIN-BATAN, Kawasan Puspiptek Serpong Cisauk-Tangerang 15314, Indonesia

Abstract

Corrosion phenomena of stainless steel in liquid leadebismuth as a coolant in nuclear fast breeder reactor are a field which is intensivelyinvestigated by the researchers in the recent year. We try to study this corrosion phenomena by computer simulation using molecular dynamicsmethods. The initial positions of the system were taken from the crystal structure data including the cell parameters and the types of the crystal.In this simulation, interatomic potential between FeeFe, PbePb, BieBi, NieNi and CreCr was assumed to follow LennardeJones potential.The LennardeJones potential parameters have been derived by fitting the data available in the literature. Nickel and chromium atoms weresubstituted into Fe crystal with the percentage of 10% and 16% to construct systems like SS 316. Molecular dynamics simulation has beendone by interfacing iron and steel with liquid lead and liquid leadebismuth in several temperatures. The result of this simulation showedthat lead atoms can diffuse into Fee10%Nie16%Cr about 1.18 A at 773 K while in Fee10%Ni and Fee16%Cr about 7.25 A and 11.08 A,respectively.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Liquid leadebismuth; Corrosion; Molecular dynamics; LennardeJones potential

1. Introduction

The investigation of using liquid leadebismuth as a coolantin nuclear fast breeder reactor (FBR) has been done inten-sively by many researchers in the recent years. The liquidleadebismuth has many excellent and unique properties in nu-clear applications. The favorable properties of liquid leadebismuth for nuclear coolant application are its melting pointat 123.5 �C and boiling point at 1670 �C with volume changeupon solidification at about 1.5%. The other advantage ofapplying liquid leadebismuth is that it is not only inert withwater and air but also able to produce 30 neutrons for every

* Corresponding author. Laboratory of Nuclear Reactor, Department of

Physics, Bandung Institute of Technology, Jl. Ganesha 10, Bandung 40132,

Indonesia.

E-mail addresses: alan.maulana@students.itb.ac.id, al_mau@yahoo.com

(A. Maulana).

0149-1970/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.pnucene.2007.11.087

incoming 1 GeV proton (Wu et al., 2002). Generally, this liq-uid leadebismuth provides good operation performance withthe structural and core materials of the FBR at high tempera-ture (Qi and Takahashi, 2003).

Unfortunately this liquid metal is known to be corrosive ininteraction with structural materials like stainless steel espe-cially at high temperatures (Sapundjiev et al., 2006; Hataet al., 2005). The corrosion rate and the corrosion depth inthese reports depend on the type of steel which means thatimpurity inside stainless steel (SS) is an important factor.Chemical composition of impurity inside stainless steel espe-cially SS 316 mainly consists of nickel and chromium atoms;therefore many experiments were conducted by the variationof this impurity.

Although many investigations of corrosion phenomena ofSS in liquid PbeBi have been carried out experimentallynevertheless there were some studies done by using computersimulation. Among those studies, contributions by the groupsfrom Tokyo Institute of Technology (Tokyo Tech) and

Table 1

LennardeJones parameters used in this study

No Element 3 (eV) s (A)

1 FeeFe �0.62 2.26

2 PbePb �0.125 2.75

3 BieBi �0.075 2.8

4 NieNi �0.6 1.9

5 CreCr �0.2 2.5

617A. Maulana et al. / Progress in Nuclear Energy 50 (2008) 616e620

University of Nevada (Wu et al., 2002; Qi and Takahashi, 2003;Takahashi et al., 2004) are important. The group from Univer-sity of Nevada in 2002 had investigated corrosion phenomenausing computational chemical kinetics and hydrodynamicsmethods. Meanwhile, the group from Tokyo Tech carried outthe studies using molecular dynamics methods. Their inter-atomic potentials for FeeFe, PbePb, BieBi, FeePb, FeeBiand PbeBi were derived by using ab initio method. The depthof the potential well was used to analyze the cohesive forcesbetween atoms. The deeper the potential depth, the higherthe cohesive forces and therefore the more difficult the diffu-sion process to occur. In 2004 the group from Tokyo Techhad published a paper about molecular dynamics simulationon interatomic interaction of Fe crystal with Pb and Bi atoms.This simulation involved only one Pb atom and one Bi atomlocated at the surface of Fe crystal. The result of this simulationshowed the motion of Pb atom and Bi atom on Fe crystalsurface.

However, since they used ab initio method that is computa-tionally very expensive, only a few atoms were involved intheir simulation. Therefore in order to accommodate moreatoms, we used simple interatomic potential to be used withmolecular dynamics methods at reasonable computationaltime. We used LennardeJones potential in this simulation.Some of Fe atoms in Fe crystal were substituted by nickeland chromium atoms to represent SS 316. Basically corrosionphenomena in this regard are indicated by the penetration ofPbeBi into stainless steel. Our simulation results showed theexistence of this penetration process.

The LennardeJones parameters of the same pair of atomswere taken by fitting the data available from the literature(Qi and Takahashi, 2003; Olsson et al., 2005; Landa et al.,1998) of which the data of interatomic potential betweenFeeFe, PbePb and BieBi were derived from ab initio calcu-lation. Although the LennardeJones potential is very simplesome researchers have used this potential in the study of metalssuch as phase transition of NieAl (Ozgen and Adiguzel, 2003),crystalline phase transition (Kastner, 2003) and solideliquidphase transition (Maeda et al., 2003).

Molecular dynamics is a method to calculate positions andvelocities of molecules or atoms as a function of time-step orphase space trajectories of atoms, which can be used to deter-mine the properties of materials. The calculations of positionsand velocities in the next time-step usually use finite differentmethods to solve the Newton second law equation. (Haile,1997; Frenkel and Smith, 2002). Result of this calculationcan be processed based on the statistical mechanics to getmacroscopic properties of materials.

This work is an attempt to understand the atomic mecha-nism of corrosion phenomena of stainless steel in liquidleadebismuth as a function of time-step and temperatures.As the first step of this study, we have developed molecular dy-namics code which can be used to simulate a system of binaryatoms. The code has been used to simulate 50%Pbe50%Bi atseveral temperatures. In this simulation the radial distributionfunction and accuracy of total energy versus time-step werecalculated and has been reported by Maulana et al. (2005a,b).

As for a system of more than two atomic types we usedMOLDY, an open source molecular dynamics software(Refson, 2000).

In this computer experiment, we investigated the interfacingeffect of pure iron with Pb, Fee10%Ni with Pb, Fee16%Crwith Pb, Fee10%Nie16%Cr with Pb and Fee10%Nie16%Cr with PbeBi in several temperatures. From this simula-tion we derived the properties of the system such as the radialdistribution function, mean square displacement in addition tothe phase space itself.

2. Methodology

The simulation requires the initial condition of positionsand velocities of all atoms involved in the system. The interac-tion between atoms as a function of distance is also given. Theinitial positions of Fe were developed based on the BCC struc-ture with lattice parameter 2.8665 A and Pb atoms are deter-mined to fill the FCC cell with lattice parameter 4.9508 A.The unit cell of the systems can be extended periodically inthree dimensions with the equation:

r!¼ r!uc þ nx a!þ ny b!þ nz c!; ð1Þ

where r!uc is the vector position of unit cell, nx, ny, nz arethe number of cells in the x, y, z directions and a!, b

!and

c! are the unit vectors in the cartesian coordinates. Nickelsand chromiums were randomly added to the pure Fe struc-ture above as impurities to develop Fe alloy structure. Somebismuth atoms also added to replace some lead atoms in thepure lead structure to develop PbeBi alloy. The initial ve-locities of each atom were generated randomly with thevalue between 1 and �1.

The LennardeJones potential parameters of FeeFe, PbePband BieBi atoms had been derived by fitting the data availablein the literature (Qi and Takahashi, 2003) and were used in theinteraction potential given by

uðrÞ ¼ 43

��s

r

�12

��s

r

�6�; ð2Þ

where r is the atomic distance, 3 the potential depth and s thepotential parameter. Meanwhile the LennardeJones parame-ters of CreCr and NieNi were obtained, respectively, by fit-ting the data from the previous publications. The former wascalculated by embedded atomic method (EAM) (Olssonet al., 2005) and the latter was calculated by the FinniseSinclair potential (Landa et al., 1998). The results are shown

Fig. 1. The atomic positions of interface after molecular dynamics simulation.

618 A. Maulana et al. / Progress in Nuclear Energy 50 (2008) 616e620

in Table 1. To calculate the parameters 3 and s for differenttypes of atoms we used the LorentzeBerthelot mixing ruleequation as follows (Zhdanov and Fakhretdinov, 2005):

sAB ¼ 0:5ðsAA þ sBBÞ; ð3Þ

3AB ¼ ð3AA3BBÞ1=2: ð4Þ

The forces based on the LennardeJones potential above are

F!¼�duðrÞ

dr¼ 24

3

s

�2�s

r

�13

��s

r

�7�

ð5Þ

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12r (Angstrom)

g(r)

Fe-FeFe-PbPb-Pb

Fig. 2. Radial distribution function of Fe and Pb system.

and

F!

i ¼XN

jsi

F!

ij ði; j ¼ 1;.;NÞ ð6Þ

where F!

i is the sum of forces F!

ij on atom i due to atom j( j s i) in the system. N is the number atoms involved in thesimulation. These atoms are assumed to be classical and there-fore obey the classical mechanics. The equation of motion canbe written as follows:

F!

i ¼ mi

v v!i

vt¼ mi

v r!i

vtði¼ 1;.;NÞ; ð7Þ

-100

0

100

200

300

400

500

600

700

800

900

0 5000 10000 15000 20000

time-step (x 0.001 pScond)

T (K

)

823 K773 K673 K

573 K

473 K373 K

0 K

Fig. 3. System temperature for simulation of Fee10%Nie16%Cr with

50%Pbe50%Bi.

Fig. 4. Molecular dynamics simulation result of interface Fee10%Nie16%Cr

with 50%Pbe50%Bi at 773 K.

619A. Maulana et al. / Progress in Nuclear Energy 50 (2008) 616e620

where mi is the atomic mass of particle i, vi and ri are, respec-tively, the velocity and position of atom i.

Eq. (7) can be solved numerically by finite differencemethod. Some methods to solve this equation among othersare the well known leap-frog algorithm, Verlet algorithm,Gear predictorecorrector, Beeman algorithm, etc. In this studywe used Beeman algorithm.

From the above simulation, one can generate the phasespace trajectories of the particles. The macroscopic quantitiesof the system such as temperature, pressure, diffusion con-stant, radial distribution function, mean square displacement,etc., can be related by these time averages over the phasespace trajectories of the particles. The relationship betweentemperature and velocities is given by

T ¼ 1

kBð3N�NCÞX

i

miv2i ; ð8Þ

where kB is the Boltzmann constant, 3N is the total number ofdegree of freedom for the system of N atoms while NC is thetotal number of independent internal constraints.

The radial distribution function equation is the following:

gðrÞ ¼ hNðr;DrÞi12NrVðr;DrÞ; ð9Þ

0

2

4

6

8

10

12

0 200 400 600 800 1000T (K)

Pen

etratio

n d

ep

th

(an

gstro

m)

PbBi

Fig. 5. The penetration depth of Pb and Bi into Fee10%Nie16%Cr.

where r is the density of atoms. The radial distribution func-tion g(r) measures how atoms organize themselves around an-other. It is proportional to the probability of finding two atomsseparated by distance r�Dr. The behavior of g(r) can be usedto help identifying the phase of simulated system. For crystal-line solids, g(r) contains deeper valleys and higher narrowerpeaks than does g(r) for fluids.

3. Results and discussion

We performed surface interaction simulation between pureiron, Fee10%Ni, Fee16%Cr and Fee10%Nie16%Cr withlead and 50%Pbe50%Bi at several system temperatures byusing MOLDY program. The simulation cell was a box withthe volume 60� 30� 30 A3 that was filled by 2864 atoms.

Fig. 1(a) shows the position of the atoms in our systemswhich consist of two types of atoms with purely crystallinestructure of Fe and Pb at temperature of 0 K. Fig. 1(b)e(d)shows our simulation results after 20,000 time-steps at773 K and the presence of Pb at the left hand side of thefigures is due to the effect of periodic boundary condition. Itobviously shows that Pb able to penetrate into the Fee10%Ni, Fee16%Cr and Fee10%Nie16%Cr. Pb goes asdeep as 7.25 A, 11.08 A and 1.18 A into those three differentmaterials, respectively. However, this is not the case when wesimulated pure Fe and Pb interface. There were no such pen-etration processes of Pb into the Fe even at temperature ashigh as 823 K.

Fig. 2 shows the radial distribution function (RDF) of inter-action between pure Fe and pure Pb at temperature 773 K. Itcan be seen that the RDF (Radial Distribution Function) ofFe is more Higher and more narrower than RDF Pb. Theseresult indicates that Fe system in the solid state while Pbsystem is in the liquid state.

The simulation has also been carried out using surface in-teraction between Fee10%Nie16%Cr with 50%Pbe50%Biat several temperatures as shown in Fig. 3. The results of thesesimulations are shown in Figs. 4 and 5. Fig. 4 shows the posi-tions of all atoms in the simulation cell after moleculardynamics simulation at temperature 773 K with 20,000 time-step. The result indicates that Pb and Bi can diffuse intoFee10%Nie16%Cr. At temperatures above the melting pointof PbeBi at 398 K, Pb and Bi atoms generally able to pene-trate into Fee10%Nie16%Cr with different penetrationdepths. The result of this simulation is shown in Fig. 5. Pbatoms able to diffuse deeper than Bi atoms into the Fe systemand the higher the temperature the deeper the diffusiontakes place. The concentrations of Pb and Bi which can pene-trate into the alloy at 773 K are 14.35% and 16.20%,respectively.

4. Conclusion

Molecular dynamics method can be used to predict thedepth of penetration of atoms Pb or Bi into Fe system withor without substitution of impurity atoms. The penetrationdepth into Fee10%Nie16%Cr of Pb was deeper than the

620 A. Maulana et al. / Progress in Nuclear Energy 50 (2008) 616e620

one of Bi at high temperatures. The higher the temperature thedeeper the penetration depth for both Pb and Bi. The concen-trations of Pb and Bi which can penetrate into the alloy at773 K were 14.35% and 16.20%, respectively.

Acknowledgement

These research supported by Ministry of Science andTechnology of Republik Indonesia under Rintisan Gelarprogramme.

References

Frenkel, D., Smith, B., 2002. Understandings Molecular Simulation. Academic

Press.

Haile, J.M., 1997. Molecular Dynamics Simulations, Elementary Methods.

John Wiley & Sons, New-York.

Hata, Koji, Hara, K., Takahashi, M., 2005. Experimental studies on steels corro-

sion in PbeBi with steam injection. Progress in Nuclear Energy 47 (1e4),

596e603.

Kastner, O., 2003. Molecular-dynamics of a 2D model of the shape memory

effect part I: model and simulations. Continuum Mechanics and Thermo-

dynamics 15, 487e502.

Landa, A., Wynblat, P., et al., 1998. Development of FinniseSinclair type

potentials for Pb, PbeBi and PbeNi systems: application to surface

segregation. Acta Materialia 46 (9), 3027e3032.

Maeda, K., Matsuoka, W., Fuse, T., Fukui, K., Hirota, S., 2003. Solideliquid

phase transition of binary LennardeJones mixtures on molecular dynamics

simulations. Journal of Molecular Liquids 102 (1e3), 1e9.

Maulana, A., Su’ud, Z., Hermawan, K.D., Khairurrijal, 2005.

Corrosion study of steels in PbeBi using molecular dynamics

method. In: COE-INES-Indonesia International Symposium, pp.

123e127.

Maulana, A., Su’ud, Z., Hermawan, K.D., Khairurrijal, 2005. Application of

molecular dynamics method to study steels corrosion phenomena in

liquid leadebismuth cooled reactors. In: Proceedings of Asian Physics

Symposium, Bandung, Indonesia, pp. 350e353.

Olsson, P., Wallenius, J., et al., 2005. Two-band modeling of a-prime phase

formation in FeeCr. Physical Review B 72, 214119.

Ozgen, S., Adiguzel, O., 2003. Molecular dynamics simulation of diffusionless

phase transformation in a quenched NiAl alloy model. Journal of Physics

and Chemistry of Solids 64, 459e464.

Qi, Y., Takahashi, M., 2003. Study on corrosion phenomena of steels in PbeBi

flows. In: Proceeding ICONE 11, 36375, Tokyo, Japan.

Refson, K., 2000. MOLDY, a portable molecular dynamics simulation pro-

gram for serial and parallel computers. Computer Physics Communica-

tions 126 (3), 309e328.

Sapundjiev, D., Van Dyck, S., Bogaerts, W., 2006. Liquid metal corrosion of

T91 and A316L materials in PbeBi eutectic at temperatures 400e600 �C.

Corrosion Science 48, 577e594.

Takahashi, M., Qi, Y., Nitta, H., Nishikawa, N., Takahisa, Ohno, 2004.

First-principle molecular dynamics simulation on interatomic interaction

of Fe crystal with Pb and Bi atoms. Science and Technology of

Advanced Materials 5, 673e676.

Wu, Chao, Dasika, K.K., Chen, Y., Moujaes, S., 2002. Numerical modelling of

lead oxidation in controlled lead bismuth eutectic systems: chemical kinet-

ics and hydrodynamic effect. In: International on Advanced Nuclear Power

Plants in Hollywood, Florida, June 9e13.

Zhdanov, E.R., Fakhretdinov, 2005. Molecular dynamics simulation of the

liquidevapor interface of binary mixtures. Journal of Molecular Liquids

120 (1e3), 51e53.