Temperature Effect on the Mechanical Properties of Pd 3 Rh and PdRh 3 Ordered Alloys

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1 Wold Academy of Scence, Engneeng and Technology Intenatonal Jounal of uclea and Quantum Engneeng Vol:5, o:6, Tempeatue Effect on the Mechancal Popetes of 3 and 3 Odeed Alloys J. Davood, J. Moad Intenatonal Scence Index, uclea and Quantum Engneeng Vol:5, o:6, waset.og/publcaton/586 Abstact The am of ths eseach was to calculate the mechancal popetes of 3 and 3 odeed alloys. The molecula dynamcs (MD smulaton technque was used to obtan tempeatue dependence of the enegy, the Yong modulus, the shea modulus, the bulk modulus, Posson s ato and the elastc stffness constants at the sobac-sothemal (PT ensemble n the ange of -35 K. The nteatomc potental enegy and foce on atoms wee calculated by Quantum Sutton-Chen (Q-SC many body potental. Ou MD smulaton esults show the effect of tempeatue on the cohesve enegy and mechancal popetes of 3 as well as 3 alloys. Ou computed esults show good ageement wth the expemental esults whee they have been avalable. Keywods - alloy; Mechancal popetes; Molecula dynamcs smulaton I. ITRODUCTIO HE -based alloys have been studed fo moe than T yeas n both fundamental and applcaton aspects. Especally, n the last thee decade themodynamc and mechancal popetes of pd-based alloys wee studed by molecula dynamcs smulaton technque [-8]. Among the -based alloys, the - alloy s of specal nteest owng to the nceased hydogen absopton capacty. Moeove, based alloys can potentally offe mpovements to pue membanes [9-]. Vey few expemental data ae avalable fo the mechancal popetes of 3 and 3 odeed alloys n the vaous tempeatues. The phase-stablty of alloys was studed by Tanus Saha and Abht Mookeee [9]. J. Luyten et al povded theoetcal estmates fo the heats of mxng of the bnay systems Pt-, Pt- and - and the tenay Pt-- systems. The heats of mxng and the phase dagams wee calculated wth Monte Calo smulatons and wth the sem-empcal modfed embedded atom method [3]. In the othe nvestgaton, M. Haa et al wee nvestgated absopton and desopton sothems of potum and deuteum wee measued fo pd-based alloys []. In ths nvestgaton, we have pefomed MD smulaton unde constant pessue, constant tempeatue condtons [, ] to calculate the mechancal popetes ncludng, Yong modulus, the shea modulus, the bulk modulus, Posson s ato and the elastc stffness constants of 3 and 3 odeed alloys. The Q-SC potental [5, 4] paametes of the pue and metals wee used as nteatomc potental paametes to calculate the cohesve enegy as well as the mechancal popetes of, pue metals and 3 and 3 odeed alloys. We have calculated mechancal popetes and J. Davood s wth the Unvesty of Zanan, Zanan, Ian (phone: ; fax: ; e-mal: davood@znu.ac.. J. Moad s wth the Unvesty of Zanan, Zanan, Ian (phone: ; fax: cohesve enegy of, elements unde atmosphec pessue fo evaluatng compute codes and the nteatomc potental. II. DETAILS OF MD SIMULATIO A. Inteatomc potental The foce expeenced by ndvdual atom n an -atom cluste was obtaned fom Q-SC nteatomc potental enegy functon. The potental enegy of the pue metals and alloys n Sutton-Chen fomalsm fo the systems of atoms s gven as follows [5, 4] U tot m / a ε ( n a c ε The fst tem n equaton ( s a two body nteacton between the atoms and, the second tem epesents the many-body coheson tem assocated wth atom, a s the length paamete scalng to the lattce spacng of the cystal, c s a dmensonless paamete scalng the attactve tems, ε s an enegy paamete detemned fom expement, and n, m ae ntege paamete wth n>m whch detemne the ange of the two component of the potental (Table I. TABLE I THE Q-SC POTETIAL PARAMETERS FOR PD-RH MODEL ALLOY SYSTEM Inteacton ε(ev a(ǻ c n m To constuct the potental fo the bnay alloy state, -, fom the coespondng Q-SC potentals fo the elemental state, we used the followng mxng ule [6, 9] ε m / ( ε ε, ( m + m / a ( aa ( ( n + n n, B. Tempeatue and pessue contol The tempeatue contol was mplemented va the ose- Hoove heat bath [5, 6] whose ntoducton modfes the (3 Intenatonal Scholaly and Scentfc Reseach & Innovaton 5(6 9 schola.waset.og/37-689/586

2 Wold Academy of Scence, Engneeng and Technology Intenatonal Jounal of uclea and Quantum Engneeng Vol:5, o:6, Intenatonal Scence Index, uclea and Quantum Engneeng Vol:5, o:6, waset.og/publcaton/586 standad velocty Velet equaton of moton [] to the followng foms [7] f + δtv + δt ζ( tv m δt f v v + ζ( tv m δt ζ ζ + m v gkbt Q (4 δt ζ( t + δt ζ + m v gkbt Q f v v + δt + δtζ( t + δt m whee ζ s the tme-dependent fcton of the heat bath. A patcula paametezaton of the Q s gven by Q gkbtτ (5 whee τ s the elaxaton tme of the heat bath, nomally of the same ode of magntude as the smulaton tme step. Ths paamete contols the speed wth whch the bath damps down the fluctuaton n the tempeatue. The numbe of degees of feedom s gven by g3(-. The standad Beendsen baostat [8] was used fo pessue contol of the system. The Beendsen baostat uses a scale facto, μ, whch s a functon of nstantaneous pessue, P, to scale lengths n the system x( μx( y( μy( z( μz( L μl whee μ s gven by δt μ + ( P P τ p / 3 Hee, τ p s the se tme of the baostat, and P s the set pont pessue. The system pessue s set towad a desed value by changng the dmensons of the smulaton cell sze dung the smulaton. C. Smulaton data Ou smulatons wee caed out usng molecula dynamcs beads on Q-SC nteatomc potental enegy functon. The smulatons nvolved clustes of and pue metals, and 3, 3 odeed alloys. The total numbe of atoms (6 (6 pesent was 48 fo both pue metals and alloys. The smulaton tme step was set to.5 fs. The peodc bounday condtons wee employed n all dectons. The clustes wee fst equlbated fo tme steps at T K, and then the tempeatue was ased by K at each tempeatue step. At each step that the tempeatue was nceased, the system was e-equlbated fo tme steps. Fgs. and show the vaaton of enegy, volume and pessue n ode to obtan the equlbum state. These quanttes appoached to constants values to make sue that the system had eached equlbum as seen n fgues. Enegy (ev Tme step Fg. Tme vaaton of the cohesve enegy fo,, 3 and 3 dung equlbaton phase Volume Volume Pessue Tme step Fg. Tme vaaton of the volume and pessue fo 3 dung equlbaton phase III. RESULTS AD DISCUSSIO The quanttes that wee calculated n these MD smulatons wee the mechancal popetes ncludng the Yong modulus, the shea modulus, the bulk modulus, Posson s ato and the elastc stffness constants of,, 3 and 3. Accodng to the hooke s law, fo small defomaton, the stess components σ ae dectly popotonal to the stan Pessue (GPa Intenatonal Scholaly and Scentfc Reseach & Innovaton 5(6 3 schola.waset.og/37-689/586

3 Wold Academy of Scence, Engneeng and Technology Intenatonal Jounal of uclea and Quantum Engneeng Vol:5, o:6, components ε. Ths elaton can be expessed n mathematcal tems as [9] 33 σ (8 Cklε kl 3 whee C kl ae the elastc stffness constants. The numbe of ndependent elastc stffness constants s educed f the cystal possesses symmety elements. Fo example, fo cubc cystal, thee ae only thee ndependent constants (C, C, C 44 [9]. In tems of the nteatomc potental, these constants can be expessed as [3] C (GPa Intenatonal Scence Index, uclea and Quantum Engneeng Vol:5, o:6, waset.og/publcaton/586 c ( E v ε ( E v ε ε ( E 4v ε c (9 c 44 whee E ( and v ae the total enegy of an atom and the volume pe atom, espectvely. We have used the Reuss assumpton to estmate the bulk modulus B, shea modulus G, young modulus E, and Posson s ato v, accodng the followng elatons [3] B ( c + c / 3 ( G 5( C C /(4C44 + 3( C C ( E 9GB /(3B + G ( v ( 3B G / (3B + G (3 Fgs. 3- show the tempeatue dependence of enegy, elastc stffness constants (C, C, C 44, Bulk modulus, shea modulus, Young modulus, and Posson s ato espectvely. Enegy pe atom (ev Fg. 3 Vaaton of enegy of,, 3 and 3 wth tempeatue Fg. 4 Vaaton of elastc stffness constant (C of,, 3 and 3 wth tempeatue C (GPa Fg. 5 Vaaton of elastc stffness constant (C of,, 3 and 3 wth tempeatue C 44 (GPa Fg. 6 Vaaton of elastc stffness constant (C 44 of,, 3 and 3 wth tempeatue Intenatonal Scholaly and Scentfc Reseach & Innovaton 5(6 3 schola.waset.og/37-689/586

4 Wold Academy of Scence, Engneeng and Technology Intenatonal Jounal of uclea and Quantum Engneeng Vol:5, o:6,.3 Bulk modulus(gpa Posson's ato Intenatonal Scence Index, uclea and Quantum Engneeng Vol:5, o:6, waset.og/publcaton/586 Fg. 7 Vaaton of Bulk modulus of,, 3 and 3 wth tempeatue Shea modulus (GPa Fg. 8 Vaaton of shea modulus of,, 3 and 3 wth tempeatue Young modulus (Gpa Fg. 9 Vaaton of Young modulus of,, 3 and 3 wth tempeatue Fg. Vaaton of Posson s ato of,, 3 and 3 wth tempeatue Ou motvaton fo computng these popetes fo the elemental mateals was to obtan an estmate of the accuacy of the Q-SC potentals. A compason of the esults fo the elemental mateals shows that ou computed values ae n easonable ageement wth the expemental data that ae avalable fo these mateals. Ths can gve an ndcaton of the qualty of the alloy potental enegy functons that we have constucted on the bass of these elemental potentals. The pecentage eos assocated wth the computed values, as compaed to the expemental data ae lsted n Table II. TABLE II THE PERCETAGE ERROR OF THE COMPUTED VALUES VIS-A-VIS THE EXPERIMETAL DATA Physcal popetes Cohesve enegy..8 C.6 3. C C B G E v The vaaton of cohesve enegy wth tempeatue fo both elemental mateals,, and the odeed alloy 3 and 3 wee plotted n the fg. 3. We see that the enegy of,, 3 and 3 ncease wth tempeatue. Ths s patculaly tue fo the vaaton of the enegy wth tempeatue. The tempeatue dependence of the mechancal popetes ae shown n fgues 4-. Fom fgues 4-9, we see that the elastc stffness constants, the bulk modulus, the shea modulus and the Yong modulus decease concuent wth the nceased tempeatue. Ths decease s essentally lnea wth tempeatue, and has ageement wth an appoxmate equaton descbng the modulus-tempeatue eltonshpe accodng the followng elaton [3] Intenatonal Scholaly and Scentfc Reseach & Innovaton 5(6 3 schola.waset.og/37-689/586

5 Wold Academy of Scence, Engneeng and Technology Intenatonal Jounal of uclea and Quantum Engneeng Vol:5, o:6, Intenatonal Scence Index, uclea and Quantum Engneeng Vol:5, o:6, waset.og/publcaton/586 T M M a ( T m whee M s the modulus at tempeatue T and M the modulus at K. The popotonalty constant a fo most cystal solds s on the ode of.5. The esults fo the Posson s ato ae shown n fg.. Ths fg. shows that the Posson s ato s appoxmately constant n the ange of -35 K. IV. COCLUSIO In ths pape, we have pefomed molecula dynamcs smulatons, based on Q-SC potental, to nvestgate the mechancal popetes of pue, metals and 3, 3 odeed alloys. The tempeatue dependence of the enegy, the elastc stffness constants, the bulk modulus, the shea modulus, the Yong modulus and Posson s ato wee obtaned fom these smulatons. The esults fom ou smulatons show that ou choce of the nteatomc potental and the mxng ule employed to constuct the alloy states lead to a good pedcton of the popetes of these mateals. Ths eseach shows the cohesve enegy ncease wth tempeatue. Moeove, the mechancal popetes, except Posson s ato, decease wth the nceased tempeatue n a lnea manne. Also, the Posson s ato emans appoxmately constant n the ange of -35 K. REFERECES [] H. H. Kat, M. Tomak, M. Uludogan, et al, Intenatonal ounal of moden physcs B. 8, 4, pp [] S. Ozdem Kat, M. Tomak, M. Uludogan, and T. Cagn, J. oncyst. Sol. 337, 4, pp. -8. [3] H. H. Kat, M. Tomak, M. Uludogan, T. Cag?n, Tuk. J. Phys. 3, 6, pp [4] J. Davood, M. Ahmad, H. Raf-Taba, Mateal scence and engneeng A 57,, pp [5] H. H. Kat, M. Tomak, M. Uludogan, T. Cag?n, Comput. Mate. Sc. 3, 5, pp [6] J. Davood, M. Ahmad, Computatonal mateals scence 49,, pp. -4. [7] S. Ozdem Kat, M. Tomak, M. Uludogan, and T. Cagn, J. oncyst. Sol. 337, 4, pp. -8. [8] H. H. Kat, M. Tomak, M. Uludogan, T. Cag?n, Mateals scence and engneeng A-Stuctual mateals popetes mcostuctue and pocessng 435, 6, pp [9] J.C. Baton, J.A.S. Geen, F.A. Lews, Tans. Faaday Soc. 6, 966, pp. 96. [] F.A. Lews, W.D. McFall, T.C. Wthespoon, Z. Phys. Chem.. F. 84, 973, pp. 3. [] Y. Sakamoto, Y. Haaguch, M. Ua, F.L. Chen, Be. Bunsenges Phys. Chem. 98, 994, pp [] Y. Sakamoto, K. Kuuma, Y. atom, J. Appl. Electochem. 4, 994, pp. 38. [3] J. Luyten, J. De Keyze, P. Wollants, C. Ceemes, CALPHAD: Compute Couplng of Phase Dagams and Themochemsty 33, 9, pp [4] B. Baanowsk, S. Machzak, T.B. Flanagan, J. Phys. Chem. 77, 973, pp. 35. [5] Y. Sakamoto, F.L. Chen, M. Ua, T.B. Flanagan, Be. Bunsenges Phys. Chem. 99, 995, pp. 87. [6] G. Mengol, M. Fabzo, C. Manduch, E. Mll, G. Zannon, J. Electoanal. Chem. 39, 995, pp. 35. [7] A.K.M. Fazle Kba, T. Kubota, A. Kagawa, Y. Sakamoto, Int. J. Hydogen Enegy 4, 999, pp [8] A.K.M. Fazle Kba, Y. Sakamoto, Int. J. Hydogen Enegy 5,, pp [9] T. Saha and A. Mookeee, J. Phys.: Condens. Matte 9, 997, pp [] M. Haa, L. Wan, M. Matsuyama, K. Watanabe, Jounal of Alloys and Compounds 48, 7, pp [] H. J. C. Beendsen, J. P. M. Postma, W. F. van Gunsteen, A. Dola, and J. R. Haak, J. Chem. Phys. 8, 984, pp [] J.M. Hale, Molecula Dynamcs Smulaton, John Wley & Sons, ew Yok, 99. [3] H. Raf-Taba, A. P. Sutton, Phl. Mag. Lett. 63, 99, pp [4] Y. Q, T. Cag?n, Y. Kmua and W.A. Goddad, Phys. Rev. B 59, 999, pp [5] W. G. Hoove, Canoncal dynamcs: Phys. Rev. A 3, 985, pp [6] S. ose, J. Chem. Phys. 8, 984, pp [7] A.P. Sutton, J.B. Pethca, H. Raf-Taba, J.A. emnen, n: D.G. Pettfo, A.H. Cottell (Eds., Electon Theoy n Alloy Desgn, Insttute of Mateals, London, 994, 9. [8] H. J. C. Beendsen, J. P. M. Postma, W. F. van Gunsteen, A. Dola, and J. R. Haak, J. Chem. Phys. 8, 984, pp [9] C. Kttel, Intoducton to Sold State Physcs, eghth ed., John Wley & Sons, ew Yok, 5. [3] A. P. Sutton and J. Chen, Phl. Mag. Lett. 6, 99, pp [3] J.F. ye, Popétés Physques des Matéaux, Dunod, 96. [3] T.H. Coutney, Mechancal behavo of metals, second ed., McGaw- Hll, ew Yok,. Intenatonal Scholaly and Scentfc Reseach & Innovaton 5(6 33 schola.waset.og/37-689/586