THE KIRKENDALL SHIFT AND FRENKEL EFFECT DURING MULTI-COMPONENT DIFFUSION PROCESS

Transcription

1 ECCOMAS Congress 2016 II European Congress on Coputatonal Methods n Appled Scences and Engneerng M. Papadrakaks,. Papadopoulos, G. Stefanou,. Plers (eds.) Crete Island, Greece, 5 10 June 2016 THE KIRKENDALL SHIFT AND FRENKEL EFFECT DURING MULTI-COMPONENT DIFFUSION PROCESS 1 1 Rzeszow Unersty of Technology, Faculty of Mechancal Engneerng and Aeronautcs, al. Powstańców Warszawy 12, Rzeszow, Poland. e-al: bwerzba@prz.edu.pl Keywords: Krkendall plane, ods, dffuson, ass conseraton. Abstract. In ths paper the phenoenologcal process related to the eoluton of the ternary and hgher syste s dscussed. The two coupled phenoena related to the dfference n dffuson coeffcents wll be presented by nuercal nestgatons - anly the Krkendall and Frenkel effects. Such a dfference leads to the: lattce shft, stress generaton and relaxaton, non-ulbru dstrbuton of acances and odng. The generalzed Darken ethod (belocty) of the ult-coponent syste s forulated. The approxaton ethod allows for deternaton the eoluton of the ods. Thus the relaxaton te for the acances n ters of ean graton length and acancy dffuson coeffcent n case of ultcoponent syste wll be forulated. Moreoer, the od eoluton n Fe-Pd syste wll be calculated.

2 1 INTRODUCTION The Naer-Stokes set of uatons s a theoretcal odel for contnuu flud dynacs where the local therodynac ulbru holds. Howeer, ths uatons encounters soe challengng dffcultes for non-contnuu flows. Ths phenoena was wdely studed n the gas county, where the gas densty s usually ery low. In recent years, non-contnuu flows hae also attracted uch attenton wth the rapd deelopent n croelectroechancal systes. Howeer, due to the fnte Knudsen nuber effect, the contnuu-ulbru assupton ay break down and the odel wll fal to work for these flows [1, 2, 3]. In the recent years Brenner proposed new odel based on the flud elocty [4, 5, 6]. He ntroduced one sngle addtonal ter nto each of the oentu and energy uatons. Generally Brenner assued that the ass elocty n the hydrodynacs uatons should be replaced by olue elocty whch relates to the flux of olue rather than ass [4, 5, 6]. On the other hand, dffuson county knows ery well an alternate way of prodng olue conseraton. Naely, t s a lattce drft (Krkendall effect) caused by the acancy flux dergence leadng to dslocaton clb and subsuently to the constructon of extraplanes n accuulaton regon and dsantlng atoc planes n depletng regon [7, 8, 9]. The drft elocty s than generated. Ths constrant eans zero dergence of oerall olue flux densty: r x 1 J 0 When the only drng force s checal potental gradent, by:, the olue flux s gen ch * J D. x In ths paper the ntrdffuson n sold state descrpton wll be forulated. The ass, heat, olue contnuty uaton, flux defnton and acancy eoluton uaton wll be presented. We wll focus on the knetc effects that are related to dfference of obltes, naely on the stress generaton and relaxaton [3,10]. Interdffuson leads to the accuulaton of atter at the sde of slower coponent of the dffuson couple. Due to the olue constrant, nature just ust fnd soe ways to reduce ths accuulaton to zero. Ths reducton, wll be realzed here by ntroducton of: 1) Krkendall effect; 2) Backstress effect and 3) non-

3 ulbru acancy dstrbuton. The last paragraph wll present the results of the odel anly, the ods eoluton durng nterdffuson process n Fe-Pd syste. 2. THE MASS AND HEAT EQUATIONS Mass conseraton law. Consder ultcoponent xture, where coponent densty. The eoluton of the densty s descrbed by the uaton: d 0 t (1) denote the -th where J s the oerall flux of the -th coponent and denote ts olue elocty (the edu elocty). We do not take nto account ass producton or consupton n ths uaton. Heat balance. Dffuse flux can nfluence the heat balance. The heat flux can be defned after Fourer (for sotropc ateral) as: q J k grad T (2) where k denote the theral conductty and T s the teperature. We can defne the enthalpy of the syste accordng to nternal energy densty as: tr h u (3) The dfferental for of the aboe uaton s: Dtr Dh Du Dtr D u (4) Dt Dt Dt Dt Dt where DT Dt denote the Lagrange derate defned oer the elocty : DT T grad T, (5) Dt t Assung, that the changng enthalpy n te s proportonal to ts specfc heat and teperature, the followng uaton can be wrtten: Dh Dt DT cp (6) Dt Fnally, by ntroducng Eqs. (2) - (6) nto the energy conseraton law and the echancal parts are neglected, than the fnal heat transport uaton can be dered:

4 DT cp d k grad T (7) Dt The fluxes. The oerall flux and elocty of the ass are defned by the olue drft and dffuson eloctes: where :, (8) drft d drft d drft d where = 1,, r and r denotes the nuber of coponents and fracton. N denote the olar The olue drft elocty. Durng an arbtrary transport process when olue s affected by the dstrbuton of eery xture coponent and the stress feld, fro the Loulle theore t follows: d d. (9) t We consder here a ultcoponent soluton n a closed syste, where partal olar olues and elastc propertes do not depend on coposton, f N N the olue can be only affected by the external forces: where: d dt dx. (10) 1,..., r. Thus, s a elocty generated by external forces (elastc deforaton). Introducng Euler relaton ( 1) and Eq. (8), (9) allow to calculate the drft elocty of the xture: drft d d d (11) Denotng the Darken elocty, D d D as: (12) The fnal for of the drft elocty can be rewrtten n the for: d drft D d (13) Equaton (13) defnes the drft elocty of the syste. In one denson the drft elocty can be expressed by analytcal functon as: drft D (14)

5 In ths paper we wll consder, addtonally four ore effects: 1) Krkendall effect, 2) Backstress 3) acancy generaton effect and 4) theral gradent The Krkendall effect eans the oeent of lattce fro slower dffusant sde towards the faster dffusant sde wth soe drft elocty, follow: drft d drft. Thus ths effect results n dffuson flux as J, (15) The dffuson flux s defned by the Nernst-Planck flux uaton [11,12] whch n general for reads: drft J B grad, (16) where s the generalzed dffuson potental of the -th coponent. Backstress effect - the dffuson potental s affected by the nternal stress effect - stress gradent appearng due to attept of atter accuulaton. Each dffusng ato of both speces s affected by coon stress force: Int Int grad grad p (17) Non-ulbru acancy dstrbuton - the dffuson potental s a dfference of coponent checal and coon acancy potentals, ch, and ualzaton of the dffuson fluxes nstead of lattce shft, s proded by the non-ulbru acancy gradent appearng due to attept of atter accuulaton. Role of effecte force here s played by the gradent of acancy checal potental, proportonal to the gradent of deaton of acant stes fracton fro ts local ulbru alue: kt ln (18) Teperature dstrbuton - the sotheral heat transtted by ong the ato n the process of jupng a lattce ste less the ntrnsc enthalpy [13, 14]: where: grad 1 T T * Q gradt (19) * Q s defned as heat of transport. Fnally, the dffuson potental s a su of coponent checal and coon acancy and stress potentals, ch Int T. Consuently, the gradent of the dffuson potental s defned as follow:

6 1 kt Int 1 * grad kt grad grad grad p Q gradt T (20) Internal pressure - the nternal pressure s a result of the dfference n the dffuson coeffcents of the coponents and dfference n the lattce n dffuson couple [15]: p t where: the oerall dffuson elocty s defned as: Int E d d, (21) d d (22) acancy exchange. The last uaton defnng the odel s the acancy exchange. The uaton s defned as: where r d B grad 0 (23) t 1 N s the acancy olar fracton, j s the acancy flux. The acancy ulbru olar fracton and relaxaton te, respectely. N and denote the 3. RESULTS. In ths secton the results of nterdffuson and ods foraton wll be shown. The ods foraton wll be analyzed n two-densonal space. For experental erfcaton the Fe- Pd bnary syste was used. The ods rad wll be estated and the results for dfferent calculaton tes wll be shown. The followng set of the uaton wll be soled: d D d 0 on \ t 1 - ass conseraton r d B grad 0 on \ - acancy exchange t 0 on t t where: grad kt grad grad 1 kt. Ths set of the uaton wll be suppleented wth boundary condtons: The subset d 0 on - dffuson flux on the boundary denote the poston of the ods. The ods are ntally ntroduced onto the

7 calculaton esh. Thus, only the growth wll be sulated. In general, the ods growth on the faster dffuson couple sde. In Fe-Pb syste the ods growth on the ron reach sde of the dffuson couple. We assue that n Fe-Pd syste the ean graton length for acancy was L The dffuson coeffcent was calculated fro Boltzann-Matano analyss (fro known experental results of dffuson n Fe-Pd syste at 1273K for 150h [16]),.e. at the concentraton pont N Fe 0.75 the dffuson coeffcents was estated as: D Fe and D Pd s. The results for dfferent calculaton tes are shown on fgures 1-2. Fgure 1. The sualton results of the a) - c) Fe concentraton and b) - d) ods foraton (acances eoluton) n Fe-Pd syste after annealng at 1273K for 200h

8 Fgure 2. The sualton results of the a) - c) Fe concentraton and b) - d) ods foraton (acances eoluton) n Fe-Pd syste after annealng at 1273K for 1000h 4. CONCLUSIONS. Present paper shows the ethod whch allow for calculaton of the physcal odels of nterdffuson. The drft elocty was calculated n case of presence of the acancy, backstress and checal force felds. Moreoer, t was shown, that the nterdffuson process can be nfluenced by the acancy and heat felds. The odel was checked by sulatng ods growth durng dffuson couple experents. It was presented, that the ods are fored at the faster dffuson sde of the dffuson couple. Moreoer the rad of the od changes parabolcally wth te. In the future the sulaton should take nto account, that the ods are ong durng the dffuson process. Acknowledgeents. Ths work has been supported by the Natonal Scence Centre (NCN) n Poland, decson nuber 2013/09/B/ST8/ REFERENCES [1] E. P. Muntz, Annu. Re. Flud Mech., 21, 387, [2] C. M. Ho, Y. C. Ta, Annu. Re. Flud Mech., 30, 579, [3] Z. Guo, and K. Xu, Ad. Appl. Math. Mech., 1, 391, [4] H. Brenner, Physca A, 349, 11, [5] H. Brenner, Physca A, 349, 60, 2005.

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