CADiff: a software and method to numerically model diffusion and reactive diffusion in multi-component systems

Transcription

1 Cff: a softwae and method to numecally model dffuson and eactve dffuson n mult-component systems Maek NIELEWSKI a, Batlomej WIERZB b Intedscplnay Cente fo Mateals Modelng, Faculty of Mateals Scence and Ceamcs GH Unvesty of Scence and Technology, l. Mckewcza 3, 3-59 Cacow, Poland a danel@agh.edu.pl, b bwezba@agh.edu.pl; Fax: bstact. We pesent the softwae and method Cff whch allows fo numecal smulaton of ntedffuson and eactve dffuson pocesses. Wth two examples we demonstate the effectveness of ou method. The known methods on the eactve and ntedffuson ae sgnfcantly extended as follows: ) the dffuson fluxes ae defned n the volume fxed efeence fame that s goously defned ) the aken method s extended and allows analyzng the non deal alloys showng dffeent mola volumes, ) the bounday condtons take nto account both the ntedffuson and eactve dffuson and v) the flux constants ae used to couple pocesses occung at dffeent tme scales. We demonstate solutons of the ntedffuson n Fe-N-Cu alloy and heteogeneous eacton between multcomponent alloy and oxdant. The ntedffuson can be computed statng fom the Fck s consttutve dffuson flux fomula to the combned aken-onsage appoach. It allows modelng pue ntedffuson n multcomponent systems as well as a wde class of the eacton-dffuson poblems assocated wth sold state chemsty, cooson, suface teatment etc. Keywods: dffuson, ntedffuson, eacton dffuson, oxdaton, chemcal ntedffuson.

2 . Intoducton. The quanttatve models of the pocesses and mateal behavo can be vey useful n testng mateal stablty, estmatng equed popetes and pedctng the lfe tme. The elable and smple smulaton softwae consdeably acceleates both the eseach and the development. It helps n desgnng and poducton of new mateals. The heteogeneous eactons n tenay system wee analyzed by Wagne [,,-,34]. These ae essentally the quas-equlbum pocesses (chemcal dffuson, and eactons at ntefaces) that take place unde the nfluence of chemcal potental gadent [5,6]. He was fst to notce that the oxdzed component entes the oxde phase as a esult of the suface eacton and of the dffuson though alloy oxde nteface. The non-eactng elements dffuse nto the nteo of the alloy [3]. The both Onsage [7,8] and aken [9] methods ae commonly used n nonequlbum themodynamcs to descbe dffuson n solds. The key aken postulate that the total mass flow s a sum of the dffuson and dft flows was appled fo the descpton of the dffuson n mult-component sold soluton [,]. The equatons of mass consevaton, the appopate expessons descbng the fluxes and the postulate of constant mola volume of the system allowed the quanttatve descpton of the dffuson tanspot pocess n the open as well as closed system,.e., n the dffuson couple of the fnte thckness. The method descbes ntedffuson when ntnsc dffusvtes depend on composton and allows ncludng actvtes of components []. In ths softwae the aken method s futhe extended to nclude the vaable and/o dffeent mola volumes, e.g., the Vegad law []. Moeove, we goously combne ntedffuson and eactve dffuson by boadenng the Wagne bounday

3 condton [3]. The method pesented hee and smulaton package allows modelng the wde ange of pocesses that can be teated as one-dmensonal dffuson wth convecton (aken dft) and eacton at ntefaces. Classcal examples shown ae: Intedffuson n the closed mult-component systems, e.g., dffuson mxng, evoluton of gadent mateals and mult-laye systems. Reacton-dffuson,.e., the coupled ntedffuson and eactve dffuson due to the chemcal eacton at the movng nteface. In what follows we consde the pocesses whee the dffusvtes may dffe by odes of magntude and numbe of components s unlmted. Consequently, the poblem exhbts dffeent tme scales, e.g., the slow chemcal dffuson (n the gowng laye) and fast ntedffuson n mult-component mxtue (alloy, ntemetallc compound, etc.). The two-scale computatonal methods of modelng wee used by many authos to model the mechancal popetes of mateals wth mcostuctue and twoscale smulatons wee poven to be effcent and elable [3,4]. In most cases system of equatons wee solved n dffeent tme scales. In ths wok we use the concept of the flux lmte [5]. The ntedffuson method, the eacton-dffuson and smulaton esults ae shown n the followng sectons.. Intedffuson n multcomponent open systems ependng on the avalablty of the data and the equed accuacy, Cff allows selectng: ) the consttutve equatons fo dffuson flux, Fg. and secton.3, ) the equaton of state, secton., ) expesson fo dffusvtes, secton.4 and v) the fom of the known o computed eacton flux between alloy and suoundng.

4 ) Laws: a) Consevaton of mass (Secton.): c + dvj =, =,..., t b) Equaton of state (Secton.) c) ffuson flux (Secton.3) d) ffusvtes (Secton.4) ) Bounday condtons and eacton flux (Sectons.5) Flux fomulae can be chosen fom the set of the followng consttutve equatons: ffuson and Convecton: aken-onsage (O) J = L X + cυ, =,..., j j j= o aken-nenst-planck (NP) J = Bc X + cυ, =,..., o o ffuson only: Onsage (O) J = L X, =,..., j j j= o Nenst-Planck (NP) J = Bc X, =,..., o aken-fck (F) J = gad c + cυ, =,..., o o Fck (F) J = gad c, =,..., Fg.. The schema showng the pesented method and the avalable flux fomulae s.

5 Fgue shows the schema of the method. The flux can be expessed as pue dffuson, J = J (the fomulae F, NP and O), o t can be coupled wth the aken d method, J J cυ d = + (F, NP and O). The bounday condtons allow couplng ntedffuson and eactve-dffuson. The followng basc methods ae avalable to model the eactve-dffuson: the quas-statonay chemcal dffuson of one o moe elements n the gowng laye, e.g., oxdaton, ntemetallc phase gowth, suface teatment; the concentaton dependent flux though the nteface (e.g., evapoaton) o abtay (postulated o measued) tme dependent flux(es) though nteface... Mass consevaton law We do not consde the chemcal and/o othe eactons wthn the dffuson zone and the equaton of mass consevaton has fom: c + dv( cυ ) =, () t whee: c = c ( t, x) and υ υ (, ) = denote the mola concentaton and the velocty of tx the -th component. Summng up Eqs. () fo all components, the global consevaton law follows: c + dv( cυ ) =. () t The most geneal fom of the flux used n ou method s the aken-onsage flux: j j j= J = L X + J, =,..., (3)

6 whee: J and J denote the oveall flux and the aken flux (dft) of the -th component, espectvely, L j ae phenomenologcal tanspot coeffcents and X j denotes the foce actng on the dffusng component. The Onsage consttutve equatons base on the defnton of the phenomenologcal tanspot coeffcents [7]: NB k k k=, k L = cb, = j, =,...,, NB k = k k (4) NB j j Lj = cb, j,, j =,...,, NB k = k k (5) whee N = N ( t, x) and B B ( N,..., N ) = denote mola ato and moblty. The tanspot coeffcents n Eq. (5) ae symmetcal,.e., L = L, j: j j NB j j NB Lj = cb = cjbj = Lj, j,, j =,...,, NB NB k k k k k= k= (6) The foces n Eq. (3), X j, ae defned as a gadent of the chemcal potental: X = gadµ ch (7) j j The themodynamc data n Eq. (7) can be ntoduced as analytcal functons o assumng dealty sweepng statement (a = c ). The mobltes n Eqs. (4)-(5), can depend on concentatons. One should notce that the esults and the accuacy of the smulatons ctcally depend on the qualty of these descptons (data).

7 .. Equatons of state The total mass densty, ρ ρ( tx, ) Ω=Ω ( tx, ), and the mola mass, M M ( t, x) elatons: =, the mola concentaton, the mola volume, =, ae defned by the followng standad ρ:= = = ρ = M c, (8) ρ ρ = c: = = = = = Ω M N Ω M N = = = c, (9) whee M s the mola mass of -th component and Total and patal mola volumes obey the followng elatons: Ω denotes ts patal mola volume. Ω dω N N N = dn = Ω dn (,,..., ), () = N = N Ωc * Ω ( N, N,..., N) = Ω dn =, at a moment t [, t], () c = = whee N Ω dn =Ω N. When mola volumes do not depend on concentaton and stesses ae neglgble, the elaton known as Vegad law follows fom Eq. () []: Ω Ωc = = Ω = N c = Ω = Ωc = () In ths wok we do consde noncompessble pocesses only. Consequently we assume that patal mola volumes ae constant, Ω = const. The two dffeent foms of the equaton of state can be used:

8 - the classcal aken postulate of constant mola concentaton c : = c = const and - the Vegad law ( tx) Ω, : = NΩ =, Eq. (). c = =.3 ffuson flux n multcomponent systems. The aken flux, J, s computed elatvely to the volume fxed efeence fame: J cυ = (3) whee: υ s aken velocty that s common fo evey component. The aken velocty, υ, s geneated dung ntedffuson due to locally unbalanced dffuson fluxes [6, 7]. The dffeent patal mola volumes mply: υ ( tx, ) = c Ω d υ. (4) = cω ependng on the avalable data, the dffuson flux can be expessed by the dffeent consttutve equatons. When dffeent patal mola volumes ae consdeed, Ω Ωj, j the dffuson fluxes that ae defned n the volume fxed efeence fame and contan the coecton facto, f Ω = [6]. The followng consttutve equatons ae Ω allowed n the pesented method, Fgue : a) The aken-onsage flux, Ω J = L X + J, =,..., (5) Ω j j j= b) The Onsage flux: Ω J = L X, =,..., (6) Ω j j j=

9 c) The aken-nenst-planck flux : Ω J = cb X + J, =,...,. (7) Ω d) The Nenst-Planck dffuson flux [8,9]: Ω J = cb X, =,..., Ω (8) e) The Fck-aken flux : J = gad c + J, =,..., (9) f) The Fck flux, Eq. (9), wthout aken velocty and when mola volumes ae equal becomes st Fck law: J = gad c, =,..., ().4 Composton dependent dffusvtes. The method allows to ntoduce constant as well as composton dependent dffusvtes fo all of the components, e.g., as analytcal functons. Othe possblty s to appoxmate the composton dependent dffusvtes fom the known self dffuson coeffcents fo each component: * Q = exp, =,... RT () Equaton (7) follows fom (3) when: L =, fo j and L cb, =,...,. j Equaton (9) follows fom Eq. (7). By assumng ) the dealty sweepng statement and ) the equal patal mola volumes of components and Nenst- Ensten elaton between self dffuson coeffcent and moblty, = BkT. One X = gad + RT ln c RT = gadc c, whee: µ s standad chemcal gets: ( µ ) potental, R and T denote the gas constant and tempeatue.

10 whee: and Q denote the pe-exponental constant and actvaton enegy fo dffuson. Followng ICTR [], n the spt of the Calphad appoach, the composton dependency of these two factos, s epesented wth a lnea combnaton of the values at each endpont of the composton space. Usng the Redlch-Kste expanson and Eq. () one gets: * Φ = RTln = RTln Q, =,... () whee: Φ can be ntepeted as fee enegy of dffuson of the -th component. The ntnsc dffusvtes n -component mxtue (sold soluton) ae computed usng the followng elaton []: whee: value of Φ = jφ + j k Φ j k = j= j= k> j p= m j p j, k p N N N ( N N ),,... (3) Φ can be ntepeted as fee enegy of ntnsc dffuson n the alloy, Φ fo pue j-th component and p j, k Φ ae bnay nteacton paametes. Thus fom the Eq. (), the ntnsc dffusvtes n the alloy ae gven by: j Φ s a alloy Φ = = exp =,.... (4) RT.5. Bounday condtons The dffuson pocesses can be smulated usng dffeent bounday condtons. The softwae can smulate both the open and closed systems. ) In closed system all fluxes at all boundaes equal zeo, ( ) ( ) λ ( ) J t, = J t, t =, =,... (5) R

11 ) The bounday condton n a case of ntedffuson n open systems (system exchangng mass wth a suoundng, Fg. ) states that the flux of the abtay element though nteface equals ts dffuson flux n alloy at nteface [], e.g., fo the ght bounday t s: (, λ ) (, λ ) J t = J t, (6) d R R whee flux though nteface, (, ) J t λ, s a known o computed functon of tme. R N ( gas) alloy X N (, x ) λ R () N alloy X X ( gas) d J ( t, x) a ox c b N (, ) t x ch J ( t) X ( t) λ R () t Fg.. The schema of the ntedffuson foced by the chemcal eacton at alloy oxde nteface.

12 In ths wok we consde the fluxes at the boundaes, J t, () t J t [3]: ch that ae a esult of chemcal eacton, ( ) ch (, ()) () R ( λ R ) and (,) J t, J t λ t = J t. (7) The eacton flux can be ntoduced by: a) the analytcal, known functon to smulate the abtay pocess, e.g., a suface teatment of the alloy. The use has a possblty to ntoduce dffeent fluxes fo evey component. b) n a case of oxdaton the fluxes can be ntoduced based on the Wagne method [3]. Wagne was fst to notce that the oxdzed component entes the oxde phase as a esult of the suface eacton and of the dffuson though alloy oxde nteface. The noneactng element dffuse nto the nteo of the mult-component alloy [3]. By analogy, ch the eacton flux of, () J t, s due to the suface eacton, ( λ ) flux of though the alloy oxde nteface, (, ) J t λ, Fg.. R c t, dλ dt, and R R ch dλr J () t = c( t, λr) + J( t, λr) (8) dt The total (volume) balance of dung eacton of the mult-component alloy showng the constant mola concentaton eques: ch dλr J () t = c (9) dt Fom Eqs. (8) and (9) the flux of n the alloy at alloy oxde nteface s gven by (, ) ( (, ) (, )) d λr J t λr = c t λr c t λr. (3) dt

13 bove equaton s dentcal wth bounday condton n the case of bnay N-Pt alloy analyzed by Wagne [3]. In a case of mult-component alloy t s necessay to compute nwad fluxes of the all non-eactng components. These dffuson fluxes ae a esult of the alloy consumpton: dλ R dt. Consequently: (, ) (, ) d λr J t λr = c t λr fo =,,..., (3) dt The fluxes n the alloy as gven by Eqs. (3) and (3), ae computed only when both dffuson and convecton ae consdeed, Fg.. The schema of the ntedffuson foced by the chemcal eacton at alloy oxde nteface s shown n Fg.. In a case of low ox nonstochomety of the gowng phase, c const., the flux of the eactng component depends on tme only (does not depend on poston) [3]. Ths flux can be expessed by the Nenst-Planck fomula [8,9] ox ch ox ox () = J t c B µ x (3) The Nenst-Ensten elaton, = BkT [], allows wtng the eacton flux n the fom: J c RT µ x ox ox ox ch () t = (33) The balance of at the oxde gas nteface (Stefan condton) eques: ch ox dx J () t = c. (34) dt By compason of the Eqs. (3) and (34), than applyng the mean value theoem, one can obtan the ate gowth equaton, whee the eacton ate s a functon of tme [3]: () () µ () t dx k t ox ox = whee k() t = dµ dt X t RT (35) µ

14 whee µ () t = µ ( alloy oxde) and µ µ ( oxde O ) = ae the local chemcal potentals of at ntefaces, Fg.. Upon assumng the local themodynamcal equlbum at the alloy oxde nteface, ( alloy) ( t, ( t) ) ( t, ( t) ) ( oxde)( t) µ λ = µ λ = µ, (36) the nstantaneous ate constant n Eq. (35) can be expessed as a functon of concentaton n the alloy, at the alloy oxde nteface: () k t µ ( t, λ ) R ox ox = dµ RT. (37) µ Equatons (34) and (35) allow computng the flux of n the oxde as a functon of the nstantaneous ate constant (chemcal potental of at nteface): ( ) () ch ox k t J () t = c = X t ox t ( ) c k t k ( τ ), (38) dτ whee () t ( τ ) X t = k dτ. The ntedffuson n the alloy and the eacton dffuson n the oxde ae coupled by the flux of eactng component though the nteface, Fg.. The total flux of eactng component, Eq. (38), though the alloy a X b nteface s a esult of eacton (alloy consumpton), Eq. (3). Combnng Eqs. (9), (3) and (38), one gets the equaton descbng the flux of oxdzed component though the nteface: () ( ) µ ( t, λ ) ox R c k t ox ox J( t, λ R() t ) = whee k() t = t d RT µ. (39) k τ dτ µ

15 .6. Intal condtons ) Intal dstbutons of the mxtue components must be known: ( ) ( ) c, x = c x fo =,,..., (4) The ntal dstbutons of the components can be ntoduced as a step functon (Heavsde functon) o any pofle gven by the set of ponts. Thus, one can analyze pocesses n gadent mateals, mult-layes, multples mateals and many othes. ) Intal poston of the ght bounday of the mxtue, ( ) the mxtue/alloy). 3) Intal thckness of the poduct laye, ( ) X = X λ R = λ (the thckness of R 3. Examples 3.. The eacton-dffuson, pocesses at dffeent tme scales Fom the techncal pont of vew, vey nteestng ae the heteogeneous eactons, e.g., the oxdaton of alloys, some CV pocesses, etc. In such pocesses thee dffeent phases ae consdeed. Fgue shows the substate (the mult-component alloy), eactng element n gas o lqud phase (oxdant X) and sold poduct laye ( a X b ). The espectve ntefaces ae plana and local equlbum s assumed. Reacton-dffuson, non statonay Wagne model. Heteogeneous eactons n many cases esult n the fomaton of the compound showng naow homogenety ange,.e., showng the low nonstochomety, a-y X b, whee y. In ths wok we analyse gowth of the bnay compound shown on Fg.. The -component of the alloy eacts wth, X-component n the gas atmosphee, X (gas). The slowest, ate contollng pocess s dffuson of and/o X n the gowng poduct laye of fnte thckness, X(t), Fg.. The compound fomaton occus accodng to eacton:

16 ( ) a (alloy) + b X gas = X. (4) a b Let us assume that the alloy s an deal sold soluton, µ = µ + RTln c. The local equlbum at () t λ R, Eq. (36), mples µ ( )( ) ln (, ( ) ) oxde t µ RT c t λ t = +. In such a case the nstantaneous ate constant n Eq. (39) becomes: o () c () t ox k t = dln c (4) () c N () t ox k t = dln N (43) N whee: N () t N( t, λr() t ) = s a mola ato of the component n the alloy at the alloy oxde nteface. Followng Wagne, the mola ato of the oxdzed component n the oxde can be expessed assumng the local equlbum wthn the scale. Consequently, the oxdzed component mole facton at the alloy oxde nteface s gven by [3]: ( ()) () 4/z e O O ( ) N t p t = p (44) ( e) whee: p () t and p denote the oxygen patal pessues at the alloy oxde and (pue O O metal ) oxde ntefaces espectvely, z s the valence of oxdzed component n oxde. Combnng Eqs. (43) and (44): po ox ln po () t z k() t = d po 4 (45) whee p O () t = po ( alloy oxde) and po po ( oxde O ) ntefaces of the gowng laye. = ae the oxygen pessues at

17 It was shown by authos, that pesented method gves bette appoxmaton of the expemental data than the Wagne quas-statonay soluton []. Reacton-dffuson, abtay tenay alloy. In ths wok we show the nfluence of the dffeent tme scales, dffeent dffuson coeffcents n alloy and oxde. The data used to smulate the oxdaton of the abtay tenay alloy ae shown n Table. We assume that t s an deal sold-solutons and that ntnsc dffusvtes n an alloy and oxde dffe (ntedffuson coeffcent n alloy depends on composton). Table. ata used to smulate the oxdaton of the -B-C alloy: N ()=.6, N B ()=., N C ()=.3 and the ntal thckness of oxdzed alloy equals 5 µm. ox() cm s cm s c ox mol 3 cm p ( e) [ atm] p [ atm] O O k C cm s =. 3 5 a) 5. B = C = 5. 5 =. 5 5 b) 5. B = C = 5. 3 Fgue 3 shows the esults obtaned fo an abtay tenay alloy, -B-C. It can be seen that when ntnsc dffusvtes n the alloy ae hghe than n the gowng laye, Fg. 3a, then the oxdaton pocess can be appoxmated by the paabolc ate law,.e., the slow dffuson n the eacton poduct s ate contollng step. One may also note elatvely hgh concentaton of eactng element n the alloy at the alloy oxde nteface. The

18 low ntnsc dffusvtes n the alloy makedly decease the eacton ate, Fg. 3b. The concentaton of eactng metal n alloy at alloy oxde nteface afte the long eacton tme deceases to the vey low values and oxdaton pocess s non-paabolc one. a) b) Fg. 3. The gowth ate of the O scale on tenay alloy and concentatons of components as a functon of dstance fom alloy/oxde nteface afte hous. The last example s the eactve-ntedffuson s agan selectve oxdaton of the abtay tenay gadent mateal (alloy). The ntal concentaton of components

19 shown n Fg. 4 s typcal fo the alloy coatng system. The data used to smulate the ntedffuson and eacton of the abtay tenay alloy ae shown n Table. Table. Tanspot popetes of the tenay alloy and oxde, X. ox() cm s cm s c ox mol 3 cm p ( e) [ atm] p [ atm] O O k C cm s = B = C = 5. 3 The ntal thckness of the alloy equals µm, oxdaton tme equals h. Fg. 4. The thckness of the O scale gowng on -B-C tenay alloy/coatng system afte hous; the concentaton of components as a functon of dstance fom the coatng/oxde nteface. It can be seen that lve tme of the mateal exceeds hous and pocess follows the paabolc ate law.

20 3.. Intedffuson In ths secton we compae the expemental esults [3] wth the computed esults usng the aken-nenst-planck flux fomulae, Eq. (7). The Fe-Cu-N system was chosen because t s a sngle phase n a wde ange of compostons and the extensve expemental data ae avalable. The followng data wee used: - the themodynamc popetes of Fe-Cu-N alloys at T = 73 [K] [4]; - the expementally measued tace dffusvtes n Cu-Fe-N alloys [5]; - the composton of dffuson couple: 4.9NCu-58.N Fe [wt.%], ts thckness: [mm] ±. and annealng tme: 7 [h] [3]; - the patal mola volumes of components, Ω, wee estmated fom the densty of the pue metals and elaton: Ω = M ρ T. The densty of alloy, T ρ, was estmated usng T T the expesson: ρ ρ m ( ) = k T T m whee T m s an alloy meltng tempeatue, T m ρ ts densty and k the coeffcent of themal expanson [6]. The patal mola volumes equal: Ω Cu = 7.9, Ω Fe = 7.66, Ω N = 7. [cm 3 mol - ]. Fgue 5 shows the compason of the expemental data [3] wth computed esults. Fgue 6 shows the esults obtaned wth the use the composton dependent dffusvtes (secton.4) and the aveage ntnsc dffuson coeffcents:. -9 Cu =,.54 - Fe = and = cm s [5]. N /

21 Ths wok Fg. 5. Compason of the expemental and calculated dstbuton of elements n the 4.9NCu-58.N Fe [wt.%] dffuson couple. Ths wok, composton ndependent ntnsc dffusvtes Ths wok, composton dependent ntnsc dffusvtes Fg. 6. The compason of dffeent appoxmaton of dffuson coeffcent

22 It s evdent that the appoxmaton gve satsfactoy ageement wth expemental esults n case of smulatng the Cu-Fe-N dffuson couple 58.N-4.9NCu 49.6N - 5.4Fe [wt. %] [3]. 4. Summay and conclusons We pesented the model of the eacton dffuson n the mult-component, twophase system. The smulaton of ntedffuson n tenay alloy and the selectve oxdaton of the bnay/tenay compound valdated model. The smulaton of ntedffuson n the Fe-N-Cu alloy show satsfactoy ageement wth expemental data. Moeove, t confms the effectveness of the appoxmaton method used to evaluate the ntnsc dffusvtes as a functon of concentaton. The model/method and the softwae can be extended to compute the moe complex pocesses. cknowledgement The authos wsh to thanks R. Flpek fo povdng themodynamcal data. Ths wok has been suppoted by the Mnsty of Scence and Hghe Educaton of Poland poject COST/47/6. Refeences. [] C. Wagne, J. Electochem. Soc. 3 (956) 57 [] C. Wagne, J. Electochem. Soc. 3 (956) 67 [3] C. Wagne, J. Electochem. Soc. 99 (95) 369 [4] C. Wagne, Z. Elektochem. 63 (959) 77

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24 [5] S.V. vnsk, F. Hske, C. Hezg, R. Flpek and M. anelewsk: Self- and Intedffuson n Tenay Cu-Fe-N lloys, efect and ffuson Foum, 5, 37-4, pp [6].R. Lde, CRC Handbook of Chemsty and Physcs, 79th Edton,