DELFT UNIVERSITY OF TECHNOLOGY

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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT - A mathematcal model for the dssoluton of stochometrc partcles n mult-component alloys F.J. Vermolen, C. Vuk and S. van der Zwaag ISSN Reports of the Department of Appled Mathematcal Analyss Delft 2

2 Copyrght 2 by Department of Appled Mathematcal Analyss, Delft, The Netherlands. No part of the Journal may be reproduced, stored n a retreval system, or transmtted, n any form or by any means, electronc, mechancal, photocopyng, recordng, or otherwse, wthout the pror wrtten permsson from Department of Appled Mathematcal Analyss, Delft Unversty of Technology, The Netherlands.

3 A mathematcal model for the dssoluton of stochometrcles n mult-component alloys Fred Vermolen, Kees Vuk and Sybrand van der Zwaag January 4, 2 Abstract A general model for the dssoluton of stochometrcles n mult-component alloys s proposed and analysed. We ntroduce the concept of mass-conservng solutons and gve a self-smlar soluton for the resultng Stefan-problem. Furthermore, we show that partcle dssoluton n mult-component alloys can under certan crcumstances be approxmated by a model for partcle dssoluton n bnary alloys. Subsequently, we propose a numercal method to solve the coupled dssoluton problem. We end wth some examples of hypothetcal applcatons from metallurgy. Keywords: mult-component alloy, partcle dssoluton, dffuson, vector-valued Stefan problem, self-smlar soluton Introducton In the thermal processng of both ferrous and non-ferrous alloys, homogensaton of the exstng mcrostructure by annealng at such a hgh temperature that unwanted precptates are fully dssolved, s requred to obtan a mcrostructure suted to undergo heavy plastc deformaton as an optmal startng condton for a subsequent precptaton hardenng treatment. Such a homogensaton treatment, to name just a few examples, s appled n hot-rollng of Al klled constructon steels, HSLA steels, all engneerng steels, as well as alumnum extruson alloys. Although precptate dssoluton s not the only metallurgcal process takng place, t s often the most crtcal of the occurrng processes. The mnmum temperature at whch the annealng should take place can be determned from thermodynamc analyss of the phases present. The mnmum annealng tme at ths temperature, however, s not a constant but depends on partcle sze, partcle geometry, partcle concentraton, overall composton etc. Faculty of Techncal Mathematcs and Informatcs, Delft Unversty of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands, e-mal: F.J.Vermolen@math.tudelft.nl, C.Vuk@math.tudelft.nl Laboratory for Materals Scence, Delft Unversty of Technology, Rotterdamse weg 37, 2628 AL, Delft, The Netherlands, emal: S.vanderZwaag@tnw.tudelft.nl

4 2 Due to the scentfc and ndustral relevance of beng able to predct the knetcs of partcle dssoluton, many models of varous complexty [9,, 7, 3, 2, 7, 6, 2, 9,, 2, 2, 8, 28, 4, 5, 8] have been presented and expermentally valdated. In recent years the smpler models coverng bnary and ternary alloys have been extended to cover multcomponent partcles [24, 26, 25]. These advanced models cover a range of physcal assumptons concernng the dssoluton condtons and the ntal mcrostructure. Furthermore, mathematcal mplcatons (such as a possble bfurcaton of the soluton, monotoncty of the soluton and well-posedness) are addressed and mathematcally sound extensons to the case of n compound partcles, wth proven theorems concernng exstence of massconcernng solutons and soluton bounds, have been derved. The current paper does not am at beng mathematcally rgorous but merely ams at beng descrptve about the mplcatons of the developed mathematcs of these more complex models. Frst we formulate the model for partcle dssoluton n mult-component alloys. Subsequently, we gve asymptotutons for both the planar and sphercal partcle. Ths asymptotuton s used to verfy numercal computatons. Furthermore, we show that the mult-component problem (a vector-valued Stefan problem) can be approxmated by a bnary problem ( scalar Stefan problem) under certan crcumstances. Next, we gve a numercal scheme to solve the mathematcal problem for more general cases. Subsequently, some test-cases are shown usng some hypothetcal expermental data. We end up wth a dscusson and some conclusons. 2 Basc assumptons n the model We consder a partcle of a mult-component β phase surrounded by a matrx of phase α, of unform or non-unform composton. The boundary between the β-partcle and α-matrx s referred to as the nterface. The metal s dvded nto cells n whch a partcle of phase β dssolves n an α-matrx. The model s based on the concept of local equlbrum,.e. the nterfacal concentratons are those predcted by thermodynamcs. In [25] we consdered the dssoluton of a stochometrcle n a ternary alloy. The hyperbolc relatonshp between the nterfacal concentratons for ternary alloys s derved usng a three-dmensonal Gbbs-space. For the case that the partcle conssts of n chemcal elements apart from the atoms that form the bulk of the β-phase, a generalsaton to a n-dmensonal Gbbs hyperspace has to be made. The Gbbs-surfaces become hypersurfaces. We expect that smlar consequences follow and that hence the hyperbolc relaton between the nterfacal concentratons remans vald for the general stochometrcle n a mult-component alloy. We denote the chemcal speces by Sp, {,..., n + }. We denote the stochometry of the partcle by (Sp ) m (Sp 2 ) m2 (Sp 3 ) m3 (...)(Sp n ) mn. The numbers m, m 2,... are stochometrc constants. We denote the nterfacal concentraton of speces by and we use the followng hyperbolc relatonshp for the nterfacal concentratons: ( )m ( 2 )m 2 (...)( n )mn = K = K(T ). ()

5 3 The factor K s referred to as the solublty product. It depends on temperature T accordng to an Arrhenus relatonshp. We denote the poston of the movng nterface between the β partcle and α phase by S(t). Consder a one-dmensonal doman,.e. there s one spatal varable, whch extends from up to M. The spatal co-ordnate s denoted by r, S(t) r M. Ths doman s referred to as Ω(t) := {r R : S(t) r M}. The α-matrx where dffuson takes place s gven by Ω(t) and the β-partcle s represented by the doman r < S(t). Hence for each alloyng element, we have for r Ω(t) and t > (where t denotes tme) c t = D r a r { r a c r }, for {,..., n}. (2) Here D and c respectvely denote the dffuson coeffcent and the concentraton of the speces n the α-rch phase. We neglect the cross-dffuson coeffcents. Hence above equaton s a smplfcaton of the more general mult-component dffuson equaton as stated by Krkaldy and Young []. The geometry s planar, cylndrcal and sphercal for respectvely a =, and 2. Let c denote the ntal concentraton of each element n the α phase,.e. we take as ntal condtons (IC) for r Ω() c (r, ) = c (r) for {,..., n} (IC) S() = S. We omt the more general case n [24] where we consder the possblty of two smultaneously dssolvng / growng partcles. At a boundary not beng an nterface,.e. at M or when S(t) =, we assume no flux through t,.e. c =, for {,..., n}. (3) r Furthermore at the movng nterface S(t) we have the Drchlet boundary condton for each alloyng element. The concentraton of element n the partcle s denoted by, ths concentraton s fxed at all stages. Ths assumpton follows from the constrant that the stochometry of the partcle s mantaned durng dssoluton n lne wth Reso et al [5]. The dssoluton rate (nterfacal velocty) s obtaned from a mass-balance. Summarsed, we obtan at the nterface for t > and, j {,..., n}: c (S(t), t) = ds dt = D c (S(t), t) r D c D (S(t), t) = r j j j c j (S(t), t). (4) r Above formulated problem falls wthn the class of Stefan-problems,.e. dffuson wth a movng boundary. Snce we consder smultaneous dffuson of several chemcal elements, t s referred to as a vector-valued Stefan problem. The unknowns n above equatons are the concentratons c, nterfacal concentratons and the nterfacal poston S(t). For a mathematcal overvew of Stefan problems we refer to the textbooks of Crank [5], Chadam and Rasmussen [3] and Vsntn [27].

6 4 3 Analyss of the model In ths secton we consder some general mathematcal propertes of the dssoluton model. For the dffuson equaton wth approprate boundary condtons there exsts exactly one soluton c that s contnuous at least up to the frst and second dervatve wth respect to respectvely tme t and poston r. Protter and Wenberger [4] prove that these smooth solutons satsfy a maxmum prncple,.e. the global extremes of c occur ether at the boundares (r = S(t), r = M) or at t =. 3. Mass conservng solutons We requre that the total mass of all chemcal elements s constant n the whole dssoluton cell,.e. over r M. Further, let c be constant over Ω(), then M c (r, t)r a dr = S a+ a + + c M a+ S a+. a + Subtracton of M c ra dr = c M a+ a + from both sdes of above equaton gves M (c (r, t) c )r a dr = ( c ) Sa+ a +. (5) All solutons of the Stefan-problem have to satsfy ths condton. We use an ntutve argument to show that some Stefan-problems do not have solutons that satsfy massconservaton and hence are ll-posed. A mathematcal theorem s rgorously proven n [24]. Suppose that c < <,.e. the nterfacal concentraton exceeds the ntal concentraton (see Fgure ). From t = the nterfacal concentraton can ncrease (buld up) only due to transport of atoms from the partcle to the nterface and matrx (snce concentraton gradents and reactons are absent ntally). Ths mples that the total number of atoms of the alloyng elements n the partcle must decrease. On the other hand from the maxmum prncple of the dffuson equaton follows that c (S(t), t) <. Hence, the total number of atoms of the alloyng element n the matrx r ncreases. Furthermore, we have <, whch mples ds >, hence the total dt number of atoms of the alloyng elements n the partcle ncreases. Ths gves a contradcton. Both the nterfacal movement due to growth and the ncrease of the total number of atoms of the alloyng element are sketched n Fgure. Mass can not be conserved for ths case. Smlar arguments can be used to show that the other case < < c also volates mass-conservaton (see Fgure 2). Ths statement can be generalsed n the followng result:

7 5 t = t > part c part c c c S() (I) S(t) (II) Fgure : The hypothetcal case c < < whch gves growth of the α-phase and volaton of the mass-balance. Left (I) shows the ntal stuaton and rght (II) shows a stuaton at some tme t >. t = t > c c S() (I) S(t) (II) Fgure 2: The hypothetcal case < < c whch gves growth of the α-phase and volaton of the mass-balance. Left (I) shows the ntal stuaton and rght (II) shows a stuaton at some tme t >.

8 6 Theorem: Let all concentratons be non-negatve, then the followng combnatons gve non-conservng solutons n the sense of equaton (5): c < cpart < < c, < (see Fgures and 2 for both cases). Ths result s used to reject possble (numercal) unphyscal solutons that result from the vector-valued Stefan problem. 3.2 An asymptotuton for the planar case Here we consder the case of an α partcle dssolvng n an unbounded doman,.e. M = and Ω(t) s unbounded at the rght sde. Furthermore, the doman s planar,.e. a = n equaton (2). The nterfacal concentratons satsfy equaton (). For completeness we start wth the dervaton of the self-smlar soluton for the one-component problem. As far as we know, Weber was the frst to derve such soluton for the freezng problem [29] The one-component problem Suppose that the nterface concentraton of a certan component s known, say c(s(t), t) = for a some component. Then, we have to solve the followng problem (we refer to ths problem as (P)): c t = D 2 c r 2 (P ) ds dt = D c (S(t), t) r c(s(t), t) = c(r, ) = c = c(, t), S() = S. Here we omt the subscrpt. As n [24] we search a self-smlar soluton for the functon c = c(r, t) and for S = S(t) we state a square-root behavour as a functon of tme. Tral of c = c( r S 2 ) shows that these expressons satsfy the dfferental equatons n (P). Dt Settng η := r S 2 gves the followng dfferental equaton for c = c(η) wth general Dt soluton ηc = c c = c(η) = A erfc(η) + B. The complementary error functon s defned as erfc(x) := erf(x) = 2 π dy. Tral x e y2 of S = S + k t, substtuton of c(s(t), t) = nto above soluton and use of the ntal

9 7 condton gves k A erfc( 2 D ) + B = csol lm x erfc(x) = B = c. Solvng for A and B gves the Neumann-soluton as derved n [23] ( ) c(r, t) = c r S erfc( k 2 )erfc 2 + c Dt t Above soluton and S = S + k t are substtuted nto equaton (4) to gve the followng expresson for the constant k: c D c sol π e k2 4D erfc( k 2 ) = k 2 D (6) Above equaton s solved for k usng a standard zero-pont teraton method. The physcal model n whch the solubltes are coupled hyperbolcally, see equaton (), s vald only for the dlute regme []. Therefore, n most cases we have c. Hence, n general we c have. From above equaton t can be shown that k s then approxmated csol by (we refer to [23] for more detal) k = 2 csol c D c sol π. (7) Ths gves the classcal formula for the nterfacal velocty that has been obtaned by Aaron and Kotler [] usng Laplace transforms ds dt = csol c D c sol π t. (8) We remark that the dervaton of above equaton usng the Laplace tranasform contans the assumpton that the nterface moves very slowly compared to the rate of dffuson,.e. t s a so-called frozen profle approach. For most cases of partcle dssoluton n sold metals and alloys, the frozen profle approach s a reasonable approxmaton. Therefore above approxmaton s used n the present paper for the extenson to the mult-component problem The mult-component problem As a tral soluton for the planar case n a sem-unbounded regon, we take the nterfacal concentratons to be constant (these concentratons are not constant n tme for other

10 8 cases). Equaton (4) has to be fullflled, hence combned wth equaton (6) one obtans the followng system of non-lnear equatons to be solved for k and for {,..., n}: c sol c D π e k 2 4D k erfc( 2 D ) = k for {,..., n}, 2 (9) ( )m ( 2 )m 2 ( 3 )m 3 (...) = K. Due to the non-lnear nature of above equatons, the soluton s n general not unque. Above set of equatons provdes an exact soluton for the vector valued Stefan problem n the parameters k and, csol 2,...csol n. Due to ts complexty we rely on numercal soluton technques to obtan ts soluton. Ths soluton s referred to as the Neumann soluton for the planar case. Instead of equaton (6) we use equaton (8) as an approxmaton for c the case that. Ths gves the followng set of equatons to be solved n k,, csol 2,..., csol n : k = 2 c D ( )m ( 2 )m 2 ( 3 )m 3 (...) = K. π for {,..., n}, c Note that above equaton s accurate whenever. For cases where ths nequalty does not hold, then the above set of equatons should be replaced by system (9). To llustrate the fact that more solutons can occur, we consder a hypothetcal ternary alloy wth m = = m 2, = 5, 2 =, c = 2 and c 2 = 3 and D 2 = 2D, then after some calculaton one obtans two solutons for the nterfacal concentratons and rate factor k { c sol = , = , 2 = , k = slow soluton 2 = , k = fast soluton. Both solutons conserve mass and hence are well-posed. Note that the fast soluton does c not satsfy and hence the use of above equatons gves an naccurate value for the fast soluton. We remark here that ths has only been gven for llustratonal purposes. Real-world alloys do not fall nto ths class snce the model s only vald n the dlute soluton regme,.e. c, {,..., n} for t >, r Ω(t). For the nterested reader, we refer to [23] where several computatons have been done to determne the two solutons for a ternary alloy. The same computatons for the exact Neumann soluton are done n [23] n whch the set of equatons (9) s used nstead of system (). Furthermore, n [24] t has been shown that the nterfacal velocty has the followng upper- and lower ()

11 9 bound: c csol D πt < ds dt < c csol c D πt. () Ths gves two easy bounds for the soluton of the nterfacal concentratons and hence the dssoluton rate can be estmated very quckly. The dlute case We consder the case that the partcle concentraton s much larger than the nterface concentraton. Furthermore, we assume that the ntal concentraton s almost equal to zero,.e. c. From the upper and lower bounds n above expresson, t follows that the nterface velocty can be approxmated by ds dt = csol D πt for {,..., n}. (2) Snce ths has to hold for all {,..., n} t follows that all nterfacal concentratons can be expressed n terms of, for nstance, the nterfacal concentraton correspondng to the frst element,.e. csol D = csol D = cpart D D We substtute all these expressons for nto the hyperbolc relaton for the nterfacal concentratons (equaton ()) to obtan a smple exponental equaton for whose nonnegatve real-valued soluton gves ( )µ ( 2 D D 2 ) m2 ( 4 D D 4 ) m4 ( n... D D n ) mn = K = cpart D [Π n = ( D ) m ] µ K ( R + ). where Π n =f := f f 2...f n and µ := m + m m n. Note agan that we consder only non-negatve and real-valued concentratons. The soluton for s substtuted nto equaton (2) to obtan the nterface velocty: ds dt = csol eff Deff πt eff wth eff := K µ, c part eff := [ ] Π n = (cpart ) m µ, D eff := [Π n = (D ) m ] µ. (3)

12 We see that for ths case partcle dssoluton n a mult-component alloy s mathematcally reduced to partcle dssoluton n a bnary alloy. The effectve parameters (partcle concentraton and dffuson coeffcent) are equal to geometrc averages wth weghts accordng to stochometry. Above dfferental equaton s solved to gve the followng dssoluton tme τ τ = π(cpart eff )2 S 2. 4( eff )2 D eff We consder an example wth three components: Let the partcle concentratons of speces and 2 be equal, = 33wt.% = 2, for the frst two speces. The partcle concentraton of the thrd component, 3, s allowed to vary. When the stochometry s unchanged for all confguratons, then the varaton of the partcle concentraton reflects the molecular weght of the thrd component. Further, let the ntal concentraton n the matrx be zero for all alloyng elements,.e. c =. The solublty product s chosen equal to one,.e. K =. We start wth a layer of thckness S = 6 m. Ths data-set gves eff =. We start wth dffusvtes D = 3 m 2 /s and D 2 = 2 3 m 2 /s. The thrd dffuson coeffcent s allowed to vary to study ts mpact on the dssoluton tme. As can be seen from substtuton nto above relaton, the dssoluton tme τ (s) vares wth the dffuson coeffcent of the thrd component accordng to a recprocal power of one thrd,.e. τ (D 3 ) /3. Fgure 3 shows the varaton of the dssoluton tme wth the value of the thrd dffuson coeffcent for relatvely low values. Both the approxmate soluton, based on equaton (2), and the exact Neumann soluton of system (9), are plotted n Fgure 3 for consecutve partcle concentratons. It can be seen that the dfference s small. From Fgure 3 t can be seen that the dssoluton tme s hghly senstve to changes of the dffuson coeffcent of the thrd component, D 3, when D 3 s small. A small value of D 3 corresponds to the addton of a slowly dffusng thrd component. Hence dssoluton tmes are long when a slowly dffusng thrd component s added. Furthermore, t can be seen from Fgure 3 that the dssoluton tme ncreases for ncreasng partcle concentraton of the thrd component. Fgure 4 shows a smlar pcture n the top-left as n Fgure 3, however, the dffuson coeffcent of the thrd component s vared over a larger range and the partcle concentraton of the thrd component s set equal to 3,.e. 3 = 3wt.%. In Fgure 4, we see that the nterfacal concentraton of the frst component ncreases for ncreasng D 3. Hence, the dfference decreases and therewth csol c ncreases and hence csol c c c s no longer true. However, the nterfacal concentraton of the second component decreases and hence csol c decreases. Ths mples that these two effects work aganst c each other, and ths supports the small dfference between the exact Neumann soluton and approxmate soluton (equaton ()), also for hgh values of D 3. Note, however, that the dfference n Fgure 4 s more sgnfcant than n Fgure 3. For completeness, we also gve the evoluton of the the velocty coeffcent, k, as a functon of the dffuson coeffcent of the thrd component.

13 When we ncrease the dffuson coeffcent D 3 suffcenty, then, as predcted by equaton () exceeds the value of. Therewth, one enters the regon of ll-posedness (mass s no longer conserved). Of course equaton () cannot be used for ths case. So ths behavour of τ (D 3 ) /3 breaks down for large D 3. Ths break-down takes place when becomes sgnfcant wth respect to. Equaton () s used as an ntal guess for the exact Neumann soluton (9), obtaned from numercal soluton of ths system. We also observed that when, as determned from approxmaton () and used as an ntal guess for the soluton of equaton (6), exceeds no convergence s obtaned when equaton (6) s solved numercally. It s shown n [23] that also no Neumann solutons exst n ths range. In Fgure 3 and 4 we see that the dssoluton tme as predcted by the Neumann soluton s smaller than for the quas-bnary approach. Ths s explaned as follows: consder equaton (6), we see that convergence to equaton (7) takes place as D 3. Snce c = and c t follows that = < csol. Use of equaton (8) shows that the Neumann soluton tme s smaller than the quas-bnary dssoluton tme for D 3 suffcently large. Hence τ as D 3. We defne the relatve error made by the use of the quas-bnary approach as ε := τ N τ qb τ N, where τ N and τ qb respectvely correspond to the dssoluton tme as predcted by the use of the Neumann soluton and the quas-bnary approach. We plot the relatve error made by the use of the quas-bnary approach as a functon of the dffuson coeffcent for dfferent partcle concentratons n Fgure 5 on a double logarthmc scale. It can be seen that the relatve error ncreases monotoncally wth D 3. Furthermore, the relatve error ncreases as the partcle concentraton of the thrd component decreases. Ths s explaned by the ncrease of the sgnfcance of the nterfacal concentraton. The slope of all lnes for consecutve partcle concentratons s the same. Ths suggests an approxmate power behavour for the relatve error ε (D 3 ).42. When the assumpton c s relaxed, both expressons n equaton () can be combned to get a polynomal equaton of order µ = m + m m n n. A numercal zero-pont method can be used to get the soluton. Ths more general case s omtted here, snce we are not able to fnd general expressons for the zeros of the resultng polynomal. 3.3 An asymptotuton for the sphercal case The dffuson equaton for the sphercal case n an unbounded doman admts a smlar self-smlar soluton, n terms of c = c(r/ t), as for the planar case. However, one obtans ncompatblty wth the nterface rate equaton. Therefore, we are not able to fnd a soluton of the type of the prevous secton. Usng Laplace transforms, Whelan [3] came

14 = 33 3 Dssoluton tme τ (s) 5 = = Thrd dffuson coeffcent D 3 Fgure 3: The dssoluton tme τ as a functon of the dffuson coeffcent of the thrd element D 3 for consecutve values of the partcle concentraton of the thrd element 3. The dffuson coeffcent and tme are respectvey gven n the unts µm 2 /s and s. The sold and dotted curves respectvely correspond to the approxmate soluton and the exact soluton. up wth the followng expresson for the nterface velocty { D S + ds dt = csol c D πt }. (4) The nterfacal poston S(t) has been treated as a constant durng Whelan s dervaton. At early stages t behaves lke Aaron and Kotler s [] soluton,.e. the second (planar) term domnates. At later stages as the second term decreases, the frst term becomes more mportant durng dssoluton. Therefore, at some stage, say t < t < t 2, we approxmate the nterfacal velocty by ds dt = csol c D S. (5) Above equaton s also obtaned after solvng of the statonary dffuson equaton and subsequent substtuton of ths soluton nto the Stefan condton. We remark that ths soluton becomes naccurate agan as blow up ( ds as S ) takes place. Smlar dt to the planar case we assume c csol. After carryng out the same analyss as before for the planar case, we see all nterfacal concentratons can be expressed n terms

15 τ D D D 3 k D 3 Fgure 4: Top-left: The dssoluton tme as a functon of the dffuson coeffcent of the thrd component for larger values. The dotted and sold lnes respectvely correspond to the exact and approxmate soluton. Top-rght: The nterfacal concentraton of the frst component as a functon of the dffuson coeffcent of the thrd component. Botton-left: The nterfacal concentraton as a functon of the dffuson coeffcent of the thrd component. Botton-rght: The velocty coeffcent k as a functon of the dffuson coeffcent of the thrd component. Calculatons have been done wth partcle concentratons of 33 for the frst two components, the partcle concentraton of the thrd component s 3.

16 4 = 3 3 Relatve error = 8 3 = 33 3 = D 3 Fgure 5: The relatve error of the dssoluton tme as predcted by the quas-bnary approach as a functon of the dffuson coeffcent of the thrd component for consecutve partcle concentratons. of the nterfacal concentraton of the frst element = cpart D for {,..., n}. D Substtuton of above relatons nto the hyperbolc relaton for the nterfacal concentratons (equaton ()) gves the followng expresson for the nterfacal concentraton of the frst alloyng element = cpart D [ Π n = ( D ] µ ) m K, wth µ := m + m m n. Above expresson s smlar to the one for planar geometry except for the absence of the square root for the dffuson coeffcents. Ths gves hence dfferent values for the nterfacal concentratons. So durng the dssoluton process the values of the nterfacal concentratons converge from the values as determned n the prevous secton to the values just mentoned. Note that ths holds for the case that the nterfacal poston moves slowly,.e. c. Let the tme be n the nterval t < t < t 2, then substtuton of above expresson nto equaton (5) gves for the nterfacal velocty ds dt = csol D S ds dt = csol eff eff D eff S,

17 5 where the effectve nterfacal concentraton, partcle concentraton and dffuson coeffcent are defned by eff := K µ, c part eff := [ ] Π n = (cpart ) m µ, D eff := [Π n = (D ) m ] µ. Soluton of above equaton s trval. Note that these effectve parameters are equal to the ones that were obtaned for the planar case. Nevertheless, the nterfacal concentratons dffer n both cases. Furthermore, t should be noted that the approxmatons hold under lmtng assumptons. For the more general case, where c does not hold necessarly and where we are n the range of tme where both terms n equaton (4) are of same order, the nterfacal concentratons are contnuous functons of tme. Ther values start at the planar soluton (see preceedng subsecton) and converges towards the soluton obtaned n ths subsecton (see above relaton). The evoluton of the nterfacal concentratons and nterfacal poston s obtaned by the use of the full equatons (4) and hyperbolc relaton () for the nterfacal concentratons. The calculaton of the nterfacal concentratons s straghtforward. 4 Numercal method Varous numercal methods are known to solve Stefan problems: front-trackng, front-fxng and fxed doman methods. Snce the concentraton at the nterface vares wth tme n our problem, we restrct ourselves to a front-trackng method. Recently a number of promsng methods are proposed for mult-dmensonal problems: phase feld methods and level set methods, such as n [9, ]. However, mposng local equlbrum condton at the nterface n such models s not as straghtforward as n front-trackng methods that are used here. A couplng between thermodynamcs and a phase feld model s presented by Grafe et al [6]. Our man nterest s to gve an accurate dscretsaton of the boundary condtons for ths Stefan problem wth one spatal co-ordnate. Therefore we use the classcal movng grd method of Murray and Lands [3] to dscretse the dffuson equatons. In ths paper we brefly descrbe the method, for more detals we refer to [24]. Dscretsaton of the nteror regon We use an mplct fnte dfference method to solve the dffuson equaton n the nner regon. An explctly treated convecton term due to grd-movement s ncluded. Snce the magntude of the gradent s maxmal near the movng nterface we use a geometrcally dstrbuted grd such that the dscretsaton near the nterface s fne and coarse farther away from the movng nterface. Furthermore, we use a vrtual grd-pont near the movng boundary. The dstance between the vrtual node and the nterface s chosen equal to the dstance between the nterface and the frst grd-node. The resultng set of lnear equatons s solved usng a trdagonal matrx solver. Dscrete boundary condtons at the nterface We defne the dscrete approxmaton of the concentraton as c j,k, where j, and k respec-

18 6 tvely denote the tme-step, the ndex of the chemcal (alloyng) element and grdnode. The vrtual grdnode behnd the movng nterface and the grdnode at the nterface respectvely have ndces k = and k =. At the movng nterface, we obtan from dscretsaton of equaton (4) D c j+, c j+, 2 r = D + c j+ + csol + +, cj+ +, 2 r, for j {,..., n }. Note that the concentraton profle of each element s determned by the value of the nterfacal concentraton. Above equaton can be re-arranged nto a zero-pont equaton for all chemcal elements. All nterfacal concentratons satsfy the hyperbolc relaton (). Combnaton of all ths, gves for {,..., n } and = n f (c j+,, cj +, ) := D (c j+, c j+, )(cpart + csol +) D + (c j+ +, cj+ +, )(cpart ) = f n (,..., n ) := ( ) m ( 2 ) m 2 (...)( n ) mn K =. To approxmate a root for the vector-functon f we use Newton s method combned wth dscrete approxmatons for the non-zero entres n the frst n rows of the Jacoban matrx. The teraton s termnated when suffcent accuracy s reached. Ths s explaned n more detal n [24]. Adaptaton of the movng boundary The movng nterface s adapted accordng to equaton (4). In [22] the forward (explct) Euler and Trapezum tme ntegraton methods are descrbed and compared. It was found that the (mplct) Trapezum method was superor n accuracy. Furthermore, the teraton step to determne the nterfacal concentratons s ncluded n each Trapezum step to determne the nterfacal poston. Hence, the work per tme-teraton remans the same for both tme-ntegraton methods. Therefore, the Trapezum rule s used to determne the nterfacal poston as a functon of tme. We termnate the teraton when suffcent accuracy s reached,.e. let ε be the naccuracy, then we stop the teraton when the nequalty n = (p + ) (p) + Sj+ (p + ) S j+ (p) < ε S j+ M holds. Here S j denotes the dscrete approxmaton of the nterfacal poston at tme-step j. The nteger p represents the teraton number durng the determnaton of the nterfacal concentratons and poston. 5 Numercal experments Ths secton contans the numercal experments done wth the Fnte Dfference method. We am at a comparson between the quas-bnary soluton and the full mult-component

19 7 soluton. Experments are done for planar and sphercal geometres. The nput-data used s hypothetcal but the order of magntude s comparable to the case of commercal alumnum alloys. 5. Planar experments We consder the quas-bnary and mult-component approach for the planar geometry. The confguraton entered here apples to a quaternary alloy. As nput-data we use the values as lsted n Table. Table : Input data Physcal quantty Value S-Unt D 3 m 2 /s D m 2 /s K c - c 2 - c 3 - m - m S 6 m M 4 m We vary the dffuson coeffcent and partcle concentraton of the thrd component (D 3 and 3 ). The results are shown n Fgure 6 where we plot the nterfacal poston as a functon of tme. For all these stuatons t can be seen that the quas-bnary soluton and full mult-component soluton agree very well (see Fgure 6). Ths agreement perssts also for the hgher dffusvtes of the thrd alloyng element. Ths s n agreement wth the result shown n Fgure 4 for the case that the regon s unbounded. Furthermore, one expects that the concentraton at the cell boundary (.e. at r = M) s larger for cases where the dffuson coeffcent of the thrd component s larger. However, both the penetraton depth and the nterfacal poston exhbt a square-root behavour wth tme. Ths mples that possbly the atoms from the alloyng elements reach the cell boundary M after complete dssoluton of the partcle. Ths depends on the cell sze and geometry. Therefore, the observed dfferences n dssoluton rate reman small for all cases where the geometrcal settngs are equal. Fgure 7 shows the same confguraton as n Table wth 3 = 33 and D 3 = 3 except all curves correspond to cellszes M = 5 5 and M = The bold lnes correspond to the full mult-component soluton. Whereas the other lnes are predcted usng the quas-bnary approach. It can be seen that the dfference between the quasbnary approach and full mult-component approach s more sgnfcant. Ths sgnfcant

20 8.9.8 nterfacal poston (µ m) II I.2. IV III Tme (s) Fgure 6: The nterfacal poston as a functon of tme. All curves correspond to the confguraton as lsted n Table. The bold and ordnary curves respectvely reflect the quas-bnary and full mult-component soluton. Curves I corresponds to 3 = 33 and D 3 =. 3. Curve II reflects the case that 3 = 3 and D 3 =. 3. Curve III dsplays the stuaton n whch 3 = 33 and D 3 = 3, whereas curve IV shows the confguraton 3 = 3 and D 3 = 3

21 9.9.8 Interfacal poston (µ m) M = M = Tme (s) Fgure 7: The nterfacal poston as a functon of tme. The bold curve corresponds to the quas-bnary approach and the other curve corresponds to the mult-component approach. dfference has also been observed when non-zero values for the ntal concentratons are taken. Fgure 8 presents the nterfacal poston as computed by the quas-bnary approach as a functon of the computed nterfacal poston by the full mult-component approach. It can be seen that the curvature of the lne ncreases for smaller cell-szes. Ths s attrbuted to the accumulaton of the atoms of the alloyng elements at the cell boundary. For the case of M = an nterestng behavour s observed: At the early stages where the atoms dd not reach the cell-boundary yet, the curve s straght. Later, as the atoms reach the cell-boundary, the curve starts to devate from a straght lne, smlar to the case of a cell-boundary at M = 5 5. Subsequently, the matrx gets saturated and dssoluton stops: the dssoluton rate converges towards zero. The equlbrum state s not effected by the use of the quas-bnary approach and the thereby the curve converges back to the straght lne. 5.2 Sphercal experments We consder the dssoluton of a sphercal partcle n the mult-component and quasbnary alloy. The confguraton entered here apples to a quaternary alloy. We use the same nput-data from Table, except for the geometry and the cell sze: M = 5, unless stated otherwse. We vary the partcle concentraton and dffuson coeffcent of the thrd component. For dfferent dffusvtes and partcle concentratons of the thrd component, the results are shown n Fgure 9. The agreement between the quas-bnary approach and the full

22 2.9.8 quas bnary nterfacal poston M = 4 M = M = mult component nterfacal poston Fgure 8: The computed nterfacal poston by the quas-bnary approach as a functon of the computed nterfacal poston usng the full mult-component approach for consecutve cell-szes. mult-component soluton s good. The dfference between the two approaches s smallest for the hgher values of the dffuson coeffcent of the thrd component. As expected, the dssoluton process takes place relatvely fast compared to the rate of penetraton of the alloyng elements from the partcle nto the matrx, although the order of magntude of the rate of both processes s smlar. Let S mc and S qb be the nterfacal postons predcted usng respectvely the mult-component and quas-bnary approach. We compare the relatve errors, defned by ɛ := S mc S qb S mc %, taken at the tmes when the mult-component soluton s nearest to S mc =.5, we see that the errors for the cases correspondng to curves I,II,III and IV n Fgure 9 are respectvely gven by 9.94, 3.9, 4.23 and 3.87 %. It can be seen that the quas-bnary approach s most accurate for cases where the dffuson coeffcent of the thrd component s large. Ths observaton s contrary to Fgure 5 where the error becomes more sgnfcant for larger dffuson coeffcents of the thrd component. Ths dscrepancy may be caused by the geometrcal dfferences between ths stuaton and the stuaton n Fgure 5 (sphere and bounded doman versus plane and unbounded). Some dssoluton curves are shown n Fgure for dfferent values of the cell radus. Here we took a low value for the dffuson coeffcent of the thrd component, beng a case where the quas-bnary approach s less accurate (compared to the case where the dffuson coeffcent of the thrd component s hgh). It can be seen that the dfference between

23 2.9.8 Intefacal poston (µ m) II I.2. III IV Tme (s) Fgure 9: The nterfacal poston as a functon of tme for a sphercal dssolvng partcle. All curves correspond to the nput-data from Table. The bold and ordnary curves respectvely correspond to the quas-bnary and full mult-component approaches. Curves I correspond to 3 = 33 and D 3 = 4. Curves II depct the case that 3 = 3 and D 3 = 4. The stuaton wth 3 = 33 and D 3 = 2 s shown by curves III. Whereas curves IV dsplay the case that 3 = 3 and D 3 = 2. the quas-bnary and full mult-component approach s reasonably small. However, for the case that M = 4 6 the dfference s large. Apparently, the dfference between both approaches s large when dssoluton tmes are large, but fnte. In the case that M = full dssoluton does not take place. The dfference between the dfferent approaches s large at early stages, but becomes less sgnfcant as tme proceeds: the curves reach the same lmt. Ths observaton s smlar to the observaton n the planar case (see Fgure 8). We remark that ths observaton follows from experment and that a more mathematcal bass s needed for a full understandng. Fnally we show the nterfacal poston as a functon of tme for dfferent ntal matrx concentratons n Fgure. For low ntal matrx concentratons, the dfference between the quas-bnary and full mult-component approach s small. For larger concentratons the dfference ncreases and hence the quas-bnary approach breaks down. Ths s attrbuted to the fact that the dfference between the nterfacal concentraton and ntal concentraton becomes more sgnfcant (see equaton (5)).

24 22.9 M = m.8 Interfacal poston (µ m) M = 4 6 m.2. M = 5 m Tme (s) Fgure : The nterfacal poston as a functon of tme for a sphercal dssolvng partcle for consecutve cell rad. The bold curves correspond to the quas-bnary approach. The confguraton s taken from Table wth 3 = 33 and D 3 = Interfacal poston (µ m) c =.5 c =.2 c = Tme (s) Fgure : The nterfacal poston as a functon of tme for a sphercal dssolvng partcle for consecutve ntal matrx concentratons. The bold curves correspond to the quasbnary approach. The confguraton s taken from Table wth 3 = 33 and D 3 = 4.

25 23 6 Conclusons A model, based on a vector-valued Stefan problem, has been developed to predct the dssoluton of partcles n general mult-component alloys. A remark has been gven concernng exstence and well-posedness of solutons of the vector-valued Stefan problem. The remark s motvated usng a physcal argument. For general cases wth one spatal co-ordnate the full vector-valued Stefan problem s solved usng Fnte Dfferences. For some cases, when the dfference between the partcle concentratons and nterfacal concentatons are large and when the ntal matrx concentraton s neglgble, the full mult-component (vector-valued) Stefan problem can be approxmated accurately usng an averagng technque for the partcle concentratons and dffuson coeffcents. Ths reduces the mult-component problem to a quas-bnary problem. Ths approxmaton s essentally usefull when more geometrc flexblty s ncluded nto the model, see for nstance [28, 6]. It also turned out that ths quas-bnary approach s accurate for the sphercal dssolvng phases. We expect ths method also to be accurate for the case of dssolvng cylndrcal phases n mult-component alloys. References [] H.B. Aaron and G.R. Kotler. Second phase dssoluton. Metallurgcal transactons, 2:65 656, 97. [2] U.L. Baty, R.A. Tanzll, and R.W. Heckel. Dssoluton knetcs of CuAl 2 n an Al-4Cu alloy. Metallurgcal Transactons, :65 656, 97. [3] J. Chadam and H. Rasmussen. Free boundary problems nvolvng solds. Longman, Scentfc & techncal Harlow, 993. [4] S.P. Chen, M.S. Vossenberg, F.J. Vermolen, J. van de Langkrus, and S. van der Zwaag. Dssoluton of β partcles n an Al-Mg-S alloy durng DSC-runs. Materals scence and engneerng, A272:25 256, 999. [5] J. Crank. Free and movng boundary problems. Clarendon Press, Oxford, 984. [6] U. Grafe, B. Böttger, J. Taden, and S.G. Fres. Couplng of multcomponent thermodynamcs to a phase feld model: applcaton to soldfcaton and sold-state transformatons of superalloys. Scrpta Materala, 42:79 86, 2. [7] J. Ågren. Dffuson n phases wth several components and sublattces. Journal of physcal chemstry of solds, 43:42 43, 98. [8] R. Hubert. Modelsaton numerque de la crossance et de la dssoluton des precptes dans l acer. ATB Metallurge, 34-35:4 4, 995.

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