3D ELECTRODE SHAPE CHANGE SIMULATION IN ELECTROPLATING

Transcription

1 3D ELECTRODE SHAPE CHANGE SIMULATION IN ELECTROPLATING MARIUS PURCAR 1, CALIN MUNTEANU, VASILE TOPA Ky words: Elctroplating, 3D modling, Elctrod shap chang, Potntial modl, Boundary lmnt mthod. This papr proposs a gnral applicabl numrical tchniqu for th simulation of thr-dimnsional (3D) lctrod shap changs obtaind during th lctroplating procsss. Th shap of th lctrod is found displacing ach msh nod proportional with, and in th dirction of th local currnt dnsity according to Faraday s law. Th local growth rat is obtaind solving numrically th potntial modl using Boundary Elmnt Mthod (BEM). An xampl daling with th 3D simulation of lctroplating nar an insulator (singularity) is prsntd and compard with masurmnts and twodimnsional (2D) simulations. 1. INTRODUCTION In th lctroplating clls, mtal dposit occurs at th cathods. Th high non-uniform currnt dnsity, du to th dg ffcts, transfrs to a pronouncd nonuniform dposition layr. In ordr to prdict and mitigat such ffcts suitabl mathmatical modl and simulating softwar ar ncssary. Dilut solution thory provid th gnral quations dscribing th lctrochmical procsss, as dscribd in dtail in [1]. Most of th publishd paprs prsnt a mor or lss simplifid mathmatical modl. For xampl, whn th lctrod ractions tak plac at low rats, such that th concntration gradints ar nglctd, th potntial distribution may b found only using Laplac s quation. Hnc, th rsulting modl dscribs th ohmic ffcts in th lctrolyt and is known as th potntial modl (PM) [2]. Svral authors applid th PM to comput th currnt dnsity distribution and hnc th lctrod growth rat for lctroplating applications. Alkir t. al. applid in [3] th finit lmnt mthod (FEM) to solv th rsulting Laplac quation, with nonlinar boundary conditions that account for lctrod charg transfr ractions. Dconinck t al. [2] discrtizd 1 Tchnical Univrsity of Cluj-Napoca, Dpartmnt of Elctrotchnics and Masurmnts, Baritiu Strt 26-28, marius.purcar@thm.utcluj.ro. Rv. Roum. Sci. Tchn. Élctrotchn. t Énrg., 58, 3, p , Bucarst, 2013

2 2 3D lctrod shap chang simulation in lctroplating 253 th quations of th PM using th BEM in ordr to comput th changing lctrod profil for nonlinar boundary conditions. Thy also prsntd a complt study of th lctrod volution function of th lctrod ractor dimnsion for diffrnt angls btwn th lctrod and an adjacnt insulator. Using th finit diffrnc mthod (FDM), Bozzini and Cavallotti [4] prsntd numrical simulations for th Cr-plating procss on cornr-shapd cathods. Most of th works prsntd abov applid an Eulr schm for th intgration of Faraday s law with rspct to th tim, combind with th displacmnt of th discrtization nods of th lctrod surfac. This mthod, rfrrd as th string thory in [5], drivs from th Lagrangian approach of changing boundary problms that implis th grid boundary to b attachd to th moving front. L t al. in [6] and Purcar t al. in [7, 8] proposd lvl st mthod (LSM) for th calculation of th lctrod shap changing. Both approachs usd th PM for th computation of th currnt dnsity distribution. LSM dos not xplicitly distort th profil of th intrfac but implicitly comput it as a solution of a scalar advction quation [5]. A limitd numbr of publications dal with thr dimnsional (3D) currnt dnsity distribution simulations in lctrochmical ractors, but in th fild of 3D lctroplating thy ar vry scarc. Applications that considr currnt dnsity distributions for mor complx 3D gomtris can b found in th fild of cathod protction and wafr plating [9, 10]. Purcar and Bortls [11 13] applid th principls of advancd CAD intgratd approach for 3D lctrochmical machining simulations. Although during th last dcad significant progrss in th dvloping of th simulation tools of th lctrochmical procss has bn mad, thr is still a high nd for ffctiv softwar and compact intgration of ths tools in th product dsign-dvlopmnt chain in ordr to minimiz th computational tim, which is xtrmly long for th 3D problms. This papr prsnts a gnral simulation approach of th 3D lctroplating applications with strong non-linar boundary conditions including raction fficincy and illustrats th basics principls of th 3D lctrod shap chang algorithm. 2. MATHEMATICAL MODEL AND NUMERICAL SOLUTION METHOD Th lctrochmical procss (.g. lctroplating, lctroforming, lctro rosion tc.) dvlops continuously in tim. In ordr to simulat th volution of th lctrod profil, th continuous procss tim must b dividd into a squnc of discrt tim stps [13]. For ach tim stp two problms should b usually solvd:

3 254 Marius Purcar, Calin Muntanu, Vasil Topa 3 1) th lctrochmical modl (a stady stat problm for calculating th local lctrod growth rat) and 2) th lctrod growth problm (a tim dpndnt problm for calculating th shap of th lctrod surfac at ach tim stp). Th lctrod growth rat, providd by Faraday s law, nsurs th coupling of th two problms ELECTROCHEMICAL MODEL In most practical cass (.g. 3D simulations), th dilut solution thory which dscribs th migration, convction, diffusion and lctrod ractions in lctrolyts [1] is subjctd to important simplifications. Hnc, considring th lctrolyt solution vry wll stirrd (situation oftn mt in th practical situations) concntration gradints can b nglctd (xcpt for a vry thin layr nar th lctrods whr th lctrochmical ractions occurs). Assuming charg consrvation in th bulk and th conductivity of th lctrolyt constant, potntial modl yilds. Th potntial modl is dscribd by th Laplac quation for th lctrolyt potntial U, whras th phnomna occurring in th diffusion layr and at th lctrod intrfac [1] (.g. th lctrochmical ractions) ar includd in th boundary conditions. 2 U = 0 J = σ U, (1) whr σ rprsnts th conductivity of th lctrolyt in [Ω -1 m -1 ], U th lctrolyt potntial in [V] and J th currnt dnsity in [A/m 2 ] BOUNDARY CONDITIONS Th boundary conditions in quation (1) ar dividd into insulators and lctrods [2, 7, 8]. As no currnt can flow through th gasous mdium in contact with th lctrolyt and lctrochmical clls walls (th normal currnt dnsity J n is zro, s Fig. 1): Fig. 1 Boundary conditions at th lctrods [10].

4 4 3D lctrod shap chang simulation in lctroplating 255 At th intrfac btwn th lctrod and th lctrolyt, known as doubl layr [1], lctrochmical ractions occur. Th lctrochmical raction driving forc is xprssd at ach point of th lctrod by a diffrnc btwn th solution U and lctrod potntial V (Fig. 1) and is calld th charg transfr ovrpotntial or polarization η [1, 2, 7, 8]. Th following xprssion of th ovrpotntial gnrally holds for a givn raction at th lctrods: η 0 = 0 ( c ) = V U E f,, (2) with U th potntial of th solution adjacnt to th lctrod, outsid to th doubl layr, V th mtal potntial, E 0 th quilibrium potntial, c 0 th surfac concntration of th ion(s) involvd in th raction in [mol/m 3 ] and J n th normal currnt dnsity at th lctrod (Fig. 1). Function of th ovrpotntial sign, oxidation (positiv η) or rduction (ngativ η) occurs at th lctrods. Study of th kintics of th lctrod raction gnrally rvals a strong non-linar bhaviour in (2). Th lctrochmical raction (2) is incorporatd in th PM as a non-linar boundary condition and in th most gnral situation is writtn as: ( η) J n J n = f. (3) Such function is usually obtaind masuring th physicochmical paramtrs of th lctrolyt and approachd with a splin function as in [2, 7, 8] ELECTRODE GROWTH RATE Th amount of matrial rducd at th cathod is dirctly proportional to th amount of th lctrical charg flowing btwn th lctrods. This can b xprssd using Faraday s law: M I T m =, (4) z F with: m th dissolvd or dpositd mass [kg], I th total currnt that flows btwn th lctrods [A/m 2 ], T th total procss tim [s], M th atomic wight of th mtal [kg/mol], F th Faraday s constant [ C/mol] and z th lctric charg [C]. Considring th dpositd mass m = ρ dv, whr ρ is mtal dnsity [kg/m 3 ] dfind on an infinitsimal dv = dh ds volum, th local lctrod growth dh can b simply computd from (4) as: M T J d h = θ n, (5) ρ z F

5 256 Marius Purcar, Calin Muntanu, Vasil Topa 5 whr J n = I/ds is th normal currnt dnsity through an infinitsimal surfac ds at th lctrod surfac. For a mor gnral dfinition of th lctrochmical ractions (3) (.g. incorporating th sid ffcts lik gas volution), th fficincy θ is introducd. In that cas th fficincy θ is blow 100 % NUMERICAL SOLUTION METHOD Th quations (1 3) dscrib gomtris with arbitrary boundary conditions which cannot b solvd analytically. Th solutions can b obtaind using diffrnt discrtization tchniqus (.g. BEM, FEM, FDM tc.), which assum th approximation of th continuous solution as a st of quantitis at discrt locations, oftn on an irrgularly spacd msh. Th computational nvironmnt, ESCFramWork V1.1, dvlopd by th authors, is objct-orintd and can handl any 3D singl or multipl domain modls. A flxibl grid gnrator [14] is usd to crat an unstructurd triangular msh sparatly on ach fac of th computational domain. Th msh is optimally rfind towards th zons whr a stp variation of fild variabls is xpctd but gnrats coars lmnts at any othr position, s Fig. 3 (right). Th computation of th quation systm (8) is implmntd both for an itrativ and for a dirct LU solvr ( BEM FOR ELECTROLYTE AND ELECTRODE REACTIONS In ordr to solv th simplifid charg consrvation quation (1), BEM is mor suitabl ovr othr discrtization mthods (FEM and FDM) bcaus th conductivity of th lctrolyt is considrd constant. On of th BEM advantags is that only th boundaris of th computational domain must b discrtisd. Anothr advantag, in particular for lctrochmical systms, is that th currnt dnsity distribution along th lctrods is a dirct unknown to th problm, rathr than a variabl that must furthr b computd from th drivativ of th potntial fild at th lctrods. Th boundary S Γ of th computational domain is mshd with N nonovrlapping lmnts dfind by nods χ i = (x i, y i, z i ) in (i = 1 N p ). Th 3 charactristic BEM quation for th contribution to a point i is [15, 16]: i c U i + N w * U d SΓ n = = 1 = 1 s N s w * Q d S Γ, (6) with w * =1/(4πr) th 3D Grn function (r bing th position rlativ to point i), Q = U/ n th inward flux on th boundary nods and c i is an intgration constant for point i. Th indx rangs ovr all lmnts of th domain and intgration is prformd ovr th surfac S of ach lmnt. Triangular lmnts with linar

6 6 3D lctrod shap chang simulation in lctroplating 257 shap functions for th unknown potntial U and flux fild Q ar usd, hnc, rstricting th unknowns to th nodal valus i. BEM quation (6) rsults into th systm of quations: [ H ] [ U ] = [ G] [ Q], (7) whr {Q} is a vctor of siz N p containing th valus of th lctric fild normal to th boundary S Γ in ach point, {U} is a vctor of siz N p containing th valu of th potntial in ach nod and [H] and [G] ar squar matrics of siz N p x N p. Including in (7) th boundary conditions on th lctrods (ovrpotntials) and insulators th systm (7) is rordrd such that all unknowns ar put to th lft hand sid: [ ] [ ] [ 0 ] [ 1 ] { 0} ( ) H G U =. (8) Q f η As for most of th lctrochmical procss th ovrpotntial η is non-linar, a Nwton-Raphson itrativ algorithm is applid to solv th systm (8). Numrical validation of th BEM modl has bn prformd in [6]. Rsults showd a vry good agrmnt with th analytical valus, with a rlativ rror blow 2 % in dirct rlation to th surfac msh rfinmnt ELECTRODE SHAPE CHANGE COMPUTATION In ordr to comput th lctrod profil, th continuous procss tim T is dividd into a discrt squnc of tim points t m (m = 0, 1,... N t ). Knowing at ach nodal point of th boundary th currnt dnsity, quation (5) rducs to N p quations, on for ach nodal point i. Th lctrod growth rat bcoms: dh( χ i, t) M = θ σ Jn( χ, i t) n d t ρ z F 1, (9) whr th unit normal 1 n is orintd outwards from th computational domain. Equation (9) is th Lagrangian formulation of th intrfac volution and rprsnts th basic quation of th nodal displacmnt mthod (NDM) [13]. For t = 0 th initial condition h (χ i, t = 0) = 0 is imposd. Rplacing th diffrntial in th formulation (9) with a simpl forward diffrnc approximation, rsults: t t t h i m = hi m + λ σ J n m, i 1 t, (10) + 1 with λ = θ M/(ρ z F) and t m = t m+1 t m th tim stp. Equation (10) rprsnts th xplicit Eulr intgration schm which is in fact th first ordr Taylor xpansion of th solution of quation (9). Th nw boundary n m

7 258 Marius Purcar, Calin Muntanu, Vasil Topa 7 is asily found displacing ach nod on th lctrod proportional with and in th dirction of th local growth rat (Fig. 2). This is asy to fulfil if th connctivity dos not chang and th surfac lmnts ar not highly distortd. Fig. 2 Displacmnt of th boundary nods. 3. ELECTROPLATING SIMULATION NEAR A SINGULARITY This xampl prsnts th 3D coppr lctroplating simulation in th vicinity of a insulator singularity. A similar xampl has bn studid in rfrncs [2, 7] for 2D numrical simulations basd on diffrnt computational mthods. Ths comparisons will b usd to validat th 3D lctrod shap chang modl prsntd in this papr MODEL DEFINITION AND PROCESS CONDITIONS Th cll in th xprimntal st-up [2] and simulations [2, 7] is givn in Fig. 3 (lft). Th distanc btwn th cathod and anod is 12 cm. Th lngth of th cathod and anod is 19.9 rspctivly 30.7 cm. Th ractor s width is 0.79 cm. Fig. 3 Gomtry of th lctrochmical cll in th vicinity of th singularity (right) and th triangular surfac msh of th computational domain (lft).

8 8 3D lctrod shap chang simulation in lctroplating 259 Th xprimnt dscribd in [2] ran for h to a constant voltag U = V btwn th cathod and anod. Th sam physicochmical input paramtrs as in [2] and [7] ar usd. For ths paramtrs th potntial modl (PM) and th lctrochmical problm can b approachd by th Laplac quation (1) with non-linar boundary conditions (3) THREE-DIMENSIONAL RESULTS Th computational domain is mshd using in triangular lmnts and points (Fig. 3 right). Around th singularity th msh is highly rfind in ordr to captur th dg ffcts. On th othr zons th msh siz is furthr coarsnd. This nabls to prform calculations on a normal PC with 8GB RAM and 2.4GHz CPU within about 10 minuts for on tim stp. Th calculatd dposition tim T = 86.9 hours is dividd into 25 constant tim stps, t = 209 minuts ( s). Th govrning Laplac quation is solvd using th 3D BEM, which yilds th normal currnt dnsity distribution at th lctrod surfac. In a scond stp using th quation (10), th nodal points ar displacd proportional with th normal currnt dnsity. Th procss is rpatd until th total procss tim is achivd. Th high non-uniform currnt dnsity du to th dg ffcts on lctrod surfacs nar to th insulating substrat, transfrs to a pronouncd non-uniform dposition layr. Succssiv simulatd layr thicknss dposition profils on th cathod, togthr with photography of th final coppr dposition ar prsntd in Fig. 4. Fig. 4 Simulation of th 3D dposition profil (lft) and th xprimntal rsult in [2] (right). Th comparison btwn th simulatd and xprimntal lctrod profils in (Fig. 5) shows a good agrmnt [2, 7]. Th maximum rlativ dviation, btwn th 3D simulations and masurmnts is 17% at th pak of th growth. Th maximum rlativ dviation, btwn th 3D and 2D simulations in [2, 7] is about 8%. Th dviation btwn th 3D and 2D simulation rsults can b du a bttr accuracy of th 2D BEM solvr vrsus th 3D BEM solvr. Anothr sourc of rrors for both 2D and 3D modls vrsus th xprimntal rsults is that th

9 260 Marius Purcar, Calin Muntanu, Vasil Topa 9 undrlying potntial modl dos considr th mass transfr phnomna that occur in th diffusion layr nar th lctrod. Ths phnomna ar mor pronouncd for th highr variations in th gradint of th fild,.g. at th nighbourhood of th cathod and insulator substrat. Fig. 5 Dformd plot rprsntation of th numrical and xprimntal rsults: masurd ( ) [2], simulatd 2D NDM ( ) [2], simulatd 2D LSM ( ) [7], simulatd 3D NDM ( ). 4. CONCLUSIONS Th principls of 3D lctrod shap chang simulations in lctroplating calculations hav bn prsntd. Th simulation tool is basd on th potntial modl solvd with a standard 3D BEM approach for calculation th currnt dnsity distribution at th lctrod and nodal displacmnt mthod to find th lctrod growth profil. In ordr to rval th mthod, an xampl daling with th 3D simulation of lctroplating nar an insulator (singularity) is prsntd and compard with masurmnts and two-dimnsional (2D) simulations from litratur. Th 3D and 2D rsults match vry wll. Diffrncs can b considrd as rsulting from numrical inaccuracis dvlopd by both th 2D and 3D approachs. Anothr sourc of rrors is that th potntial modl cannot tak into account th mass transfr phnomna that occur in th diffusion layr. Th us of th 3D simulation tool allows calculating diffrnt st-ups in a fast and chap way.

10 10 3D lctrod shap chang simulation in lctroplating 261 Th mthodology dvlopd hr provd its robustnss whn daling with and combining distinctiv stags in th lctrod growth simulations procss such as currnt dnsity computation and nodal displacmnt. ACKNOWLEDGEMENTS This work was supportd within th rsarch program posdru/89/1.5/s/ Rcivd on January 29, 2013 REFERENCES 1. J. Nwman, Elctrochmical Systms, 2'nd dition Englwood, Nw Jrsy, Prntinc-Hall, 1991, p J. Dconick, Currnt distribution and lctrod shap chang in lctrochmical systms- A boundary lmnt approach, Lctur Nots in Enginring Springr-Vrlag, 1992, p R. Alkir, T. Brgh, TL. Sani, Prdicting Elctrod Shap Chang with Us of Finit Elmnt Mthods, J. Elctrochm. Soc., 125, p (1978). 4. B. Bozzini, PL. Cavallotti, Numrical modlling of th growth of lctrodpositd chromium on cornr-shapd cathods, Int. J. Matr. Prod. Tc., 15, 1 2, p. 34 (2000). 5. J. A. Sthian, Lvl St Mthods and Fast Marching Mthods: Evolving Intrfacs in Computational Gomtry, Cambridg, Cambridg Univrsity Prss, 1999, p J. L, JB. Talbot, Simulation of Particl Incorporation during Elctrodposition Procss, J. Elctrochm. Soc., 152, 10, pp. C706-C715 (2005). 7. M. Purcar, J. Dconinck, B. Van dn Bossch, L. Bortls, Elctroforming simulations basd on th lvl st mthod, EPJAP, 2, 39, p. 85 (2007). 8. M. Purcar, V. Topa, C. Muntanu, R. Chrchs, A. Avram, L. Grindi, Optimisation of th layr thicknss distribution in lctrochmical procsss using th lvl st mthod, IET Scinc, Masurmnt & Tchnology, 6, 5, pp (2012). 9. M. Purcar, B. Van dn Bossch, L. Bortls, J. Dconinck, P.Wsslius, Numrical 3D Simulation of a CP systm for a Burid Pip Sgmnt Surroundd by a Load Rliving U-shapd Vault, Corrosion, 59, pp (2003). 10. M. Purcar, B. Van dn Bossch, L. Bortls, J. Dconinck, Thr-Dimnsional Currnt Dnsity Distribution Simulations for a Rsistiv Pattrnd Wafr, J. Elctrochm. Soc., 151, 9, pp. D78-D86 (2004). 11. M. Purcar, L. Bortls, B. Van dn Bossch, J. Dconinck, 3D lctrochmical machining computr simulations, J. Matr. Procss. Tchnol., 149, 1 3, pp , (2004). 12. L. Bortls, M. Purcar, B. Van dn Bossch, J. Dconinck, A usr-frindly simulation softwar tool for 3D ECM. J. Matr. Procss Tchnol., 149, 1 3, pp (2004). 13. M. Purcar, A. Dorochnko, L. Bortls, J. Dconinck, B. Van dn Bossch, Advancd CAD intgratd approach for 3D lctrochmical machining simulations, J. Matr. Procss. Tchnol., 203, pp (2008). 14. A. Dorochnko, A. Athanasiadis, L. Bortls, H. Dconinck, Intgration of grid gnration and lctrochmical nginring simulation softwar in a CAD nvironmnt, EUA4X 2005, Annual Confrnc at TCN CAE 05, Lcc, Italy, Octobr, 2005.

11 262 Marius Purcar, Calin Muntanu, Vasil Topa C.A. Brbbia, Boundary Elmnt Tchniqus Thory and Applications in Enginring, Springr-Vrlag, Brlin, Hidlbrg, M. Maricaru, P. Minciunscu, I. R. Ciric, M.Vasilscu, A Nw Vctor Boundary Elmnts Procdur For Inductanc Computation, Rv. Roum. Sci. Tchn. Élctrotchn. t Énrg., 56, 1, pp (2011).