JOURNAL OF OPTOELECTRONICS AND ADVANCED MATERIALS Vol. 8, No. 1, Ferury 2006, p. 271-279 Numericl simultion of ohmic heting in idelized thin-lyer electrodeposition cells P. BARVINSCHI West University of Timisor, Fculty of Physics, Bvd. V.Prvn 4, 300223, Timisor, Romni The ohmic (Joule) heting in n electrolysis cell during electrochemicl deposition (ECD) with inry electrolyte could e significnt since the conductivity of the solution without supporting electrolyte is low. Here we report some results concerning the simultion of the therml field due to ohmic heting in idelized 2D nd 3D thin-lyer ECD cells. The mthemticl model is sed on Lplce-type eqution for the electricl potentil nd time-dependent eqution for the het conduction. The coupled equtions system is solved using finite element method. The numericl results re compred with experimentl ones nd qulittive greement is found. (Received Decemer 15, 2005; ccepted Jnury 26, 2006) Keywords: Thin-lyer electrodeposition, Therml field, Finite element method 1. Introduction In electrochemicl systems het my e generted due to mss-trnsport phenomen nd chemicl rections [1]. The comined set of grdients nd of het- nd msstrnsfer rtes re coupled nd they could modify the locl properties within given phse of the system nd, s consequence, they re influencing the course of some phenomen or the results of some properties mesurements nd their interprettion. In region of complex chemicl composition like thin-lyer electrodeposition cell the mentioned effects my e even more importnt ecuse in such smll volume the electric potentil, density or temperture grdients, respectively, could hve very lrge vlues. It is known lso tht in thin-lyer ECD cells rnched ggregtes of qulittively distinct morphologies re oserved s the deposition prmeters re vried [2]; is lso importnt to estlish the role of therml effects in the morphology selection. We hve found only two references devoted to some fetures of therml effects in thin-lyer ECD. In the first one [3] there is mention on the concentrtion mesurement error due to temperture vrition in n interferometric studies of rnched electrodeposition. The other pper [4] is entirely devoted to n experimentl study of the ohmic heting influence on the flow field in thin-lyer ECD. In [4] thin-lyer ECD cell ws filled with n ZnSO 4 queous solution nd the electrodeposition ws performed t constnt electricl potentil. The deposit t the cthode ws elonging to the homogenous or dense rnched morphology (DBM). The temperture field due to Joule heting in the electrolyte solution ws mesured y mens of n infrred cmer. As mentioned in [4], the direct comprility of the results reported in this work with experiments performed in stndrd ECD cells is not possile ecuse the ECD cell ws of slightly modified type. However, some of the results reported in [4] re pproprite to our purpose. These results re presented in figures 2-4 of the mentioned reference, giving n exmple of two-dimensionl thermogrphy in the neighorhood of the nodic zinc wire, nd the temporl evolution of the temperture field t the cthode nd the node s function of the distnce to the electrodes, respectively. The properties of the temperture field correspond closely with those of the concentrtion field in the cell (see ellow) nd this oservtion motivte our pproch. A detiled description of the dense morphology in prllel geometry electrodeposition cn e found in Ref. [5]. In this type of growth the electrodeposit consist in n rry of qusi-regulrly spced porous trees (rnches or filments), s is shown in Fig. 1. This dense rnched ggregte is ounded y flt liner front which invdes the cell t constnt velocity nd remins prllel to the electrodes. In the sence of convection nd due to the fst electrodeposition of the ctions ner nd etween the tips, one cn ssume tht the zone comprised etween the cthode nd the tips of the trees is totlly depleted of ctions. Correspondingly, the nions must e expelled from the tip region while the tip dvnce, otherwise they would crete lrge spce chrge region. The nions must leve this region t the sme velocity s the front progress in the cell. The growth cn thus e regrded s the dvnce of the front of n rry of rnches t speed equl to the drift velocity of the nions; the growth front push wy depletion lyer of oth nions nd ctions (concentrtion oundry lyer or diffusion lyer) whose size remins constnt. As the totl mount of nions must remin constnt the nions must ccumulte ner the node, where their chrge is lnced y nodic genertion of n equl mount of ctions, resulting oundry lyer of high concentrtion nd low resistivity. Between the two oundry lyers (cthodic nd nodic) there is neutrl region whose concentrtion is equl to its initil vlue. The filments re supposed to e highly conducting, therefore they keep the region etween the cthode nd the tips equipotentil; the ggregte ehve s moving cthode. The cthodic electrodeposition tht elongs to the dense prllel morphology cn e more or less regulr. For high enough potentil vlues, regulrly columnr or chnnel like deposit is formed [2,3].
272 P. Brvinschi the temperture distriution in the ECD cells. The numericl results re compred with experimentl mesurements reported in [4] nd we point out on some new results tht could e useful in forthcoming studies devoted to this topics. 2. Theoreticl nd numericl model Fig. 1. Illustrtion of dense growth y ECD in thin cell. Fig. 2. Schemtic digrm of the prolem spce nd the oundry conditions for the electricl nd therml prolems. A mcroscopic (>0.1 mm) chrcteriztion of dense rnched ggregte ws proposed in [5]. The nlysis relies on the computtion of correltion function which provides sttisticl estimte of the men distnce λ etween rnches nd of the width w of the rnches. If one defines the occupncy rtio θ s the proportion of the cell width occupied y the deposit then, in first pproximtion, we cn write w = θλ nd the deposit cn e imgined s rectiliner eqully spced set of filments (see Fig. 2). Oviously, this pproximtion is very good for chnnel like deposits. In this pper we present model for the simultion of the therml field in idelized 2D nd 3D thin-lyer ECD cells y solving the prolem of simultneous electric nd time-dependent therml conduction in two or three dimensions. The mthemticl model is the sme s in [6] ut in the present work we extend the clcultions to rmified deposits, mking rigorously prmetric study of In the following we present numericl study of the temperture distriution in idelized 2D nd 3D ECD cells solving the prolem of comined electric nd het conduction. In our pproch we consider only the contriution of the dissipted ohmic het nd neglect ny other form of energy dissiption. In ccordnce with the description of the ECD process in section 1, we consider the ctive prt of the electrochemicl cell s thin rectngulr foil composed of two prllel electrodes mde of the sme metl (Zn), dilute solution of slt of this metl (in our cse ZnSO 4 ), the electrodeposit nd two diffusion lyers, one in front of the deposit nd the other ner the node. The electrodeposit is supposed to e of the sme metl s the electrodes. In the idelized 2D cells the thickness of the foil lying in the (x,y) plne is supposed to e infinitesimlly smll. The more relistic 3D cell is composed of finite thickness lyer of the sme electrolyte solution, two diffusion lyers nd deposit sndwiched ll of them etween two prllel glss pltes. The geometry of the thin lyer in the (x,y) plne etween the glss pltes is the sme s in the 2D cse. The thickness of the 3D cell is mesured in the z direction. Becuse the two electrodes re very thin nd cn conduct het much etter thn the fluid nd the glss, we ssume tht they hve negligile therml resistnce in oth the y- direction nd z-direction. As result, the electrodes cn e ssumed to e trnsprent for the flux of het nd re removed from the model for the clcultion of the temperture field. Ech numericl experiment is performed considering fixed shpe electrodeposit (no deposit growth). We consider the ulk solution nd the two diffusion lyers to e ech of them homogeneous nd t rest. It is ssumed tht electroneutrlity is mintined in ll the cell, nd the diffusive nd convective effects re sent. We focus on the study of the temperture distriution in such idelized 2D or 3D cells t constnt voltge pplied etween the two electrodes (potentiosttic conditions). Clerly, the model descried so fr is first step towrd more relistic description of the ohmic heting in thin-lyer electrodeposition, in which electroneutrlity ssumption is removed nd full ion trnsport is tken into ccount. The nlysis of the full prolem will e crried out in forthcoming pper. Bsed on previous considertions we cn clculte the temperture distriution in the electrochemicl cell s follows. Supposing electro-neutrlity holds nd in the sence of diffusive nd convective trnsport of chrge crriers, ech region of the solution in the cell ehves like
Numericl simultion of ohmic heting in idelized thin-lyer electrodeposition cells 273 n ordinry metl nd the electricl current density j flowing into the cell is relted to the field strength E y j = σe = σ Φ where σ is the electric conductivity nd Φ is electric potentil. Under the sme conditions, the conservtion of electricl chrge requires tht (1) j = 0 (2) Comining the reltions (1) nd (2) we otin j = ( σe) = ( σ Φ) = 0 This is PDE of the Lplce type where the conductivity σ my vry in spce. The current density cuses dissiption of het t rte j E per unit volume. This electric power will pper s source term h in the timedependent eqution for het conduction T f + ρc p = h (4) t where T is the solute temperture, f = k T the het flux density, k the therml conductivity, ρ the mss density nd c p the specific het cpcity. In order to clculte the temperture distriution in the ECD cell we must solve simultneously eqution (3) for the electricl potentil nd eqution (4) for temperture with pproprite initil nd oundry conditions. Ech region of the 2D or 3D cell is chrcterized y its own electricl nd therml conductivity, mss density nd specific het cpcity. Some dt re tken from [7] nd [8] nd ll of them re summrized in Tle 1. For the purpose of the present study these properties re ssumed temperture invrint. Tle 1. Physicl nd chemicl property dt. -1 m -1 k Wm -1 K -1 c p Jm -3 K -1 Deposit (Zn) 121 1.5 10 7 2.78 10 6 Cthodic diffusion lyer 0.6 0.005 4.0 10 6 (0.001 M/l) Neutrl solution (0.1 M/l) 0.4 0.456 4.5 10 6 Anodic diffusion lyer 0.6 10 4.6 10 6 (0.25 M/l) Glss pltes 0.78 10-11 2.268 10 6 The initil conditions re lwys T = 300K for the temperture nd Φ = 0 for the electricl potentil. The oundry conditions re of Dirichlet nd Neumnn type nd they must e specified only on the oundry seprting the system nd its surroundings: Φ = Φ 0 nd (3) T / n = 0 on the cthode side, Φ = +Φ 0 nd T / n = 0 on the node side nd Φ / n = 0 nd T = T 0 on the lterl oundries of the cell, where Φ 0 is hlf the voltge pplied to the ECD cell nd n is the outwrd norml t the oundry of the domin. For the 3D cell dditionl oundry conditions were specified on the externl fces of the glss pltes: for the lterl ones the conditions re the sme s for the corresponding sides of the 2D cell, nd for the ottom nd up fces we put Φ / n = 0 nd T / n = 0. In our prolem no oundry conditions re needed for Φ nd T t the interfces etween two different regions ecuse the progrm tret s continuous the electricl potentil nd the temperture cross the interfces. In the cse of Joule heting, there re vriety of possile electrode oundry conditions for the therml prolem. An inspection of the experimentl temperture profiles shown in Figs. 3 nd 4 of Ref. [4] my suggests tht the therml oundry conditions re chnging in time. For exmple, we think tht on the cthode side there is first period when the oundry conditions must e of the Neumnn type ut with non-vnishing vlue of the outwrd surfce-norml flux T / n, followed y regime when the oundry conditions re of the Dirichlet type (T is some constnt temperture). For the ske of simplicity, in our clcultions we hve dopted the mentioned oundry conditions ut study of other choices for this type of prolem could e interesting. For the numericl solution of the equtions we hve used the student version of the finite element PDE solver FlexPDE [10]. A 2D prolem spce is divided into tringulr elements nd the vriles re pproximted y second or third order polynomils; in 3D prolem the spce is divided into tetrhedrons. The progrm employs n dptive mesh refinement technique to improve the ccurcy of the solution. 3. Results nd discussion 3.1 Therml field in idelized 2D cells As first pproch to the nlysis of the therml field in simplified thin-lyer ECD cells we tke 2D rmified deposit with ten equidistnt spikes hving compct rectngulr shpe. The prolem spce is shown in Fig. 2. The length nd width of the cell re the sme in ll numericl experiments, L y = 40 mm nd 2L x = 20 mm respectively. The distnce etween the xis of two successive spikes is λ = 2 mm. Bsed on experimentl findings concerning the homogeneous electrodeposition from ZnSO 4 queous solution, in our model we suppose tht etween spikes there is pure wter. A numer of prmeters in the prolem cn e vried: the pplied electric potentil Φ 0, the width δ c nd δ of the cthodic nd nodic diffusion lyers respectively, the physicl properties of the regions in the cell ( σ, k, ρ, c p ), the length l nd the width w of the spikes in the deposit. We
274 P. Brvinschi re le thn to mke prmetric study of the ohmic heting phenomenon in our simplified ECD cells. We performed clcultions for vrious vlues of Φ 0, δ c, l nd w; ecuse the width of the cell hs constnt vlue, chnging w mens vrition of the occupncy rtio θ. For θ = 1 the spikes touch ech other nd the deposit ecomes compct one. lmost vnish in the rest of the cell, hving very smll vlues etween the spikes. The clculted temperture distriution in the entire cell T(x,y) for the sme vlues of prmeters s in Fig.3 is shown y contours lines in Fig. 4. A pinted imge of this temperture distriution resemle quite well the thermogrphy in Fig. 2 of reference [4]. The temporl evolution of the temperture distriution long the y direction in the cell, for the sme vlues of prmeters, is shown in Fig. 5. The generl ehvior of the simulted temperture distriution is qulittively similr to tht found experimentlly nd reported in Ref. [4] (see Figs. 3 nd 4 of this reference), ut the temperture increse is not the sme. The most importnt fetures of the temperture profile is the existence of locl temperture mximum whose position correspond to tht of the cthodic diffusion lyer, followed y plteu of sptilly qusi-constnt temperture extended in region etween the two diffusion lyers. Fig. 3. Equipotentil lines for the solved electricl potentil ner the deposit; the vlues of the contours re given on the right in volts (see text for detils). Fig. 5. Clculted temperture profiles long the 2D cell (see text for the vlues of prmeters). Fig. 4. Isotherml lines for the clculted therml field in the entire ECD cell rising from Joule heting; the vlues on the right re in Kelvin (see text for detils). We hve clculted the two-dimensionl electricl potentil Φ(x,y) nd the electric field E (x, y) in the cell. A typicl result for Φ(x,y) otined for the vlues of prmeters l = 2 mm, Φ 0 = 10 V, δ c = δ = 0.8 mm, θ = 0.73 t the moment t = 400 s is shown in Fig. 3. We see tht the equipotentil lines re deformed in the usul mnner due to the presence of spikes. The electric field (not shown here) is concentrted on the tips of spikes nd Fig. 6. The dependence of the clculted temperture long the 2D cell on the voltge pplied to the cell (see text for the vlues of the other prmeters). We mke step further in our nlysis focusing on the dependence of the temperture profile long the cell, T(y), on some prmeters involved in the model. First of ll, one would expect n increse of the temperture in the entire cell s result of n increse of the pplied electric
Numericl simultion of ohmic heting in idelized thin-lyer electrodeposition cells 275 potentil Φ 0. Our clcultion show tht this ssertion is true, s one cn see in Fig. 6 where we hve represented the T(y) distriution for three vlues of Φ 0 (10 V, 15 V, nd 20 V) t t = 400 s nd keeping ll the other prmeters t the sme vlues (l = 2 mm, δ c = δ = 0.8 mm, θ = 0.73). It is lso remrkle tht for vrious vlues of Φ 0 the shpe of the T(y) distriution is the sme. Turning out to the dependence of the temperture profile T(y) on the occupncy rtio θ, the result of the clcultion for l = 2 mm, Φ 0 = 10 V, δ c = δ = 0.8 mm t t = 400 s nd three vlues of θ (0.23, 0.60 nd 1.00) re shown in Fig. 7. Trying to explin this ehvior of the temperture we hve performed clcultion of the electricl potentil Φ(y) long the cell for the sme vlues of the prmeters nd the results re shown in Fig. 7. There is remrkle resemlnce etween the profiles in this figure nd the profiles for the electric potentil otined y Chzlviel (see Figs. 2 nd 3 in Ref. [9]). We oserve tht the most importnt prt of the potentil drop in the cell is confined in the very nrrow region in front of the deposit (the cthodic diffusion lyer). As is seen in Fig. 7, slightly dependence of Φ(y) with the occupncy rtio is oservle for higher vlues of θ nd more importnt one for smll vlues of θ. In our pure ohmic model this ehvior of Φ(y) cn e explined y the increse of the potentil drop long the deposit s result of n incresing of the electricl resistnce when θ decrese. A motivtion for the dependence of the temperture distriution T(y) on the occupncy rtio cn now e offered. It is clerly seen tht in the deposit region the temperture is higher for smll vlues of θ nd tht in front of the cthodic diffusion lyer the sitution is reversed (the temperture increse for high occupncy rtio). These oservtions re in ccord with the electricl potentil dependence on θ discussed previously: the potentil drop long the deposit is more importnt for smll vlues of θ, nd the temperture in this region will increse when θ decrese ecuse the electricl power hs the sme ehvior s the potentil. In the sme time, the potentil drop in the ulk solution is incresing for higher vlues of θ, so the temperture in this region increses when the occupncy rtio increses too. Fig. 7. Dependence of the clculted temperture profile T(y) on the occupncy rtio θ (), nd the corresponding electricl potentil profiles long the cell (). Fig. 8. Dependence of the clculted temperture profile T(y) on the cthodic diffusion lyer width δ c (), nd the corresponding electricl potentil profiles long the cell (). Another cler effect on the temperture profile T(y) is due to the vrition of the cthodic diffusion lyer width δ c. The clculted T(y) profiles for l = 2 mm, Φ 0 = 10 V, δ = 0.8 mm, θ = 0.73 t t = 400 s, for three vlues of δ c (0.4 mm, 0.8 mm nd 1.2 mm), re shown in Fig. 8. The corresponding electricl potentil profiles (Fig. 8) give once gin the motivtion for the ehvior of the temperture distriution in the cell: n increse of the potentil drop in the ulk solution follows the reduction of the diffusion lyer width nd this implies n increse of the temperture in the ulk solution region.
276 P. Brvinschi Fig. 9. Dependence of the clculted temperture profile T(y) on σ, k nd ρc p in the deposit; the indicted vlues hve the sme units s in Tle 1. As it is known, the density of metl in the electrodeposit region is very low (see section 1) nd the previous ssumption of homogeneous nd compct pure metllic deposit is very crude pproximtion. Despite the fct tht the deposit is considered to hve very good electricl nd therml conductivity, we think tht the pproprite verge vlues for σ, k, ρ nd c p for kind of homogeneous nd compct deposit re closer to those of n queous solution rther thn those of metl. For this reson we hve performed numer of simultions tking for σ, k nd ρ c p in the deposit region some fictitious vlues rnging etween the corresponding ones for wter nd metllic zinc, the other prmeters eing the sme: l = 2 mm, Φ 0 = 10 V, δ c = δ = 0.8 mm, θ = 1.00. The resulting T(y) profiles t t = 400 s re shown in Fig. 9. We see tht the vlues of temperture in the deposit region re incresing if the vlues of σ nd k ecome more nd more closer to tht of wter. In the sme time, the T(y) profile in the rest of the cell is roughly unchnged. From the results presented so fr it is clerly seen tht he temperture profile long the cell, T(y), hs roughly the sme shpe whtever the θ my e, in other words for ll morphologies elonging to he homogeneous growth. However, importnt differences etween the shpes of the temperture profiles in the x direction (prllel to the growth front) re seen when we chnge the occupncy rtio nd implicitly the morphology of the deposit. Such temperture distriutions re shown in Figs. 10 d for l = 2 mm, Φ 0 = 10 V, δ c = δ = 0.8 mm nd three occupncy rtio (θ = 0.23, 0.60 nd 1.00) t t = 400 s. The T(x) profiles re clculted t the distnces y = l - δ c, y = l, y = l + δ c nd y = l + 2δ c from the cthode. These profiles suggest tht importnt grdients of temperture my exist etween the spikes nd in the tip region. The possile influence of such temperture grdients on the cthodic convective rolls is topic tht will e crried out in forthcoming pper. T(x) profiles clculted for different vlues of the cthodic diffusion lyer width δ c, not shown here, conduct to similr results: when δ c increse, T(x) decrese in the deposit re ut increse in front of the deposit; s compred to the occupncy rtio influence, this time the temperture grdients re smller. c d Fig. 10. Clculted temperture profiles in the direction prllel to the growth front t the distnces y = l - δ c (), y = l (), y = l + δ c (c) nd y = l + 2δ c (d) from the cthode, for the occupncy rtio θ = 0.23 (1), θ = 0.60 (2), nd θ = 1.00 (3). (See text for the other prmeters).
Numericl simultion of ohmic heting in idelized thin-lyer electrodeposition cells 277 3.2 Therml field in simplified 3D cells We next consider the cse of more relistic ECD cell, composed of thin lyer of the sme electrolyte, metllic deposit (Zn) nd two diffusion lyers (cthodic nd nodic) confined etween two prllel glss pltes. Due to the limited numer of nodes in the student version of the FlexPDE progrm, the geometry of the thin lyer etween the glss pltes in the (x,y) plne is very simple: the deposit is supposed to e homogeneous nd compct rectngle (occupncy rtio θ = 1), hving the length l nd the width 2L x. Becuse the study in the present work is qulittive one nd tking into ccount the results otined in the previous section, the cse θ = 1 is pproprite for our purposes. We hve performed the sme numericl experiments s in the 2D cells ut with two dditionl prmeters, the thickness t w of the glss pltes nd the thickness t s of the solution etween the pltes. The clculted electricl potentil Φ(y) in the middle plne of the cell hs the sme profile s in the idelized 2D cells nd is not shown here. In wht follows we focus only on the temperture distriution in the cell nd its dependence on some prmeters involved in the prolem. We egin y presenting the two-dimensionl temperture distriution T(x,y) in the middle plne of the cell nd on the externl side of the glss plte t t = 400 s (Fig. 11 nd 11), clculted for the prmeters vlues l = 3 mm, Φ 0 = 10 V, δ c = δ = 1.0 mm, t w = 2 mm nd t s = 0.5 mm. The temperture distriution T(y,z) in the cross section of the cell for x = 0 nd the sme vlues of prmeters is shown in Fig. 11c. More illustrtive for our study re the profiles of the temperture distriution long the cell, T(y), for some given x nd z coordintes nd different vlues of the prmeters involved. We will consider the T(y) profiles in the middle plne of the cell (x = 0, z = t w + t s /2) nd on the externl side of the glss plte (x = 0 nd z = 0 or z = 2t w + t s ). c Fig. 11. Isotherml lines for the clculted therml field in the 3D cell in the middle plne of the cell (), on the externl side of the glss plte () nd in verticl cross section (c); the vlues on the right re in Kelvin. (See text for detils). The clculted T(y) profiles in our 3D cell for the sme prmeters s those in Fig. 11 re shown in Fig. 12, together with the T(y) profile for the sme vlues of l, Φ 0, δ c, δ nd t the sme time ut in the 2D cell with compct deposit. First of ll we oserve tht the T(y) profile in the 3D cell hs the sme shpe s in the 2D cell. Oviously, for the sme vlues of the prmeters the temperture in the 2D cell is higher thn the temperture in the 3D cell ecuse the electric power is the sme ut the heted volume is much lrger in the cse of 3D cell. We lso oserve n importnt difference in the vlues of the mximum temperture in the middle plne of the cell nd on the externl side of the glss plte. In the sme time, we note tht the shpe of the T(y) curves in the two plnes of the cell re different for y etween 0 nd roughly l + 2δ c : s norml, the glss plte hs decresing nd smoothing effect on the temperture profile for the y coordinte lying in this rnge. Due to this smoothing effect of the glss plte on the temperture profile, the resemlnce etween our clculted T(y) profile in 3D cells nd the corresponding experimentl profile in Ref. [4] is etter thn in the cse of 2D cells.
278 P. Brvinschi Fig. 12. Clculted temperture profiles long the 2D nd 3D cells; int nd ext mens the temperture in the middle plne of the cell nd on the externl side of the glss plte, respectively, in the 3D cell. (See text for detils). d Fig. 13. Dependence of the clculted temperture profiles long the 3D cell on the pplied electricl potentil (), cthodic diffusion lyer width (), wll thickness (c), nd electrolyte thickness (d). Other clculted T(y) profiles in the two plnes of the 3D cell showing the dependence of temperture on Φ 0, δ c, t w nd t s re presented in in Figs. 13 d. Keeping t the sme vlues ll the other prmeters nd incresing only the pplied electricl potentil Φ 0, the electricl power ecomes lrger in the sme volume nd the temperture increses too, s is seen in Fig. 13. The explntions for the temperture dependence on the vrition of the cthodic diffusion lyer width δ c (Fig. 13) is the sme s in the cse of 2D cell nd we don t repet them here. New fcts in the cse of 3D cell re the dependences of T(y) on the glss pltes nd solution thickness (Fig. 13c nd 13d, respectively). For constnt vlue of the solution thickness nd decresing the glss pltes thickness, the temperture in the entire cell is incresing. This cn e esily understood: the het source eing the sme, for smller volume of the system its temperture ecomes higher. In the sme time, the difference etween the mximum temperture in the solution nd on the externl side of the glss wll is oviously smller when t w decrese. We hve n opposite sitution when the solution thickness is decresed, keeping the sme vlue for the glss pltes: the electricl power density is the sme ut the volume of the heting source is smller, so the temperture of the system decreses when t s is smller. c Fig. 14. Comprison of the temperture profiles long two 3D cells hving the wlls mde of glss nd Plexigls, respectively, the other prmeters eing the sme (see text).
Numericl simultion of ohmic heting in idelized thin-lyer electrodeposition cells 279 We sved n very importnt oservtion for the end of this section: the y coordinte of the mximum temperture in the solution is lwys smller thn the y coordinte of the mximum temperture on the externl side of the glss wll. A closer inspection of Figs. 12-13 revels tht the shift y etween the coordintes of the two mxim depend on the vrious prmeters involved in the prolem (Φ 0, δ c, etc). Replcing glss with Plexigls (σ = 10-13 S m -1, k = 0.20 W m -1 K -1, ρ = 1190 kg m -3, c = 1470 J kg -1 K -1 ) of the sme thickness, the shift y is even lrger, s is seen in Fig. 14. This fct cn e understood compring the therml diffusion coefficient = k / ρc p for the medi in the cell ( is roughly 4.0 10-7 m 2 s -1 for glss, 1.0 10-7 m 2 s -1 for Plexigls nd 1.4 10-7 m 2 s -1 for wter). Becuse the penetrtion time [11] vries s 1/, the existence of difference etween the position of the two temperture mxim nd its dependence on mteril properties is esy to understnd. 4. Conclusions We hve performed numericl study of the ohmic heting in idelized 2D nd 3D thin-lyer electrochemicl deposition cells. Bsed on experimentl dt, the domin used for the simultion is composed of four regions: deposit, cthodic diffusion lyer, ulk electrolyte solution nd nodic diffusion lyer. We don t tke into ccount the growth of the deposit nd the trnsport of ions y diffusion nd convection, our model eing sttic nd pure ohmic one. The mthemticl model consist of two PDE, Lplce-type eqution for the electricl potentil nd the time-dependent eqution for het conduction. They re simultneously solved y the finite element method using progrm clled FlexPDE. We hve computed the electricl potentil, electricl field strength, electricl current density, nd the temperture distriution in the entire 2D nd 3D cells. We were le to mde prmetric study of the temperture distriution, our results eing confirmed y some experimentl fcts nd pointing out on other spects elonging to thin-lyer electrodeposition topic. The min result of our simultions is qulittive one nd consists in the resemlnce of the clculted temperture profiles long the cell nd the corresponding profiles otined experimentlly in [4]. The fetures of the clculted profiles re the sme for ll the vlues of the prmeters we hve used nd this mens tht the temperture distriution long the cell is due to the specific structure in four regions of n ECD cell. The temperture increses otined in our clcultions re smller thn those mesured nd reported in [4]. This is not surprising ecuse our model is very simplified version of the rel world: we hve not considered the growth process, the trnsport y diffusion nd convection, nd ny other het genertion process thn the ohmic heting. It is rel chllenge to include in model wht is missing in the simultion we hve performed until now ut the work is in progress. Even with our very simple model we hve otined some new results: the temperture profile T(x) in the direction prllel with the growing front, the shift of the temperture mximum position in 3D cell, nd the dependencies of T(y) nd T(x) profiles on vrious prmeters involved in the clcultion. The confirmtion of these results using more sophisticted models my hve some implictions on the experimentl interprettion of concentrtion, flow, therml nd electricl mesurements performed in such systems. All these will lso help in future studies devoted to the ohmic effects on morphology selection in thin lyer electrochemicl deposition. Acknowledgments The uthor is thnkful to PDE Solutions Inc. for providing the student version of the FlexPDE solver. References [1] J. Newmn, Electrochemicl Systems, J.Wiley & Sons, New York (2004). [2] F. Sgues, M. Q. Lopez-Slvns & J. Clret, Physics Reports 337, 97 (2000). [3] D. P. Brkey, D. Wtt, Z.Liu & S.Rer, J. Electrochem. Soc.141, 1206 (1994). [4] M. Schrötter, K. Kssner, I. Reherg, J. Clret & F. Sgues, Phys.Rev. E 66, 026307 (2002). [5] C. Leger, J. Elezgry & F. Argoul, Phys. Rev. E 61, 5452 (2000) nd references therein. [6] P. Brvinschi, Anlele Universittii de Vest din Timisor, Seri Fizic, 45, 158 (2004). [7] D. Doos, Electrochemicl Dt, Elsevier, New York (1975). [8] R. C.West (Ed.), CRC Hndook of Chemistry nd Physics, CRC Press Inc., (1977). [9] J. -N. Chzlviel, Phys. Rev. A 42, 7355 (1990). [10] PDE Solutions Inc. USA; www.pdesolutions.com. [11] F. White, Het nd Mss Trnsfer, Addison-Wesley Pulishing Compny, Reding, Msschusetts (1988). * Corresponding uthor: prvi@physics.uvt.ro