Cyclic voltammetry simulation at microelectrode arrays with COMSOL Multiphysics Ò

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1 J Appl Eletrohem (009) 39:9 63 DOI 0.007/s ORIGINAL PAPER Cyli voltmmetry simultion t miroeletrode rrys with COMSOL Multiphysis Ò Alessndro Lvhi Æ U. Brdi Æ C. Borri Æ S. Cporli Æ A. Fossti Æ I. Perissi Reeived: Septemer 008 / Aepted: Jnury 009 / Pulished online: Jnury 009 Ó Springer Siene+Business Medi B.V. 009 Astrt The present pper reports the results otined pplying the generl purpose softwre COMSOL Multiphysis Ò to the finite elements simultion of Cyli Voltmmetries (CV s) t miroeletrodes rrys (MEA). CV s t inlid miro disk eletrode rrys hve een simulted enhmrking our results with those otined y Compton with the finite differene method. Then the influene of meshing on the qulity of the simulted dt hve een investigted showing tht d meshing my provide shpes with no physil mening. Simultions hve lso een performed on reessed miro disk rrys in order to show the effet of the depth of the reess on the voltmmetri wve shpe. We found tht COMSOL Multiphysis Ò provides flexile nd strightforwrd route to the simultion of eletrohemil systems with omplex geometry. Keywords Cyli voltmmetry Miroeletrodes rrys (MEA s) Finite elements COMSOL Multiphysis Ò Introdution The use of miroeletrode rrys (MEA s) in eletrohemistry is quikly rising. Min resons re etter signl to noise rtio nd lower pitne ompred to tht of A. Lvhi (&) U. Brdi C. Borri S. Cporli A. Fossti I. Perissi Deprtment of Chemistry, University of Firenze, Vi dell Lstrui, 3, 009 Sesto Fiorentino, FI, Itly e-mil: lessndro.lvhi@unifi.it lrge plnr eletrode. Sttionry voltmmetri wveforms re esily otined for CV s even t high potentil sn rtes. These fetures mke MEA s n ttrting hoie for eletronlytil sensors [, ]. The theoretil determintion of the eletrohemil response of MEA s is not s strightforwrd s tht of simple plnr geometries for the onvolution of physil nd geometri effets. Only nlytil pproximted expressions hve een derived for the predition of sttionry CV s urrent in the se of miroeletrodes [3]. Sine the optimum signl is otined under rdil diffusion onditions the simultion plys here key role. The overlp of the diffusion louds from neighorhood eletrode my destroy the rdil ehvior moving progressively to liner diffusion onditions. The result is lowering of the urrent density nd the ourrene of pek shped CV s. In the MEA s design suh ondition hs to e voided. The lultions performed in the present pper im t ssessing the use of COMSOL Multyphysis Ò s tool for the simultion of CV s t MEA s. The min purpose is to provide n id to the design of MEA s y n priori nlysis of the diffusion ehvior of the systems s funtion of miroeletrodes rdius nd intereletrodi distnes. An ssessment of the method in terms of the qulity of the results is provided, ompring our findings with those otined y Compton et l. for nlogous inlid MEA s []. Meshing is usully ritil in the finite elements espeilly when diffusion t edges is involved. The effet of mesh refinements with speil fous on eletrode edges hs een tken into ount to determine the numer of elements whih gurntee suffiiently urte results within the lower omputtion time. More omplex geometry re often enountered in MEA s. Tht is the se of reessed miroeletrode. Here CV s n e suessfully simulted with numeril methods.

2 60 J Appl Eletrohem (009) 39:9 63. Equtions All the simultion performed in the pper ssumed the following eletrohemil retion: A þ e B: ðþ Chrge lne hs not een tken into ount ording to the supporting eletrolyte hypothesis whih will e disussed lter. For the () we ssumed E 0 = 0 V. The whole mss trnsport prolem relted to retion () is desried y ouple of Nernst Plnk equtions. o i ot þr ð D ir i u i z i F i r/ þ i u~ Þ ¼ R i ; ðþ where i is the onentrtion of A nd B; D i the diffusion oeffiient of A nd B; u i the moility of A nd B; z i the hrge of A nd B; F the Frdy s onstnt (96,87 C mol - ); / the eletri potentil; u~ the veloity field; R i the retion term for A nd B. The terms due to the migrtion nd onvetion re negleted here. The ssumption is justified y the presene of supporting eletrolyte nd the sene of onvetion, oth ommon fts in dynmi eletrohemistry investigtions. Also the retion term hs een negleted, sine no hemil retions of the eletrotive speies hve een ssumed for the ulk of the eletrolyte. Under these onditions Eq. redues to the time dependent diffusion lw s desried y ouple of Fik s seond equtions: o i ot þr ð D ir i Þ ¼ 0: ð3þ Diffusion oeffiients were set to 0-9 m s - for oth A nd B. The effet of the hrge trnsfer kineti ws inluded in the mthemtil formultion of the prolem in the oundry onditions. The () is ssumed to e hrge trnsfer ontrolled retion, so the Butler Volmer eqution hold (): i ¼ i 0 exp AnF RT g exp CnF RT g ; ðþ where i 0 is the exhnge urrent density; g the overpotentil (V - E 0 ); n the numer of exhnged eletrons; A the nodi hrge trnsfer oeffiient; C the thodi hrge trnsfer oeffiient; R the gs onstnt; T the temperture. Eqution is not suitle for the use s flux oundry ondition for mss trnsport prolem, sine it is expressed in terms of urrent densities. To use it in our model we need to rerrnge it to Eq. : M ¼ C ox Kf C red K; ðþ where M is the mss flow density expressed s the numer of moles rossing the unit surfe in the time unit nd Kf nd K re given y Eqs. 6 nd 7, respetively: Kf ¼ K het e A ð nfg RT Þ ; ð6þ K ¼ K het e C ð nfg RT Þ : ð7þ In our simultions K het ws set to 0 - ms -, A nd C were set to 0. while the numer of exhnged eletrons n nd the temperture were fixed to nd 98 K, respetively. All the simultions hve een performed for single CV yle etween 0. nd -0. V with sn rte of 0. V s -. Computtion The simultion of CV s t squre rrys of oth inlid nd reessed mirodisk hve een onsidered here. To redue the prolem from 3D to D xil symmetry to limit the omputtion time, we pplied the domin wll pproximtion desried in [ 8] s shown in Fig.. A sketh of the ross setion of the simultion ell is shown in Fig. for the inlid MEA s nd Fig. for the reessed MEA s. Aording to the domin wll pproximtion the reltion etween the inter-eletrode distne d nd the R 0 prmeter is the following: R 0 ¼ 0:6d: ð8þ The COMSOL Nernst Plnk without eletroneutrlity pplition mode hs een seleted. Two equtions hve een set to ount for the trnsport of oth A nd B. Referring to Fig., insultion/symmetry oundry Fig. The diffusion domin pproximtion for regulr rry of miroeletrodes with ui geometry, s desried in [ 8]

3 J Appl Eletrohem (009) 39: Z mx R R 0 3 R h R 0 3 Fig. Sketh of the simultion geometries: inlid nd reessed. Z mx is the height of the simulted ell nd h is the reess depth onditions hve een set for oundries, 3,, for the inlid eletrode nd, 3,,, 6 for the reessed se. Eqution hs een set s flux oundry ondition for the oundry for oth the inlid nd reessed mirodisk. In order to void effets due to the finiteness of the domin, the simultion ell height ws set lrger thn the mximum diffusion lyer thikness. Aording to wht reported in the literture [3] we defined the ell height Z mx s: p Z mx ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffi ; ð9þ 6Dt mx where t mx is the time required for the forwrd prt of the CV sn nd D is the diffusion oeffiient. The domin hs een meshed with Lgrnge tringulr qudrti elements. Tringulr elements hve een seleted to llow lol mesh refinement. The simultions of ny of the presented CV s hve een performed with suessive mesh refinement until no ppreile hnges ourred in the simulted urves. All the simultions hve een rried out on linux PC equipped with G of rm nd 3.60 GHz Intel Pentium proessor. The COMSOL Multyphysis relese used for the present investigtion ws the Results 3. Inlid mirodisk eletrodes Figure 3 report the results of the simultion of the CV urves t three MEA s, respetively, with R of 0.,, nd 0 lm nd R 0 fixed t 0 lm. Figure 3 shows tht MEA s with 0. lm miroeletrode provide CV shpe whih is sttionry wveform. As the eletrode size inreses, the shpe of the urves moves progressively to pek shped CV inditing the ourrene of liner diffusion (Fig. 3, ). The ft is explined onsidering tht diffusion louds overlp when the rtio etween R nd R 0 eomes lrger. Suh superimposition provides diffusion to propgte lmost orthogonlly to the eletrode surfe pprohing the limit ondition of n infinitely lrge plnr Fig. 3 Cyli Voltmmetry simulted on miroeletrode rry with R = 0. lm nd R 0 = 0 lm, R = lm nd R 0 = 0 lm nd R = 0 lm nd R = 0 lm eletrode. If the R /R 0 rtio is smll diffusion ours in rdil wy with negligile overlp. Simultions hve een performed t onstnt eletrode rdius (R = 0 lm) nd vrying the distne mong the eletrode enters. Here we notied tht the CV is dominted y the liner diffusion for R 0 = 0 lm (Fig. ), nd progressively pproh to sttionry wveform when R 0 is inresed to 0 (Fig. ) nd 00 lm (Fig. ). In our simultions we found tht the meshing is ritil espeilly for the smller eletrode size (0. lm) s expeted. Bd meshing n result in urve with no physil

4 6 J Appl Eletrohem (009) 39: x Potentil (V) Fig. Cyli Voltmmetry simulted on miroeletrode rry with () R = 0 lm nd R 0 = 0 lm, () R = 0 lm nd R 0 = 0 lm nd () R = 0 lm nd R 0 = 00 lm mening s those showed in Fig. nd. Here the urrent rises up even in the kwrd sn in the rnge etween -0.3 nd 0 V. A lol mesh refinement in the eletrode s region (oundry ) is needed to orretly solve the prolem. The shpe of the urve in ft modifies pssing from,07 elements, to,636 elements nd finlly to 3,9 elements (Fig., ). The ltter simultion produes orret sttionry urve s reported in Fig.. Further refinement of the mesh size in the oundry does not provide ny signifint vrition of the urrent, while inreses drmtilly the omputtion time. Tle lists the numer of elements generted for eh of the performed simultion nd the omputtion times otined with our onfigurtion. Tle reports the mximum urrent densities otined in this investigtion nd those otined y Compton et l. [] with finite the differene method for R 0 = 00 lm nd R set to 0,, nd 0. lm. Results re in good greement nd demonstrte the effetiveness of the use of multipurpose softwre for suh simultions. 3 0 x Potentil (V) Fig. Cyli Voltmmetry simulted on miroeletrode rry with R = 0. lm nd R 0 =0lm; () Numer of elements:,07; () Numer of elements:,636; () Numer of elements: 3,9 3. Reessed mirodisk eletrodes The prmeter investigted here is the depth of the reess. R 0 nd R were fixed t 0 nd 00 lm, respetively. Figure 6 shows the result of the simultion of reessed mirodisk eletrode with reess depth of 0 lm. The wveform here is lose to tht of Fig.. The reson is trivil. The depth of the reess is just 0 lm nd the diffusion loud quikly exit from it rehing the rdil diffusion regime. The sitution hnges lot when the depth inreses to 33 lm. The CV (Fig. 6) is lerly pek shped. At higher overpotentil the response pprohes to sttionry one. Tht is euse the pek ours when the diffusion loud propgtes inside the reess, ut very losed to top of it. So just few moments fter rehing the pek the loud strts to propgte in rdil wy providing gin sttionry nswer. Figure 6 reports the CV wveform t reessed mirodisk eletrode rry with reess depth of 00 lm. The voltmmetri wve shows here pronouned pek nd no mssive trnsition to rdil diffusion regime ours. During the whole simulted time the diffusion front propgtes mostly inside the reess gin in liner diffusion fshion. The min differene etween CV from lrge plnr eletrodes ours t very lrge overpotentil. The slope of the urve here is less pronouned inditing tht the ontriution of the rdil diffusion outside the reess is not negligile. Tle shows tht there re not ppreile differene in the omputtionl s funtion of the depth of the reess, nd tht the solution times re quite in the sme rnge of those otined for the inlid MEA s. Conlusions We proved tht the finite element engine of COMSOL Multiphysis Ò n solve the eqution governing diffusion omplex eletrohemil systems. First the method hs een ssessed y the simultion of systems whih CV response hs just een reported in the literture finding omplete greement nd stressing the importne of meshing. We found tht d meshing n led to urve with no physil mening. In order to get urte results refinement of the mesh, until no further hnges in the shpe of the simulted urve is oserved, hs to e performed. To void n exessive demnding of omputtionl resoures the improvement of meshing my our trough lol mesh refinement on ritil points suh s those representing the eletrode edges. Applying suh hnges in the COMSOL grphi user interfe is strightforwrd tsk. We lso showed tht the pplition of the model n e extended to the simultion of even

5 J Appl Eletrohem (009) 39: Tle Time of omputing for different geometries with numer of mesh elements Geometril prmeters Numer of elements Solution time (s) R = 0. lm, R 0 = 0 lm (Fig. ),07 3 R = 0. lm, R 0 = 0 lm (Fig. ),636 7 R = 0. lm, R 0 = 0 lm (Fig. ) 3,9 3 R = 0 lm, R 0 = 0 lm 3,787 R = 0 lm, R 0 = 0 lm,0 38 R = 0 lm, R 0 = 00 lm,7 00 R = 0 lm, R 0 = 00 lm, h = 0 lm 3, R = 0 lm, R 0 = 00 lm, h = 33 lm 3,70 0 R = 0 lm, R 0 = 00 lm, h = 00 lm 3,69 36 R = 0. lm, R 0 = 00 lm,3 87 R = lm, R 0 = 00 lm 0,86 83 R = 0 lm, R 0 = 00 lm 0,08 87 Tle Comprison of the limiting urrents (I p ) for some geometril prmeters with the ones otined y Compton et l. [] Geometril prmeters I p (A) I p [] (A) R = 0. lm, R 0 = 00 lm R = lm, R 0 = 00 lm R = 0 lm, R 0 = 00 lm x Potentil (V) Fig. 6 Cyli Voltmmetry simulted on reessed miroeletrode rry with R = 0 lm, R 0 = 00 lm nd reess depth: () h = 00 lm; () h = 33 lm; () h = 0 lm more omplex geometries suh s reessed mirodisk MEA s, providing vlule tool for the estimtion of the eletrohemil ehvior of system whih nnot e urtely desried y pproximted nlytil solution. Here we explored the effet of the reess on the shpe of the CV showing tht rdil or liner diffusion ours for smll (0 lm) or lrge (00 lm) reess depth ompred to the diffusion length nd finding the shpe of the CV for n intermedite se (33 lm). In the end we found tht COMSOL Multyphysis Ò is n exellent nd flexile tool for the simultion of omplex eletrohemil systems whih n esily help to explore the ehvior of rel systems. The more omplex is the geometry nd the lrger is the dvntge in using suh softwre. Under suh ontext work with the min purpose of ssessing the ury of MEA s sed eletronlytil determintion due to the geometri tolernes of the prodution proess is urrently under progress nd will e the sujet for future pulition. Referenes. Aoki K (993) Eletronlysis :67. Arrign DWM (00) Anlyst 9:7 3. Britz D (00) Digitl simultions in eletrohemistry. Springer, Berlin. Dvies TJ, Compton RG (00) J Eletronl Chem 8:63. Amtore C, Sve 0 nt J-M, Tessier D (983) J Eletronl Chem 7:39 6. Dvies J, Wrd-Jones S, Bnks CE, del Cmpo J, Ms R, Muñoz FJ, Compton RG (00) J Eletronl Chem 8: 7. Gvghn DJ (996) J Eletronl Chem 0:7 8. Dvies TJ, Bnks CE, Compton RG (00) J Solid Stte Eletrohem 9:797