Except fom the Poceedngs of the OMSOL onfeence 9 Mlan lasscal Models of the Inteface between an Electode and an Electolyte E. Gongadze *, S. Petesen, U. Beck, U. van Renen Insttute of Geneal Electcal Engneeng, Unvesty of Rostock Insttute of Electonc Applances and cuts, Unvesty of Rostock *oespondng autho: Albet-Ensten-St., 859 Rostock, Gemany, ekatena.gongadze@un-ostock.de Abstact: An electcal double laye (EDL) plays a majo ole n undestandng the nteface between a chaged suface (e.g. an mplant) and onc lquds (e.g. body fluds). The thee classcal models of the EDL (elmholtz, Gouy hapman and Sten Model) ae numecally solved fo a flat suface electode n the D Electostatcs mode of omsol Multphyscs.5a Softwae. The values of the electc potental dop nea the electode s suface ae compaed and t s shown the aea of valdty. The double laye capactance s computed analytcally, numecally and measued by Electochemcal Impedance Spectoscopy (EIS) and t has been shown that the classcal models do not agee wth the expemental measuements. Keywods: Electcal double laye, capactance, FEM, smulaton. Electcal Double Laye The pesent pape s concened wth the compason of the classcal models of the EDL, whch consde the as a contnuum delectc solvent and dlute onc soluton. The electode s pefectly polazed and any chemcal eactons on t ae neglected. A ccula plate condensato wth electode adus R nm and potentals of ±5 mv, flled wth Sodum hlode (Nal) wth concentaton of. and. M s egaded hee.. elmholtz Model The fst and smplest double laye model, ceated n 879 by elmholtz [4], consdeed the concept of the chage sepaaton at the nteface between a metallc electode and an soluton.. Intoducton The clncal success of an mplant depends on the pofound knowledge of the nteacton between the bomateal and the cells. The contact between the mplant and the body fluds esults nto fomaton of an EDL. Ths double laye conssts of a laye of electons (f the nonelectolytc phase s a metal o electonc conducto), a laye of adsobed ons, and a dffuse double laye wth an onc atmosphee, whee ons wth a sgn opposte to the electode suface ae found to be pesent n excess compaed to the bulk. The EDL s fomed smultaneously afte the contact of the electode wth the and esults nto a fall of the electc potental, assstng potens adheson and the esultng cell speadng. Theefoe, EDL has a huge mpact on the analyss and smulaton of electcal nteactons of mplants wth the bosystem. Fgue. elmholtz Model The electode holds a chage densty (σ M ) asng fom ethe an excess (-σ M ) o defcency (+σ M ) of electons at the electode suface. The chage on the electode s balanced by edstbuton of the ons n the soluton by an equal but
oppostely chaged amount of ons. The esult s two layes of opposte chage sepaated by some dstance ld/ lmted to the adus d/ of the attacted ons and a sngle laye of solvaton aound each on (fg.). The lne dawn though the cente of such ons maks the bounday known as the Oute elmholtz Plane (OP) and the egon wthn t the electcal double laye. The potental n the elmholtz laye s descbed by the Posson s equaton n D, whch elates the potental wth the chage dstbuton ϕ ρ( x), () whee φ s the electc potental, ρ s the chage densty, x s the dstance fom the electode, s the pemttvty of vacuum, s the elatve pemttvty of the medum. The appoach teats the ons as pont chages and ths allows us to ewte Eq-n () between the two layes to ϕ. () Ths behavou s compaable to the classcal poblem of a paallel-plate capacto.e. EDL s capable of stong electc chage. Theefoe, the double laye capactance pe unt aea s gven as:, () l whee l s the thckness of the double laye. Fo 8.854 F / m, 78. 5 and 9 l. m, we get cm.4 /. (4) The model does not account fo the dependence of the measued capacty on potental o concentaton. Anothe dawback s the neglect of nteactons that occu away fom the OP.. Gouy-hapman Model Gouy and hapman [, ] wee the fst to consde the themal moton of ons nea a chaged suface. They pctued a dffuse double laye (DDL) consstng of counteons (.e. ons of opposte chage to the suface), whch ae attacted to the suface and co-ons epelled by t embedded n a delectc contnuum descbed by the Posson - Boltzmann (PB) dffeental equaton. Fgue. Gouy-hapman Model The dstbuton of ons s descbed by the Boltzmann dstbuton: zeϕ n n exp, (5) kt whee n s the concentaton of on n the bulk, e s the unt chage, z - chage on the on, k Boltzmann constant, T absolute tempeatue. The total chage densty pe unt volume fo all onc speces s the sum ove all ons: zeϕ ρ ( x) n z e n z eexp. (6) kt ombnng Eq. () and Eq. (6) leads to the Posson-Boltzmann equaton: ϕ e zeϕ n z exp. (7) kt By usng the popety of devatves, ϕ (8)
PB equaton can be solved as e zeϕ d n z exp dϕ. (9) kt Fo the followng bounday condtons, x x ϕ ϕ ϕ () whee φ s the potental at the electode, the ntegaton yelds to n kt zeϕ exp kt. () Fo a symmetcal (z:z), Eq. () has the fom / 8kTn zeϕ snh. () kt The chaactestc dstance fo the dffuse laye thckness s gven as / kbt κ. () n z e The chage densty of the dffuse laye s used as In ode to assess the valdty of the model, the followng paametes φ 5 mv, z, c. M ae chosen and the dffeental capactance s detemned as G 77.6 / cm. (6) The Gouy-hapman Model s a contnuum meanfeld-lke appoach assumng pont-lke ons n themodynamc equlbum and neglectng statstcal coelatons. Fo low concentaton s, ths theoy has been successful n pedctng onc pofles close to plana and weakly cuved sufaces and the esultng foces. oweve, t s known to oveestmate stongly onc concentatons close to chaged sufaces. In patcula, ths shotcomng of the PB theoy s ponounced fo hghly chaged sufaces and multvalent ons.. Sten Model In 94 Sten [6] smply developed the doublelaye theoy by suggestng a moe ealstc way of descbng the physcal stuaton at the nteface. e combned the two pevous models by adaptng the compact laye of ons used by elmholtz and next to the dffuse laye of Gouy- hapman extendng nto the bulk soluton. e took nto account the fact that ons have fnte sze, and consequently the closest appoach of OP to the electode wll vay wth the onc adus. σ M x / zeϕ ( 8kTn ) snh kt. (4) By dffeentatng, the dffeental capactance s obtaned as G dσ M dϕ z e n kt. (5) zeϕ cosh kt / Fgue. Sten Model
In mathematcal tems the dffeental capactance of the double laye s s equvalent to two capactos n sees o and espectvely fo the above descbed models we mplement the.h.s. of Eq-ns (), () and a combnaton of them n the Sten Model. s +, (7) G whee s the capactance of the chages held nsde the OP and G s the capactance of the dffuse laye. By substtutng the aleady eceved values, we obtan s 57.9 / cm. (8). Bounday ondtons Symmety φ φ contnuty Electc potental. Numecal Smulatons wth OMSOL Multphyscs. Geomety and Subdoman Paametes The D Electostatc mode of the A/D module s used to smulate the thee classcal models of the EDL as an deal paallel-plate capacto. In coespondence wth the expemental setup, the capacto has a cylndcal fom wth a damete and length of nm. Addtonal plates n a dstance of. nm coespondng to the adus of a hydated on ae ncluded. φ φ φ -φ -φ -φ Symmety Fgue 5. Bounday condtons The bounday condtons fo the elmholtz Model was chosen so that the electc potental of the electode φ φ and of the OP at a dstance of. nm to be set to contnuty meanng that the nomal component of the electc dsplacement s contnuous acoss the nteo bounday. Smlaly, the Gouy-hapman Model was mplemented, but ts nne plate was placed n a к - dstance. Sten model s smply a combnaton of the latte ones. The oute boundaes ae chosen n the way that the nomal component of the electc dsplacement to be zeo. ø nm Electolyte: Nal Electode. Mesh Geneaton nm. nm. nm Fgue 4. The model geomety In the D Electostatcs Mode, the followng equaton s solved ( ϕ) ρ (9) Fgue 6. Mesh geneaton fo the used geomety
Fo the models, a tangula mesh (Fg. 6) was used wth a fne mesh between the electode and the OP and a coase one between the two opposte electodes. The numbe of degees of feedom solved fo all models s aound and the soluton tme nealy 4 s. suface and the OP and the numecal smulaton confms ths esult. The electc potental dop n the Gouy-hapman Model has an exponental fom (Fg. 8), whch was valdated by vayng the Nal concentaton between. and. M..4 Postpocessng The capactance fo all models s calculated by fndng the chage obtaned by ntegatng ove aea A the electc dsplacement multpled by the nomal vecto (nomd_emes) D nda () Γ () Δϕ Fo the elmholtz Model, the paametes ae obtaned as ϕ.5v ϕ.485v.978 8 and the dffeental capactance s.69 / cm. () ϕ ϕ Ths value concdes absolutely to the analytcal esult as obtaned n Eq. (4). Fgue 8. Electc potental dstbuton fo Gouy- hapman Model wth φ 5 mv and Nal concentaton of. and. M. The paametes used hee ae ϕ.5v ϕ.57v 5.765 G thus 8 G G 7.74 / cm. () ϕ ϕ As the analytcal value takes nto account the whole, we need to add the effect of the est chage futhe away fom the Debye length к - as ϕ.57v ϕ ϕ 4 electolyt e 5.454-9 Fgue 7. Electc potental dstbuton fo elmholtz Model wth φ 5 mv and Nal concentaton of. and. M. The electc potental n the elmholtz Model (Fg.7) has a lnea fall between the electode s.9 / cm. (4) ϕ ϕ4 Summng up Eq. () and Eq. (4) gves G 77. / cm. (5) As we have seen that appoxmaton s easonable and povdes a consstent esult wth the theoy and Eq. (6).
In Sten Model, the electc potental has a lnea fall wthn the OP and an exponental one n the dffuse laye (Fg. 9). and takng nto account the capactance n the est of the by summng up Eq. (8) and (9), the s s obtaned as s 57.657 / cm () whch s a good appoxmaton of Eq. (8). The compason between the models (Fg.) shows that fo low concentaton s lke. M Nal, the Sten Model has smla behavou to Gouy-hapman Model. Fgue 9. Electc potental dstbuton fo Sten Model wth φ 5 mv and Nal concentaton of. and. M. The paametes fo the Sten Model ae as follows: ϕ.5v ϕ.4v ϕ.98v ϕ ϕ 4 G 7.4756 4.5644 yeldng to 8 8 5.49.7 / cm (6) ϕ ϕ -9 Fgue. ompason of the elmholtz, Gouy- hapman and Sten Model fo Nal concentaton of. M 4. Expemental Measuements The electochemcal expements [5] wee pefomed by a thee-electode technque n a glass contanng 8 cm of a phosphate-buffeed salne PBS (p 7.) soluton. A T specmen seved as wokng electode and as efeence system a satuated calomel electode KE- (Sensotechnk Mensbeg, Gemany). The counte electode conssted of a platnum sheet placed n a -mm dstance plana to the wokng electode. All measuements wee pefomed at oom tempeatue (+- ). G G 68.88 / cm (7) ϕ ϕ 4.5 / cm (8) ϕ ϕ4 Fnally, the dffeental capactance n Sten Model s equal to s + 5.957 / cm (9) G Fgue. Expemental setup fo EIS
Electochemcal mpedance spectoscopy (EIS) [] was pefomed wth the electochemcal measung system IM6e (ZANER, Gemany). We measued n the fequency ange of mz to kz n the sngle sne mode wth an ac ampltude of mv wth espect to open-ccut potental (OP). The EIS data wee analysed usng TALES softwae fom ZANER. Voltammetc expements wee also pefomed wth the Zahne system to get a quasstatonay cuent-potental cuve. The potental scans wee caed out fom -.5 V (SE) to V (SE) n anodc decton wth a scan ate.5 mv/s. Values fo OP, cooson cuent ( co ), cooson esstance (R co) and cathodc Tafel slope wee obtaned by classcal Tafel analyss of the cathodc banch. A chonoampeometc expement was used to detemne the amount of chage equed fo eloadng of the double laye capacty fo a mv potental jump. The esultng cuent tansents wee ecoded fo s wth a esoluton of ms and ntegated. The capactance obtaned fo a polshed T s aound 6μ F / cm. 5. Dscusson The elmholtz Model of the double laye seems to be appopate fo polasable electodes n suffcently hgh concentatons of (> M). At lowe concentatons (<. M), new featues appea n the measuement of the double laye capactance as a functon of potental whch cannot be explaned by the elmholtz Model. The Gouy- hapman Model s a contnuum meanfeld-lke appoach assumng pont-lke ons n themodynamc equlbum and oveestmates stongly onc concentatons close to the chaged suface. Table : Dffeental capactance values of the EDL Even though Sten made a key mpovement of the model, the capactance value eceved by measuement s much smalle than that obtaned analytcally and numecally (Table ). 6. oncluson Geneally, the classcal models descbe the fundamental behavou of the ons nea the electode s suface fomng the double laye. oweve, they gnoe key factos as on-on coelatons, electostatc mage nteactons, stec effects, ovelappng of ons leadng to nconsstency wth the expemental esults. Theefoe, as the am of ou futue wok s the mplementaton of an EDL model on non-plana electodes wth well-pedefned geometcal stuctues, we need anothe appoach, whch wll ncopoate all these factos. 7. Refeences. Bad, A., Faulkne, L., Electochemcal methods: Fundamentals and Applcatons, John Wley and Sons, Inc. (). hapman, D. L., A contbuton to the theoy of electocapllaty, Phlos. Mag. 6, (9). Gouy, M. G., Su la consttuton de la chage electque a la suface d'un, J. Phys. Radum, 457 468 (9) 4. elmholtz,., Studen übe elektsche Genzschchten, Ann. Phys., 7-8 (879) 5. Kbs, A., Lange, R., Nebe, B., Rychly, R., Baumann, A., Neumann,.-G., Beck, U., Methods fo the physcal and chemcal chaactesaton of sufaces of ttanum mplants, Mateals Scence and Engneeng, 45-49 () 6. Sten, O., Z. Elektochem.,, 58 (94) 8. Acknowledgements We thank DFG fo fundng ou poject n Reseach Tanng Goup 55/ welsa. dl [µf/cm ] analytcal dl [µf/cm ] numecal elmholtz Model Gouy- hapman Model Sten Model Expement.4 77.6 57.9 6.69 77. 57.6 6