Two-Dimensional Lithium-Ion Battery Modeling with Electrolyte and Cathode Extensions

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1 Advance n Chemcal Engneerng and Scence, 2012, 2, Publhed Onlne October 2012 ( Two-Dmenonal Lthum-Ion Battery Modelng wth Electrolyte and Cathode Extenon Glyn F. Kennell, Rchard W. Evtt Department of Chemcal and Bologcal Engneerng, Unverty of Sakatchewan, Sakatoon, Canada Emal: GlynKennell@uak.ca Receved July 3, 2012; reved Augut 6, 2012; accepted Augut 18, 2012 ABSTRACT A two-dmenonal model for tranport and the coupled electrc feld appled to mulate a chargng lthum-on cell and nvetgate the effect of lthum concentraton gradent wthn electrode on cell performance. The lthum concentraton gradent wthn electrode are affected by the cell geometry. Two dfferent geometre are nvetgated: extendng the length of the electrolyte pat the edge of the electrode and extendng the length of the cathode pat the edge of the anode. It found that the electrolyte extenon ha lttle mpact on the behavor of the electrode, although t doe ncreae the effectve conductvty of the electrolyte n the edge regon. However, the extenon of the cathode pat the edge of the anode, and the poblty for electrochemcal reacton on the flooded electrode edge, are both found to mpact the concentraton gradent of lthum n electrode and the current dtrbuton wthn the electrolyte durng chargng. It found that concentraton gradent of lthum wthn electrode may have tronger mpact on electrolytc current dtrbuton, dependng on the level of completene of cell charge. Th becaue very dfferent gradent of electrc potental are expected from mlar electrode gradent of lthum concentraton at dfferent level of cell charge, epecally for the L x C 6 cathode nvetgated n th tudy. Th lead to the predcton of gnfcant electrc potental gradent along the electrolyte length durng early cell chargng, and a reduced rk of lthum depoton on the cathode edge durng later cell chargng, a een expermentally by other. Keyword: Lthum-Ion Cell; Mathematcal Modelng; Cathode Extenon; Electrolyte Extenon; Current Dtrbuton; Electrc and Concentraton Feld 1. Introducton Lthum-on cell tore and releae energy va the emon, tranport and nerton of lthum-on from/nto electrode materal at dfferent electrochemcal potental. Th dfference n potental may be becaue the electrode are compred of dfferent materal, becaue of an externally appled electrc potental, and/or may alo be becaue of the tochometrc coeffcent of lthum already preent n the electrode. At the end of electrode are edge. Lthum-on may be produced and conumed at thee edge regon f they are n contact wth the electrolyte, uch a when the electrolyte flooded. At flooded electrode edge the geometry of the edge may caue mult-dmenonal effect, uch a concentraton gradent n the electrolyte and the electrode, and alo electrc potental gradent n the electrolyte. Thee effect are the focu of th paper. It ha prevouly been found that negatve conequence to cell performance may are due to the concentraton gradent aocated wth the flooded electrode edge. Thee conequence nclude the ncreaed rk of lthum depoton at the cathode edge regon. Therefore, a cathode edge may be extended pat the anode edge to reduce the lkelhood of lthum depoton at the cathode edge regon; however, th may reult n other problem. Some of thee were expermentally oberved by Scott et al. [1,2], and nclude a relatvely large electrc potental drop along the length of the electrolyte, parallel to the electrode, and aocated wth the extended cathode edge. Wet et al. [3] developed a one-dmenonal model accountng for the tranport n the electrolyte and electrode phae of a cell wth porou nerton electrode and a lqud electrolyte. It wa demontrated how electrolyte depleton wa the prncpal factor lmtng the dcharge capacty of the ytem. Doyle et al. [4] preented a model for a lthum-on cell that wa mplemented by conderng one-dmenonal tranport for a galvanotatc current. It wa found that the decreaed lthum concentraton n the compote cathode llutrated the need for hgher lthum concentraton. Th model wa expanded by Fuller et al. [5] who condered a porou nerton anode ntead of a lthum fol anode.

2 424 G. F. KENNELL, R. W. EVITTS Tranport wa condered one-dmenonally. Arora et al. [6] ued the model of Fuller et al. [5] for one-dmenonal lthum-on battery predcton. They concluded that lthum depoton may occur n cell wth lower exce negatve electrode capacty. Tang et al. [7] preented a two-dmenonal model for the nvetgaton of lthum depoton. They utlzed a COMSOL Multphyc model (baed on dlute oluton theory) to explan why extendng the cathode edge may decreae the tendency for lthum depoton durng cell chargng. Some of the aumpton made by Tang et al. were: contant and unform electrolyte concentraton and conductvty, unform anode concentraton wth repect to poton, lnearzed Tafel knetc, old flm electrode, and electrolyte electroneutralty. Tang et al. howed that for ther model a cathodc extenon of 0.4 mm uffcent to prevent the onet of lthum depoton. Eberman et al. [8] ued a two-dmenonal model baed on concentrated oluton theory to model the effect of a cathode under-lap (the oppote to a cathode extenon). Eberman et al. ued th model to conduct a entvty analy of varou factor on the rk for lthum depoton. They found the three mot gnfcant factor affectng the rk of depoton to be: the opencrcut potental, the ze of the underlap, and the charge rate. Kennell and Evtt [9] preented a two-dmenonal model for the concentraton, current dtrbuton, and electrc feld a a functon of tme, n a lthum-on cell. They demontrated that t poble to predct not only the lthum depoton at the cathode edge at later chargng tme, but alo the large electrc gradent that were expermentally oberved by Scott et al. [1,2] along the electrolyte durng early chargng. It can be noted that all of the model decrbed above ncorporate mplfed nerton knetc when compared wth model that focu on the nerton and tranport of lthum nde electrode partcle [10]. The reearch preented n th current paper ue the model of Kennell and Evtt [9] to contnue the tudy of a lthum-on cell and numercally predct the effect aocated wth equal and extended electrode/ electrolyte. 2. Theory and Model Implementaton Th paper focued on reult from a numercal model mplemented ung C++. Lthum-on cell were modeled ung two governng equaton where flud bulk velocty ha been neglected [9]: C 2 zuf C D C S (1) t F 2 zd C zs F F t zc (2) Equaton (1) may be ued to decrbe the tranport of pece due to dffuon and electro mgraton. Equaton (1) alo contan a term for the ource or nk of pece due to reacton. Becaue Equaton (1) wa developed ung the Control Volume technque, for ue wth an up-wndng cheme, t omt one term that would be preent n an equaton developed for ue wth alternatve C. Th applcaton of Equaton (1), ung the Control Volume technque, alo enure the conervaton of charge and ma due to tranport. Equaton (2) decrbe the Laplacan of potental due to dffuon potental, patally eparated anodc and cathodc reacton, and charge denty. When th equaton appled over a tme nterval an aumpton that the electrc feld wll promote electroneutralty ncorporated that enure a ytem of + 1 equaton are avalable for olvng for pece concentraton and the electrc feld. Th ytem of equaton advantageou when compared to equaton et contanng Poon equaton, 2 method: zuf F zc, becaue Equaton (2) not nu- mercally tff. The valdty and development of thee equaton were demontrated by Kennell who appled thee equaton to everal dfferent electrochemcal ytem [11]. Thee equaton were olved ung C++ and numercal technque decrbed elewhere [11]. The effectve dffuon coeffcent wa calculated [9]: D D 0 Value for dffuon coeffcent and other model parameter are preented n Table 1. Conductvty wa aumed non-unform wth tme and poton nde the electrolyte: (3) F zuc (4) For the calculaton of electro moblte the Nernt- Enten equaton wa ued: D RTu (5) Predcton were conducted for a lthum-on cell that conted of an electrolyte andwched between two old electrode. Fgure 1 how the cell geometry and the apect that were modfed n the mulaton; the extenon of the cathode edge beyond the anode edge, and the electrolyte edge beyond the cathode edge, were vared. Thee edge extenon were conducted n the x-dmenon, or along the length of the cell. The numercal doman wa plt nto three part: the cathode, the anode, and the electrolyte. Thee three doman were olved concurrently, where charged pece were aumed to ext n the electrolyte, but not n the old electrode. Thu Equaton (1) wrtten for the nteror of an electrode a:

3 G. F. KENNELL, R. W. EVITTS 425 Table 1. Cell parameter. Electrode parameter L x C 6 L x CoO 2 Lthum nerton rate contant, k, m 2.5 mol [6] [12] Intal tochometrc coeffcent 0.01 [7] 1 [7] Maxmum concentraton, C t, mol m 3 30,540 [7] 56,250 [7] Dffuon coeffcent, D, m [12] [12] Tranfer coeffcent, α a, α c 0.5 [7] 0.5 [7] Electrolyte parameter Volume fracton, ε 0.55 [13] LPF 6 ntal concentraton, mol m [7] L + dffuon coeffcent n lqud phae, D 0, m [14] Fgure 1. Cell confguraton (not to cale). The x-dmenon correpond to the length of the cell and the y-dmenon correpond to the cell heght. C t 2 D C S (6) The electrc potental feld nde each electrode wa aumed to be unform and equal to an appled value, or a c. Conductvty wa alo aumed contant nde each electrode. The boundary condton for all boundare of each of the three numercal doman for Equaton (1) or (6) wa: C 0 (7) x and the boundary condton ued for Equaton (2) at all electrolyte boundare wa: 0 (8) x Durng charge and dcharge, charge and ma were calculated to move between the three numercal doman va the ource term n Equaton (1), (2) and (6). The followng development wll decrbe how the ource term were calculated a beng dependent on electrochemcal reacton, how charge and ma were conerved, and how the overall cell potental wa determned. The rate of electrochemcal reacton were aumed to follow Tafel knetc for the anode and cathode: a ak, F o, a exp a e Ua (9) RT F U RT ck, c o, c exp c e c (10) where e repreent the electrc potental of the electrolyte adjacent to the electrode. e wa calculated from Equaton (2), and wa not aumed unform wth tme and poton. U a and U c repreent the equlbrum potental of the anode and cathode repectvely, a a functon of lthum tochometrc coeffcent. The equlbrum potental n the anode wa calculated by [12]: U a C C C ta, C ta, 3 4 C C C ta, Ct exp C 4 C C ta, C ta, 2 (11) The equlbrum potental n the cathode wa calculated by [6]: U c C C Ct Ct C C Ct Ct C exp Ct C exp C t a c a o t Fk C C C C 0.5 (12) The exchange current denty from Equaton (9) and (10) wa calculated by [7]: (13) Fgure 2 how the equlbrum potental of the elec-

4 426 G. F. KENNELL, R. W. EVITTS trode decrbed by Equaton (11) and (12) for an anode fabrcated from L y CoO 2 and for a cathode from L x C 6. The old lne n Fgure 2 repreent the porton of the equlbrum potental that correpond to the tochometrc coeffcent that lkely to ext n a lthum-on cell. In other word, the old part of the lne for L y CoO 2 correpond to the tochometrc coeffcent of between 0.99 and 0.58 (n the anode) and the old part of the lne for L x C 6 correpond to the tochometrc coeffcent of between 0.01 and 1 (n the cathode). The electrochemcal reacton were treated a ource term. If t aumed that at the urface of the electrode the current decrbed by Equaton (9) and (10) are perpendcular to the electrode urface, the current vector decrbng the electrochemcal reacton rate,, produced. Th current vector may be converted nto a ource term for ue wth Equaton (1), (2), and (6): S (14) zf d L Ietlelectrode electrode Therefore, the pece produced by electrochemcal reacton were ntroduced nto the numercal procedure va the ource term n Equaton (1), (2) and (6). The conervaton of charge and ma acro numercal boundare wa guaranteed by enurng the um of each electrochemcal reacton along the length of the electroactve urface wa equal to a precrbed current: (15) Equaton (15) wa atfed by modfyng the rate of electrochemcal reacton by varyng the appled electrc potental, a and c. Then, the overall cell potental wa calculated by: V (16) cell a c 3. Reult and Dcuon Lthum-on cell depend upon the tranport of L +. It may be tranported through an electrolyte becaue of dfferent potental of electrode. Dfferent potental may be caued by dfferent equlbrum potental and dfferent potental appled to each electrode. Equlbrum potental depend on electrode materal and on the tochometrc coeffcent of nerted lthum. The two electrode materal nvetgated n th paper are L y CoO 2 and L x C 6. The equlbrum potental correpondng to dfferent tochometrc coeffcent n each of thee materal are preented n Fgure 2. Smulaton preented n th paper nvetgate both the electrc potental etablhed between the anode and cathode of a lthum-on cell (along the y-dmenon), and alo the electrc potental that may ext parallel to the electrode along the cell length (x-dmenon). The cell geometre nvetgated n the mulaton preented n th paper are hown n Fgure 1. The electrode are condered to be old flm electrode and the eparator cont of an electrolyte wth the followng compoton: 1.2 M LPF 6 n a 1:2 v/v mxture of ethylene carbonate and dmethyl carbonate (EC:2DMC). Further detal are gven n Fgure 1, ncludng the locaton of electrode edge and nteror urface Electrolyte Extenon In th ecton predcton for lthum-on cell wth a flooded electrolyte extended pat the edge of the electrode are preented. In th ecton, the anode and cathode edge are located at the ame cell length (x-dmenon). Th geometry expoe the edge of both the anode and cathode to the electrolyte. Secton explore mulaton where no electrochemcal reacton occur on thee electrode edge and Secton explore the cae where electrochemcal reacton occur on the electrode edge n contact wth the flooded electrolyte Equal Length Electrode wthout Edge Reacton Fgure 3 how the predcted electrc feld for the cae where the anode and cathode were of equal length (xdmenon) and the electrolyte length wa extended pat the edge of the electrode by 25 µm and the cell underwent 4.37 Am 2 chargng for 60 econd. The length of the electrode ncorporated nto the mulaton wa 70 µm. It wa found that th length encompaed completely the mult-dmenonal edge effect, for th cae. It wa aumed that only the nteror urface of the electrode were electro actve. In other word, the edge of the electrode dd not emt or nert any lthum-on. Th mean that the ncreaed urface area due to electrode edge dd not have an mpact on the overall rate of electrochemcal reacton around the edge. Therefore, n Fgure 2. Equlbrum potental of electrode a a functon of tochometrc coeffcent, x or y. Sold lne ndcate tochometrc coeffcent range aumed n th paper.

5 G. F. KENNELL, R. W. EVITTS 427 the electrc potental adjacent to the electrode wa reduced n the proxmty of the edge regon, ncludng on the nteror electrode urface, due to the ncreaed effectve conductvty of the extended electrolyte regon. In other word, an ncreaed effectve conductvty due to an extended electrolyte may decreae the potental gradent acro the electrolyte (n the y-dmenon) and th may caue an ncreae n electrochemcal reacton rate on the nteror urface near the electrode edge. Th effect wa een n the mulaton. However, th effect wa extremely mall, a can be een n Fgure 5, whch how the concentraton of lthum nerted nto the cathode after a full hour of 1 C chargng; the ncreae n lthum concentraton near the cathode edge/tp wa o mall that t unobervable n th fgure. Hence, t can be concluded that the electrolyte extenon doe not have a percevable effect on electrode concentraton at the condton examned, but t doe ncreae the effectve conductvty near the edge regon. Fgure 3. Predcted electrc potental feld for the cae of equal length flooded electrode wthout edge reacton after 60 econd a (A) a urface plot and (B) a contour plot wth cell geometry overlay. The heght of the cathode wa 8 µm. Smulated electrode length to bulk condton wa 70 µm. th cae, the man edge effect wa to ncreae the effectve conductvty of the electrolyte toward the edge, caued by the electrolyte length (x-dmenon) extenon. The effect of th electrolyte extenon on the predcted electrc current dtrbuton hown n Fgure 4. Th fgure demontrate how the electrc current tended to move n the y-dmenon acro the electrolyte drectly from the anode to the cathode n the bulk nteror of the cell (at larger dtance from the edge); however, n the electrolyte nearer the edge of the electrode t hown that the electrc current dd not take the hortet path from the anode to cathode. Intead, the current tended to pread out along the cell length (x-dmenon) nto the extended electrolyte regon that would otherwe have contaned no current denty. In other word, the extended electrolyte regon had the effect of ncreang the effectve conductvty of th area. Th ncreae n effectve conductvty and reduced current dente near the edge regon lead to a lower electrc potental gradent, a een n Fgure 3. The rate of electrochemcal reacton (Equaton (9) and (10)) occurrng on the electrode nteror urface were dependent upon the electrc potental adjacent to the electrode,. A decrbed above, Fgure 3 how how e Fgure 4. Predcted electrc current dtrbuton near electrode edge for the cae of equal length flooded electrode wthout edge reacton after 60 econd. Fgure 5. Predcted cathode concentraton for the cae of cathode heght of 8 µm and equal length flooded electrode after 1 hour of chargng and no edge reacton.

6 428 G. F. KENNELL, R. W. EVITTS Equal Length Electrode wth Edge Reacton Fgure 6 how the predcted electrc feld for the cae of two equal heght (5 µm n y-dmenon) and equal length electrode wth an electrolyte length extenon of 25 µm after 60 econd of 4.37 Am 2 chargng. For th mulaton, t wa aumed that the electrochemcal reacton occurred along the complete urface of the electrode n contact wth the electrolyte, ncludng the electrode nteror urface and electrode edge. It mportant to note that the cale change along the abca n Fgure 6. From th fgure t may be een that the effect of the edge reacton (when the heght of both electrode wa 5 µm) wa to rae the electrc potental near the edge above that n the bulk regon. The reaon for th ncreae wa aocated wth gradent of lthum tochometrc coeffcent n the electrode toward the edge. Th phenomenon explaned n the next paragraph. The electrc potental wa elevated along the electrolyte length toward the electrode edge becaue the overall rate of anodc half reacton n th regon wa predcted to be greater than the overall rate of cathodc reacton. In other word, n th edge regon, the anode wa producng more electrc current than the cathode wa conumng. Th exce electrc current then had to mgrate along the length of the electrolyte, parallel to the electrode, toward the bulk cell. Th predcted electrc current dtrbuton hown n Fgure 7. Th fgure how electrc current emanatng from the anode nteror urface and the anode edge. A gnfcant porton of th electrc current flowed nto the extended electrolyte regon, takng advantage of the ncreaed effectve conductvty n th area. Th electrc current flowed toward the cathode nteror urface and edge, where lthum wa nerted nto the cathode. The relatve magntude of the current aocated wth the tp of the anode and cathode edge can alo be oberved n Fgure 7. In Fgure 7, the arrow depctng the current flowng/nertng nto the cathode tp approxmately half of the ze of the arrow depctng the current emanatng from the anode. The dfference n magntude of thee two current due to the dependence of the rate of electrochemcal reacton on the equlbrum potental, U a and Uc. When the chargng of the cell tarted, the tochometrc coeffcent of lthum n the anode very cloe to unty and repreented by the rght-hand porton of the old lne n Fgure 2. Th repreent only a lght gradent of potental for a gradent n tochometrc coeffcent. In other word, th equlbrum potental doe not change gnfcantly for a concentraton gradent nde the electrode, and electrochemcal reacton rate wll not change dratcally for a gradent n potental. Therefore, for th cae a hgher current wll emanate from the anode tp that more dependent on the tp urface area than t by the concentraton gradent wthn the electrode. The oppote true of the cathode tp. At the tart of cell chargng, the tochometrc coeffcent Fgure 6. Predcted electrc potental feld for the cae of equal length flooded electrode wth edge reacton after 60 econd of chargng a (A) a urface plot and (B) a contour plot wth cell geometry overlay. Fgure 7. Predcted electrc current dtrbuton near electrode edge for the cae of equal length flooded electrode wth edge reacton after 60 econd. The heght of both electrode 5 µm.

7 G. F. KENNELL, R. W. EVITTS 429 nde the cathode mall and hown by the left de of the L x C 6 old lne n Fgure 2. A evdent, there a very large gradent of potental aocated wth a mall change n tochometrc coeffcent n th regon. Therefore, the rate of electrochemcal reacton would be greatly affected by a gradent of lthum concentraton n the electrode. Although the larger urface area at the cathode tp may promote an ncreaed rate of lthum nerton nto the tp area, the large change n equlbrum potental that th would caue prevent th ncreae. Intead, the extra current produced at the anode tp and edge mut flow along a dfferent path, along the electrolyte length (n the x-dmenon) adjacent to the cathode, n a manner that balance tranport and concentraton gradent n the electrolyte wth concentraton and equlbrum potental n the electrode. The reultng concentraton gradent wthn the electrode are dcued below. Fgure 8 how the concentraton gradent n the regon of the cathode edge after 60 econd of cell chargng. The lengthwe gradent were retrcted to wthn 5 µm of the edge. Th a maller regon than for the gradent n the anode, hown n Fgure 9, where the lengthwe gradent extended 25 µm from the anode edge. Th renforce the concept that tochometrc gradent occur n the anode durng early cell chargng, but not n the cathode becaue of the large equlbrum potental gradent that th would caue. In other word, the electrochemcal reacton rate of lthum producton at the anode not greatly affected by the anodc lthum tochometrc coeffcent; the oppote true of the cathode (durng early chargng). Becaue the numercal model ued a non-unform meh, the length of the bulk cell that wa modeled wa long enough o that the mcrocopc phenomena occurrng at the electrode edge would not have a gnfcant effect on the macrocopc bulk cell and overall cell potental. Becaue the model balance the two-dmenonal potental feld at the edge wth the bulk cell potental, the model predct where the exce current from the anode edge nerted nto the cathode. Fgure 10 how the current denty that occurred on the urface of both the anode and cathode a a Fgure 8. Predcted cathode concentraton for the cae of equal length flooded electrode wth edge reacton after 60 econd of 4.37 Am 2 chargng. Fgure 9. Predcted anode concentraton for the cae of equal length flooded electrode wth edge reacton after 60 econd of 4.37 Am 2 chargng. Fgure 10. Predcted electrochemcal reacton rate of lthum doluton or nerton along urface for the cae of equal length flooded electrode wth edge reacton after 60 econd of 4.37 Am 2 chargng. functon of dtance from the electrode edge (potve dtance correpond to locaton on the electrode nteror urface, away from the edge, and negatve dtance correpond to dtance away from the electrode tp, along the electrode edge telf). Fgure 10 how a reducton n the cathodc current denty at the electrode tp, of approxmately 65% of the bulk value, and th neceary n order to avod large tochometrc lthum gradent n the cathode (a decrbed above). The anodc reacton rate uffer only a mall reducton at the tp, a hown n Fgure 10. However, nce the overall producton and conumpton of L + mut be equal for the entre cell, th exce L + produced at the electrode edge mut be conumed elewhere. Fgure 10 how at dtance of approxmately 0.01 mm to 0.02 mm from the edge the anodc reacton rate wa predcted to be le than the cathodc reacton rate. Therefore, t n th regon that the exce current from the edge regon wa nerted nto the cathode. Th avoded teep lthum concentraton gradent toward the cathode edge, and ntead balanced thee gradent wth R drop and concentraton effect n the electrolyte.

8 430 G. F. KENNELL, R. W. EVITTS Fgure 11 how the predcton for a cell wth mlar geometry to the one ued n the prevou predcton, but wth a cathode heght of 8 µm. Th ncreae n cathode heght decreae the concentraton gradent wthn the cathode caued by lthum dffuon from the urface. The predcton howed that an ncreae n cathode heght decreae the value of electrc potental toward the electrode edge. Fgure 12 how the electrc current dtrbuton for th cae. Fgure 12 dplay the current that wa drawn parallel to the electrode (along t length) nto the edge area to compenate for the addtonal lthum nerted nto the cathode n th area. Hence, durng early cell chargng, the rate of lthum nerton nto the cathode prmarly determned by the concentraton gradent wthn the cathode, and the rate of lthum emon from the anode determned by anode urface area and electrolyte potental gradent Extended Cathode Large potental drop due to concentraton gradent wthn the cathode have been een expermentally when Fgure 11. Predcted electrc potental feld for the cae of equal length flooded electrode wth edge reacton after 60 econd of chargng a (A) a urface plot and (B) a contour plot wth cell geometry overlay. The heght of the cathode wa 8 µm. The electrolyte length wa extended pat the edge of the electrode by 25 µm. Fgure 12. Predcted electrc current dtrbuton near electrode edge for the cae of equal length flooded electrode and 8 µm heght cathode after 60 econd of chargng. the cathode edge extended (n the x-dmenon) gnfcantly pat the anode edge (Scott et al. [1,2]). The cathode edge may be extended n order to prevent hgher level of lthum concentraton at the tp/edge that may be detrmental to the cell. The model preented n th paper doe predct thee damagng level of lthum concentraton, and the reultng lthum depoton, at the cathode tp/edge; however, thee hgh concentraton of lthum are more lkely to occur n the cathode toward the end of cell chargng when the tochometrc coeffcent of lthum n L x C 6 almot unty, rather than at the begnnng of cell chargng. The relatonhp between the equlbrum potental and tochometrc coeffcent of lthum n L x C 6 of an almot completely charged cell hown n Fgure 2 (for tochometrc coeffcent approachng unty). The potental gradent caued by the L x C 6 tochometrc coeffcent gradent much le for coeffcent approachng unty (a fully charged cell) than for coeffcent approachng zero (an uncharged cell). Thee dfferent equlbrum potental gradent for an uncharged and charged cathode reult n the poblty for larger lthum concentraton gradent n a cathode approachng a full charge. In other word, f the dfference n tochometrc coeffcent of lthum n an electrode caue a large equlbrum potental gradent, a large electrc potental gradent may be apparent n the electrolyte, a een by Scott et al. [1,2]. If the cell n a tate of charge whereby a large dfference n tochometrc coeffcent (wth length) doe not caue a large equlbrum potental gradent, then large electrc potental gradent may not be een n the electrolyte; however, large concentraton gradent n the electrode may then be poble, along wth electrode over-aturaton and lthum depoton at regon of hgh urface area, a een n the numercal mulaton of Tang et al. [7]. The model preented n th paper predct both uch phenomena. For

9 G. F. KENNELL, R. W. EVITTS 431 example, Fgure 13 how the concentraton profle for the cathode of 6 µm heght and equal electrode length cell havng undergone 4.37 Am 2 chargng for one hour. Fgure 13 how that although much of the cathode concentraton wa gnfcantly below the maxmum concentraton of 30.5 M, the tp regon wa above th concentraton and lthum depoton here lkely. Fgure 14 how the predcted electrc feld for the cae where the length of the cathode wa extended pat the edge of the anode by 1.75 cm, after 100 econd of chargng at 2 Am -2, correpondng to the current denty utlzed by Scott et al. [1]. The heght of the cathode wa 8 µm and the ntal tochometrc coeffcent n the cathode wa Fgure 14 how a predcted electrc feld that ha a mnmum of approxmately 0.4 V wth repect to the bulk value. Th about half of the maxmum expermental value een by Scott et al. [1] after 90 econd for a lthum on cell wth a cathode extenon undergong chargng of a mlar current denty. However, many cell parameter were unpublhed by Scott et al. and the predcton gven here are for old electrode, not porou one. Of nteret n th cae not a drect comparon wth expermental data, but ntead, the trend caued by lthum gradent n the electrode and reultng effect are examned below. For the mulaton preented n Fgure 14 the concentraton of lthum n both electrode changed wth poton and tme. Becaue the concentraton of lthum non-unform wth poton, potental gradent were evdent n the electrolyte. The effect of the lthum electrode gradent on the cell are nvetgated through an examnaton of the rate of electrochemcal reacton predcted to occur on electrode urface. Fgure 15 how the rate of anodc current that emanated from the anode from the mulaton preented n Fgure 14 a a functon of poton at dfferent tme. Potve dtance value repreent the nteror urface of the anode and negatve value the mall edge regon. Becaue at larger dtance along the cell length, away from the anode edge, the current wa predcted to reman contant, th data wa not preented a part of th fgure. Fgure 15 how that at early chargng tme (100 and 500 ) the hghet rate of current were drawn from regon cloe to the anode edge. The anode emanated cur- Fgure 13. Predcted cathode concentraton for the cae of equal length flooded electrode wth edge reacton after 1 hour of 4.37 Am 2 chargng. The cathode heght wa 6 µm. Fgure 14. Predcted electrc potental feld (V) n the electrolyte for the cae of a 1.75 cm cathode extenon after 100 econd of chargng a (A) a urface plot and (B) a contour plot wth cell geometry overlay. The cathode heght wa 8 µm and the cathode length extenon wa 1.75 cm. Fgure 15. Predcted electrc current emanatng from anode urface at dfferent tme for the cae of a 1.75 cm cathode extenon. Potve dtance value repreent the nteror anode urface and negatve dtance value repreent the dtance along the anode edge telf, away from the anode tp. rent from th regon for two reaon: th regon wa the cloet to the extended cathode and th regon had a greater urface area due to the anode edge. Alo, from Fgure 2 t evdent that gnfcant concentraton gradent were poble n the ntally chargng anode that

10 432 G. F. KENNELL, R. W. EVITTS dd not caue gnfcant gradent of equlbrum potental. Fgure 16 how a fgure mlar to Fgure 15, but decrbng the current nerted nto the cathode. At the early chargng tme of 100 and 500, Fgure 16 how that current dente of approxmately 0.4 and 0.2 A/m 2 were nerted along the extended regon of the cathode, repectvely. At early chargng tme the gradent of lthum n the cathode caued gnfcant gradent of equlbrum potental whch provded the drvng force behnd the large potental drop n the electrolyte (hown n Fgure 14), and caued thee gnfcant cathodc reacton rate toward the edge (hown n Fgure 16). Over an ntal perod of tme, mall amount of lthum were nerted nto the extended cathode regon and the concentraton of lthum n the extended cathode regon ncreaed uch that a large dfference n equlbrum potental no longer exted between the edge regon and the bulk cell regon (ee Fgure 2). Th decreae n the dfference n equlbrum potental decreaed the avalable drvng force for the mgraton and nerton of lthum along the length of the extended cathode and vble n Fgure 16 that how how the current drawn by the extended cathode decreaed for tme of 1000 and later. Th decreae n current drawn by the extended cathode alo affected the current emanatng from the anode. Fgure 15 how that after 1000 the exce current drawn from the regon toward the edge of the anode had decreaed and ome other effect were vble a the hook hape n the reacton rate toward the edge regon. Th hook hape wa lkely becaue of the decreaed current beng drawn toward the extended cathode and the decreaed Ohmc drop n the electrolyte makng t more poble for current to be drawn from the anode urface further nto the cell. Th current drawn from the anode further nto the cell took advantage of the fact that the anode edge became more depleted of lthum durng the early chargng when the extended cathode wa drawng gnfcant quantte of current. Fgure 15 how that a tme progreed further, le and le exce current wa produced toward the anode edge regon, and ntead, becaue the extended cathode wa no longer drawng gnfcant current, the concentraton gradent prevouly etablhed n the anode became the domnant phenomenon mpactng the rate of anodc reacton. Th wa becaue, a the cell became more charged and the tochometrc coeffcent n the anode decreaed, a lght electrochemcal potental gradent evdent toward the left of the correpondng old lne n Fgure 2. Th gnfcant lthum concentraton gradent n the anode after 1 hour hown n Fgure 17. Fgure 18 how the lthum concentraton gradent n the cathode after 1 hour. It can be een that the concentraton of lthum n the extended regon wa approxmately one tenth of the maxmum value een n the bulk cell. Fgure 16. Predcted electrc current nerted nto cathode urface at dfferent tme for the cae of a 1.75 cm cathode extenon. Potve dtance value repreent the nteror cathode urface and negatve dtance value repreent the dtance along the cathode edge telf, away from the cathode tp. Fgure 17. Predcted anode concentraton for the cae of a 1.75 cm cathode extenon after 1 hour of chargng 2 A/m 2. Fgure 18. Predcted cathode concentraton for the cae of a 1.75 cm cathode extenon after 1 hour of chargng at 2 A/m Concluon Th paper explore the edge effect of electrode n lthum-on cell undergong chargng, and the effect of tochometrc coeffcent gradent wthn electrode. It wa predcted, for the cae examned, that the ncreae n effectve conductvty aocated wth a flooded electrolyte that extended pat the electrode edge doe not have an apprecable effect on the rate of anodc or cathodc reacton near the edge regon. However, t wa

11 G. F. KENNELL, R. W. EVITTS 433 predcted that lthum concentraton gradent nde the cathode mpact the rate of cathodc reacton gnfcantly and concentraton gradent nde the anode do not gnfcantly mpact the rate of anodc reacton, both durng early cell chargng. Intead, the rate of anodc reacton are gnfcantly affected by the urface area of the anode contactng the electrolyte, and not the concentraton gradent of lthum n the anode. It wa alo predcted that durng later tage of cell chargng, when the gradent of equlbrum potental due to a gradent n cathodc tochometrc coeffcent le teep, concentraton gradent wthn the cathode (for equal electrode length) are more lkely and mght lead to a poblty for lthum depoton at the cathode edge regon. Smulaton were conducted for the cae where the cathode edge wa extended pat the anode edge to reduce the poblty for lthum depoton at the cathode edge regon. The mulaton ndcate that the tochometrc coeffcent of lthum n an extended cathode edge would be reduced n value; however, th extenon may caue a large electrc potental drop along the electrolyte length (durng early cell chargng) that correpond to the lthum tochometrc coeffcent gradent n the extended cathode and alo wth the Ohmc loe and concentraton gradent wthn the electrolyte telf. It wa oberved that th equlbrum potental gradent would decreae a chargng of the cell proceeded, caung a reducton n the rate of cathodc reacton occurrng along the extended cathode regon. Th reducton n the rate of cathodc reacton along the extended cathode regon reduce the rk for lthum depoton at the cathode edge regon, a dered by many cell manufacturer. 5. Acknowledgement The author thank the Unverty of Sakatchewan for computng faclte and the Natonal Scence and Engneerng Reearch Councl for a Canada Doctoral Scholarhp. REFERENCES [1] E. Scott, G. Tam, B. Anderon and C. Schmdt, Anomalou Potental n Lthum Ion Cell: Makng the Cae for 3-D Modelng of 3-D Sytem, The Electrochemcal Socety Meetng, Orlando, 13 October [2] E. Scott, G. Tam, B. Anderon and C. Schmdt, Obervaton and Mechanm of Anomalou Local Potental durng Chargng of Lthum Ion Cell, The Electrochemcal Socety Meetng, Par, 29 Aprl [3] K. Wet, T. Jacoben and S. Atlung, Modelng of Porou Inerton Electrode wth Lqud Electrolyte, Journal of the Electrochemcal Socety, Vol. 129, No. 7, 1982, pp do: / [4] M. Doyle, T. F. Fuller and J. Newman, Modelng of Galvanotatc Charge and Dcharge of the Lthum/Polymer/Inerton Cell, Journal of the Electrochemcal Socety, Vol. 140, No. 6, 1993, pp do: / [5] T. F. Fuller, M. Doyle and J. Newman, Smulaton and Optmzaton of the Dual Lthum Ion Inerton Cell, Journal of the Electrochemcal Socety, Vol. 141, No. 1, 1994, pp do: / [6] P. Arora, M. Doyle and R. E. Whte, Mathematcal Modelng of the Lthum Depoton Overcharge Reacton n Lthum-Ion Battere Ung Carbon-Baed Negatve Electrode, Journal of the Electrochemcal Socety, Vol. 146, No. 10, 1999, pp do: / [7] M. Tang, P. Albertu and J. Newman, Two-Dmenonal Modellng of Lthum Depoton durng Cell Chargng, Journal of the Electrochemcal Socety, Vol. 156, No. 5, 2009, pp. A390-A399. do: / [8] K. Eberman, P. M. Gomadam, G. Jan and E. Scott, Materal and Degn Opton for Avodng Lthum-Platng durng Chargng, ECS Tranacton, Vol. 25, No. 35, 2010, pp do: / [9] G. F. Kennell and R. W. Evtt, Charge Denty n Non- Iotropc Electrolyte Conductng Current, The Canadan Journal of Chemcal Engneerng, Vol. 90, No. 2, 2012, pp [10] W. Dreyer, M. Gabercek, C. Guhlke, R. Huth and J. Jamnk, Phae Tranton n a Rechargeable Lthum Battery, European Journal of Appled Mathematc, Vol. 22, No. 3, 2011, pp do: /s [11] G. F. Kennell, Electrolytc Tranport, Electrc Feld, and the Propenty for Charge Denty n Electrolyte, Ph.D. Dertaton, Unverty of Sakatchewan, Sakatoon, [12] M. Doyle and Y. Fuente, Computer Smulaton of a Lthum-Ion Polymer Battery and Implcaton for Hgher Capacty Next-Generaton Battery Degn, Journal of the Electrochemcal Socety, Vol. 150, No. 6, 2003, pp. A706-A713. do: / [13] J. Chrtenen, V. Srnvaan and J. Newman, Optmzaton of Lthum-Ttanate Electrode for Hgh-Power Cell, Journal of the Electrochemcal Socety, Vol. 153, No. 3, 2006, pp. A560-A565. do: / [14] S. G. Stewart and J. Newman, The Ue of UV/V Aborpton to Meaure Dffuon Coeffcent n LPF6 Electrolytc Soluton, Journal of the Electrochemcal Socety, Vol. 155, No. 1, 2008, pp. F13-F16. do: /

12 434 G. F. KENNELL, R. W. EVITTS Nomenclature C pece concentraton (mol/m 3 ) C old phae lthum concentraton (mol/m 3 ) C t maxmum lthum concentraton (mol/m 3 ) D effectve dffuon coeffcent (m 2 /) D 0 dffuon coeffcent of lqud phae (m 2 /) F Faraday Contant (96,487 C/mol) current denty (A/m 2 ) current denty vector (A/m 2 ) L current denty of nerton/doluton (A/m 2 ) k lthum nerton rate contant (m 2.5 mol ) I et appled current denty (A/m 2 ) l length (m) lel ectrode length of electrode (m) R ga contant (8.314 JK 1 mol 1 ) S ource term (mol/m 3 ) poton (m) t tme () T temperature (K) u moblty (m 2 mol/j ) U equlbrum potental (V) x poton along the x-dmenon (m) z charge number Greek Letter tranfer coeffcent volume fracton electrc potental (V) conductvty (C/Vm) Subcrpt/Supercrpt a anode c cathode e electrolyte pece