IV. Transort Phnomna Lctur 35: Porous Elctrods (I. Surcaactors) MIT Studnt (and MZB) 1. Effctv Equatons for Thn Doubl Layrs For surcaactor lctrods, convcton s usually nglgbl, and w dro out convcton trms hr. Lt s focus on ffctv quatons govrnng th transorts and lctrostatcs n lctrolyt. Fgur 1. Flat Elctrod Surfac Scs consrvaton quatons, Nrnst-Plank flux consttutv quatons, and Posson quaton mak u Posson-Nrnst-Plank (PNP) st of quatons (bold fonts ndcat that th varabls ar n vctor quantty): c t F 0 (1) F Dc 2 zc (2) (3) Elctrostatc constran and flux constrans on th surfac scfy boundary condtons. Elctrostatc constran can b ntrrtd dffrntly, gvn dffrnt scfd varabls. Whn thr s a scfd amount of surfac charg, w can hav Gaussan law satsfyng th lctrostatc constran:
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant nˆ ( ) qs (4) On th othr hand, whn gvn a scfd surfac otntal, w can us th followng aroxmaton, Equaton (5), nstad of Equaton (4), satsfyng th lctrostatc constran. Ths boundary condton could b mor smlfd assumng nglgbl caacty n Strn layr. (x = 0) nˆ (x = 0) (x = 0) s (5) W can us thr Equaton (4) or (5) to satsfy th lctrostatc constran, dndng on scfd varabls at th surfac. In addton, flux constrans of scs scfy th rst of ncssary boundary condtons: nf ˆ R (6) W now aly th abov st of quatons (PNP) as wll as boundary condtons to th orous lctrod wth doubl layr thcknss far thnnr than th or lngth scal. Fgur 2. Thn Doubl Layr n a Por Whn th or lngth scal s far largr than th lngth scal of doubl layr,, w hav saraton of lngth scals. Th mathmatcal structur of thn doubl layr roblm s wll undrstood from th rsctv of sngular rturbaton analyss, n whch ach of two rgons rqurs a dffrnt aroxmaton. Two dffrnt aroxmatons ar constrand by matchd asymtotc xansons. W frst assgn notatons for dffrnt varabls. From now on throughout ths lctur, w us th followng notatons. : Concntraton of Scs n Doubl Layr : Concntraton of Scs n Bulk Elctrolyt
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant : Chmcal Potntal of Scs n Doubl Layr : Chmcal Potntal of Scs n Bulk Elctrolyt Fgur 3. Varabls n Two Dffrnt Rgons In quas-nutral bulk lctrolyt, w can us th quas-nutral aroxmaton, and conclud wth zro dvrgnc of currnt dnsty. z c 0 (7) j z F 0 (8) As statd abov dscusson, asymtotc xansons of aroxmatd varabls n two dffrnt rgons should b matchd wth th corrsondng ar rsctvly. In a or, varabls ar wll aroxmatd by th quas-qulbrum aroxmaton, and chmcal otntals n th two rgons ar aroxmatly constant across th or. lmcˆ lmc xˆ ˆ x0 (9) (10) Now, condtons constranng th varabls n doubl layr can b found by dfnng surfac varabls as shown n Fgur 3 (Strn layr caacty gnord): cˆ lm ˆ 0 c dx (11) x F D S S (12) Thn th ffctv boundary condton usng th varabls abov s (Chu and Bazant, 2007): S S ˆ S F R n F (13) t
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant whr : Excss Surfac Concntraton of Scs r Ara : Surfac Flux of Scs : Dmnsonlss Chmcal Potntal of Scs : Nt racton rat of Scs Usng Bulk Varabls ( Total dffus charg dnsty can b calculatd wth xcss surfac concntratons. q z (14) 2. Porous Elctrods Formal drvaton by volum avragng (homognzaton) gos from th thn doubl layr quatons nsd ors to macroscoc artal dffrntal quatons. An arorat modl would nvolv volum avrags of varabls ovr a volum lmnt small comard to th ovrall dmnsons (L), but larg than th or structur lngth scal (h ). Hnc, to hav ths modl vald, th followng condton s ncssary. L h D (15) Fgur 4. Lngth Scals n Porous Elctrod Not always, but for surcaactors, w do not dlt th salt ons n lctrolyt. Thus t s a vald aroxmaton to assum that concntratons ( and otntals ( and ) ar varyng slowly n th macroscoc vwont. Ths assumton justfs th volum-avragng or homognzaton. In ths macroscoc tratmnt, w do not consdr th actual gomtrc dtals of th ors. Rathr, w dfn macroscoc otntal n lctrolyt, otntal n sold matral, and on concntratons to b contnuous and wll-dfnd functons of sac coordnats. As a rsult, th orous lctrod n ths modl s rrsntd by th suroston of two contnuous mda
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant wthout mcrostructur, on corrsondng to th lctrolyt soluton and th othr corrsondng to th sold matral matrx. In ths modl both mda ar dfnd n th whol doman. Thrfor, n ths macroscoc modl, otntal n lctrolyt ( ) as wll as otntal n conductng sold matral ( ) ar dfnd n th whol doman of sac, whras thy wr only dfnd n ach has formrly. Volum-avragd concntratons n macroscoc vwont ar dffrnt from and rlatd to th concntratons n bulk lctrolyt and n doubl layrs as shown n th followng quaton (n formr lcturs, was usd for th concntratons n rsrvors or n nlt flows of ful. In ths lctur, s usd for volum-avragd concntratons n macroscoc vwont). Th macroscoc on concntratons ar dfnd throughout th whol volum as wll. c c a (16) Th frst trm on rght hand sd corrsonds to th contrbutons from bulk lctrolyt, and th rd scond trm s from xcss concntratons n doubl layrs. Nwman s book (3 Ed, 2004) dos not tak account of th scond trm. Howvr, ths trm may hav sgnfcant ffcts scally whn w consdr surcaactors whos chargs ar mostly stord n th doubl layrs. Wth th varabls dfnd wth th macroscoc modl, w can construct th scs consrvaton quatons, as wll as th flux consttutv quatons: c x F R (17) F Dc (18) W could bttr modl th macroscoc flux quaton, ncludng th transort n doubl layrs as shown n Equaton (19) and Fgur 5. Transort n doubl layr (surfac transort) would b sgnfcant whn th stord dffus charg s larg. F D c D a S (19) Fgur 5. Transort n Doubl Layr (Surfac Transort)
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant For th macroscoc volum-avragd racton trm ( ), w could modl t by usng th surfac racton trm basd on th bulk concntratons and otntals: S R a R ( c,, ) (20) whr : Dmnsonlss Chmcal Potntal of Scs : Effctv Dffusvty n orous mdum (s Lctur 34) ϵ a : Effctv Surfac Dffusvty : Macroscoc Porosty : Surfac Ara Dnsty Wth th macroscoc concntratons and fluxs abov, w can also dfn macroscoc currnt dnsty as wll as charg dnsty. For th charg dnsty, only xcss surfac concntratons mattr, snc bulk lctrolyt ks quas-nutralty. j zf z c a z a q D (21) (22) Du to macroscoc nutralty,. Ths macroscoc charg dnsty s consrvd by th followng quaton. Th ngatv sgn n Equaton (23) s du to th dfnton of currnt dnsty, whch dfns anodc currnts to b ostv. Now, w dfn lctronc currnt dnsty n a orous lctrod. j j F zr (23) t j (24) Du to macroscoc charg consrvaton, th dvrgnc of total currnt s zro. j j 0 (25) Wth th govrnng quatons dscussd abov, w can now undrstand what th solutons look lk for dffrnt alcatons. In ths lctur, w focus on gttng and undrstandng th soluton for surcaactors. Th solutons for battrs and ful clls wll b dscussd n th nxt lctur (Lctur 36). In surcaactor alcatons, w do not hav sgnfcant faradac ractons on th lctrod or surfacs. Thus, w can gt th soluton, sttng th faradac
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant transfrnc currnt to b zro ( ). On th othr hand, for battrs and ful clls, w can nglct caactanc of doubl layrs, and focus on faradac contrbutons. Also, t s rqurd to modl racton roducts, n addton to th ractng ons and lctrons. 3. Surcaactors In a modl for surcaactors, w can nglct faradac transfrnc currnt ( ) and changs n salt concntraton du to caacty chargng of doubl layrs (c.f. salt dlton du to doubl layr chargng s consdrabl n caacty dsalnaton alcatons). Thn, w can dfn ffctv onc conductvty (, aroxmatd to b constant. j (26) Th charg consrvaton quaton, Equaton (23), thn bcoms: s j (27) t t Combnng Equaton (24) and (25), j j 2 2 (28) W nd to mloy a doubl layr caactanc modl (C D ), such as DH modl or GC modl, to rlat th charg dnsts to th otntals. q t D C (,, c,...) t D (29) Pluggng n Equaton (22) and (28) to th abov Equaton (29), w now hav an quaton govrnng th otntals n lctrolyt and sold matral. ac 2 2 D t (30) Equaton (30) contans two govrnng artal dffrntal quatons for otntal n lctrolyt ( ) and otntal for sold matral ( ). To solv th govrnng quatons, w nd boundary condtons and ntal condtons. Th boundary condtons ar shown n Fgur 6. On th surfac contactng wth a sarator, lctron flux s not allowd and th normal comonnt of otntal gradnt n sold matral s zro. In smlar mannr, on th surfac contactng wth a currnt collctor, on flux s not allowd and th normal comonnt of otntal gradnt n lctrolyt s zro. Th otntal n lctrolyt at th surfac contactng a sarator s st to b th zro rfrnc valu. And w st otntal (V), may b a functon of tm, n conductng sold matral at th surfac contactng a currnt collctor.
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant Fgur 6. Boundary Condtons for Macroscoc Potntals For ntal condtons, lt s start from charg-otntal rlatonsh. Th charg dnsty (q) s rlatd to th caactanc and th otntal dffrnc wth th followng quaton: q( x, t) C (31) W consdr startng from th qulbrum voltag (on crcut voltag, V o ). Ths lads us to hav a unform ntal charg dnsty throughout th orous lctrod. In gnral, ths ntal condton can b xrssd usng Equaton (32) and (33). q( x,0) unform CVo ( x,0) ( x,0) V o (32) (33) Th govrnng quatons, Equaton (30), could b solvd for surcaactor alcatons wth th boundary condtons shown n Fgur 6, and th ntal condton, Equaton (33). Th scfc solutons wll b dscussd n th nxt lctur, Lctur 36. 4. Transmsson Ln Modl In fact, th govrnng quatons, Equaton (30), could b quvalntly obtand by usng th transmsson ln modl. Ths can b asly sn by manulatng xrssons of conductvty and caactanc, usng rsstancs and caactanc r lngth.
Lctur 35 10.626 Elctrochmcal Enrgy Systms (2011) Bazant Fgur 7. Macroscoc Cross-sctonal Ara R R 1 A C Aa C 1 A D : Rsstanc to Elctron Currnt r Lngth : Rsstanc to Ion Currnt r Lngth : Total Caactanc of Doubl Layr r Lngth Wth th nwly dfnd aramtrs abov, w can rarrang th govrnng quaton, Equaton (30), and show that ths quaton s quvalnt to th transmsson ln modl. By dong so, w can now undrstand whch hyscal bass th crcut lmnts ar basd on. 1 1 C R R t 2 2 (34) Fgur 8. Transmsson Ln Equvalnt Crcut Modl In nxt lctur, Lctur 36, w wll solv th govrnng quatons, Equaton (30) or quvalntly Equaton (34), usng th boundary condtons shown n Fgur 6, and th ntal condton, Equaton (33), for surcaactor alcatons.
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