Chapter 2 Theoretical Framework of the Electrochemical Model

Transcription

1 Chaptr 2 Thortcal Framwork of th Elctrochmcal Modl Th basc prncpls of th lctrochmcal modl for L on battry s dvlopd from fundamntals of thrmodynamcs and transport phnomna. Th voluton of th lctrochmcal modl and th nhrnt assumptons ar dscussd. Th dscussons and drvatons ar slf-consstnt and complt. Mathmatcal modl for ach procss n th L-on cll s constructd n a stpws mannr to volv th complt lctrochmcal modl. 2.1 Introducton Th framwork for modlng lthum-on battrs from a physcs prspctv ssntally nvolvs th spcfcaton of th mass and charg consrvaton quatons n th varous rgons of th battry at a suffcntly coars scal. Th basc componnts of th lthum-on cll ar th sold postv and th ngatv lctrods sandwchng a sparator whch s an lctron nsulatng matral. Th whol rgon of th battry s thn flld wth a lqud lctrolyt (a majorty of th commrcal clls stll us lqud lctrolyts). As a mattr of trmnology, actv componnts rfr to thos that partcpat n th ssntal physcochmcal procsss that convrt th chmcal nrgy to lctrcal nrgy and vc vrsa. Thus, th fllrs or bndng matral ar not drctly consdrd. Th lctrods consttutng th battry ar porous n natur. Ths s an artfact of th synthss procss whch grossly nvolvs mxng of th actv matral n a solvnt and thn allowng th solvnt to dry. Dpndng on th ntnsty of th crushng of th matral as wll as th packng dnsty, th porous structurs can hav a por sz dstrbuton typcally of th ordr of a mcron. Ths porous structurs nhanc th contact surfac ara of th actv matral wth th lctrolyt and ncras th probablty of th charg transfr racton. Sprngr Intrnatonal Publshng AG 2018 K. S. Harharan t al., Mathmatcal Modlng of Lthum Battrs, Grn Enrgy and Tchnology, 13

2 14 2 Thortcal Framwork of th Elctrochmcal Modl Fg. 2.1 Schmatc dagram of th lthum-on battry. Durng charg, th lthum ons mov from th postv to th ngatv lctrod. Durng dscharg, th nvrs drcton s followd + Dscharg L + Charg Ngatv Sparator Postv Ths allows th basc framwork n whch th lctrods ar modld by appalng to th stablshd porous lctrod thory. Rfrnc 32] Th basc pctur of an lctrod s a porous sold flld wth lctrolyt n ts ntrstcs. At th lvl of th lngth scal of th porous lctrod thory, howvr, th dffrntaton btwn a por and th sold s not mad. Ths allows both th lctrod as wll as th lctrolyt phas to b modld as contnuous mda, mplyng that both th phass ar assumd to xst at all ponts n spac. Addtonally, any pont n spac s consdrd lctrcally nutral. Thus, th modl trats th systm at a lngth scal largr than ndvdual charg carrrs, and smallr than th lctrod dmnsons. A schmatc of a lthum-on clls s gvn n Fg Th prncpal procss durng dscharg s th transfr of lthum on from th postv lctrod to th ngatv lctrod through th lctrolyt, and n th rvrs drcton durng charg. Each ndvdual lctrod scton conssts of th sold phas consstng of th actv matral, and th lctrolyt, through whch th lthum s transfrrd. Th combnd pctur of a cll has a contnuous lctrolyt mdum, ntrposd wth sold lctrod partcls at th rspctv lctrod sctons. As a cll s a closd systm, mass consrvaton s appld to th lthum that xsts n th onzd stat n th lctrolyt. As th lthum on carrs a unt charg wth t, charg s also smultanously consrvd. Smlar to th lctrolyt, ths consrvaton laws also apply to th lctrod. Th ntracton btwn th sold partcls of th lctrod and th lqud lctrolyt happn at th ntrfac, whr charg s transfrrd. Ths charg transfr racton coupls th lctrod and th lctrolyt phass. From th abov dscrpton, t can b sn that any gvn lctrod has to satsfy a coupl of consrvaton laws on for th mass of lthum and th othr for th charg n ach of th sold and th lctrolyt phass, wth th charg transfr racton couplng both of thm. Each lctrod rgon can, thrfor, b rprsntd by a total of fv quatons, and th sparator rgon by two quatons as ths rgon s charactrzd by th sngl lctrolyt phas wth no charg transfr racton.

3 2.1 Introducton 15 Ths st of 12 quatons forms th cor of th lctrochmcal modl or th macrohomognous modl dvlopd basd on th contnuum pctur for th cll. Th mathmatcal modl nvolvs svral lvls of abstracton namly: 1. Th soluton mthodology rprsnts th porous sold phas as a collcton of partcls that ar ndvdually surroundd by th lctrolyt phas. 2. Th framwork rprsnts th lctrod as a homognous systm of lctrolyt and partcls at vry pont. 3. Du to th frst abstracton, th dffuson of lthum n sold phas btwn partcls s nglctd. Dffuson n th sold phas s modld wthn ach partcls that ar typcally assumd to b sphrs. Th dffuson quaton, whch s th mass consrvaton quaton n th sold phas, s solvd n sphrcal coordnats to obtan th lthum surfac concntraton n th sphrs. As th sold phas s also contnuous, th surfac concntraton s obtand at vry pont n th lctrod rgon. Th surfac concntraton s th drvng forc for charg transfr racton. Th racton flux, n turn, appars as th boundary condton for th sold phas dffuson quaton. Thus, th modl rsults n a st of quatons that ar fully coupld. It s to b notd that all th quatons n th modl ar solvd as a functon of th thcknss of th lctrod, xcpt th dffuson n th sold phas whch s solvd n th sphrcal coordnats. Du to ths ffctv mappng of th output of th sphrcal dffuson quaton onto othr fld quatons that ar solvd n Cartsan coordnats, ths modl s also calld as a Psudo-2D (P2D) modl 32]. 2.2 Consrvaton Equatons Th govrnng quatons of th P2D modl can b drvd by appalng to th charg and mass consrvaton quatons n th varous rgons of th battry. Ths ar drvd n th followng subsctons. Th dmnsons of th LB modl lctrods and sparator ar llustratd n Fg Th thcknss of postv lctrod, sparator, and ngatv lctrod ar gvn by L p, L s and L n, rspctvly. Fg. 2.2 Dmnsons of th battry modl componnts L n L s L p x =0 x = L cll

4 16 2 Thortcal Framwork of th Elctrochmcal Modl Mass Consrvaton n th Elctrod (Sold) Rgons In ths scton, th mass consrvaton quaton n th sold phas of th actv matrals for both th lctrods s drvd. Th porous lctrod consttuts of ndvdual partcls rprsntd as sphrs that ntract through th lctrolyt that surrounds ach partcl. Lthum dffuss nto th actv matral partcls aftr ganng an lctron by th charg transfr racton at th lctrod lctrolyt ntrfac, t s consdrd as nutral lthum. Durng th chargng or dschargng procsss, th lthum on undrgos dffuson nsd th sold porous lctrods. Th porous lctrod rgon s abstractd to b composd of ndvdual partcls, assumd to b sphrs of fnt radus, that ntract wth th lctrolyt that surrounds ach partcl. Lthum on undrgos th charg transfr racton at th lctrod lctrolyt ntrfac, bcoms nutral by ganng an lctron, and dffuss nto th actv matral partcls Hnc, th mass consrvaton n th sold phas, n th most smplstc rprsntaton, s provdd by th Fck s law of dffuson n th sphrcal coordnats. Ths contnuty quaton s gvn by, c s t = 1 ( D r 2 s r 2 c ) s, (2.1) r r whr c s s th concntraton of lthum n th sold partcls, wth = p, n provdng an ndx for thr th postv or ngatv lctrod. Th sphrcal symmtry rsults n th boundary condton at th cntr of th sphr c s r = 0. (2.2) r=0 Th concntratons of th sold and th lctrolyt phass ar connctd by th condton at th surfac of th partcls. At th surfac, th flux of lthum s gvn by th por wall flux, gvn by th charg transfr racton. D s c s r = j. (2.3) r=rp Mass and Charg Flux from Soluton Thrmodynamcs A basc dscrpton of mass and charg flux n th soluton phas s rqurd to drv th mass and charg balanc quatons n th lctrolyt rgon.th framwork s drvd n ths subscton. Drvatons for th mass and charg consrvaton quatons rsultng n th lctrochmcal modl for L-on battry ar prsntd n th nxt subsctons. Basd on th concntratd soluton thory, th flux s dfnd as

5 2.2 Consrvaton Equatons 17 N = M c μ, (2.4) whr M s th moblty of on, and s gvn by th Enstn rlaton to b M = D k B T, (2.5) whr D s th mass dffusvty, k B th Boltzmann constant, and T s th absolut tmpratur. Corrspondng to th mass flux, a currnt flux can b dfnd as J = z N. (2.6) For lctrochmcal systms, μ s constructd ncludng th contrbuton from th actvty a, charg on th on z and th lctronc charg, as wll as th potntal φ, as gvn blow μ = k B T ln(a ) + z φ (2.7) Actvty s rlatd to concntraton c through th actvty coffcnt γ va th rlaton a = c γ. Thus, μ = k B T(ln(c ) + ln(γ )) + z φ (2.8) and μ = k B T ln(c ) + k B T ln(γ ) + z φ (2.9) μ = k B T c c Basd on ths, th flux s wrttn as 1 + ln(γ ] ) + z φ = k B T c 1 + ln(γ ] ) + z φ. ln(c ) c ln(c ) (2.10) N = M c μ = D ( k B T c k B T c 1 + ln(γ ] ) ) + z φ c ln(c ) (2.11) On smplfcaton, N = D 1 + ln(γ ] ) c D ln(c ) k B T c z φ = D I c z k B T D c φ (2.12) Th lctrc conductvty σ s dfnd as Thus, th mass and charg flux can b wrttn as σ = (z ) 2 M c = (z ) 2 c D k B T. (2.13)

6 18 2 Thortcal Framwork of th Elctrochmcal Modl N = D c z k B T D c φ (2.14) J = z D c σ φ. (2.15) Th total currnt dnsty n th lctrolyt s obtand by th sum of th currnt fluxs of ons = J = z D c σ φ. (2.16) Dfnng th conductvty of th lctrolyt as th sum of conductvts of th ons, σ = σ. Elctro nutralty s nvokd to dntfy that c = c + = c = c. Th currnt dnsty s = c = c = ln(c) k BT z D σ φ (2.17) z D 1 + ln(γ ) ln(c ) σ z 1 + ln(γ ) ln(c ) ] σ φ (2.18) ] σ φ. (2.19) Th fnal rsult s an xprsson for th total currnt dnsty n th lctrolyt n trms of th conductvty. Plas not that th sam xprsson, n trms of th dffusvts, s = c z D 2 c k B T φ z 2 D. (2.20) Th frst xprsson for th currnt flux s usd to drv th currnt balanc quaton n th lctrolyt, whras th scond s usd to drv th mass consrvaton quaton Mass Consrvaton n th Elctrolyt (Lqud Phas) To obtan th mass consrvaton quaton, w start from = c z D 2 c k B T φ z 2 D. (2.21) From ths quaton, w can obtan xprsson for th lctrolyt potntal. φ = + c z D z2 D 2 c k B T (2.22)

7 2.2 Consrvaton Equatons 19 Th abov xprsson s usd to lmnat for th lctrolyt potntal from quaton for th mass flux. ( N = D c z k B T D c + c z D ) z2 D (2.23) 2 c k B T whch on smplfcaton gvs N = D c + 2 c k B T z k B T D c z2 D ( + c z D ) (2.24) Th consrvaton quaton for concntraton stats that th accumulaton s accountd by th dvrgnc of th flux and th rat of gnraton by charg transfr racton. c t = N + R (2.25) For a porous lctrod, th consrvaton quaton s appld to th fractonal volum occupd by th lctrolyt. As th concntraton s dfnd n trms of th pur lctrolyt, th consrvaton quaton s modfd to b ε c t = N + R (2.26) Th corrspondng quaton for th mass flux s gvn by N = εm c μ (2.27) Ths modfcaton vntually rsults n N = ε D c + 2 c k B T z k B T D c z2 D ( + ε c z D ) (2.28) Th mass flux has th corrspondng dvrgnc gvn by ( ( ) z D N = ε D c + z2 D ) + Ths can b furthr smplfd for unvalnt ons to gv ( N = ε c D z D z D D )] ( ε cz D z D ) z2 D ( ) z D + z2 D (2.29) (2.30)

8 20 2 Thortcal Framwork of th Elctrochmcal Modl ( N = ε c D z D z D D )] ( ) z t + (2.31) As w ar ntrstd n th L + ons, th quaton rsults n )] ( ) t+ N + = ε c (t D + + t + D + (2.32) ) Dfnng th total chmcal dffusvty to b (t D + + t + D = D, and t, t + ar th transfrnc numbrs of th ons. ( ) N + = ε D c + ( ) t+ (2.33) Substtutng abov quaton n th mass consrvaton quaton, ε c ( ) ( ) t = t+ ε D c R + (2.34) Th consrvaton quaton can b wrttn n molar unts as ε c t ( ) t+ = (εd c ) F R + (2.35) In ordr to avod prolfraton of varabls s consdrd to b n molar unts, A/m 2. Th rat of consumpton s gvn by th Faradays law R + = a p,n j p,n = F (2.36) W wll dscuss th rlvanc of a p,n n a latr scton. Th mass consrvaton quaton s, fnally ε c t = (εd c ) + (1 t + ) a p,n j p,n F t + (2.37) Applcaton to Lthum-on Clls Th analyss starts from th mass consrvaton, n for th lctrolyt. Th varabl of ntrst s th lthum-on concntraton n th lctrolyt, and c rprsnts ths varabl. In any scton of thr of th lctrods, th local accumulaton of lthumon concntraton and transport du to dffuson s balancd by th rat of formaton/dsspaton of by th charg transfr racton 32]. Th mass balanc s gvn by,

9 2.2 Consrvaton Equatons 21 ε c t = (εd c ) + (1 t + ) a p,n j p,n F t + (2.38) In Eq. 2.38, D s th dffusvty of th lctrolyt n th bulk soluton, and can b a functon of th lctrolyt concntraton. Ths quaton accounts for th varaton n th concntraton lthum on n th x-dmnson. Ths dmnson ndcats th thcknss of th cll, starts at th currnt collctor of th ngatv lctrod (anod durng dscharg) and nds at th postv lctrod currnt collctor. Thus, ths modl s a 1D modl that accounts for concntraton gradnts across th cll thcknss. Th gnral mass consrvaton quaton gvn by Eq nds to b modfd to b of us n battry applcaton. In any lctrod rgon, a fracton of volum flld by th lctrolyt s gvn by th porosty, μ. To account for th compost natur of th lctrod rgon, th ffctv dffusvty of th lctrolyt s computd from th bulk valu n th followng mannr: D ff = D ε b. (2.39) Hr, ε b th Bruggmann factor, typcally assgnd th valu of 1.5. Although th charg transfr racton that occurs at th surfac of th lctrod and th lctrolyt, and should ntr th mass consrvaton quaton as a boundary trm. Th porous lctrod trats th lctrod and lctrolyt as contnuum, ths trm ntrs th consrvaton quatons as a sourc trm. Th mass flux of lthum du to ths racton s dfnd n trms of th surfac ara of th partcls of th actv matral. To ntroduc ths quantty nto th consrvaton quaton that s dfnd for a unt volum of th lctrod, t s multpld by th spcfc surfac ara pr unt volum of th lctrod (dnotd by a p,n ), computd by assumng that all th partcls ar sphrs of th sam radus r p, a p,n = 4πr2 p N 3(1 ε) (1 ε) =. (2.40) N 4 3 πr3 p Ths consrvaton quaton dscussd abov appls to both th postv as wll as th ngatv lctrod rgons ndvdually. For thr of th lctrods, th porosty, and hnc th ffctv dffusvty, th transfrnc numbr as wll as th racton rat ar dffrnt. Convntonally, subscrpt p and n ar usd for th postv and ngatv lctrod rgons rspctvly, and th quatons for ach of ths sctons ar ε c t = (εd ff,p,n c ) + (1 t+ ) a p,n j p,n F t + (2.41) In th abov quaton, D ff,p and D ff,n ar th ffctv dffusvts n th postv and ngatv lctrod rgons. As no racton taks plac n th sparator rgon, th quaton smplfs to c ε s = (D s c ), (2.42) t r 2 p

10 22 2 Thortcal Framwork of th Elctrochmcal Modl whr th porosty of sparator rgon corrsponds to th partal volum occupd by th lctrolyt wthn th pors of th sparator matrx. It nds to b mntond, howvr, that addtonal dpndncs can b addd to ths bas st of quatons dscussd n ths scton. For xampl, on can hav a transfrnc numbr or dffusvty that s dpndnt on concntraton. Addtonally, for hgh concntraton solutons, th varaton of ths quantts can also b consdrd. As lthum ons do not lav or ntr th cll, th boundary condton for th lthum-on mass consrvaton quaton s zro mass flux at th currnt collctor boundars, c x = 0; x=0 c x = 0, (2.43) x=lcll whr L cll s th thcknss of th cll. Th contnuty of concntratons and fluxs ar also mposd at th postv lctrod sparator (x = L n ) as wll as th sparatorngatv lctrod ntrfacs (x = L n + L s ) Charg Consrvaton n th Elctrolyt (Lqud Phas) Corrspondng to th mass flux, a currnt flux can b dfnd as Th lctrc conductvty σ s dfnd as Thus, charg flux can b wrttn as J = z N. (2.44) σ = (z ) 2 M c = (z ) 2 c D k B T. (2.45) J = z D c σ φ. (2.46) Th total currnt dnsty n th lctrolyt s obtand by th sum of th currnt fluxs of ons = J = z D c σ φ. (2.47) Dfnng th conductvty of th lctrolyt as th sum of conductvts of th ons, σ = σ. Elctro nutralty s nvokd to dntfy that c = c + = c = c. Th currnt dnsty s

11 2.2 Consrvaton Equatons 23 = c = c = ln(c) k BT z D σ φ (2.48) z D 1 + ln(γ ) ln(c ) σ z 1 + ln(γ ) ln(c ) ] σ φ (2.49) ] σ φ. (2.50) Th fnal rsult s an xprsson for th total currnt dnsty n th lctrolyt n trms of th conductvty. Now = ln(c) k BT σ z 1 + ln(γ ] ) σ φ. (2.51) ln(c ) Th transfrnc numbr s dfnd n trms of conductvts as t = σ /σ and satsfs th rlaton t + + t = 1. Th currnt dnsty can b rwrttn as = ln(c) k BT t z 1 + ln(γ ] ) σ φ. (2.52) ln(c ) For a bnary lctrolyt, lk n th cas of lthum-on clls, z + = z = 1. = ln(c) k BTσ On smplfcaton = ln(c) k BTσ ( (1 t + ) 1 + ln(γ ] ) t ln(γ ]) ) σ φ. (2.53) ln(c ) ln(c ) ( 1 + ln(γ ) 2t ln(γ+ ) + t + + ln(γ ]) ) σ φ. (2.54) Th actvty coffcnt of th lctrolyt can dfnd n trms of th rspctv onc valus usng γ ± = γ + γ. Ths lads to furthr smplfcaton as gvn blow = ln(c) k BTσ ( 2(1 t + ) 1 + ln(γ ] ±) 1 + ln(γ ]) +) σ φ. (2.55) Ths rsults n th fnal xprsson for th currnt dnsty n th lctrolyt = σ φ + 2(1 t + ) k BTσ 1 + ln(γ ±) ] ln(c) k BTσ 1 + ln(γ +) ] ln(c). (2.56)

12 24 2 Thortcal Framwork of th Elctrochmcal Modl Th last trm can b wrttn n trms of th mass dffusvty as = σ φ + 2(1 t + ) k BTσ 1 + ln(γ ] ±) ln(c) c D ln(γ ] +). (2.57) t + For bnary lctrolyts wth unt chargs, th transfrnc numbr can also b dfnd as t = D / D = D /D. Hnc = σ φ + 2(1 t + ) k BTσ 1 + ln(γ ] ±) ln(c) cd 1 + ln(γ ] +). (2.58) Dfnng a chmcal potntal to b μ = k B T ln(γ c ), followng smplfcaton, = σ φ + 2(1 t + ) k BTσ 1 + ln(γ ] ±) ln(c) c k B T D μ. (2.59) In obtanng th lctrolyt currnt balanc 32], on nds to assum that th flux gnratd du to couplng btwn th total mass dffusvty of th lctrolyt and th chmcal potntal gradnts ar nglctd, n comparson to th othr contrbutons. Thus, th quaton smplfs to = σ φ + 2(1 t + ) k BTσ 1 + ln(γ ] ±) ln(c). (2.60) Th abov quaton s wrttn for th lctrolyt, thr n th lctrod rgon or th sparator. Ths quaton s furthr rfnd on dntfyng /(k B T) = F/(R G T), = σ φ + 2(1 t + ) R GTσ F 1 + ln(γ ] ±) ln(c). (2.61) Applcaton to Lthum-Ion Clls Th lctrolyt currnt s computd usng th concntratd soluton thory as dscussd abov. Convntonally, Eq s wrttn as ( 2R G T = κ φ + κ F (1 t ) 1 + ln f ) ln c, (2.62) ln c whr κ s th ffctv conductvty of th lctrolyt computd from th porosts of thr th postv or th ngatv lctrod rgon, usng th Bruggmann rlaton (Eq. 2.39). Ths quaton s appld for ach of th lctrods to solv for th lctrolyt potntal, φ. As cll potntal s arbtrary up to a constant, a boundary condton s to ground th ngatv nd of th cll (x = 0),.., φ = 0. As th currnt ntrs and lavs th cll through th sold partcls n contact wth th currnt collctors, an nsulaton boundary condton s st at th othr nd of th cll,

13 2.2 Consrvaton Equatons 25 φ x. Altrnat s to st nsulaton boundary condton at both x = 0 and x=lcll x = L cll. Snc all th currnt flows rght through th sparator rgon, th quaton bcoms ( 2R G T I = κ s φ + κ s F (1 t s) 1 + ln f ) s ln c, (2.63) ln c whr I s th total currnt that ntrs or lavs th cll Charg Consrvaton n th Elctrod (Sold) Rgon Th consrvaton of currnt n th sold phas of th lctrod rgon s provdd by th gnralzd Ohm s law wrttn as s = σ s φ s. (2.64) Throughout th ngatv lctrod lthum ons ar gnratd du to th charg transfr racton and ar consumd n th postv lctrod. Faradays law gvs th quvalnt currnt gnratd du th producton of lthum ons. For on mol of unvalnt lthum on, th followng rlaton btwn th dvrgnc of th sold phas currnt and th rat of gnraton of lthum ons holds s = Fa j. (2.65) Combnng th Faradays law wth th Ohms law rlats th potntal n th lctrod phas wth th rat of racton. σ 2 φ s = Fa j. (2.66) Th abov quaton s solvd for th postv and ngatv lctrods to obtan th potntal of th sold phas n th lctrod. As th currnt ntrs th cll at x = 0 and lavs at x = L cll, boundary condtons ar σ n φ s x=0 = I; σ n φ s x=lcll = I. (2.67) An altrnat boundary condton for th sold potntal s φ s x=0 = 0. (2.68) Ths boundary condton can b usd to st th datum n th potntal, f nsulaton boundary condtons at usd for th lctrolyt potntal, φ l ar x = 0 and x = L cll. At x = L n, th currnt lavs th sold phas and s th total appld currnt s carrd

14 26 2 Thortcal Framwork of th Elctrochmcal Modl through th lctrolyt n th sparator, and th rvrs happns at x = L n + L s. Thus, th sold phas currnt s zro at ths ntrfacs σ n φ s x=ln = 0; σ p φ s x=ln +L s = 0. (2.69) Th sold phas currnt consrvaton quaton s solvd for both th lctrods to obtan th potntal n th sold phas at both th lctrods. Th soluton also nabls to study th varaton of ths quantty at varous cll thcknss. Soluton of ths quaton s of spcal sgnfcanc, as th cll voltag, th fnal output of any mathmatcal modl for th battry s dfnd as th dffrnc n sold phas potntal btwn th nds of th cll. V cll = φ s x=lcll φ s x=0. (2.70) 2.3 Th Charg Transfr Racton Th charg transfr racton nvolvs an qulbrum componnt as wll as a dynamc componnt. Th qulbrum componnt basd on thrmodynamcs rlats th opncrcut potntal to th actvty or quvalntly th concntratons, and s gvn by th Nrnst quaton.th dynamc componnt rlats th drvng forc for th charg transfr racton to th racton flux or th currnt transfrrd. W wll start wth th dscusson of th Nrnst quaton Nrnst Equaton: Equlbrum Componnt and ts Thrmodynamc Connct For any charg transfr racton at qulbrum, R + n P (2.71) Th Nrnst quaton rlats th qulbrum lctrod potntal V 0 to th standard cll potntal V 0 and th compostons of th ractants and th products. V 0 = V 0 + R GT ln Z (2.72) nf

15 2.3 Th Charg Transfr Racton 27 Th compostons ar adquatly rprsntd Z, th rato of th actvts of th products to th ractants, or smply, th rato of thr concntratons, Z = a p a R = c P c R. (2.73) To obtan th Nrnst quaton, th frst stp s to dntfy th fundamntal conncton btwn lctrochmcal and thrmodynamc quantts. Onc ths s dntfd, fundamntal thrmodynamc rlatons ar subsquntly usd. Th most ntrgung thrmodynamc quantty s th Gbbs fr nrgy, G. Ths quantty bng dfnd undr condtons of constant tmpratur and prssur, asly achvabl xprmntal condtons, maks t th most usd (and msundrstood!) thrmodynamc varabl. Th Gbbs fr nrgy s dfnd as G = H + TS. (2.74) W wll go through a smpl thrmodynamc rlaton and s what masur G corrsponds to. Th nthalpy H s rlatd to ntrnal nrgy E, and prssur P and volum V,va H = E + PV. (2.75) Th chang n nthalpy for a constant prssur procss s gvn by, ΔH = ΔE + Δ(PV) = ΔE + PΔV. (2.76) Th chang n ntrnal nrgy can also b rlatd to hat transfr and th rvrsbl work, ΔE = ΔQ ΔW = TΔS ΔW rv, (2.77) wth S bng th ntropy. From th arlr rlatons, th chang n Gbbs fr nrgy at constant tmpratur s ΔG = ΔH Δ(TS) = PΔV ΔW rv. (2.78) Total rvrsbl work conssts of xpanson work gvn by PΔV and all othr typs of work. Thus, th dffrnc btwn th quantts n th rght sd of th abov quatons gvs th nt non-xpanson work that s drvd out of th systm. In th cas of th lthum-on cll t s lctrcal work, ΔW c. Thus, t can b sn that ΔG = ΔW c. (2.79) Chang n Gbbs fr nrgy gvs th rvrsbl lctrcal work n a lthum-on cll. Elctrcal work s also dfnd n trms of transfrrng nf lctrons across an qulbrum voltag of V 0 as, ΔW c = nfv 0. (2.80)

16 28 2 Thortcal Framwork of th Elctrochmcal Modl Thus, w arrv at th fundamntal rlatonshp btwn thrmodynamcs and lctrochmstry that rlats th qulbrum potntal and Gbbs fr nrgy, ΔG = nfv 0. (2.81) For chmcal racton thrmodynamcs, chang n Gbbs fr nrgy s rlatd to concntratons as, ΔG = ΔG(0) + R G T ln c P c R. (2.82) whch agan can b drvd from dntfyng that ΔG s th dffrnc of chmcal potntals of th product and th ractant. ΔG = μ P μ R. (2.83) As sn arlr, chmcal potntal s dfnd for modrat concntratons as μ P = μ P + R GT ln c P, μ R = μ R + R GT ln c R. (2.84) Substtutng ths n th arlr quaton, on gts th rlaton for chang n Gbbs fr nrgy n trms of concntratons. Usng our nwfound rlaton btwn Gbbs fr nrgy and qulbrum potntal and dvdng th abov quaton by nf, w gt th Nrnst quaton, V 0 = V 0 (0) + R GT nf ln c P c R. (2.85) Th Nrnst quaton s th bass for obtanng any dpndnc of qulbrum potntal of postv or ngatv lctrod wth th concntratons. Ths quaton, howvr, obtand from thrmodynamcs, gvs only an dal cas. Howvr, matral varatons mpart unqu faturs n th OCV curv. For ralstc scnaros, varous modfcatons ar proposd. In Fgs. 2.3 and 2.4, th opn-crcut potntals of common matrals ar shown. Th prsnt crop of commrcal battrs prdomnantly us som blnd of carbon (grapht, cok, hard carbon, or msocarbon mcrobads (MCMB)). Thr ar mor varty for th cathod lctrods and t s dpndnt on th applcaton and th rqust nrgy dnsty.wth ths subscton, w obtan th qulbrum part n th charg transfr racton. Th dynamc part s subsquntly drvd.

17 2.3 Th Charg Transfr Racton 29 Fg. 2.3 Th opn-crcut potntal of commonly usd anod matrals Fg. 2.4 Th opn-crcut potntal of commonly usd cathod matrals Butlr Volmr Equaton: Th Dynamc Componnt For a charg transfr chmcal racton as shown blow, R s rducd and P s oxdzd. Th formr s trmd as an anodc procss, and th lattr cathodc. Th chmcal racton rsults n an quvalnt currnt dnsty gvn by, j c = k c c R and, j a = k a c P. (2.86) whr j c and j a ar trmd as th cathodc and anodc currnt dnsts, k c and k a ar th rat constants for th rspctv ractons. Ths ractons happn at th lctrod surfacs, and th nt racton rat dtrmns f th rvrsbl procss s oxdaton or rducton. In th ngatv lctrod, th nt racton durng dscharg s oxdaton and th nt racton at th postv lctrod s rducton. Durng chargng howvr, th procsss ar rvrsd. As a rsult rducton racton occurrs at th ngatv lctrod and oxdaton at th postv lctrod. Th nt racton rat that rsults n th nt currnt dnsty s gvn by = Fj a Fj c = Fk a c P Fk c c R (2.87)

18 30 2 Thortcal Framwork of th Elctrochmcal Modl Each of ths procsss s actvatd procss. Ths mans that th procsss ar trggrd onc th molculs undrgo a thrshold ntrnal nrgy chang rqurd at a gvn tmpratur. Thus k = k 0 xp ΔE/R GT = k 0 xp { ΔH/R GT} (2.88) Bcaus, from frst law of thrmodynamcs, ΔH ΔE, wth nglgbl chang n P or V. Ths quaton s modfd and usng th scond law, Gbbs fr nrgy agan appars k = k 0 xp { (ΔH TΔS)/R GT} = k 0 xp { ΔG/R GT}. (2.89) Thus, th rat constants of both th lctrodc procss can b xprssd n trms of Gbbs fr nrgy as, k c = k c0 xp { (ΔG c)/r G T}, k a = k a0 xp { ΔG a/r G T}. (2.90) It can b nfrrd from our arlr dscusson arlr, Gbbs fr nrgy prtans to th lctrcal work ndd to swtch on ths procsss. Th nt currnt dnsty at any lctrod s gvn by = Fk a0 xp { ΔG a/r G T} c P Fk c0 xp { (ΔG c)/r G T} c R. (2.91) Th abov quaton, th nt rat of a rvrsbl racton that s actvatd, s th basc form of Butlr Volmr racton. Th ractons ar howvr, not at qulbrum, whch was our basc prms for movng byond Nrnst typ quaton. In lthum-on battrs, a lthum-on taks up only on lctron durng charg transfr racton,th racton s lk, R + P. If th racton nvolvs a chang of potntal gvn by ΔΦ = V V (0), th xtra lctrcal work rqurd, th Gbbs fr nrgy chang s gvn by ΔG = ΔG(0) + FΔΦ. (2.92) If th nrgy stat at th ntal stat (dnotd as 0 ) s tslf nough for th racton to occur, howvr, ΔG = ΔG(0). In a ralstc scnaro, th nrgy chang rqurd s takn as an avrag of ths two xtrms, and th constant, a transfr coffcnt α s ntroducd, ΔG = ΔG(0) + αfδφ. (2.93) Th anodc and cathodc procsss ar th forward and rvrs ractons of th rvrsbl lctrod racton. If w consdr th abov scnaro for th cathodc procss, corrspondng quaton for th anodc procss gvs ΔG = ΔG(0) (1 α)fδφ. (2.94)

19 2.3 Th Charg Transfr Racton 31 Insrtng th abov xprssons for our nt rat of racton, and absorbng th trms nvolvng ΔG(0) nto th constant, = Fk a0 xp {(1 α)fδφ/r G T} c P Fk c0 xp { αfδφ/r G T} c R. (2.95) For a systm at qulbrum, ΔΦ = V 0 V (0), and th nt currnt s zro. Ths rsults n both th currnt dnsts bng th sam, trmd as th xchang currnt dnsty, 0. Lt us also ntroduc f = F/R G T 0 = Fk a0 xp {(1 α)fδφ/r G T} c P,EQ = Fk c0 xp { αfδφ/r G T} c R,EQ. (2.96) At ths stat of qulbrum, on can us Nrnst quaton to s th dpndnc btwn concntratons, xp {f (V 0 V (0))} = C P,Eq, C R,Eq xp { αf (V 0 V (0))} = Thus, th xchang currnt dnsty s wrttn as ( CP,Eq C R,Eq ) α. (2.97) 0 = Fk 0 c α P,Eq c(1 α) R,Eq (2.98) To lmnat th ffct of th datum potntal, th nt rat s dvdd by th xchang currnt, Fka0 c P xp {(1 α)f (V 0 V (0))} = 0 Fk a0 c P,Eq xp {(1 α)f (V 0 V (0))} Fk ] c0c R xp { αf (V 0 V (0))} Fk c0 c R,Eq xp { αf (V 0 V (0))} (2.99) Evntually, th most common form of th Butlr Volmr racton flux s obtand as ] c P = 0 xp {(1 α)f (V 0 V (0))} c R xp { αf (V 0 V (0))}. (2.100) c P,Eq c R,Eq In many cass, a local qulbrum of concntraton s consdrd, whch furthr smplfs th quaton to gv, = 0 xp {(1 α)f (V0 V (0))} xp { αf (V 0 V (0))} ] (2.101) Applcaton to Lthum-Ion Clls Lthum ons n th lctrolyt wth concntraton c, and potntal φ ntrcalat nto th lctrod partcls. In ths cas, th concntraton of th fr actv sts also dtrmns th rat of racton. In an lctrod partcl wth sold concntraton c s

20 32 2 Thortcal Framwork of th Elctrochmcal Modl and th solublty c s,max, th concntraton of unoccupd sts s gvn by c s,max c s. Also, du to th lctrolyt potntal, Δφ = φ s φ V (0). Accountng for ths factors, th fnal form of Butlr Volmr quaton that s usd to solv th lctrochmcal modl rads as, = 0 xp {(1 α)f (φs φ V (0))} xp { αf (φ s φ V (0))} ], (2.102) Wth th xchang currnt dnsty gvn by, 0 = Fk 0 c α s ( cs,max c s ) 1 α c (1 α). (2.103) To ncorporat n th lctrochmcal modl, ths quaton has to b solvd for both th lctrods. Whl solvng for ralstc cas scnaros, th rat constants of th lctrods ar oftn adjustd to match th xprmntal rsults.

21