Time constant τ = RC:

Transcription

1 Tme ontnt τ C: Z jωτ / jωc ωτ Semrle n the mpene plot Y ωc ωτ / jωc ωτ j Semrle n the mttne plot If n equvlent rut h everl tme ontnt loe to eh other, the eprton of rut element ffult. C, C, Smple moel for PEFC: m t, t, ften t,» t, τ» τ n only one em-rle pper n the mpene plot. Cthoe n noe ptne C, n C, mut uully e reple y ontnt phe element epree em-rle; th wll e ue hortly.

2 Aorpton ften eletrotve pee or frt on the eletroe n eletron trnfer follow orpton. Mot often orpton nlyze through Lngmur type proe. In Lngmur type orpton only one lyer of n orte llowe n there no nterton etween the orte. The mxmum mount of ore moleule n monolyer enote y Γ mx n the urfe overge y Γ/Γ mx where Γ the mount of ore mount n mol/m. A typl vlue of Γ mx 9 mol/m. The net equton of orpton t where the orpton n eorpton rte ontnt, n the onentrton mol/m 3 of the orte t the eletroe. From the ove equton t een tht the unt of m 3 /mol n tht of /. The frt term on the rght hn e tell tht orpton n te ple only on free urfe, repreente y the ftor. At equlrum /t nturlly zero n the urfe onentrton equl to the ul onentrton. Hene: K eq eq eq ; K K The frme equton the fmou Lngmur orpton otherm; K the orpton equlrum ontnt. At low onentrton whh proly the e n eletronlytl wor the lner form of the otherm,.e. the numertor lone uffe: eq K. At tey-tte, /t ut mut e lulte from the tey-tte trnport equton.

3 Let trt the nly wth n extreme e where reox ouple mmolze on the urfe. Th n e one y grftng the urfe wth, e.g. n the e of gol eletroe wth thol, or n the e of ron eletroe wth zonum lt. The fgure elow ept the tuton. We ume tht Γ Γ Γ mx or,.e. the urfe fully overe y n. There thu no mterl exhnge etween the oluton n the urfe monolyer. Eletr urrent enty mply nf Γ mx t t Anlogouly to prevou tretment, the lnerze urrent-overpotentl equton η T nf, eq, eq The pelty here tht jutng the eletroe potentl to ny vlue, fter trnent phe, the urfe overge ettle orng to the Nernt equton eue the mount of rete pee not reple y trnport from the oluton:, eq, eq nf θ exp T ' E E eq, eq θ θ ;, eq θ

4 The exhnge urrent enty nlogouly α α α θ nf Γ Γ Γ, eq, eq nf mx nf θ θ α Γ mx α θ θ Note tht the unt of now / n we hve fferent t eh potentl! Strghtforwr nly n Lple omn proee follow: nf Γ mx, eq, eq,eq nf Γ,eq ;,eq naγ,eq η T nf nf Γ, Z ω mx eq, eq n F AΓ ω j C eq eq ω,, T nfi T mx j t t n F T AΓ mx θ θ α ; C n F T AΓ mx θ θ

5 In the plot e, the potentl epenent prt of t n C re plotte. Now the ptne h mxmum t E. Inertng Γ mx 9 mol/m gve C mx mf/m. Sne the re of porou eletroe n e of the orer of m /g ptor of very hgh hrge n e prepre.e. uper ptor. Tng the mnmum hrge trnfer retne. 53 Ω m,.e. the retne very mll wth hgh urfe re. Let oner next more onventonl e where metl ton M z frt ore on the eletroe n then reue: nf re Γ M e α f ' EE Γ M '' θ α θ.3. C. θ ' C ' θ We hve thu gnore the no prt of the Butler-Volmer equton ne we re epotng metl E «E, n urrent negtve; lo, z n. The urfe onentrton of the ton, Γ M, n nturlly e wrtten Γ mx. The m lne of the ore ton, umng Lngmur type orpton t nf Γ mx onumpton n eletroe reton < M orpton M M nf Γ eorpton mx

6 Let mplfy the prolem negletng m trnfer,.e. whh men tht orpton the rte etermnng tep. At M tey-tte /t n from the m lne t otne: / nf Γ mx 3 Lnerzng eq. n trnformng t to Lple omn gve nf Γ mx re [ α f E ] 4 where enote evton from tey-tte ue to the gnl. Eq. lrey lner eue or o not epen on potentl. Thu t Lple trnform nf Γ mx 5 6 nf Γ mx Now we pply for the frt tme the rule of fnng the perol omponent. The lt term on the rght not perol eue t ontn nether nor E. We throw t wy n nert the ret nto eq. 4. After ome lger:

7 α fz nfaγ mx re where oth e hve een ve y. After ome rerrngement n replng wth jω t otne: Z T α n F AΓmx re jω The frt term n the mpene oe not epen on frequeny: hene t the hrge trnfer retne. The eon term omnton of two element. Ponerng for whle t ovou tht t prllel omnton of retor n ptor. Ang the lwy preent eletroe oule lyer ptne n the oluton retne the mpene plot loo le th: Comnng eq. n 3, n e olve re

8 Let wrte own the explt expreon of the fr mpene element: T t α n F AΓmx re α n F T AΓ mx re C α n F AΓ mx T re Thee element re otne from non-lner ft, n the vlue of the phyl prmeter re otne from them: t re C C / t re Impene metho thu gve the vlue of the eletroheml rte ontnt, re, expltly, ut fnng the vlue of or requre meurement t vryng oluton onentrton.

9 Let oner next the e where reox ouple h ore ntermete n orpton very ft, oeyng Lngmur otherm. Aorpton thu uner m trnfer ontrol. The generlze urrent ounry onton t x D t x D nf x x Γ Γ Lnerze Lngmur otherm: θ Γ Γ θ K K K K * mx Λ Γ Γ Γ t t K t K K t mx mx, / A e A x x D [ ] Λ Λ Λ D A A D A D A nf n Λ Λ Λ D nf D nf D nf A. extr term Λ, n the enomntor! >> K

10 Z T nfi T n F A D jω Λ jω D jω Λ jω C W W C, n F T A Λ jω t C, C, But th equvlent rut ntnguhle from the nle rut, emontrtng tht more thn one equvlent rut gve mlr mpene plot, n reerher mut tuy the ytem frt wth, e.g. yl voltmmetry to ee wht the ytem le. If my tool hmmer, I ee ll the prolem nl.

11 Moel for n eletroe wth n orpton tep C C Ω C µf t 5 Ω C 5 µf Ω orpton net retne W σ Ω / t 8 -Z''/Ω Z'/Ω

12 Heterogenety of the eletroe urfe Generl element Q, the Contnt Phe Element, CPE In relty, omplete em-rle re elom heve ut they re flttene. Th trete mthemtlly y replng ptor wth ontnt phe element, the mttne of whh Y Y jω α. Below multon wth Ω, Y 7 Ω n α,.9,.8: the lower the vlue of α, the fltter the em-rle Q Z rel / Ω

13 el meurement, humn ver n: Z mg / Ω et em-rle meure 3 4 Z rel / Ω ft Feture: - elongton long the rel x - flttenng of the em-rle Q The lower C rnh orrepon to polr relxton tng ple n the memrne mtrx wth the hrtert relxton tme τ C. Sn lp??? C K. Konttur, L. Murtomä, Phrm. e. 9, 994, J.. Monl, Impene Spetroopy, John Wley & Son, New Yor 987, p. 49.

14 The orgn of CPE n the peron of tme ontnt. An C rut h only one tme ontnt τ C, ut n relty, n eletroe urfe heterogeneou wth te of vrle retvty. Inte of ngle C rut, the urfe houl e llutrte networ of C rut, formng pn of tme ontnt over everl ee. The nly of CPE n lo e one through frtl geometry: α D D F frtl menon F D F. D.3 F Cro eton of two frtl urfe wth D F. n.3, orreponng to α.9 n.77. K. Konttur, et l., Phrm. e.,, 993, S.H. Lu, Phy. ev. Lett

15 Alo, n the Wrurg mpene the urfe heterogenety een the evton of the phe ngle from 45. Wrurg mpene ~ jω / reple y CPE ~ jω α wth α <.5 for prtlly loe urfe n α >.5 for rough urfe. In th e, the relton to the frtl menon α D F / D F α. In potentl tep experment I ~ t α. Thee relton hve een prove orret wth tlor-me eletroe wth unmguouly efne frtl menon. D F.5 Impene of Koh eletroe: T. Pjoy, L. Nyo, J. Eletrohem. So

16 Potentl tep t Serpn get: T. Pjoy, L. Nyo, Eletrohm. At Cottrell ehvor: Serpn get. D F.585