pubs.acs.rg/jacs An Interfacal and Bulk Charge Transprt Mdel fr Dye-Senstzed Slar Cells Based n Phtandes Cnsstng f Cre Shell Nanwre Arrays Justn J. Hll,, Nck Banks, Kelly Haller, Mark E. Orazem, and Krk J. Zegler*,,z Department f Chemcal Engneerng, Unversty f Flrda, Ganesvlle, Flrda 32611, Unted States z Center fr Surface Scence and Engneerng, Unversty f Flrda, Ganesvlle, Flrda 32611, Unted States bs Supprtng Infrmatn ABSTRACT: Dye-senstzed slar cells (DSSCs) based n rdered phtande mrphlges, such as nantubes and nanwres, are wdely ganng attentn because these gemetres are beleved t enhance nterfacal charge transfer and bulk charge transprt. Unfrtunately, expermental results have yet t shw substantal mprvement t cnversn effcency ver nanpartcle-based DSSCs. A mdel s develped t characterze the perfrmance f an dealzed phtande based n an rdered array f transparent cnductve nanwres cated wth an anatase ttana shell. The rle f the nterfacal electrc feld n nanwre-based DSSCs s explred cmputatnally by turnng electrn mgratn ON r OFF. The results shw that back-reactn rates are mst strngly nfluenced by the electrc feld. These electrn lss mechansms can be reduced by several rders f magntude, leadng t mprvements n shrt-crcut current, pencrcut vltage, and fll factr. INTRODUCTION Dye-senstzed slar cells (DSSC) are tradtnally based n flms f ttana nanpartcles (NP-DSSC). 1 These devces mmc phtsynthess and currently have acheved cnversn effcences n excess f 11%. 2 They can be fabrcated at lwer csts than slcn-based phtvltacs, makng them an attractve alternatve fr prducng lw-cst, clean energy. 3 5 Addtnally, DSSCs have been shwn t utperfrm slcn-based phtvltacs n cludy days. 6 The nterfacal knetcs and energy levels f the hetergeneus cmpnents f tradtnal NP-DSSCs are desgned t ptmze perfrmance. 1,4,7 The maxmum utput vltage s determned by the dfference between the flat-band ptental f the semcnductr and the electrchemcal ptental f the electrlyte, E REDOX. Once an electrn s phtexcted frm the hghest ccuped mlecular rbtal f the dye (HOMO) t the lwest unccuped mlecular rbtal (LUMO), the electrn can ether relax back t the grund state, be njected nt the semcnductr layer, r back-react wth the electrlyte. Mst electrns shuld be njected nt the semcnductr regn t acheve maxmum perfrmance; hwever, lss f phtgenerated electrns t the electrlyte s energetcally favrable. Therefre, the knetcs f electrn njectn nt the semcnductr must be much faster than electrn lss t the electrlyte (10 13 vs 10 2 s 1 ). The electrlyte can be mdfed t ncrease the REDOX ptental, but ths typcally ncreases the rate f back reactn. 8 Therefre, the requrement that reactns be cntrlled by knetcs rather than thermdynamcs results n lwer utput ptental f the cell. When the nterface f a DSSC s llumnated, an excess f pstve charge wll buld up n the space-charge layer f the n-type semcnductr. Ths accumulatn n the space-charge layer helps establsh an nterfacal electrc feld that separates charge. Hwever, Fgure 1a shws that ths feld wll dsappear at a fnte, and nnzer, nanpartcle dameter. Ths feld becmes neglgble fr partcle szes belw 30 nm 9 such that typcal DSSC systems have the flat-band ptental llustrated n Fgure 1b. In addtn, Fgure 1a shws that any charge drven nt the center f the partcle by ths feld experences a barrer t transprt ut f the nanpartcle. Therefre, electrn transprt n the prus, nancrystallne phtande f the NP-DSSC s gverned by trap-lmted dffusve transprt f cnductn band electrns n the surface f the nanpartcle. 10 15 Unfrtunately, the cmpetng relatnshp between electrn dffusn and lfetme lmts the electrn dffusn length t abut 10 μm wthn a nanpartcle flm. 11 Attempts t ncrease phtcurrent by fabrcatng nanpartcle phtandes wth dmensns n excess f 10 μm are ften futle because almst nne f the electrns generated n ths regn are cllected. 11,16 18 The lmts f NP-DSSCs have led researchers t nvestgate nanwre-based DSSCs (NW-DSSCs) as an alternatve cnfguratn fr the phtande. In these systems, an nterfacal electrc feld s establshed that s nt present n NP-DSSCs. Ths feld ads nterfacal charge transfer and mnmzes electrn lss. 19 22 Hwever, the effect f the electrc feld dmnshes as Receved: May 19, 2011 Publshed: September 07, 2011 r 2011 Amercan Chemcal Scety 18663 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety Fgure 1. Schematc cmparsn f the energy level dagrams fr the semcnductr dye electrlyte nterface f DSSCs based n (tp) ttana nanpartcles and (bttm) ttana cre shell nanwres. The energy levels wthn the nanpartcle and shell are shwn n (a) and (c), respectvely. The resultng energy level dagrams fr the devces are shwn n (b) and (d), respectvely. the dameter f the nanwre decreases, leadng several researchers t nvestgate cnductve cre semcnductve shell nanwres n rder t enhance axal transprt f electrns.21 24 Fgure 1c shws that these cre shell nanwre systems exhbt the same charge buldup at the semcnductr electrlyte nterface. Hwever, the drect electrcal cntact f the cre t the current cllectr allws the charge t be remved frm the ttana layer wthut gng thrugh the ptental energy barrer that NP-DSSCs experence. Therefre, an electrc field s establshed that helps separate the charge at the nterface and drve the electrns t the nanwre cre, as shwn n Fgure 1d. Even n the absence f ths field, the prbablty f radal electrn transprt t the cnductve cre fllwed by axal transprt s hgher than axal transprt f electrns slely thrugh the ttana shell r n ts surface t the current cllectr. Ths preferental drectn n transprt s especally mprtant fr electrns njected at lengths greater than an electrn can dffuse t the current cllectr.11,16 18 Therefre, electrns are expected t preferentally flw radally tward the nanwre cre. Due t the rapd transprt f electrns n the cnductve cre, t s beleved that njected electrns n even small cre shell nanwres wll be drven t the cre. Ths effectvely decuples charge cllectn dstance frm phtande thckness and allws fr phtactve surface area (and phtgenerated current) t be ncreased beynd the current transprt-drven lmtatns f NP-DSSCs. Mathematcal mdels f NP-DSSCs that accunt fr nterfacal reactn dynamcs and energy levels have been develped n cnjunctn wth dynamc respnse measurements.8,16,25,26 Bth dscrete and cntnuum-based cmputatns ndcate that the transprt f electrns wthn the nanpartcle phtande s purely dffusve.11,13 15 Furthermre, the verall perfrmance f NP-DSSCs has been accurately predcted thrugh bth a cntnuum mdel f the phtande15 and a crcut mdel f the phtelectrchemcal cell.12 Althugh the perfrmance and dynamcs f NP-DSSCs can be calculated, n cmparable mdels exst fr the NW-DSSC. T ur knwledge, ths paper descrbes the first mdel that can predct the verall behavr f DSSCs based n nanwre phtandes cmpsed f a cnductve cre and a semcnductve shell. The mdel s used t llustrate hw the nterfacal prcesses are affected by the electrc field and cnductve cre present n cre shell NW-DSSCs and, thus, nly ncludes the dmnant rutes f charge transprt n the Fgure 2. (a) Sde-vew and (b) tp-vew schematc dagrams f the nanwre phtande. The nanwres are cated wth a semcnductr shell and surrunded by an electrlyte. Only a cylndrcal shell f electrlyte s cnsdered. radal drectn. The electrc field n the system s nt artfically mpsed, but arses slely as a result f charge carrer dstrbutns n the semcnductr thrugh dffusn and mgratn. As such, the thckness f the shell sgnficantly alters the electrc field gradents. The effect f the field and cnductve cre s quantfied by elmnatng mgratnal cntrbutns t electrn transprt and cmparng the changes t nterfacal reactns and verall perfrmance. The results shw that back-reactn rates can change by as much as 4 rders f magntude n the presence f an electrc field. MODEL DEVELOPMENT The structure f the phtande, shwn n Fgure 2a, cnssts f transparent cnductve nanwres wth hgh aspect rats. The nanwres are cated wth a cnfrmal semcnductve layer and a phtactve dye. The nanwre array s mmersed n a REDOX electrlyte, whch fr the purpses f these cmputatns nly ncludes a cylndrcal shell f electrlyte. Therefre, there s a small regn between the nanwres that s neglected, as s shwn n Fgure 2b. The caxal nanwre has a cnductve cre f radus r and length l, a semcnductng shell (sh) that s bunded between r [r, r + τsh] and z [0, l] and a 18664 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety surrundng electrlyte regn (el) bunded between r [r + τ sh, r + τ sh + τ el ] and z [0, l ], where τ sh s the semcnductr shell thckness and τ el s half f the nanwre nanwre spacng. The dstance separatng the current cllectr frm the cunter electrde s gven by L. Because the array cnssts f clsely packed nanwres wth hgh aspect rats, any cntrbutns t phtvltac perfrmance frm the tps f the nanwres r the semcnductr TCO substrate (.e., the bttm f the phtande) were neglected snce the planar surface area cntrbutes nly a mnr fractn f the ttal phtactve surface area. The electrlyte was taken t be an dde/trdde system (I /I 3 ), wth L + as the cuntern. Key cmpnents f the NW-DSSC, such as carrer cncentratn, ptental, flux f charge carrers, and electrlyte cncentratn, are dependent n the radal and axal dmensns. Hwever, scalng arguments smplfy the mathematcs and decrease cmputatnal expense. The spatal crdnates were scaled such that ~Z =z/l, ~R el =(r r τ sh )/τ el,and~r sh =(r r )/τ sh, where ~Z s the dmensnless length scale n the z-drectn and ~R el and ~R sh are the dmensnless length scales n the radal drectn fr the electrlyte and semcnductr, respectvely. In ths study, the lengths f the nanwres (l ) were n the rder f 1 10 μmandthe semcnductng (τ sh ) and electrlyte (τ el ) regns were n the rder f 10 nm. Therefre, fr any functn, F (~R j, ~Z), the dfferental peratr rb = ^r( / r) + ^z( / z) can be apprxmated by ~ F ¼ 1 τ j ^r j þ 1 a j ^z ~Z! F = 1 τ j ^r j F ð1þ and smlarly 3 2 becmes! 2 F ¼ 1 τ 2 þ 1 j 2 j a 2 F = 1 2 j ~Z 2 τ 2 F ð2þ j 2 j where a j = l /τ j s the aspect rat asscated wth the semcnductr and electrlyte regns. Equatns 1 and 2 prvde reasnable apprxmatns f the gradent f the state varables n the axal drectn are n the same rder r smaller than changes t thse varables n the radal drectn. The dstance ver whch large changes t the state varables (.e., ptental, cncentratn, etc.) ccur s determned by the regn where charge densty vares dramatcally, whch s crrelated t the Debye length. Fr an electrlyte, the Debye length, λ D,el, s gven by 2 31=2 ε el RT λ D, el ¼ 4 F 2 z 2 C 5 ð3þ, where C,, ε el, and z are the bulk (reference) cncentratns f the electrlyte ns, the electrlyte permttvty and the charge f the electrlyte speces, respectvely. Typcally, the value f the charge densty appraches zer at a dstance f rughly 10 20λ D,el frm the nterface. Usng reasnable values fr the electrlyte, eq 3 yelds λ D,el = 0.95 nm. Thus, the dffuse regn f charge wll have a thckness f apprxmately 20 nm. Smlarly, the Debye length fr the semcnductr, λ D,sh, can be wrtten as " # 1=2 ε sh RT λ D, sh ¼ ð4þ F 2 ð ^N d þ ^N a Þ where ^N d and ^N a are the dnr and acceptr cncentratns, respectvely, and F s Faraday's cnstant. Equatn 4 yelds λ D,sh = 28.2 nm. The wdth f the space-charge regn n the semcnductr s a factr f (2FV/RT) 1/2 larger than the Debye length. Ths factr s equal t 6.84 fr an nterfacal ptental drp f 0.6 V (the dark nterfacal ptental dfference f a NP-DSSC), 10 yeldng a space-charge regn thckness f apprxmately 200 nm n the semcnductve shell f the nanwre. These values are cnsstent wth the expectatn that the depletn regn s much larger n a semcnductr than an electrlyte when the tw are n electrcal cntact. The Debye lengths calculated frm eq 3 and 4 suggest that the apprxmatns gven by eq 1 and 2 may nt be vald n the semcnductr wthn 200 nm f the ITO current cllectr r wthn 20 nm f the ITO n the electrlyte. Hwever, these regns nly encmpass 1% f the ttal phtactve surface area fr nanwres wth lengths f 10 μm and dameters f 200 nm. An andc half-cell DSSC encmpasses three dmans: () the semcnductr dye electrlyte nterface, () the semcnductr, and () the electrlyte. Therefre, the mdel ncludes the equatns that gvern the physcs f these three dstnct regns. An nterfacal knetc mdel presented belw shws the nteractn f charge carrers wth the dye, electrlyte, and semcnductr nterfacal states. The bulk charge carrer transprt thrugh the semcnductr shell and electrlyte are then mdeled as havng a pstn-dependent dstrbutn f charge carrer densty, whch can gve rse t an electrc feld that sweeps charge carrers preferentally t the current cllectr (cnductve nanwre cre). T evaluate the current vltage prfle, the nterfacal mdel s cupled wth the equatns f bulk charge transprt and ptental dstrbutn wthn bth the semcnductr and electrlyte. A lst f varables and cnstants used fr ths study are presented n Tables S1 S5 (Supprtng Infrmatn [SI]). INTERFACIAL KINETICS The generatn and transfer f charge at the semcnductr dye electrlyte nterface nvlves a cmplex arrangement f seres and parallel reactn pathways. As shwn n Fgure 3a, t s assumed that the ttana cvers cmpletely the cnductve cre f the nanwre. Hwever, the phtactve dye may nt cver the entre surface f the ttana layer. Therefre, charge transprt can ccur between the electrlyte and ether the dye r ttana. Charge transfer reactns medated thrugh phtexctatn f charge wthn the dye ether: (1) prduce the desred andc reactn (electrn njectn nt the semcnductr) r (2) serve as a lcalzed cathde, resultng n n usable cnversn f slar energy (.e., back react). The varus states that exst n each regn f the DSSC and the reactns between them are llustrated n Fgure 3b. The dye exsts at the nterface and has bth a snglet (S / 1 ) and trplet excted state (S / 3 ). Electrns n the semcnductr regn may have energes asscated wth the valence (VB) and cnductn bands (CB) as well as trap states (t). The nner surface state (ISS) represents a plane f charge asscated wth electrns ccupyng surface states, and the uter surface state (OSS) represents a plane f clsest apprach fr electrns n the space-charge regn. An analgus descrptn s appled fr the electrlyte, where the nner Helmhltz plane (IHP) s asscated wth adsrbed ns and the uter Helmhltz plane (OHP) s the plane f clsest apprach fr hydrated ns n the dffuse regn f charge. 18665 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety The generc rate f reactn, rl, has frward and backward reactn rate cnstants, kf,l and kb,l, as well as the andc and cathdc expnental cnstants, ba =(1 α)nf/rt and bc = αnf/rt. The reactn energetcs are assumed t be symmetrc abut the actvatn energy (.e., α = 1/2 s that ba = bc = 19.47n V 1 ). The reactn rders, p,l and q,l, are related t the stchmetrc ceffcents, s,l, whch are defned by 27 ( s, l ¼ p, l fr s, l > 0 q, l fr s, l < 0 Fgure 3. (a) Schematc dagram fr mdelng the charge transprt n NW-based DSSCs wth an electrc feld. The electrn s njected nt the ttana layer whle the hle s njected nt the electrlyte. (b) At the nterface f the electrlyte and nanstructures, numerus reactns can ccur between the dye, electrlyte, and semcnductr regns. The paths shwn n blue are the knetc reactns mdeled n ths study. States are defned as: OHP/IHP = Outer/Inner Helmhltz Plane; OSS/ ISS = Outer/Inner Surface States; Ox/Red = Oxdzed/Reduced Electrlyte Atm; S 1 / /S 3 / = excted dye states; S g /S + = dye n grund/ reduced state; VB/CB = valence/cnductn band state; a = adsrbed speces. Multple charge-transfer reactns can ccur between these energy states as shwn by the lnes cnnectng each state n Fgure 3b. Hwever, nly reactns 1 7 (marked n blue n Fgure 3b) are cnsdered n the present wrk because these reactns prduced a gd crrelatn wth expermental data fr NP-DSSCs. 14,15 The specfc reactns ccurrng between these states are presented n Table 1. The reactns lsted n Fgure 3b fllw thse develped by Penny et al., 14 wth the exceptn that the effect f surface area s taken nt accunt. They are utlned here fr clarty usng general ntatn. 27 Reactns r 2, r 3a, and r 3b d nt depend n ptental as they ccur wthn a gven plane f charge; therwse, the rate f a generc reactn can be wrtten as r l ¼ k f,l e b aδϕ l Y C p, l k b, l e b cδϕ l Y C q, l ð5þ Ths nterfacal mdel neglects the surface charge due t adsrbed charge carrers and ns that wuld be descrbed n a mdel f the duble layer at the electrlyte and semcnductr nterface, respectvely. The nly adsrbed charge speces ncluded s the xdzed state f the dye. In addtn, ths study dd nt cnsder surface trap states r the cnductn f hles. The cncentratn f vacances n the cnductn band was balanced by the densty f cnductn band states and the cncentratn f electrns, C γ = Γ C e. Ths cnstrant ensures that the number f electrns des nt exceed the number f avalable states. In wrtng ths equatn, t s assumed that phtexctatn wthn the semcnductr s neglgble when cmpared t electrn njectn by the senstzer. The fllwng dscussn hghlghts the fundamental cmpnents t ad cmprehensn f the mdel as a whle as well as hghlght the prmary departures frm the nterfacal mdel prevusly reprted by Penny et al. 14 The prmary dfference between the rates lsted n Table 1 frm prr wrk s that these rates are dependent n the nanwre length, l, rather than the surface area t vlume rat, S V. Therefre, the ttal rates f reactn ncrease wth lnger nanwres. Hgher denstes f nanwres als crrespnd t an ncrease n the ttal rates f reactn. The ttal current densty, í, was then btaned by ntegratng ver the nanwre length. Reactn 1a (Table 1) represents the dark current (electrlyte-semcnductr charge transfer) that passes acrss the fractn f expsed semcnductr surface gven by 1 θ, where θ s the fractnal semcnductr surface cverage f the phtactve dye. Reactn 2 (Table 1) accunts fr the phtexctatn f a grund state dye mlecule, S g, t the snglet excted state f the dye, S 1 / The rate f ths reactn s a functn f the wavelengthdependent phtn flux, h (λ), mlar extnctn ceffcent f the dye mlecule, ε^(λ), and quantum yeld f the dye mlecule, Φ(λ). The slar radatn n terms f h (λ) was btaned frm the Natnal Renewable Energy Labratres database, the extnctn ceffcent f N719 dye was measured expermentally, and the quantum yeld was btaned frm Gratzel. 1 The rate f ths reactn decays alng the nanwre length accrdng t Beer s Law and t s a functn f the avalable dye n the grund state, S g. It s mprtant t nte that ths absrbance mdel neglects the absrbance f the nanwre cre and shell materal as well as the electrlyte that permeates the phtande. In addtn, the expressn fr r 2 des nt nclude phtnc r plasmnc nteractns that culd ccur n rdered structures. 28 Reactns 3a and 3b (Table 1) represent the relaxatn f the excted state f the dye mlecule thrugh an ntermedate trplet state, S 3 /. Reactns 4a and 4b (Table 1) represent the njectn f an electrn frm ether S 1 / r S 3 / nt the cnductn band f the semcnductr. Hwever, phtgenerated electrns can als be lst t the electrlyte thrugh reactns 6a and 6b. (Table 1) The rates asscated wth reactns 4a and 4b, 6a and 6b (Table 1) are dependent nly n the cncentratn f the respectve excted 18666 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety Table 1. Interfacal Reactns and Ther Crrespndng Rates reactn n. reactn rate! (1a) A Red þ nγsf Rs kf,1a A! n B Ox þ ne 1a C Red C γ ¼ exp½b a ðϕsh ϕel ϕ 1aÞŠ C kb, 1a 1a, ref C Red, C γ, Ox C Ox, (2) S g þ hv f S / Z 1 r 2 ¼ ΦðλÞε^ðλÞC Sg l lnð10þh ~ ðλþ exp½ ε^ðλþc Sg lnð10þl ZŠdλ! B C e C e,! n exp½ b c ðϕsh ϕel ϕ 1a ÞŠ (3a) S / 1 k3a sf S / 3 r 3a ¼ k 3a l C S 1 (3b) S / 3 k3b sf r S 3b ¼ k 3b l C S g 3 (4a) (4b) (5) S / 1 þ γ k4a sf S þ þ e r 4a ¼ k 4a l C S 1 S 3 þ γ k4b sf S þ þ e r 4b ¼ k 4b l C S 3 S þ þ e k5b sf γ þ S g r 5 ¼ k 5 l C S þ C e C ST exp½b a ðϕ sh ϕ S 1 ÞŠ exp½b a ðϕ sh ϕ S 3 ÞŠ exp½ b c ðϕ sh ϕ S þ ÞŠ (6a) B Ox þ ns 1 kb, sf 6a ns þ þ A Red r 6a ¼ k b, 6a l C n S exp½ b c ðϕ 1 S ϕ 1 el ÞŠ (6b) B Ox þ ns 3 kb, sf 6b ns þ þ A Red r 6b ¼ k b, 6b l C n S exp½ b c ðϕ 3 S ϕ 3 el ÞŠ (7) A Red þ ns þ! kf,7 A sf B Ox þ ns g r7 ¼ k f,7 l C n C Red S exp½b þ a ðϕ S C ϕ þ el ÞŠ Red, state f the dye (S / 1 r S / 3 ) snce the ultra fast electrn knetcs cause these reactns t be lmted by the cncentratn f the crrespndng speces. Rates fr reactns 4a and 4b (Table 1) were assumed t be nly andc n drectn wth the ptental drvng frce gven by the dfference between the nterfacal ptental f the semcnductr and ether the S / 1 r S / 3 electrnc state,.e., (ϕ sh ϕ S1 *) and (ϕ sh ϕ S3 *), respectvely. Smlarly, rates fr reactns 6a and 6b are cnsdered t be purely cathdc and represent tw lss mechansms t the phtgenerated current that are dependent n the change n nterfacal ptental,.e., (ϕ S1 / ϕ el ) and (ϕ S3 / ϕ el ), respectvely. Reactn 5 (Table 1) s the nly ther lss mechansm cnsdered n ths study. Upn njectn f the electrn frm the excted state f the dye nt the cnductn band f the semcnductr, the electrn can be lst t the xdzed state f the dye, S +. Ths reactn rate s dependent n bth the nterfacal cncentratn f electrns as well as the cncentratn f S +. The rate f reactn 5 (Table 1) s taken t be purely cathdc n drectn, and the reactn s drven by the dfference n nterfacal ptental,.e., (ϕ sh ϕ S + ). Reactn 7 (Table 1) descrbes the regeneratn f the xdzed state f the dye, S +, by electrn dnatn frm the reduced-state n, Red, n the electrlyte. The rate f reactn 7 (Table 1) s dependent n bth the cncentratn f Red and S + and s cnsdered t be purely andc. Once agan, the drvng frce f ths reactn s the dfference between the ptental f each state,.e., (ϕ S + ϕ el ). Under steady-state cndtns, balances n the ndvdual electrnc states f the dye yeld and 0 ¼ r 2 þ r 3b r 5 þ nr 7 ð6þ 0 ¼ r 2 r 3a r 4a þ nr 6a ð7þ 0 ¼ r 3a r 3b r 4b þ nr 6b ð8þ fr S g, S 1 /, and S 3 /, respectvely. Addtn f eqs 6 t 8 yelds a mre cnvenent result that equates the flux f charge n ether sde f the nterface,.e., 0 ¼ r 4a þ r 4b þ r 5 nðr 6a þ r 6b þ r 7 Þ ð9þ Thus, the cncentratns f each dye state can be calculated frm eqs 7 9 nce the nterfacal values fr ϕ sh, ϕ el, C e, C Ox, and C Red are knwn. BULK CHARGE CARRIER TRANSPORT IN THE SEMICONDUCTOR AND ELECTROLYTE REGIONS Equlbrum n the system can be defned by equatng the electrchemcal ptental f charge carrers n the semcnductr t the ns n the electrlyte. The electrchemcal ptental, μ, fr each charged speces,, sdefned as μ ¼ μ ref þ RT lnðc γ^þ þz Fϕ j ð10þ where j dentes the regn (.e., semcnductr r electrlyte) n whch the ptental ϕ j s beng evaluated. The actvty ceffcent, γ^, n eq 10 s assumed t be unty n ths study. The flux f speces s prprtnal t the prduct f the cncentratn and the gradent f the electrchemcal ptental,.e., NB ¼ u C ~ μ ð11þ where the charge carrer/nc mblty, u, can be related t the dffusn ceffcent by the Ensten Smluchwsk relatn wrtten n macrscpc unts as u = D /RT. Because f the hgh degree f cnfnement asscated wth the semcnductve shell, ths regn s nt electrcally neutral. In ther wrds, the shell thckness s much less than the depletn regn thckness defnedbythedebyelengthfthe 18667 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety semcnductr. Thus, the ptental s gven by Pssn s equatn, 2 ϕ j ¼ F ε j F j ð12þ where F j s the charge densty gven by F j ¼ z, j C, j ð13þ Fnally, the gradent f the flux wthn the semcnductr s dstrbuted accrdng t the net rate f hmgeneus prductn and/r cnsumptn f each charged speces, ~ 3 NB ¼ R net, ð14þ The rb peratrs n eqs 11, 12, and 14 are scaled t a nedmensnal dervatve accrdng t eqs 1 and 2. Thus, the equatns gvernng charge transprt and the ptental dstrbutn wthn the semcnductr are gven as 2 ϕ sh 2 N e, ~R ¼ D e τ sh and ¼ τ 2 F sh ð ^N a C e Þ ε sh C e þ fc ϕ sh e ð15þ ð16þ N e, ~R ¼ τ shr net, e ð17þ where N e, ~R s the flux slely n the radal drectn and f = F/RT. Once agan, hle transprt was nt cnsdered, and the cncentratn f vacant cnductn band states was taken t be C γ = Γ C e s that R net,e can be descrbed by a smple band-t-band knetc mdel. 29 Equatns 10 14 are utlzed agan fr the electrlyte utsde the dffuse duble layer wth the exceptn that the regn was assumed t be electrcally neutral. Thus, the ptental dstrbutn s gven by ~ ϕj ¼ í k F z D k ~ C where the current s gven by í ¼ z NB and the cnductvty s gven by k ¼ F 2 z 2 u C ð18þ The equatns that gvern cncentratn and ptental dstrbutn wthn the dffusn layer f the electrlyte are ϕ el ¼ τ í el k F k N Ox, ~R ¼ D Ox τ el N Red, ~R ¼ D Red τ el C Ox C Red z D C þ fc Ox þ fc Red ϕ el ϕ el ð19þ ð20þ ð21þ N Ox, ~R ¼ 0 ð22þ N Red, ~R ¼ 0 ð23þ and C C þ fc ϕ el C ¼ 0 ð24þ where C C s the cncentratn f the cuntern. Transprt f ns n the axal drectn (~Z-drectn) was neglected n these scaled equatns. A subsequent study wll nclude cntrbutns t the flw f ns n the ~Z-drectn, whch becmes mre mprtant when cnsderng phtandes wth lnger nanwres. BOUNDARY CONDITIONS Twelve bundary cndtns were needed t slve the abve dfferental equatns. Fr the semcnductr regn, the ptental at the semcnductr metal cntact s the appled ptental, and the cncentratn f electrns was assumed t be prprtnal t the rat f the electrn flux and the electrc feld,.e., C e ð~r sh ¼ 0Þ ¼ τ shrtn e, ~R FD e ϕ sh ð25þ whch can be btaned frm the electrn dffusn mgratn equatn, eq 16, under the assumptn that the dffusn cntrbutn t electrn flux s neglgble. The metal semcnductr nterface was assumed t be an deal hmc cntact wth n accumulatn f charge. The flux f electrns thrugh the semcnductr regn was defned by the net rate f prductn frm the nterfacal knetc mdel. As seen n Fgure 3, the net reactn rate enterng the cnductn band s gven by reactns 1a, 4a, 4b, and 5 (Table 1) such that N e, ~R ¼ ½ð1 θþnr 1a þ θðr 4a þ r 4b þ r 5 ÞŠ ð26þ Lkewse, the fluxes f the charge carrers at the nterface n the electrlyte are N Ox, ~R ¼ð1 θþbr 1a þ θðr 6a þ r 6b þ r 7 Þ ð27þ N Red, ~R ¼ ½ð1 θþar 1a þ θðr 6a þ r 6b þ r 7 ÞŠ ð28þ Furthermre, the current at any pnt n the axal drectn, ~Z,s gven by í ~Z, V ¼ nn e, ~R ¼ nn Ox, ~R ¼ nn Red, ~R Gauss Law hlds that the ptental gradent must be cntnuus acrss bundares that d nt represent planes f charge. Thus, at the OSS, ϕ sh R ~ sh f 1 ϕ ¼ τ nt ϕ OSS sh sh δ sh and, at the OHP, ϕ el ϕ OHP el ϕ ¼ τ nt el ~ R el f 1 δ el ð29þ ð30þ 18668 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety Table 2. Cnstants Used fr the Numercal Mdel cnstant value ntes C e, /N^a 1.66 10 8 ml/cm 3 bulk electrn cncentratn/dpant level 10,30 C Ox, 5 10 5 ml/cm 3 Ox cncentratn (prvded by Slarnx) C Red, 4.5 10 4 ml/cm 3 Red cncentratn (Prvded by Slarnx) C ST 7.5 S 1 V 10 5 ml/cm 2 ttal dye cncentratn D e 5cm 2 /s electrn dffusn ceffcent n sngle-crystal anatase TO 2 D Ox, D Red 5 10 5 cm 2 /s Ox and Red dffusn ceffcent 5 10 13 ma/cm 2 reactn 1a (Table 1) exchange current densty 5 í 1a,ref k 3a k 3b k 4a k 4b k 5 k 6a k 6b k 7 6.67 10 12 s 1 standard rate cnstant fr r 3a 31 1.67 10 6 s 1 standard rate cnstant fr r 3b 32 1.0 10 13 s 1 standard rate cnstant fr r 4a 5,31 5.0 10 10 s 1 standard rate cnstant fr r 4b 31 5.0 10 4 s 1 standard rate cnstant fr r 5 5 1.0 10 2 s 1 standard rate cnstant fr r 6a 33 1.0 10 2 s 1 standard rate cnstant fr r 6b 33 1.0 10 8 s 1 standard rate cnstant fr r 7 33 Γ 1.125 10 3 ml/cm 3 densty f states fr anatase TO 2 34 δ sh 0.2 nm duble layer thckness f semcnductr shell δ el 1 nm duble layer thckness f electrlyte ~N nw 10 10 cm 2 nanwre densty j 0.5 prsty f phtande ϕ S1 / 1.3 V vs SCE snglet excted state energy level f dye 33 ϕ S3 / 0.75 V vs SCE estmated trplet excted state energy level f dye based n ref 35 ϕ S + 0.5 V vs SCE xdzed energy level f dye 33 where τ j n each equatn arses frm scalng ~R j, δ j s the duble layer thckness, and the ptentals are referenced t the OSS and OHP fr the semcnductr and electrlyte, respectvely. The ISS, asscated wth electrns n surface states, and the IHP, asscated wth adsrbed nc speces, were assumed t be lcated at the same pstn and were assgned the ptental ϕ nt. The ISS, OSS, nterfacal, IHP, and OHP planes are shwn n Fgure 3. The charge held n the nterfacal plane s related t ptental thugh Gauss Law,.e., ε sh ðϕ OSS sh ϕ δ nt Þ ε el ðϕ sh δ nt ϕ OHP el Þ¼ F C S þ ð31þ el S V The bundary cndtns n the electrlyte are that the ptental at the symmetrc bundary between nanwres s zer. In addtn, all electrlyte cncentratns have ther bulk values at the uter bundary f the dffusn layer. COMPUTATIONAL DETAILS The parameters used n the mdel are lsted n Table 2. It s mprtant t nte that the standard rate cnstants lsted n Table 2 are assumed nt t devate frm thse measured n NP-DSSC systems fr Ru-based dyes. Therefre, the standard rate cnstants are ndependent f the energy levels n the dye semcnductr electrlyte nterface, whle the rates n Table 1 are dependent n these dfferences. The mdel and bundary cndtns were dscretzed usng a central dfference apprxmatn and slved usng an algrthm that vared the rt-fndng scheme between a Newtn s methd and the methd f steepest descent. Ths allwed the search-drectn step sze t vary, dependng n hw far the system was frm the cnverged slutn. The system f equatns was brken nt three sets: thse descrbng the nterfacal mdel, eqs 7 9, and the bulk charge transprt n the semcnductr, eqs 15 17, and electrlyte, eqs 19 24. Addtnally, the state varables n each system were scaled t the rder f 1 thrugh an egenvalue matrx. The nterfacal bundary cndtns cupled each set f the charge transprt equatns. As each system cnverged t an acceptable tlerance (10 20 ), a fxed pnt teratn was perfrmed between the tw systems f equatns. The fxed pnt teratn cntnued untl bth the nterfacal bundary cndtns were satsfed and the shared nterfacal varables cnverged t equvalent values abut each fxed pnt t a tlerance f 10 10. Furthermre, ntal guesses were chsen n the bass f ether knwn fxed cnstants, such as bulk cncentratns, r assumptns abut the antcpated devce perfrmance. Multple ntal guesses were used t avd lcal mnma. The current at each vltage was btaned by usng a varable pnt Gaussan quadrature methd t ntegrate the lcal current at any pnt ~Z, í(~z,v), ver the length f the nanwre íðvþ ¼S V j Z 1 0 íð~z, VÞd~Z ð32þ where j s the phtande prsty. Furthermre, the phtactve surface area t cell vlume rat, S V, s related t the nanwre number densty, ~N nw by S V ¼ 2πðr þ 2τ sh Þ ~N nw þ 1 l 0 = 2πðr þ 2τ sh Þ ~N nw ð33þ where the tw terms n eq 33 accunt fr the surface area f the nanwre and the prjected surface area. Once a slutn s fund at a desgnated appled ptental, ths slutn s used as the ntal guess fr subsequent ptentals. Cnvergence was better when the ntal ptental calculated was slghtly negatve. As descrbed n the SI, the ptental dstrbutns and cncentratn prfles fr the relevant speces were checked n each regn t ensure the slutn met the requred bundary cndtns. Fgure S1(SI) shws that NW-DSSCs have ptental dfferences acrss the semcnductr regn that can generate large electrc felds. Furthermre, Table S6 (SI) 18669 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety shws the effect f varus cnstants n shrt-crcut current. The slutn s senstve t the values f k 3a, k 3b, k 4a, k 4b, k 5, and D Red. As shwn n Fgure 3, these rate cnstants are respnsble fr the cncentratn f dye n the excted state and the njectn f the electrn nt the cnductn band. Whle the slutn was mst senstve t the prmary njectn rate cnstant (k 4a ), ths rate cnstant has been well studed, 5,31 and the value shuld nt devate sgnfcantly frm that lsted n Table 2. EXPERIMENTAL DETAILS Andc alumnum xde (AAO) templates were fabrcated drectly n ITO. 36 The pres were wdened t remve the barrer layer, facltatng the adhesn f the nanwres durng electrdepstn. In-dped ZnO (IZO) nanwres were then depsted nsde the prus template. 37 - The IZO nanwres were electrdepsted usng an aqueus slutn f 0.1 M hydrated Zn(NO 3 ) 2 and 0.75 1.0 mm InCl 3 that was phadjusted t 2.56 wth HCl. After electrdepstn, the AAO template was selectvely etched, usng ur prevusly reprted dryng methd aded by electrstatc repulsn. 38 The nanwre array was then placed nt a custm, atmc layer depstn chamber (Planar Systems, Inc.) fr grwth f the shell arund each nanwre. The ttana shells were depsted by cyclng TCl 4 and H 2 O pulses separated by dry ntrgen purges, where 1 cycle represents the sequence cnsstng f 1 s H 2 O/1 s purge/1 s TCl 4 /1 s purge. The N719 rganc dye (Slarnx) was chemsrbed nt the cre shell nanwre array by heatng the nanwre array t 200 C and mmersng t n 0.3 mm N719/ethanl fr 12 h. The substrates were subsequently rnsed wth dry ethanl and stred n an argn glvebx untl used. The cell was cmpleted by mmersng t n an dde REDOX cuple (Idlyte, Slarnx) and cnnectng t t a thermally platnzed cunter electrde. Ht melt spacers wth a thckness f 40 μm (Surlyn 1702, DuPnt) were used t sandwch and seal the cunterelectrde t the phtande. The electrlyte was ntrduced va capllary actn. The devces were tested mmedately after electrlyte mmersn under AM 1.5 smulated sunlght (Slar Lght, Inc.) usng a ptentstat (Prncetn Appled Research, Versastat 3). RESULTS AND DISCUSSION The numercal results btaned by the mdel were cmpared t expermental plarzatn curves. In addtn, the rle f the nterfacal electrc feld, whch s nly present n NW-based DSSCs, was explred by cmparsn f numercal results btaned wth and wthut the mgratn term n the flux expressns fr electrns. Calculated Devce Perfrmance and Expermental Cmparsn. A cmparsn between the calculated and expermental plarzatn curves fr DSSCs based n bth IZO and ZnO nanwres 22 s presented n Fgure 4. Bth f these nanwre systems have a ttana shell thckness f 10 nm, whch was fund t be the ptmal thckness n the expermental devces. The perfrmance f the DSSCs based n ZnO arrays s better, but ths phtande has much larger surface area snce the nanwres are 6 tmes lnger. The dye uptake was qute gd fr the IZO nanwre arrays, and t was assumed that the surface cverage, θ, was equal t 0.85. On the ther hand, a smaller value f 0.77 was assumed fr the ZnO-based DSSC snce Law et al. attrbuted ts pr perfrmance t lw dye cverage. The mdel was able t capture sme f the qualtatve and quanttatve features f the expermental plarzatn curves. In bth devces, the shrt-crcut current, í sc, was well represented by the mdel. The small dsparty n í sc culd be due t errrs n expermental measurement f the devce area. Nte that the Fgure 4. Cmparsn f expermental and calculated plarzatn curves fr NW-DSSCs based n ttana-cated (a) 2.5 μm lng IZO and (b) 15 μm lng ZnO nanwres. 22 physcs needed t descrbe the lwered shunt resstance (estmated by the nverse slpe at í sc ) fund n the IZO-based DSSCs f Fgure 4a s nt ncluded n the mdel. Whle the mdel was able t predct the pen-crcut vltage, V c, fr the NW-DSSCs based n ZnO (wthn 2%), a sgnfcant devatn was bserved fr the NW-DSSCs based n IZO. The large devatn n V c fr these devces s lkely due t the prly crystallne ttana shell, whch was als bserved n sme ZnO NW-DSSCs. 22 The majr dfference n the plarzatn curves f Fgure 4 s the fll factr, whch s a measure f devce perfrmance. In bth nanwre devces, the mdel predcts hgh fll factrs f 0.74 fr bth IZO and ZnO NW-DSSCs. Hwever, the expermental data has lwer fll factrs f 0.62 and 0.58 fr IZO and ZnO NW-DSSCs, respectvely. These devatns n fll factr are lkely asscated wth the fact that the mdel s assumng a cnductve rather than semcnductve nanwre cre. Thus, the electrns njected nt the semcnductr regn are quckly beng swept away n the mdel results, allwng better charge transprt and devce perfrmance. Interestngly, the seres resstance f the curves fr the NW-DSSCs based n IZO are nearly dentcal. The smlarty s lkely due t the fact that IZO s mre cnductve than ZnO. Influence f Interfacal Electrc Feld. A key dfference between NP- and NW-DSSCs s the presence f an nterfacal electrc feld, whch many researchers beleve enhances the charge transfer and transprt physcs n the latter system. The dfferent gemetres f the systems and the nablty t remve the effect f the electrc feld n experments has made t dffcult t quantfy ts mprtance n devce perfrmance. Hwever, elmnatn f the electrn mgratn term n eq 16 effectvely turns the electrc feld OFF n the mdel. Therefre, cmparsn 18670 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety Fgure 5. Cmparsn f the nterfacal reactn rates fr the ON and OFF states f the electrc feld. The enhancement rat relates the ON and OFF states (wrtten n the frward drectn) t characterze the effect f the nterfacal electrc feld n the perfrmance f NW-DSSCs. Therefre, values greater than 1 ndcate mprved charge transprt. f the ON and OFF states f the electrc feld allws the benefts frm a feld t be quantfed, especally the changes t speces cncentratns and the rates f charge transfer/transprt prcesses. T depct succnctly the effect f the nterfacal feld n charge transfer reactns, each reactn was wrtten n terms f an enhancement rat. Ths rat s defned n the drectn f current flw s the rat fr back reactns (rat f OFF/ON states) s ppste f the frward reactns (ON/OFF). An enhancement rat f unty ndcates that the feld has n effect n the prcess, whle values greater than unty ndcate mprvements t charge transfer due t the nterfacal electrc feld. The enhancement rat fr each reactn due t the electrc feld s shwn n Fgure 5. The mst crtcal prcess wthn all DSSCs s the njectn f the electrn frm the snglet state t the cnductn band, whch s gven by r 4a. As can be seen frm Fgure 5, the prmary njectn rate frm the snglet state remans essentally unchanged. The lack f changes s due t the fact that the dye s desgned t heavly favr njectn ver back reactn. On the ther hand, the back-reactn rate frm the snglet state (r 6a ) decreases by a factr f abut 100 n the presence f an electrc feld. Ths ndcates that the back reactn f prmary electrns s 100 tmes slwer n the presence f an nterfacal electrc feld, prvdng sgnfcant benefts t charge transprt. The decrease n electrn njectn frm the trplet state (r 4b )s surprsng. Hwever, charge transprt frm the trplet state s less mprtant than the prmary dye state snce the majrty f the charge s cncentrated n the snglet state. Further, ths decrease s nly 100 tmes smaller whle the crrespndng back reactn rate (r 6b ) shws a 10,000-fld mprvement t charge transfer. Ths dfference effectvely results n a 100-fld mprvement t the charge transprt f phtgenerated electrns frm the trplet state f the dye t the semcnductr. The excted states f the dye are als subject t relaxatn mechansms (r 3a and r 3b ). The changes fr these reactns fllw the back reactn rates (r 6a and r 6b ) very clsely. The smlarty s asscated wth the fact that the cncentratn f S 1 / and S 3 / dctate these rates. The mprvement f these lss mechansms s lkely due t the mprved charge transprt that keeps the cncentratn f S 1 / and S 3 / lwer. Fr ths reasn, the back-reactn rate fr the drect lss f an electrn frm the cnductn band t the electrlyte (r 1a ) dsplays smlar behavr. It shuld be nted that dentcal Fgure 6. Cmparsn f the plarzatn curves fr NW-DSSCs fr the ON and OFF states f the electrc feld. enhancement rats d nt mean that the reactn rates are the same but nstead ndcate that the effect f the electrc feld n the reactn s smlar. Fr example, r 1a, r 3a, and r 6a have dentcal enhancement rats but the rates are apprxmately 10 11,10 35, and 10 15 s 1, respectvely. Anther electrn-lss mechansm s the drect transfer f electrns frm the semcnductr cnductn band t the xdzed state f the dye (r 5 ). As shwn n Table 1, ths rate s a functn f the cncentratn f cnductn band electrns at the nterface f the semcnductr and electrlyte. Because f the mprved transprt f electrns, t was expected that the rate wuld decrease. Hwever, ths electrn lss rate ncreases when an nterfacal electrc feld s present wthn the semcnductr, as shwn n Fgure 5. The change t the rate s qute substantal at lw appled ptentals but the lss mechansm reaches a mnmum at an appled ptental f 0.25 V and then becmes less affected. The change n behavr s due t the cmpetng changes t electrn and S + cncentratn between the ON and OFF states. At lw ptentals, the dfference n electrn cncentratn between the tw states s hgh, but the effect starts t plateau at 0.3 V. At 0.4 V, the cncentratn f S + starts t drp n the ON state, whch results n better rates fr ths back reactn. Fnally, the regeneratn f the dye s descrbed by reactn rate r 7. Ths reactn rate remans unaffected by the electrc feld untl appled ptentals are greater than 0.4 V. At these hgher ptentals, the electrc feld causes the reactn t be slwer. Ths behavr s due t a reductn n S + cncentratn when the feld s ON. The varus reactn pathways descrbed abve can be summarzed by lkng at the changes t devce perfrmance. The smulated current vltage characterstcs f a NW-DSSC wth the feld ON r OFF s shwn n Fgure 6. The shape f the curves are smlar; hwever, the devce wth the feld ON has small but sgnfcant ncreases t shrt-crcut current and pencrcut vltage. The feld als mprves the fll factr f the devce (0.72 vs 0.68). Cmbnng these changes t the current vltage characterstcs leads t apprxmately a 15% ncrease n effcency,.e., η ON 1.15η OFF. It shuld be nted that the feld OFF state cannt be drectly related t NP-DSSCs snce the mdel des nt accunt fr dffusn thrugh the entre ttana flm. Whle the presence f an electrc feld n NW-DSSCs yelds an apprecable gan n perfrmance, these devces have substantally lwer phtactve surface area than NP-DSSCs 18671 dx.d.rg/10.1021/ja2044216 J. Am. Chem. Sc. 2011, 133, 18663 18672
Jurnal f the Amercan Chemcal Scety (ften mre than 75% lwer). Therefre, the benefts frm the electrc feld wll nt lkely mprve perfrmance f NW- ver NP-DSSCs wthut cmparable phtactve surface area. CONCLUSIONS Ths reprt ntrduces a cntnuum-based mathematcal mdel f DSSCs based n nanwre arrays as the phtande. The mdel results were n quanttatve and qualtatve agreement wth expermental plarzatn curves. The ablty t neglect artfcally the effect f electrn mgratn n the mdel,.e., gnre the nterfacal electrc feld present n nanwre systems, allwed ts mprtance t specfc nterfacal charge-transfer reactns t be evaluated. The calculatns shwed that the electrc feld dramatcally decreases phtexcted electrn back reactns wthut a sgnfcant decrease t electrn njectn. The enhanced transprt frm the electrc feld leads t better shrt-crcut current, pen-crcut vltage, and fll factr f the devce. Whle these results shw mderate mprvements t perfrmance, the cmpnents f NW-DSSCs can be further ptmzed t take full advantage f the benefts frm the electrc feld. ASSOCIATED CONTENT b S Supprtng Infrmatn. Varable lsts, cncentratn prfles, ptental dstrbutns, and senstvty analyss. Ths materal s avalable free f charge va the Internet at http:// pubs.acs.rg. AUTHOR INFORMATION Crrespndng Authr kzegler@che.ufl.edu Present Address Manstream Engneerng, Rckledge, Flrda 32955, Unted States ACKNOWLEDGMENT We acknwledge the supprt f the Dnrs f the Amercan Chemcal Scety Petrleum Research Fund, the Unversty f Flrda Opprtunty Fund, and the Natnal Scence Fundatn (CBET-1033736) fr supprt f ths research. REFERENCES (1) Gratzel, M. J. Phtchem. Phtbl., A 2004, 164, 3 14. (2) Ga, F.; Wang, Y.; Sh, D.; Zhang, J.; Wang, M.; Jng, X.; Humphry-Baker, R.; Wang, P.; Zakeeruddn, S.; Gratzel, M. J. Am. Chem. Sc. 2008, 130 (32), 10720 10728. (3) Gratzel, M.; Frank, A. J. J. Phys. Chem. 1982, 86, 2964 2967. 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