Simulation of Polarization Curves for Oxygen Reduction Reaction in 0.5 M H 2 SO 4 at a Rotating Ring Disk Electrode

Transcription

1 A816 Journal of The Electrochemcal Socety, 15 8 A816-A /2007/158/A816/10/$20.00 The Electrochemcal Socety Smulaton of Polarzaton Curves for Oxygen Reducton Reacton n 0.5 M H 2 SO at a Rotatng Rng Dsk Electrode Qngbo Dong,* Shrram Santhanagopalan, and Ralph E. Whte**,z Department of Chemcal Engneerng, Unversty of South Carolna, Columba, South Carolna 29208, USA A cylndrcal two-dmensonal model based on the Nernst Planck equatons, the Naver Stokes equaton, and the contnuty equaton s used to smulate the oxygen reducton reacton n 0.5 M H 2 SO at a rotatng rng dsk electrode. Concentraton dstrbutons and a potental profle are obtaned as a functon of the axal and radal dstances from the center of the electrode surface. Polarzaton curves are smulated to nterpret expermental results by studyng varous reacton mechansms,.e., the four-electron-transfer reducton of oxygen, the two-electron-transfer reducton of oxygen, a combnaton of the above two reactons, mechansms wth reducton of peroxde to water, and/or the heterogeneous chemcal decomposton of peroxde. Specal attenton s devoted to the effect of peroxde The Electrochemcal Socety. DOI: / All rghts reserved. Manuscrpt submtted October, 2006; revsed manuscrpt receved March 26, Avalable electroncally June 26, Peroxde generated n the cathode of a proton exchange membrane fuel cell PEMFC can cause the falure of the membrane. 1-3 Peroxde can be generated n the oxygen reducton reacton ORR, whch occurs n the cathode of a PEMFC. Because t s dffcult to observe the presence of peroxde n the cathode of the fuel cell drectly, the rotatng dsk rng electrode RRDE s used for studyng the peroxde generaton n the ORR n dfferent acdc electrolytes such as sulfurc, perchlorc, hydrochlorc, and organc acd solutons. -7 The mert of the RRDE technque n ths applcaton s that the amount of peroxde generated on the dsk can be quantfed by usng the rng current. 8 In an acdc envronment, speces are consumed or generated on the dsk and the rng through four possble reactons whch can be summarzed by the followng overall equatons 9 O 2 +H + +e 2H 2 O U 1 = V 1 O 2 +2H + +2e H 2 O 2 U 2 = V 2 H 2 O 2 +2H + +2e 2H 2 O U 3 = V 3 H 2 O O 0 2 +H 2 O G 298 = kj/mol The U j s the standard electrode potental for the charge transfer reacton j n volts. Note that all of the potentals mentoned n ths 0 work are wth respect to the standard hydrogen electrode. G 298 s the standard Gbb s free energy change n a chemcal process at 298 K n kj/mol. Accordng to the nature of the actve catalyst and the operatng condtons, these reactons take place at varous rates. The drect e reducton of oxygen Reacton 1 s the prmary reacton n the ORR and t can be used to descrbe the system satsfactorly n cases such as when some low-ndex sngle-crystal surface platnum for example, Pt110 electrodes are used. 10,11 The 2e reducton Reacton 2 may also occur solely n an acdc envronment. When an S-modfed platnum electrode s used, the oxygen reducton takes the 2e pathway and peroxde s the stable fnal product n the system, as shown n the work of Mo et al. 12 But n general, the ORR s a combnaton of Reactons 1 and 2, and the generated peroxde s consumed by electrochemcal reducton, as shown by Reacton 3 and/or spontaneously decomposed to form oxygen as shown by Reacton. When two or more of the reactons occur smultaneously, varous nterestng phenomena show up n the polarzaton curves obtaned expermentally wth the RRDE. 5,6,11,13-19 The polarzaton curves ether for the dsk or the rng often show a tal at the lower potental sde the absolute value of the dsk current decreases and the value of the rng current ncreases when the potental s below 0.2 V and/or a hump n the potental at V. The absolute value of the dsk current n ths regon decreases and then ncreases back to the orgnal value. Both the tal and the hump ndcate the presence of peroxde, and ths work attempts to explan these anomales n terms of the reacton mechansm. The hydrodynamc aspects of an RRDE were studed n depth by prevous researchers, and analytcal seres solutons and onedmensonal numercal solutons were obtaned. 8,20-22 In ths work, the hydrodynamcs at an RRDE s solved wth a cylndrcal twodmensonal model based on the Naver Stokes equaton. 20 The Nernst Planck equaton s used to smulate the mass transport near the RRDE. 21,23,2 All the basc transport terms, ncludng dffuson and convecton, and the mgraton term are retaned n the Nernst Planck equaton to ensure accuracy. The knetc equatons used for the boundary condtons on the dsk and the rng where the electrochemcal reactons occur G 298 are based on the Butler Volmer 0 equaton The smulaton was carred out usng the commercal software, COMSOL MultPhyscs COMSOL MP. The model equatons subject to the assgned boundary condtons are solved wth the fnte element method readly, and polarzaton curves are smulated for cases n whch Reactons 1- occur to varous extents. Concentraton dstrbutons of oxygen and peroxde and the potental profle near the surface of the electrode are presented. Model Equatons A sketch of the cross secton of the RRDE and the smulated doman adjacent to the electrode surface s shown n Fg. 1. The * Electrochemcal Socety Student Member. ** Electrochemcal Socety Fellow. z E-mal: whte@engr.sc.edu Fgure 1. Schematc of an RRDE and the modelng doman.

2 Journal of The Electrochemcal Socety, 15 8 A816-A A817 varable z s used to represent the axal coordnate for whch the orgn s set at the surface of the electrode. The radal coordnate s represented by r and ts orgn s set at the axs of the electrode. Accordng to the work of F. M. Whte, the velocty changes are neglgble when the dmensonless dstance s greater than The dmensonless dstance s defned by = z//, where s the rotatng speed of the electrode n rad/s, s the knematc vscosty n mpa s, and s the densty of the electrolyte n g/cm 3. Bard and Faulkner also suggested that a regon of n the axal drecton should be used for materal balance. 8 Therefore, z = cm z,v 10 s selected as the smulaton doman. Numbers n talcs n the schematc shown n Fg. 1 represent the boundares as referenced n the followng sectons. Momentum balance (swrl flow model). The followng assumptons are made n ths model: the electrolyte s a Newtonan flud wth constant densty and vscosty, the physcal propertes of the electrolyte 0.5 M H 2 SO saturated wth pure oxygen at 1 atm and 298 K can be approxmated by those of water, and the system has axal symmetry and s at steady state. The generalzed equatons of moton and contnuty n cylndrcal coordnates are n the followng form 21,28 u t u + u T + u u + P =0 u =0 6 where P s the pressure n Pa, and u s the velocty vector n cm/s. Wth the assumpton of axal symmetry and steady-state flow, the dervatves wth respect to tme t and angular coordnate are all equal to zero. The densty and vscosty are assumed to be constants. Equatons 5 and 6 can then be smplfed and wrtten n the expanded form as follows 8,20,21 Contnuty equaton 1 r r ru r + z u z =0 7 r component u r u r r u 2 r + u u r z = P z r + 1 r rr u r 2 u r r r u r z component u u r r + u ru u + u z r z = 1 r rr u u r r u z component u z u r r + u u z z = P z z + 1 r rr u z + 2 u z r z 2 z 2 10 where u r s the radal component of the velocty n cm/s, u s the angular component of the velocty n cm/s, and u z s the axal component of the velocty n cm/s. The swrl flow problem s a twodmensonal problem wth only two ndependent varables r and z, even though all three velocty components n u r, u, and u z are modeled as shown n Eq These equatons can be smplfed further by ntroducng dmensonless varables. 20 However the orgnal varables are retaned n ths work for two reasons: retanng the orgnal varables helps us to readly compare the results wth expermentally measured varables, and makng the varables dmensonless does not sgnfcantly mprove the computatonal effcency because COMSOL MP automatcally scales the varables Mass balance (Nernst Planck equatons). The followng assumptons are made for the mass balance: the system s assumed to be at steady state; there are no homogeneous reactons; the axal symmetry condton s applcable; and the concentratons and lqud phase potental do not change at postons far away from the electrochemcal reacton stes. The general form of the Nernst Planck equaton used for the mass balance s as follows 21 D c z c F D RT + c u =0 11 where c s the concentraton of speces n mol/cm 3 =1,2,3,and represent HSO,H 2 O 2,H +, and O 2, respectvely, D s the dffuson coeffcent of the speces ncm 2 /s, z s the charge on speces, F s Faraday s constant 9687 C/equv, and s the potental n the electrolyte n V. Under condtons of axal symmetry, Eq. 11 can be expanded as follows c u r r + u c z z = D 2 c z c r r c r + z D + 2 r c r r r + c RT F c 2 z r z 2 + z 12 Equaton 12 can be wrtten for each speces correspondng to =1, 2, 3, and ; but there are fve varables ncludng four concentratons c 1 and the potental n the lqud phase that need to be solved. In the calculaton procedure, one of the concentratons c 1, the concentraton of HSO s obtaned from the electroneutralty condton z c =0 13 =1 and c 2, c 3, c, and are solved wth the four equatons represented by Eq. 12. Boundary condtons for velocty and pressure. Boundary 1 s at the axs of the cylndrcal coordnate, and axal symmetry condtons are applcable u r =0,u =0, atr =0, z 1 Boundary 7 s far away from the axs of the cylndrcal coordnate, and s treated as free surfaces.e., the vscous force s zero u + u T n =0 atr = 0.75 cm, z 15 or, n the expanded form 2 u r r =0, r r u r =0, u r z + u z r =0, atr = 0.75 cm, z 16 Adjacent to the electrode, the no-slp condtons are assumed to apply, hence the r and z components of the velocty are set equal to zero, whle the component of the velocty s set equal to the angular velocty of the electrode. So the veloctes at boundares 2, -6 are gven by u r =0,u z = 0 and u = r at z =0, r 17 Only the frst-order dervatve of the pressure P exsts n the governng equatons Eq. 3-6, whch means that only one boundary condton n the z drecton for pressure P s necessary. 20,21 The pressure P s arbtrarly set to be zero at boundary 3. The dervatve of velocty n z drecton s equal to zero snce the velocty has a constant value far from the surface of the electrode boundary 3. 20,21 Also u r and u can be set equal to zero at boundary 3 snce there s no vscous effect far from the electrode surface except an

3 A818 axal nflow. 20 Accordngly, the followng condtons for boundary 3 are gven u r =0,u =0,P =0, u z z =0, atz = 0.12 cm, r 18 Boundary condtons for concentratons and potental. Axal symmetry s used to set the boundary condtons at boundary 1, whch s located at r =0 c r =0, r =0 atr =0, z 19 The concentratons at boundary 3, far away from the surface of the electrode, are the bulk concentratons c bulk,nmol/cm 3, and the potental s set equal to the potental of the reference electrode at the operatng condtons RE,nV c = c,bulk, = RE at z = 0.12 cm, r 20 At boundares 2 and 5 whch are adjacent to the dsk and the rng, respectvely, the reactons occur, and a jump materal balance gves the followng equatons dc D dz + z c F D 3 d s = RT dz,j j j=1 n j F + s,r s for = 2,3, dc F z D =1 dz z c F D d = RT dz t at r =00.25 or r = cm, z =0 21 where s j s the stochometrc coeffcent of speces n reacton j, j s the current densty for reacton j na/cm 2, t s the total current densty n A/cm 2, n j s the number of electrons transferred n reacton j, r s s the chemcal reacton.e., Reacton rate at the electrode surface n mol/cm 2 s, R s the gas constant, 8.31 J/mol K, and T s the absolute temperature n K. In the frst expresson n Eq. 21, the left hand sde s the mass flux of each speces, and the rght hand sde s the generaton or consumpton of the respectve speces due to chemcal and/or electrochemcal reactons. In the second expresson n Eq. 21, the left hand sde s the net flux of charge n the electrolyte adjacent to the electrode surface, whle the rght hand sde s the total current flow. At boundares and 6, the current wll be zero, snce there are no reactons occurrng dc 0=D dz + z c F D d for = 2,3, RT dz Journal of The Electrochemcal Socety, 15 8 A816-A j = oj,ref c p,0,jexp c,ref a,jf j RT exp c,jf RT j c q,j,0 c,ref 2 where 0j,ref s the exchange current densty due to reacton j atthe reference concentratons n A/cm 2, c,0 s the concentraton of speces adjacent to the surface of electrode n mol/cm 3, c,ref s the reference concentraton of speces nmol/cm 3, aj s the anodc transfer coeffcent for reacton j, cj s the cathodc transfer coeffcent for reacton j, p j s the anodc reacton order of speces n reacton j, q j s the cathodc reacton order of speces n reacton j, and j s the overpotental of reacton j n V, and t s measured wth respect to a reference electrode of a gven knd n a soluton at the reference concentratons. The open crcut potental of reacton j at the reference concentratons relatve to a standard reference electrode of a gven knd s expressed as follows 26,27 U j,ref = U j RT n j F s,j ln c,ref U RE + RT n RE F s,re ln c,re 25 where s,re s the stochometrc coeffcent of speces nthe reacton occurrng at the reference electrode, U j,ref s the open crcut potental of the reacton j at the reference concentratons relatve to a standard reference electrode of a gven knd n V, U RE s the potental of the standard reference electrode, c,re s the concentraton of speces at the reference electrode n mol/cm 3, n RE s the number of electrons transferred n the reacton that occurs at the reference electrode. The overpotental for electrochemcal reacton j, j n Eq. 2 s gven by j = met RE 0 RE U j,ref 26 where 0 s the potental n the soluton adjacent to the electrode surface n V, met s the potental of workng electrode n V. The reacton orders p,j and q,j n Eq. 2 are related to s,j by p,j = s,j q,j =0 fs,j 0 27 p,j =0 q,j = s,j f s,j 0 The apparent transfer coeffcents for reacton j sum up to the number of electrons transferred n that reacton, that s a,j + c,j = n j 28 The total current densty s the sum of the partal current denstes 3 t = j=1 j 29 dc 0= z D =1 dz z c F D d at r = RT dz or r = cm, z =0 22 Boundary 7 s far away from the axs of the cylndrcal coordnate, and the followng condtons are appled dc 0=D dr + z c F D d for = 2,3, RT dr dc 0= z D =1 dr z c F D d at r = 0.75, z 23 RT dr Note that z s equal to zero for neutral speces O 2 and H 2 O 2 = 1 and 2, respectvely n Eq Knetc equatons. The current denstes n Eq. 21 can be obtaned from the knetc equatons for the electrochemcal reactons at the electrode surface based on the Butler Volmer expresson The rate of the catalytc decomposton of peroxde at the electrode surface s expressed as p r s = k h c H2 O 2,0 30 where the reacton order p can be a fracton or a whole number, and t s assumed to be 1 n ths work. The rate constant k h s assumed to be ndependent of the appled potental E appl,or met RE. Summares of the governng equatons and the boundary condtons ncludng the knetc equatons are lsted n Tables I and II, respectvely. Results and Dscusson The governng equatons Eq. 7-10, 12, and 13 subject to the gven boundary condtons Eq are solved numercally usng COMSOL MP. The knetc parameters, reacton propertes, and physcal propertes of the speces used n ths smulaton are shown n Table III. The constants, soluton propertes, and the operatng condtons are lsted n Table IV. The smulatons were carred out for ndvdual Reactons 1 and 2 as well as combnatons of Reac-

4 Journal of The Electrochemcal Socety, 15 8 A816-A A819 Table I. Summary of the governng equatons. Governng equatons for velocty and pressure Governng equatons for concentraton and potental 1 r r ru r + z u z =0 u u r r r u 2 u r + u r z z = p r + 1 r r ru r r u r r u r z 2 u u r r + u ru u + u r z z = 1 r r ru r u r u z 2 u u z r r + u u z z z = P z + 1 r r ru z r + 2 u z z 2 c u r r + u c z z = D R 2 c z c r r c r + z D RT F c 2 z r r r + c r + c z r + z where = 1, 2, 3, wth respect to HSO,H 2 O 2,H +, and O 2 z c =0 tons 1- as lsted n Table V. These reactons or reacton combnatons are used to capture phenomena observed n polarzaton curves obtaned expermentally n the lterature. Case (): Four-electron transfer reducton (Reacton 1). Reacton 1 s the basc reacton of the ORR at an RRDE n an acdc electrolyte, and t can be used to smulate the expermentally obtaned polarzaton curves approxmately. The sold lnes marked wth open crcles n Fg. 2 are smulaton results for Reacton 1 only wth parameters 0,1 = A/cm 2 and c1 = 1. There s no current on the rng snce there s no peroxde generated. The curve for dsk current follows the trends for that of sngle reacton systems. 27 Ths means the polarzaton curve shfts towards more cathodc potentals f 0,1 gets smaller as the overall reacton rate s slower. On the other hand, when 0,1 s ncreased, t shfts towards more anodc potentals. The potental drop n the ohmc regon wll be drastc when the transfer coeffcent c1 s large, and the drop s mld when c1 s small. Case (): Two-electron transfer reducton (Reacton 2). In cases where peroxde s a stable product, Reacton 2 can be used to smulate the polarzaton curves. A set of smulated polarzaton curves for Reacton 2 are shown n Fg. 2 and are represented by lnes marked wth trangles. The current gathered on the rng s postve due to the anodc reacton and the current gathered on the dsk s negatve due to the cathodc reacton. Reacton 2 s reversble under the gven operatng condtons. The oxygen transferred to the dsk surface s reduced to peroxde at a rate dependng on the appled potental E appl and the mass transfer lmtatons. When the peroxde s transferred to the rng on whch a constant potental of 1.2 V s appled, t s oxdzed back to oxygen. The collecton effcency N of an RRDE s defned by N = I R 31 I D where I R s the lmtng current collected on the rng n A, I D s the lmtng current collected on the dsk n A, when a sngle reversble reacton s occurrng on the dsk and all the product collected on the rng can be converted back to the reactant. The value of N for the RRDE wth dmensons shown n Fg. 1 obtaned wth the analytcal calculaton method developed by Albery et al. s ,30 The value of N obtaned n the smulaton for Reacton 2 only n ths work s The expermental result publshed by Markovc et al. was These values for N are n good agreement. The lmtng current predcted by Levch equaton I L = 0.620nFAD 2/3 O2 1/2 1/6 c O2 bulk 32 s A. However, the lmtng current obtaned n ths smulaton work s A. The dscrepancy arses from the truncated seres solutons for the veloctes 22 of the order of z 3 for u z and of z 2 for u r used n dervng the Levch equaton. The velocty profle obtaned n ths work usng the swrl flow model s consstent wth the numercal soluton of the one-dmensonal model gven by F. M. Whte. 20 We also verfed that the smulatons wth truncated seres solutons for the veloctes and the Nernst Planck equaton for materal balance wll result n the same lmtng current value as the Levch equaton predcton.e., I L = A. Case (): Competton between the four-electron transfer reducton and the two-electron transfer reducton (Reactons 1 and 2). There are three possbltes when Reactons 1 and 2 compete wth each other, as shown n Fg. 3a-c, correspondng to c,2 c,1, c,2 = c,1 and c,2 c,1, respectvely. The cathodc transfer coeffcents c are n the exponental terms n the Butler Volmer equaton Eq. 2, whch means that they have a greater effect on the current than the exchange current denstes 0,j do when the overpotental j becomes suffcently large. When c,2 c,1, as n Fg. 3a, the absolute values of the dsk currents n the mass transport lmtng regon decrease as the appled potental shfts towards 0 V. They decrease to dfferent extents and start from dfferent potentals, dependng on the value of the exchange current densty for Reacton 2 0,2. These phenomena show up n the mass transport lmtng regon n whch the oxygen flux to the electrode surface s constant and s at the maxmum value. The reacton rates of the two reactons change whle the total avalable reactant oxygen s constant when the appled potental E appl decreases from about 0.7 V to 0 V. When the appled potental E appl s hgh about 0.7 V, Reacton 1 predomnates and all the avalable reactant wll be converted to water by a four-electron transfer process. When the appled potental E appl s lower, Reacton 2 occurs at an observable rate and a part of the reactant O 2 s converted to peroxde by two-electron transfer process. The total charge transferred n the process decreases and the absolute value of the dsk current decreases, leadng to a tal n the polarzaton curve for the dsk. In the meantme more peroxde s generated on the dsk and oxdzed back on the rng, causng the rng current to ncrease. Ideally, f the appled potental E appl s contnuously lowered, Reacton 2 wll domnate, eventually all of the oxygen wll go to peroxde, and the polarzaton curve of the dsk wll reach a constant value agan correspondng to the value of the lmtng current of Reacton

5 A820 Journal of The Electrochemcal Socety, 15 8 A816-A Table II. Summary of the boundary condtons. Boundary 1 Boundary 7 Boundares 2, -6 Boundary 3 Boundary 1 Boundary 3 Boundares 2 and 5 u r =0,u =0,atr = 0, all z 2u r r =0, r r u r =0, u r z + u z r =0,atr = 0.75 cm, z u r =0,u z =0,u = r, atz = 0, all r u r =0,u =0,P =0atz = 0.12 cm for all r c r =0, =0atr = 0, all z r c = c,bulk, = re at z = 0.12 cm, all r 3 s,j j n j=1 j F + s dc,r s = D dz + z c F D d for = 2,3, RT dz t = F =1 dc z D dz z c F D d at r = or r = cm, z =0 RT dz Boundares and 6 Boundary 7 Butler Volmer equaton Overpotental Reference electrode potental 0= D dc dz + z c F D d RT dz 0= =1 for = 2,3, dc z D dz + z c F D d at r = or r = cm, z =0 RT dz 0= D dc dr + z c F D d RT dr 0= =1 for = 2,3, dc z D dr z c F D d at r = 0.75, z RT dr j = oj,ref c p,0,jexp c,ref a,jf j RT j = met RE 0 RE U j,ref U j,ref = U j RT n j F s,j ln c,ref U RE + c q,0,jexp c,ref c,jf RT j RT n RE F s,re ln c,re Reacton orders p,j = s,j q,j =0 fs,j 0 p,j =0 q,j = s,j f s,j 0 Chemcal reacton Reacton p r s = k h c H2 O 2,0 2 only. The ORR cannot be operated practcally at potentals below 0 V, where other reactons such as hydrogen evoluton preval. Hence, our smulatons were stopped at 0 V and so the tals do not reach constant values n Fg. 3a. When c,2 = c,1, as the appled potental E appl drops, the absolute value of the dsk current decreases and the value of the current at the rng ncreases when the open crcut potental OCP of Reacton 2 s reached, and then they reach a constant value as shown n Fg. 3b. However ths constant value does not correspond to the lmtng current for Reacton 2; t holds a value between the lmtng currents of Reactons 1 and 2 as determned by the rato of the exchange current denstes 0,1 and 0,2. The current does not change wth the appled potental E appl n the mass transport lmtng regon, whch mples that the rates of Reactons 1 and 2 do not change and a constant fracton of the reactant oxygen goes to peroxde. The case of c,2 c,1 s shown n Fg. 3c. When 0,2 has a value bg enough about A/cm 2 n ths case, the current for the rng wll rse and then drop down as the potental decreases, formng a hump on the curve. A smlar phenomenon occurs on the dsk current, although the sze of the hump s smaller vsually due to the scale of the ordnate n the fgure. As shown n Fg. 3c, larger values for exchange current densty for Reacton 2 0,2 lead to a larger hump. The hump n the polarzaton curves due to Reacton 2 s vsble when ts rate s comparable wth the rate of Reacton 1 n the potental regon around 0.6 V to 0. V. It s worth notng that

6 Journal of The Electrochemcal Socety, 15 8 A816-A A821 Table III. Knetc parameters, reacton propertes, and physcal propertes of the speces used for smulatng the ORR n 0.5 M H 2 SO. Knetc parameters Reacton 1 Reacton 2 Reacton 3 Reacton cj o,ref A/cm U j V n j 2 2 k h mol/s mol/cm p 1 Reacton propertes HSO H 2 O 2 H + O 2 s, s, s, s, z Soluton propertes HSO H 2 O 2 H + O 2 c,ref mol/cm D cm 2 /s 21, ths phenomenon can only occur n ths potental regon around 0.6 V to 0. V, whch s just below the OCP of Reacton 2. The exponental term n the cathodc part of the Butler Volmer equaton Eq. 20 for Reacton 1 ncreases faster than that of Reacton 2, and the rate of Reacton 2 wll not be comparable wth that of Reacton 1 when the appled potental E appl shfts further to cathodc values. In polarzaton curves obtaned expermentally, humps and tals show up together frequently. It s possble that more than one materal n the catalyst prompts the ORR. 5,31,32 For example, platnum s a known effectve catalyst for ORR and t s often made n nanometer sze partcles and supported on carbon partcles, such as the commercalzed Pt/Vulcan catalyst powder by E-TEK. Carbon s suspected to have catalytc actvty 5,13 to prompt the two-electron transfer reacton Reacton 2, and can cause the hump startng at about 0.6 V. To smulate the exstence of addtonal catalytc actve materal for Reacton 2, t s treated as two separate reactons O 2 +2H + +2e O 2 +2H + +2e catalyst a catalyst b H 2 O 2 H 2 O 2 wth knetc parameters 0,2a, c,2a 33 wth knetc parameters 0,2b, c,2b 3 The currents for Reactons 33 and 3 are wrtten ndvdually usng Eq. 2. The combnaton of Reactons 1, 33, and 3 s smulated and a set of the results are shown as sold lnes n Fg.. The dashed and dotted lnes n Fg. are smulaton results for the combnaton of Reactons 1 and 33 and the combnaton of Reactons 1 and 3, respectvely. The values on the sold lne for the rng are just a lnear summaton of values on the dashed lne and the dotted lne for the rng. Case (v): Competton nvolvng peroxde reducton (Reactons 1-3). Fgure 5 shows the effect of Reacton 3 on the competton between Reactons 1 and 2. The cathodc transfer coeffcents for Reactons 1 and 2 are equal n ths smulaton so the polarzaton curves should be the same as the curves marked wth trangles n Fg. 3b f there were only Reactons 1 and 2. When Reacton 3 s nvolved, the lmtng current shows a hump as shown n Fg. 5. Ths s because Reacton 3 converts the peroxde generated by Reacton 2 to water by a two-electron transfer process. Reacton 2 followed by Reacton 3 s equvalent to Reacton 1. The apparent result s that Reacton 1 s enhanced and Reacton 2 s weakened, and the polarzaton curves are smlar to those shown n the case of c,2 c,1 see Fg. 3c. Reacton 3 can reduce the sze or change the postons of the humps and the tals as shown n Fg. 3a and c for smlar reasons. The polarzaton curves marked wth squares n Fg. 5 are used for studyng the contrbutons of Reactons 1-3 to the total current, Table IV. Constants, soluton propertes and operaton condtons used for the smulatons. F 9687 C/mol R 8.31 J/K-mol T K U RE 0V kg/cm cm 2 /s 900 rpm Appled potental on rng 1.2 V Table V. Lst of reactons and reacton combnatons smulated. Case number Reactons nvolved 1 2 1, 2 v 1, 2, 3 v 1,2, v 1, 2,, and 1, 2, 3, Fgure 2. Polarzaton curves for sngle reactons: Comparson of fourelectron transfer reacton and two-electron transfer reacton. Parameters used: 0,1 = A/cm 2, c,1 = 1.0 for Reacton 1 only, 0,2 = A/cm 2, c,2 = 1.0 for Reacton 2 only.

7 A822 Journal of The Electrochemcal Socety, 15 8 A816-A Fgure. Polarzaton curves for competng Reactons 1, 33, and 3. Parameters used: 0,1 = A/cm 2, c,1 = 1.0. Case (v): Competton nvolvng chemcal decomposton of peroxde (Reactons 1, 2, and, or 1-). Fgure 7 shows the effect of Reacton, whch s a heterogeneous chemcal reacton n whch peroxde s oxdzed back to oxygen on the surface of the dsk wthout nvolvng charge transfer. The result s to shft the lmtng current towards the one related to Reacton 1 snce less peroxde s captured on the rng. Snce ths reacton s not related to potental, t wll not change the shape of the polarzaton curve substantally, except to lower the flat lnes n the lmtng current regon, reduce the sze of humps, or move the tals to more cathodc potentals. The extent of these changes depends on the rate of Reacton. Reactons 1- may occur smultaneously n the ORR system. 9,2 Ths comprehensve stuaton s smulated n Fg. 8. All four reactons are effectve and the rates of Reactons 1-3 are fxed, whle the rate of Reacton s changed by varyng the rate constant k h. The effect of Reacton s as dscussed n the above paragraph. The hump sze reduces when the rate of Reacton ncreases snce peroxde s consumed. The essental shape of the polarzaton curves wll depend on the knetc parameters for Reactons 1-3. The mechansm nvolvng Reacton 3 and/or Reacton can also be used to explan the hump and the tal phenomena. It s not evdent from the expermental polarzaton curves whether Reacton 3 and/or Reacton are nvolved when a hump and/or a tal shows. Fgure 3. Polarzaton curves for competng Reactons 1 and 2. Parameters used: 0,1 = A/cm 2, c,1 = 1.0, case a: c,2 = 1.2 for c,2 c,1 ; case b: c,2 = 1.0 for c,2 = c,1 ; case c: c,2 = 0.8 for c,2 c,1. and the results are shown n Fg. 6. The total dsk current s just the summaton of the currents for the three ndvdual Reactons 1-3. Under the gven smulaton condtons, only Reacton 2 occurs on the rng and the current from ths reacton s the only component of the rng current. Fgure 5. Polarzaton curves for competng Reactons 1-3. Parameters used: 0,1 = A/cm 2, c,1 = 1.0, 0,2 = A/cm 2, c,2 = 1.0.

8 Journal of The Electrochemcal Socety, 15 8 A816-A A823 Fgure 6. The contrbutons of Reactons 1-3 to the total current. Parameters used: 0,1 = A/cm 2, c,1 = 1.0, 0,2 = A/cm 2, c,2 = 1.0, 0,3 = A/cm 2, c,3 = Fgure 8. Polarzaton curves for competng Reactons 1-, Parameters used: 0,1 = A/cm 2, c,1 = 1.0, 0,2 = A/cm 2, c,2 = 1.0, 0,3 = A/cm 2, c,3 = Although there are expermental results that show the coexstence of more than one reacton n the lterature, 9,21 no conclusve evdence has been reported that favors one mechansm over the others. Ths makes the ORR complcated. Oxygen and peroxde concentraton dstrbutons and potental dstrbuton. Investgatng the concentraton and potental dstrbutons can help us to gan some nsght nto the phenomena that occur durng oxygen reducton n an RRDE system. Examples of concentraton dstrbutons of oxygen and peroxde are shown Fg. 9. The parameters used are the same as those n Fg. 6, except that the appled potental E appl s fxed at 0.5 V. Adjacent to the surface of the dsk r = cm, z =0, the concentraton of oxygen c O2 s near zero due to ts consumpton by the ORR. Away from the dsk r 0.3 cm, z =0, c O2 rses up snce oxygen s not consumed anymore and t s contnuously transported toward the surface of the electrode from the bulk soluton. Especally, adjacent to the surface of the rng r = cm, there s a jump n the concentraton of oxygen because of the generaton of oxygen at the rng. Far away from the dsk along the r drecton, c O2 reaches the same value as n the bulk. The dstrbuton of peroxde concentraton c H2 O 2 can be analyzed n a manner smlar to that of oxygen, takng nto consderaton that peroxde s generated on the dsk and consumed on the rng. Potental dstrbuton n the soluton phase of the RRDE system s shown n Fg. 10. The parameters used are the same as those used n Fg. 9. The potental s zero at z = cm, as gven by the boundary condtons, and t ncreases as the dstance from the electrode surface decreases. The potental reaches ts peak value on the surface of the dsk r = cm, z =0 where the ORR occurs. It drops down quckly beyond the dsk r 0.3 cm, z =0, and even qucker adjacent to the surface of the rng r = cm where the anodc reacton occurs. Far away from the dsk n the r drecton, the lqud potental drops contnuously untl t reaches the potental of the reference electrode whch s set to 0 V n ths work, as shown n Fg. 10. The potental dstrbuton along the z drecton s also gven n the nserted fgure n Fg. 10. Lnes for r = 0.2, 0.3, 0., and 0.6 cm are chosen as they are n the regon near the dsk, the separator, the rng, and beyond the rng, respectvely. The potental s 0 V far away from the electrode surface n the z drecton z = 0.12 cm, and t ncreases as the dstance from Fgure 7. Polarzaton curves for competng Reactons 1, 2, and. Parameters used: 0,1 = A/cm 2, c,1 = 1.0, 0,2 = A/cm 2, c,2 = 1.0. Fgure 9. The concentraton dstrbuton of O 2 and H 2 O 2 along the r drecton at the surface of the electrode. Parameters used: 0,1 = A/cm 2, c1 = 1.0, 0,2 = A/cm 2, c2 = 1.0, 0,3 = A/cm 2, c3 = 0.35, E appl = 0.5 V vs SHE.

9 A82 Journal of The Electrochemcal Socety, 15 8 A816-A Fgure 10. The potental profle along the r and the z drecton at the surface of the electrode. Parameters used: 0,1 = A/cm 2, c1 = 1.0, 0,2 = A/cm 2, c2 = 1.0, 0,3 = A/cm 2, c3 = 0.35, E appl = 0.5 V vs SHE. the electrode surface decreases. The potental ncreases dramatcally near the dsk shown by the lne correspondng to r =0 where the cathodc reactons occur. It does not change near the separator and the regon beyond the rng shown by the curves at r = 0.3 and 0.6 cm where no reacton occurs, and t decreases near the rng as shown by the lne correspondng to r = 0. cm. Conclusons The expermentally observed phenomena n polarzaton curves obtaned wth RRDE, such as tals and humps, can be explaned as the result of the competton between the four-electron transfer reacton and the two-electron transfer reacton, whle the peroxde electrochemcal reducton and peroxde heterogeneous chemcal decomposton reacton can also affect the presence or the sze and the poston of the humps and the tals. The polarzaton curves for several combnatons of reactons were smulated. The concentraton profle of the speces nvolved and potental dstrbuton near the surface of the rng dsk electrode were also obtaned. Further work s requred to attrbute the orgn of the humps and tals to one mechansm aganst another. Acknowledgment The authors are grateful for fnancal support provded by the U.S. Department of Energy under cooperatve agreement no. DE- FC36-0GO1232. The Unversty of South Carolna asssted n meetng the publcaton costs of ths artcle. Lst of Symbols A area of the dsk, cm 2 c concentraton of speces = 1, 2, 3, and represent HSO, H 2 O 2,H +, and O 2, respectvely, mol/cm 3 c,0 concentraton of speces at the surface of the electrode, mol/cm 3 c,bulk concentraton of speces n the bulk soluton, mol/cm 3 c,ref reference concentraton of speces, mol/cm 3 c,re concentraton of speces at the reference electrode, mol/cm 3 D dffuson coeffcent of speces, cm 2 /s F Faraday s constant, 9687 C/equv 0j,ref exchange current densty for reacton j at the reference concentratons, A/cm 2 0j,data exchange current densty for reacton j at the reference concentratons, A/cm 2 I L lmtng current calculated usng the Levch equaton, A I R lmtng current collected on the rng, A I D lmtng current collected on the dsk, A Greek I j current generated by the reacton j, A I dsk current on the dsk, A I rng current on the rng, A t total current densty defned n Eq. 29, A/cm 2 j current densty of the reacton j, A/cm 2 k h rate constant for the chemcal reacton.e., Reacton at the electrode surface, mol/s mol/cm 3 n j stochometrc number of electrons nvolved n the electrode reacton j n RE stochometrc number of electrons nvolved n the reacton that occurs at the reference electrode N collecton effcency of an RRDE P pressure n the electrolyte, Pa p reacton order of the non charge transfer reacton.e., Reacton p j anodc reacton order of speces nreacton j q j cathodc reacton order of speces nreacton j r radal dstance from the axs of the dsk, cm r s rate of chemcal reacton.e., Reacton at electrode surface, mol/cm 2 s R gas constant, 8.31 J/mol K s,j stochometrc coeffcent of speces nthereacton j s,re stochometrc coeffcent of speces nthereactonatthe reference electrode T absolute temperature, K u r radal component of the velocty, cm/s u z axal component of the velocty, cm/s u angular component of the velocty, cm/s s,re stochometrc coeffcent of speces n the reacton at reference electrode U j standard electrode potental for the charge transfer reacton j, V U j,ref open crcut potental of the reacton j at the reference concentratons relatve to the reference electrode, V U RE standard potental of the reference electrode relatve to SHE, V u velocty vector, cm/s z axal dstance, cm z,v axal dstance consdered to be suffcently far from the electrode surface to be consdered to be at nfnty n the doman for the momentum balance, cm z,m axal dstance consdered to be suffcently far from the electrode surface to be consdered to be at nfnty n the doman for materal balance, cm z charge on speces a,j anodc transfer coeffcent for reacton j c,j cathodc transfer coeffcent for reacton j angular coordnate, rad densty of the electrolyte, g/cm 3 knematc vscosty of the electrolyte, mpa s potental n soluton phase, V 0 potental n the soluton adjacent to the electrode surface, V RE potental of the reference electrode at the expermental condtons, V met potental of the workng electrode, V re potental of the reference electrode at the expermental condtons, V rotatng speed of the electrode, rad/s j overpotental of reacton j corrected for ohmc drop n the soluton and measured wth respect to a reference electrode of a gven knd n a soluton at the reference concentratons, V 0 G 298 standard Gbbs free energy change n a chemcal process at 298 K, kj/mol Subscrpts speces ndex, = 1, 2, 3, and represent HSO,H 2 O 2,H +, and O 2, respectvely j reacton ndex, j = 1, 2, 3, 33, 3 correspond to Reactons 1, 2, 3, 33, 3, respectvely bulk propertes or varables evaluated at the bulk soluton References 1. 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