A

ece©

K

1

tewsmtaatnaabcfcm

a(t

0d

Available online at www.sciencedirect.com

Electrochimica Acta 53 (2008) 3566–3578

A comparison of diffusion-limited currents at microelectrodesof various geometries for EC′ reactions

R. Senthamarai, L. Rajendran ∗SMSV Higher Secondary School, Karaikudi 630 001, Tamilnadu, India

Received 18 August 2007; received in revised form 10 December 2007; accepted 15 December 2007Available online 24 December 2007

bstract

′

The steady-state and non-steady-state diffusion-limited currents for EC reactions for all ultramicroelectrodes (circular disc, circular ring,lliptical disc, elliptical ring, band, hemisphere, hemicylinder, hemi-oblate and hemi-prolate electrodes) are compared. In this paper we present aritical assessment of the influence of geometry on the behavior of the transient current. In the research presented here, the surface area of all thelectrodes was fixed. The chronoamperometric currents for all times and for all K values are given in tabular and graphical form.

2007 Elsevier Ltd. All rights reserved.

-state

mul

1

aaomptatfsf‘

eywords: Reaction/diffusion; Ultramicroelectrodes; Transient current; Steady

. Introduction

If the geometric dimensions of a voltammetric working elec-rode become progressively smaller, then the behavior of thelectrode begins to quit diverge from that of a large electrode,hich can be approximated by an electrode of infinite dimen-

ion. These differences are caused by changing conditions of theass transport from the bulk of the solution towards the elec-

rode, and have several important practical implications, such asdecreased ohmic drop of potential, IR, fast establishment ofsteady-state signal, a current increase due to enhanced mass

ransport at the electrode boundary and an increased signal-to-oise ratio [1,2]. These effects make sufficiently small electrodesdvantageous in many areas of electroanalytical chemistry. Thepplication of small-sized electrodes (ultramicroelectrodes) haseen further enhanced by increasing demands from analyti-al chemistry (e.g., the need for electrodes in miniature cellsor detection for high-performance separations or in electro-hemical sensors) and biochemistry (in situ electroanalytical

easurements in living organisms) [1,2].However, many electrode processes at ultramicroelectrodes

re complicated; for example, the catalytic electrode processEC′ reaction) is one of them. With the increasing interest inhese reactions, there is a need to develop methods for the deter-

∗ Corresponding author. Tel.: +91 4565220126.E-mail address: raj sms@rediffmail.com (L. Rajendran).

eotfrte

013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2007.12.050

; EC′ mechanism

ination of the kinetics of the coupled chemical reaction and tonderstand the relationship between the rate constant, K and theimiting current density.

.1. Previous work

Much work has been carried out to study the EC′ mechanismt ultramicroelectrodes using analytical approximations [3–14]nd numerical methods [15–22]. Space permits us to mentionnly a few of them below. Delmastro and Smith [4] solved theass transport equations at a dropping mercury electrode for the

seudo-first-order reaction. Fleischmann et al. [5] adapted thiso give an expression for a spherical electrode. Dayton et al. [6]lso derived the spherical response using Neumann’s integralheorem. Denuault et al. [7,8] derived an analytical expressionor the steady-state current at a spherical electrode due to aecond-order EC′ process. Oldham [9] has derived the currentor pseudo-first-order kinetics at microdisc electrodes using theequivalent sphere’ approximation.

Bender and Stone [10] tackled the problem of finding an exactxpression for steady-state current for disc electrodes by meansf an integral equation. Zhuang and Chen [11] have derivedhe current equation for the EC′ reaction at spherical electrodes

or steady-state conditions using the concept of the steady-stateeaction-diffusion layer. Phillips [12] derived a pair of asymp-otic equations for pseudo-first-order EC′ behavior at a microdisclectrode without resorting to a spherical approximation. Most

ctroc

rrRataab

u[htfifm(mTaebmats

ddoiiiBsiDtw

smcctt

2a

lr

hccifista

∇wticitbsc

ww(cms

w

a

wlrgfiaicir

f

Wt

R. Senthamarai, L. Rajendran / Ele

ecently, Galceran et al. [13] obtained the current of inlaid andecessed microdisc electrodes using the finite element method.ajendran and Sangaranarayanan [14] have derived an exactnalytical expression for steady-state current for all K usinghe two-point Pade approximation. Most recently Galceran etl. [13] presented three new formulas for steady-state current,mong which they highlighted one equation with an accuracyetter than 0.27%.

The application of numerical solutions and digital sim-lations to electrochemical problems are discussed in Refs.15,16]. Numerical solutions of currents for EC′ reactionsave also been obtained by Lavagnini et al. [17], employinghe hopscotch method, and by Galceran et al. [13] using thenite element method. Numerical simulation of currents,or disc electrodes can also be obtained from the hopscotchethod [17], the adaptive moving-grid technique [18], the ILU

Incomplete LU factorization) preconditioned Krylov subspaceethod [19] and the adaptative finite element method [20].hese involve considerable computational effort. Abercrombiend Denuault [21] introduced apparently for the first time inlectrochemistry, the use of gradient recovery methods foroth error estimation and accurate current calculations for allechanisms and electrode geometries. Gillow et al. [22] used

n hp-adaptive discontinuous Galerkin finite element algorithmo compute the current flowing at a microdisc electrode forteady-state E and EC′ reaction mechanisms.

Cope and Tallman [23] compared the transient behaviors ofisc, ring and band microelectrodes for the E reaction. Bothisc and band electrodes may be considered limiting formsf the ring electrode, with the disc being a ring having zeronterior annulus and the band being a ring having an infinitenterior annulus. The mathematical theory for electrical capac-tance problems is analogous to diffusion problems. Recently,ruckenstein and Janiszewska [24] obtained the expression for

teady-state diffusion currents and local current densities of var-ous microelectrode geometries via this analogy. More recently,udko et al. [25] presented approximate analytic expressions for

he time-dependence of the current at any inlaid microelectrodeith a simply connected shape of finite area.However, to the best of this author’s knowledge, the steady-

tate and transient currents for the EC′ reactions of allicroelectrodes have not been compared. In this paper, we

ompare the steady and non-steady-state currents for all ultrami-roelectrodses for EC′ reactions. To facilitate this comparison,he current variable is based on electrode area without referenceo geometrical shape.

. Mathematical formulation of the problem andnalysis (steady-state)

As a representative example of the reaction-diffusion prob-ems considered, the standard pseudo-first-order catalyticeaction scheme [26]

A + e → B

B + Zk−→A + products

(1) I

himica Acta 53 (2008) 3566–3578 3567

as been chosen, with initial and boundary conditionsorresponding to the potential step method for all ultrami-roelectrodes. Here, A is regenerated electroactive species, Bs reacting species and Z is electroinactive species. Pseudo-rst-order kinetics is achieved when the concentration of Z isufficiently large. The initial boundary value problem which haso be solved in this case can be written in dimensionless formss follows [26]:

2c(r, z) − Kc(r, z) = 0 (2)

here c refers to the dimensionless concentration of the elec-roactive species B, K denotes the dimensionless reaction rate,.e., K = ka2/D and ‘a’ denotes the characteristic length asso-iated with the geometry under consideration (‘a’ may bedentified as the radius for the disc). The condition pertainingo Eq. (2) is c = 0 when r → ∞. Some theoretical work has alsoeen published for the first-order EC′ scheme at disc and hemi-pherical electrodes. The dimensionless chronoamperometricurrent for the diffusion-limited case is evaluated from Ref. [26]

i(K)

4nFDC0a= ±π

2

∫ 1

b/a

(∂c

∂z

)z=0

r dr (3)

here the sign ‘plus’ corresponds to a reduction process (n = 1)hile the sign ‘minus’ corresponds to a oxidation process

n = −1), F is the Faraday constant and D is the diffusion coeffi-ient of species A. The chronoamperometric current [27] for allicroelectrodes (except hemicylinder and band electrodes for

mall K) for small and large K values is of the form

here

0 = l0

4a, a1 = l20

8πa2 , b0 = A

4a2 and b1 = P

8a(6)

here A is the area, P is the perimeter of the electrode and0 is the steady-state of the diffusion-limited current for the Eeaction. The values of A, P and l0 for various electrodes areiven in Table 1. The dimensionless steady-state current f(K)or small and large K values for various electrodes are reportedn Table 2. For small K values, the currents for band (Eq. (B7))nd hemicylinder (Eq. (B11)) electrodes are given in Table 2. Its of great interest to devise a simple analytical expression forurrent which is accurate for all values of K. A simple functions constructed such that the coefficients of Eqs. (4) and (5) areeproduced. In this case, the steady-state current is given by [27],

(K) = i(K)

4nFDC0a= A1 + B1

√K + C1 exp(−D1

√K) (7)

hen current is normalized with respect to the area of the elec-rode, the steady-state current is defined by

ss(K)= i(K)a

4nFDC0A=a2

A

[A1 + B1

√K + C1 exp

(−D1

√K)](8)

3568R

.Senthamarai,L

.Rajendran

/Electrochim

icaA

cta53

(2008)3566–3578

Table 1Area (A), perimeter (P) and steady-state current (l0/a) of various microelectrode geometriesMicroelectrode shape A P l0/a

πa2 2πa 4 (A1)

πab 2∫ π

0

√a2 cos2 θ + b2 sin2 θ dθ 4π∫ π

0dθ/

√cos2 θ+(b/a)2 sin2 θ

(A2)

π(a2 − b2) 2π(a + b)(

l0a

)exact

= π2(1+b/a)ln(16(1+b/a)/(1−b/a)) (A3a)(

l0a

)empirical

= π2(1+b/a)

ln[(32(b/a)/(1−b/a))+exp(π2/4)](A3b)

ab 2 a (active perimeter) (actual perimeter is 2(a + b))2√

πa/b

ln(64(a/πb)t) When a/b = π, 2πln[(64t)] (quasi-steady-state) (A4)

2πa2 – 2π (A5)

R.Sentham

arai,L.R

ajendran/E

lectrochimica

Acta

53(2008)

3566–35783569

πab –2√

a/b

ln(4 e−2γ (a/b)t)When a = b, 2

ln(4 e−2γ t)(quasi-steady-state) (A6)

πa2 + πab2 cosh−1(a/b)√a2−b2

– 2π(1−(b/a)2)1/4

cos−1(b/a)(A7)

πa2 + πab2 cos−1(a/b)√b2−a2

– 2π((b/a)2−1)1/2

cosh−1(b/a)(A8)

π((a + r)(b + r) − ab) when a =bn, πa2

((1 + r

a

)((1n

)+(

ra

))− 1

n

)(n > 0)

2a

(∫ π

0

√cos2 θ +

(1n2

)sin2 θ dθ

+∫ π

0

√(1 + r

a

)2cos2 θ +

(1n + r

a

)2sin2 θ dθ

) 4π1/3

[(1 + r

a

)((1n

)+ r

a

)−(

1n

)]×(∫ π

0

√cos2 θ +

(1n2

)sin2 θ dθ +

∫ π

0

√(1 + r

a

)2cos2 θ +

((1n

)+ r

a

)2sin2 θ dθ

)1/3

(A9)

A P(

32π2a3 AP

)1/3(A10)

3570R

.Senthamarai,L

.Rajendran

/Electrochim

icaA

cta53

(2008)3566–3578

Table 2Steady-state current f(K) = I/4nFDC0a for the EC′ reactions of various geometriesElectrode f (K) = I

4nFDC0a

Small K Large K

Circular disc f (K) = 1 +(

2

π

)K1/2 +

(4

π2− 1

3

)K (B1) f (K) = π

4K1/2 + π

4+ π

16K−1/2 (B2)

Elliptical disc f (K) = (l0/a)

4+ (l0/a)2

8πK1/2 (B3) f (K) = π(b/a)

4K1/2 +

(1

4

)∫ π

0

√cos2 θ +

(b

a

)2

sin2 θ dθ (B4)

Circular ring f (K) = (l0/a)

4+ (l0/a)2

8πK1/2 (B5) f (K) = π(1 − (a/b)2)

4K1/2 + π(1 + a/b)

4(B6)

where a and b are the inner and outer radii for a circular ring electrode

Band f (K) = πK

2

∫ ∞

o

[e−Ku

ln(22739.57u)

]du (B7) f (K) = (b/a)

4K1/2 + 1

4when a/b = π, f (K) = 1

4πK1/2 + 1

4(B8)

where a and b are the length and width of the band electrode.

Hemisphere f (K) = π

2+ π

2K1/2 (B9) f (K) = π

2K1/2 + π

2(B10)

Hemicylinder f (K) = πK

2

∫ ∞

0

[e−Ku

ln(1.2609u)

]du (B11) f (K) = π(b/a)

4K1/2 + π

8(at a = b) f (K) = π

4K1/2 + π

8(B12)

where w and R0 are the length and radius of the electrode.

Hemi-oblate f (K) = (l0/a)

4+ (l0/a)2

8πK1/2 (B13) f (K) = π

4

[1 +(

b

a

)2

cosh−1(

a

b

)(1 −(

b

a

)2)−1/2

]K1/2 + π

4

[(b

a

)+ cos−1

(b

a

)(1 −(

b

a

)2)−1/2

](B14)

where a and b are the lengths of the two semi-axes andthe third axis of the hemi-oblate.

Hemi-prolate f (K) = (l0/a)

4+ (l0/a)2

8πK1/2 (B15) f (K) = π

4

[1 +(

b

a

)2

cos−1(

a

b

)((b

a

)2

− 1

)−1/2]

K1/2 + π

4

[(b

a

)+ cosh−1

(b

a

)((b

a

)2

− 1

)−1/2]

(B16)

where a and b are the lengths of the two semi-axes andthe third axis of the hemi-prolate.

Elliptical ring f (K) = (l0/a)

4+ (l0/a)2

8πK1/2 (B17) f (K)= π

4

[(1+ r

a

)(( 1

n

)+ r

a

)−(

1

n

)]K1/2 + 1

4

[∫ π

0

√cos2 θ+

(1

n2

)sin2 θ dθ+

∫ π

0

√(1+ r

a

)2cos2 θ +

((1

n

)+(

r

a

))2

sin2 θ dθ

](B18)

where a, b are the inner, outer radii and r is the width ofthe elliptical ring.

where b = a/n. (In Table 3 the diffusion-limited current for elliptical rings is calculated when n = 2)

ctroc

w

A

Fa

A

Fa

F

A

C

Fb

2co

F

F

F

A

F

A

F

A

wdca

2

c

aoEctiav

b

R. Senthamarai, L. Rajendran / Ele

here

1 = b1, B1 = b0, C1 = a0 − b1 and D1 =∣∣∣∣ (b0 − a1)

(a0 − b1)

∣∣∣∣(9)

or circular disc electrodes, the numerical values of A1, B1, C1nd D1 are

1 = 0.7854, B1 = 0.7854, C1 = 0.2146 and D1 = 0.6934

(10)

or elliptical disc electrodes, the values of A1, B1, C1 and D1re

A1 = 1

4

∫ π

0

√cos2 θ +

(b

a

)2

sin2 θ dθ, B1 = π

4

(b

a

)

C1 = l0

4a− A1 and D1 =

∣∣∣∣∣ (B1 − (l0/a)2/8π)

C1

∣∣∣∣∣(11)

or circular ring electrodes, the values of A1, B1, C1 and D1 are

1 = π

4

(1 + a

b

), B1 = π

4

(1 −

(b

a

)2)

,

1 =(

l0

4a

)− A1 and D1 =

∣∣∣∣∣B1 − ((l0/a)2/8π)

C1

∣∣∣∣∣ (12)

or elliptical ring electrodes, the values of A1, B1, C1 and D1ecomes

A1 = 1

4

⎛⎝∫ π

0

√cos2 θ +

(1

n2

)sin2 θ dθ +

∫ π

0

√(1 +

( r

a

))

B1 = π

4

[(1 + r

a

)((1

n

)+( r

a

))−(

1

n

)],

C1 =(

l0

4a

)− A1 and D1 =

∣∣∣∣∣B1 − ((l0/a)2/8π)

C1

∣∣∣∣∣or hemi-oblate electrodes the values are

A1 = π

4

⎡⎣(b

a

)+ cos−1

(b

a

)(1 −

(b

a

)2)−(1/2)

⎤⎦

B1 = π

4

⎡⎣1 +

(b

a

)2

cosh−1(a

b

)(1 −

(b

a

)2)−(1/2)

⎤⎦

C1 =(

l0)

− A1

4a

D1 =∣∣∣∣∣B1 − ((l0/a)2/8π)

C1

∣∣∣∣∣(14)

Tet

himica Acta 53 (2008) 3566–3578 3571

s2 θ +((

1

n

)+( r

a

))2

sin2 θ dθ

⎞⎠ ,

(13)

or hemi-prolate electrodes the values are

A1 = π

4

⎡⎣(b

a

)+ cosh−1

(b

a

)((b

a

2)

− 1

)−(1/2)⎤⎦

B1 = π

4

⎡⎣1 +

(b

a

)2

cos−1(a

b

)((b

a

)2

− 1

)−(1/2)⎤⎦

C1 =(

l0

4a

)− A1

D1 =∣∣∣∣∣B1 − ((l0/a)2/8π)

C1

∣∣∣∣∣

(15)

or hemisphere electrodes the values are

1 = π

2, B1 = π

2, C1 = 0 (16)

or hemicylinder electrodes (for large K) the values are

1 = π

8, B1 = π

4, C1 = 0, D1 = 0 (17)

or band electrodes (for large K) the values are

1 = 1

4, B1 = 1

24π, C1 = 0, D1 = 0 (18)

hen K = 0, Eq. (7) becomes the steady-state current of theiffusion-limited E reaction. Eq. (8) represents the steady-stateurrent iss(K) for the EC′ reactions for all electrode geometriest all values of K.

.1. Discussion

Tables 3A–3C indicates the dimensionless steady-statehronoamperometric currents f(K) for the EC′ reactions for

ll ultramicroelectrodes. Fig. 1(a–d) shows the comparisonsf steady-state currents f(K) at all ultramicroelectrodes usingq. (7) for all values of K and all values of b/a. For theseurves, current is normalized with respect to the characteris-ic length (a) associated with the geometry under consideration,.e., the ordinate is f(K) = i(K)/4nFDC0a. From Tables 3A–3Cnd Fig. 1(a–d), the values of the steady-state currents for largealues of K are in the following order:

and < hemicylinder < circular disc < circular ring

< elliptical disc < hemi-oblate

< hemisphere < elliptical ring < hemi-prolate (19a)

he steady-state currents f(K) for hemicylinder and circular disclectrodes are equal (Refer Fig. 1(c)). For elliptical disc elec-rodes, the value of the current increases when b/a increases (or

3572 R. Senthamarai, L. Rajendran / Electrochimica Acta 53 (2008) 3566–3578

Table 3AComparison of steady-state current f(K) for the EC′ reactions for various microelectrodes

log K K Hemicylinder Circular disc Elliptical disc Hemisphere

b/a = 1 disc (e = 0) b/a = 0.9 (e = 0.44) b/a = 0.5 (e = 0.9) b/a = 0.1 (e = 0.995)

−4 0.0001 0.0079 1.0064 1.0064 0.9551 0.7318 0.4261 1.5865−3 0.001 0.0614 1.0202 1.0202 0.9676 0.7391 0.4286 1.6205−2 0.01 0.4817 1.0642 1.0642 1.0072 0.7623 0.4364 1.7279−1 0.1 0.6411 1.2061 1.2061 1.1349 0.8364 0.4606 2.0675

0 1 1.1781 1.6781 1.6781 1.5587 1.0768 0.5330 3.14161 10 2.8764 3.2930 3.2930 3.0075 1.8772 0.7356 6.53812 100 8.2467 8.6396 8.6396 7.8160 4.5339 1.2920 17.27883 1000 25.2292 25.6219 25.6219 23.1008 13.0238 2.9904 51.24394 10000 78.9327 79.3254 79.3254 71.4367 39.8755 8.3580 158.6508

Table 3BComparison of steady-state current f(K) for the EC′ reactions for various microelectrodes

log K K Hemi-oblate b/a < 1 Hemi-prolate b/a > 1 Band

b/a = 0.01 b/a = 0.5 b/a = 0.99 b/a = 1.5 b/a = 2 b/a = 10

−4 0.0001 1.0157 1.4052 1.5865 1.8461 2.0930 5.3947 0.0020−3 0.001 1.0357 1.4255 1.6205 1.8919 2.1517 5.7663 0.0154−2 0.01 1.0987 1.4907 1.7279 2.0369 2.3374 6.9168 0.1028−1 0.1 1.2954 1.7064 2.0675 2.4952 2.9245 10.3497 0.2752

0 1 1.8973 2.4293 3.1416 3.9444 4.7781 19.9756 0.32961 10 3.6867 4.7703 6.5381 8.5248 10.6202 47.2706 0.5017

78843951

et(rtva

depotd

dfwdsTstnoe

TC

l

2 100 9.0913 12.1824 17.23 1000 26.0830 35.6215 51.24 10000 79.8138 109.7424 158.6

ccentricity decreases). In hemi-oblate and hemi-prolate elec-rodes, the value of current f(K) increases when b/a increasesRefer Table 3B). In circular ring electrodes, the value of the cur-ent decreases when b/a increases. In elliptical ring electrodes,he value of the current increases when a/r decreases. For smallalues of K, the current values tend towards the same value forll electrodes (see the left hand side of Fig. 1(a–d).

For all electrodes, the current at short times is directly depen-ent on the area of the electrode, and therefore, there is nolectrode geometry advantage for this condition because trans-ort will be completely planar. Normalizing the current based

n the radius or the square root of the area ignores the facthat approaches to true diffusional steady-states are depen-ent on the characteristic lengths of the electrode in three

Uhd

able 3Comparison of steady-state current f(K) for the EC′ reactions for various microelectr

og K K Circular ring

b/a = 0 disc b/a = 0.25 b/a = 0.5

−4 0.0001 1.0064 0.9975 0.9621−3 0.001 1.0202 1.0116 0.9750−2 0.01 1.0642 1.0583 1.0157−1 0.1 1.2061 1.2148 1.1447

0 1 1.6781 1.7180 1.55231 10 3.2930 3.3101 2.84062 100 8.6396 8.3447 6.90853 1000 25.621 24.2656 19.72744 10000 79.3254 74.6117 60.0794

22.9963 29.0116 132.0273 1.046068.7268 87.0688 400.0135 2.7672213.3242 270.6480 1247.4603 8.2100

imensions being small compared to the time-dependent dif-usion length (Dt)1/2. Even a cylindrical or a band electrodeith a small radius with short characteristic lengths for twoimensions of the electrode will not obtain a true diffusionalteady-state, but will reach a quasi-steady-state (Refer Table 2).he influence of the catalytic mechanism that enables a steady-tate current to be observed needs to be demonstrated withouthe complications introduced by an inappropriate dimensionalormalization. Efficiency based on the current density hasften been used as a criterion for the comparison of differ-nt reaction rate/time domains and geometries for electrodes.

sing the current normalization (Iss(K) = i(K)a/4nFDC0A) mayelp to better generalize electrode size and shape over largeomains of time and reaction rate. Hence in Eq. (8) current

odes

Elliptical ring

b/a = 0.9 a/r = 1 a/r = 10 a/r = 50

0.8263 1.8711 0.6427 0.36280.8355 1.9180 0.6483 0.36460.8644 2.0655 0.6660 0.37030.9524 2.5236 0.7212 0.38821.2007 3.9214 0.8878 0.44341.7861 8.1951 1.3505 0.60682.9742 21.6220 2.4644 1.03146.2104 64.0783 5.2617 1.8843

16.4123 198.337 13.8576 3.6154

R. Senthamarai, L. Rajendran / Electrochimica Acta 53 (2008) 3566–3578 3573

Fig. 1. (a) Steady-state current f(K) for the EC′ reactions for various values of K for hemicylinder, circular disc, elliptical disc, hemisphere and elliptical ringelectrodes. (b) Steady-state current f(K) for the EC′ reactions for various values of K for circular disc, circular ring, band, hemi-oblate and hemi-prolate electrodes.( hemicr nd ci

isTsfo

f

TC

l

c) Steady-state current f(K) for the EC′ reactions for various values of K foreactions for various values of K for circular disc, hemicylinder, elliptical ring a

s normalized with respect to the area (A) of the electrode,uch that the vertical ordinate is Iss(K) = i(K)a/4nFDC0A.

ables 4A–4C and Fig. 2(a–d) indicate dimensionless steady-tate chronoamperometric currents Iss(K) for the EC′ reactionsor all ultramicroelectrodes for all values of K. The valuesf steady-state current iss(K) for large values of K are in the

h

able 4Aomparison of steady-state current Iss(K) for the EC′ reactions for various microelec

og K K Hemicylinder Circular disc Elliptical disc

b/a = 1 disc (e = 0) b/a

−4 0.0001 0.0025 0.3204 0.3204 0.3−3 0.001 0.0195 0.3248 0.3248 0.3−2 0.01 0.1533 0.3387 0.3387 0.3−1 0.1 0.2041 0.3839 0.3839 0.4

0 1 0.3750 0.5342 0.5342 0.51 10 0.9156 1.0482 1.0482 1.02 100 2.6251 2.7502 2.7502 2.73 1000 8.0310 8.1560 8.1560 8.14 10000 25.126 25.251 25.2510 25.2

ylinder and circular disc electrodes. (d) Steady-state current f(K) for the EC′rcular ring electrodes.

ollowing order

emicylinder < hemi-prolate

< hemisphere < circular disc < hemi-oblate < band

< elliptical disc < circular ring < elliptical ring (19b)

trodes

Hemisphere

= 0.9 (e = 0.44) b/a = 0.5 (e = 0.9) b/a = 0.1 (e = 0.995)

378 0.4659 1.3565 0.2525422 0.4705 1.3643 0.2578562 0.4853 1.3891 0.2750014 0.5324 1.4660 0.3290513 0.6856 1.6967 0.4999637 1.1951 2.3413 1.0404643 2.8865 4.1124 2.7494700 8.2915 9.5185 8.1540649 25.3865 26.6036 25.2447

3574 R. Senthamarai, L. Rajendran / Electrochimica Acta 53 (2008) 3566–3578

Table 4BComparison of steady-state current Iss(K) for the EC′ reactions for various microelectrodes

log K K Hemi-oblate b/a < 1 Hemi-prolate b/a > 1 Band

b/a = 0.01 b/a = 0.5 b/a = 0.99 b/a = 1.5 b/a = 2 b/a = 10

−4 0.0001 0.3231 0.3240 0.2525 0.2182 0.1948 0.1088 0.0062−3 0.001 0.3294 0.3287 0.2578 0.2236 0.2003 0.1163 0.0484−2 0.01 0.3495 0.3437 0.2750 0.2408 0.2176 0.1395 0.3229−1 0.1 0.4121 0.3935 0.3290 0.2949 0.2722 0.2087 0.8646

0 1 0.6035 0.5602 0.4999 0.4662 0.4448 0.4028 1.03551 10 1.1727 1.1000 1.0404 1.0076 0.9886 0.9532 1.57632 100 2.8919 2.8092 2.7494 2.7181 2.7006 2.6622 3.28623 1000 8.2969 8.2140 8.1540 8.1234 8.1049 8.0659 8.69394 10000 25.3885 25.3057 25.2447 25.2146 25.1935 25.1538 25.794

Table 4CComparison of steady-state current Iss(K) for the EC′ reactions for various microelectrodes

log K K Circular ring Elliptical ring

b/a = 0 disc b/a = 0.25 b/a = 0.5 b/a = 0.9 a/r = 1 a/r = 10 a/r = 50

−4 0.0001 0.3204 0.3387 0.4082 1.3843 0.2382 1.2786 3.7992−3 0.001 0.3248 0.3434 0.4137 1.3997 0.2442 1.2899 3.8179−2 0.01 0.3387 0.3594 0.4310 1.4481 0.2630 1.3252 3.8775−1 0.1 0.3839 0.4125 0.4857 1.5955 0.3213 1.4349 4.0645

0 1 0.5342 0.5833 0.6587 2.0115 0.4992 1.7664 4.64301 10 1.0482 1.1239 1.2054 2.9920 1.0433 2.6869 6.3535

.9314

.3708

.493

TcIiceIii

3a

ff

wtic(g

I

wc

g

Trvis(gand Table 5 show in more detail, the degrees of the functionK1/2g(K,t) to constant value (=0). When t > 1 for all ultramicro-electrodes, K1/2g(K,t) = 0. Now, IT ∼= Iss(K) (Refer Table 5 andFig. 3(a)).

Table 5Values of K1/2g(K,t) for various values of K and t

log t t K1/2g(K,t)

K = 1 K = 10 K = 100

−3 0.001 16.8591 14.8571 9.5962−2 0.01 4.6982 3.0346 0.5025−1 0.1 0.9596 0.1589 0.0001

2 100 2.7502 2.8334 23 1000 8.1560 8.2391 84 10000 25.2510 25.3336 25

he steady-state currents for hemicylinder, hemisphere, cir-ular disc and band electrodes are equal (Refer Fig. 2(c)).n elliptical disc electrodes, iss(K) increases when eccentric-ty increases [24]. In hemi-oblate and hemi-prolate electrodes,urrent decreases when b/a increases, whereas in elliptical ringlectrodes, steady-state current increases when a/r increases.n all the electrodes, steady-state current increases when Kncreases. The steady-state current iss(K) is relatively slow vary-ng when log K ≤ 1 for all ultramicroelectrodes.

. Mathematical formulation of the problem andnalysis (non-steady-state)

The initial boundary value problem which has to be solvedor the case of non-steady-state can be written in dimensionlessorm as follows

∂c(r, t)

∂t= ∇2c(r, t) − Kc(r, t) (20)

here c refers to the dimensionless concentration of the elec-roactive species B, K denotes the dimensionless reaction rate,.e., K = ka2/D and t denotes dimensionless time. ‘a’ denotes theharacteristic associated with the geometry under consideration‘a’ may be identified as the radius for discs and spheres). The

eneral analytical expression of current is given by [28]

T = i(t)a

4nFDC0A= Iss(K) + 1

4K1/2g(K, t) (21)

4.9824 2.7526 4.9029 10.79910.4037 8.1576 10.4682 19.72927.4940 25.2495 27.5698 37.855

here A is the area of the electrode, Iss(K) is the steady-stateurrent for the EC′ reaction (Eq. (8)) and

(K, t) = (πKt)−1/2 e−Kt − erfc(Kt)1/2 (22)

he first term of the Eq. (21) represents the steady-state cur-ent for the EC′ reaction. The steady-state currents Iss(K) ofarious electrodes are given in the Eq. (8). This approximations more accurate for large times and large K values than forhort times and small K values [28]. When t → ∞, g(K,t) → 0Since for large t, erfc(Kt)1/2 ≈ (πKt)−1/2 e−Kt). Fig. 3(a) givesraphs of K1/2g(K,t) for the indicated t and K values. Fig. 3(a)

0 1 0.0503 0.0000 0.00001 10 0.0000 0.0000 0.00002 100 0.0000 0.0000 0.00003 1000 0.0000 0.0000 0.0000

R. Senthamarai, L. Rajendran / Electrochimica Acta 53 (2008) 3566–3578 3575

Fig. 2. (a) Steady-state current Iss(K) for the EC′ reactions for various values of K for circular disc, elliptical disc, hemisphere and elliptical ring electrodes. (b)Steady-state current Iss(K) for the EC′ reactions for various values of K for circular disc, hemicylinder, circular ring, hemi-oblate and hemi-prolate electrodes. (c)S micyI ng an

3

cri

f

TC

l

teady-state current Iss(K) for the EC′ reactions for various values of K for he

ss(K) for the EC′ reactions for various values of K for circular disc, elliptical ri

.1. Discussion

Tables 6A–6E indicates the dimensionless non-steady-statehronoamperometric currents IT = i(t)a/4nFDC0A for the EC′eactions for all ultramicroelectrodes. From Tables 6A–6E, it isnferred that the values of non-steady-state currents are in the

h

able 6Aomparison of non-steady-state current IT for the EC′ reactions for various microelec

og t t Hemicylinder Circular disc Elliptical

K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 a/r = 1

K = 1

−3 0.001 4.5897 4.6298 5.0242 4.7489 4.7625 5.1510 4.7140−2 0.01 1.5496 1.6742 2.7508 1.7087 1.8069 2.8775 1.6738−1 0.1 0.6149 0.9553 2.6251 0.7741 1.0879 2.7519 0.7391

0 1 0.3876 0.9156 2.6251 0.5468 1.0482 2.7518 0.51181 10 0.3750 0.9156 2.6251 0.5342 1.0482 2.7518 0.49922 100 0.3750 0.9156 2.6251 0.5342 1.0482 2.7518 0.49923 1000 0.3750 0.9156 2.6251 0.5342 1.0482 2.7518 0.4992

linder, hemisphere, circular disc and band electrodes. (d) Steady-state currentd circular ring electrodes.

ollowing order for all times:

emi-cylinder < hemi-prolate < hemi sphere < circular disc

< hemi-oblate < band < elliptical disc < circular ring

< elliptical ring (23)

trodes

ring

a/r = 10 a/r = 50

K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100

4.7576 5.1518 5.9801 6.4003 7.3015 8.8599 10.0698 13.19921.8019 2.8782 2.9406 3.4453 5.0284 5.8183 7.1126 10.92441.0830 2.7526 2.0062 2.7266 4.9029 4.8831 6.3933 10.79881.0433 2.7526 1.7789 2.6869 4.9029 4.6557 6.3535 10.79881.0433 2.7526 1.7664 2.6869 4.9029 4.6430 6.3535 10.79881.0433 2.7526 1.7664 2.6869 4.9029 4.6430 6.3535 10.79881.0433 2.7526 1.7664 2.6869 4.9029 4.6430 6.3535 10.7988

3576 R. Senthamarai, L. Rajendran / Electrochimica Acta 53 (2008) 3566–3578

Table 6BComparison of non-steady-state current IT for the EC′ reactions for various microelectrodes

log t t Elliptical disc Hemisphere

b/a = 1 b/a = 0.9 b/a = 0.5 b/a = 0.1 K = 1 K = 10 K = 100

K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100

−3 0.001 4.7489 4.7625 5.1510 4.7662 4.7782 5.1635 4.9000 4.9092 5.2854 5.9105 6.0549 6.5111 4.7146 4.7545 5.1486−2 0.01 1.7087 1.8069 2.8775 1.7259 1.8223 2.8899 1.8601 1.9537 3.0121 2.8710 3.0997 4.2380 1.6744 1.7990 2.8750−1 0.1 0.7741 1.0879 2.7519 0.7912 1.1034 2.7643 0.9254 1.2348 2.8865 1.9365 2.3811 4.1124 0.7398 1.0801 2.7495

0 1 0.5468 1.0482 2.7518 0.5638 1.0637 2.7643 0.6982 1.1951 2.8865 1.7092 2.3413 4.1124 0.5125 1.0404 2.74941 10 0.5342 1.0482 2.7518 0.5513 1.0637 2.7643 0.6856 1.1951 2.8865 1.6967 2.3413 4.1124 0.4999 1.0404 2.74942 100 0.5342 1.0482 2.7518 0.5513 1.0637 2.7643 0.6856 1.1951 2.8865 1.6967 2.3413 4.1124 0.4999 1.0404 2.74943 1000 0.5342 1.0482 2.7518 0.5513 1.0637 2.7643 0.6856 1.1951 2.8865 1.6967 2.3413 4.1124 0.4999 1.0404 2.7494

Table 6CComparison of non-steady-state current IT for the EC′ reactions for various microelectrodes

log t t Hemi-oblate

b/a = 0.01 b/a = 0.5 b/a = 0.9 b/a = 0.99

K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100

−3 0.001 4.8186 4.8873 5.2912 4.7748 4.8140 5.2082 4.7352 4.7637 5.1579 4.7263 4.7560 5.1503−2 0.01 1.7782 1.9314 3.0176 1.7347 1.8586 2.9348 1.6951 1.8082 2.8845 1.6862 1.8001 2.8767−1 0.1 0.8435 1.2124 2.8919 0.8001 1.1397 2.8092 0.7605 1.0893 2.7589 0.7516 1.0812 2.7511

0 1 0.6161 1.1727 2.8919 0.5727 1.1000 2.8092 0.5332 1.0495 2.7589 0.5243 1.0415 2.75111 10 0.6035 1.1727 2.8919 0.5602 1.1000 2.8092 0.5206 1.0495 2.7589 0.5117 1.0415 2.75112 100 0.6035 1.1727 2.8919 0.5602 1.1000 2.8092 0.5206 1.0495 2.7589 0.5117 1.0415 2.75113 1000 0.6035 1.1727 2.8919 0.5602 1.1000 2.8092 0.5206 1.0495 2.7589 0.5117 1.0415 2.7511

Table 6DComparison of non-steady-state current IT for the EC′ reactions for various microelectrodes

log t t Hemi-prolate Band

b/a = 1.5 b/a = 2 b/a = 10 K = 1 K = 10 K = 100

K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100

−3 0.001 4.6812 4.7221 5.1175 4.6595 4.7028 5.0997 4.6177 4.6675 5.0613 5.2489 5.2895 5.6846−2 0.01 1.6409 1.7663 2.8437 1.6193 1.7472 2.8262 1.5773 1.7118 2.7878 2.2098 2.3347 3.4119−1 0.1 0.7061 1.0474 2.7182 0.6847 1.0283 2.7006 0.6427 0.9929 2.6622 1.2753 1.6160 3.2862

0 1 0.4788 1.0076 2.7181 0.4573 0.9886 2.7006 0.4154 0.9532 2.6622 1.0481 1.5763 3.28621 10 0.4662 1.0076 2.7181 0.4448 0.9886 2.7006 0.4028 0.9532 2.6622 1.0355 1.5763 3.28622 100 0.4662 1.0076 2.7181 0.4448 0.9886 2.7006 0.4028 0.9532 2.6622 1.0355 1.5763 3.28623 1000 0.4662 1.0076 2.7181 0.4448 0.9886 2.7006 0.4028 0.9532 2.6622 1.0355 1.5763 3.2862

Table 6EComparison of non-steady-state current IT for the EC′ reactions for various microelectrodes

log t t Circular ring

b/a = 0 b/a = 0.25 b/a = 0.5 b/a = 0.9

K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100 K = 1 K = 10 K = 100

−3 0.001 4.7489 4.7625 5.1493 4.7977 4.8379 5.2323 4.8736 4.9198 5.3306 6.2273 6.7072 7.3820−2 0.01 1.7087 1.8069 2.8758 1.7578 1.8825 2.9590 1.8333 1.9640 3.0571 3.1863 3.7509 5.1081−1 0.1 0.7741 1.0879 2.7502 0.8232 1.1637 2.8334 0.8986 1.2451 2.9314 2.2514 3.0318 4.9824

0 1 0.5468 1.0482 2.7502 0.5959 1.1239 2.8334 0.6713 1.2054 2.9314 2.0240 2.9920 4.98241 10 0.5342 1.0482 2.7502 0.5833 1.1239 2.8334 0.6587 1.2054 2.9314 2.0115 2.9920 4.98242 100 0.5342 1.0482 2.7502 0.5833 1.1239 2.8334 0.6587 1.2054 2.9314 2.0115 2.9920 4.98243 1000 0.5342 1.0482 2.7502 0.5833 1.1239 2.8334 0.6587 1.2054 2.9314 2.0115 2.9920 4.9824

R. Senthamarai, L. Rajendran / Electroc

FK

sIiivciaca[c

etvF

4

dtvidtmfactkhaoierod

A

afi(tSHaolD

ig. 3. (a) Plot of the function K1/2g(K,t) versus log t for various values K (K = 1,= 10 and K = 100). (b) Plot of Kt versus g(K,t).

The non-steady-state currents IT for hemicylinder, hemi-phere and circular disc electrodes are equal (refer Table 6A).n all electrodes, the value of the current increases when Kncreases. In elliptical ring electrodes, the value of the currentncreases when a/r increases (or perimeter decreases) for allalues of K. In elliptical disc electrodes, the non-steady-stateurrent IT increases when b/a decreases (or eccentricityncreases or perimeter decreases). Elliptical disc electrodes aredvantageous under potentiodynamic conditions compared toomparable circular disc electrodes. The larger eccentricity of

n elliptical disc electrode, the greater is its analytical sensitivity24]. In hemi-oblate and hemi-prolate electrodes, the value of theurrent decreases when b/a increases, whereas in circular ring

a

A

Symbol Meaning

a Characteristic length associated with the geometry under consRadius for disc electrodeLength of semi-major axis for elliptical ring electrodeOuter radius for circular ring electrodeLength of band electrodeRadius of hemisphere electrodeLength of hemicylinder electrodeLength of equal horizontal two semi-axes for hemi-oblate aprolate electrodes

himica Acta 53 (2008) 3566–3578 3577

lectrodes, the current increases when b/a increases (or perime-er increases). When Kt ≥ 10, the current reaches its steady-statealue (constant value) for all electrodes (since g(K,t) = 0 (Referig. 3(b))).

. Conclusions

Closed form expressions for steady and non-steady-stateiffusion-limited currents for the EC′ reactions at all microelec-rodes are given. The steady-state current for small and largealues of K, for all microelectrodes are given in Table 2. Numer-cal tables and figures are given that allow for comparisons ofiffusion currents at all microelectrodes. Of the nine microelec-rode geometries investigated, the elliptical ring and circular ring

icroelectrodes appear to be the superior geometries for per-orming electroanalytical measurements for both steady-statend non-steady-state EC′ reactions. Hemicylinder, hemisphere,ircular disc, hemi-prolate and band electrodes are confirmedo be the worst geometries for analytical and electrochemicalinetic studies. Thus, elliptical ring and circular ring electrodesave greater analytical sensitivity than all other microelectrodes,nd hence these electrodes are recommended for investigationsf electrode kinetics. The extension of this procedure (compar-son of results) to other reaction mechanisms (ECE and DISP,tc.) and other geometries, such as rotating disc (RDE) and theotating ring-disc (RRDE) electrodes, etc., apart from the studyf scanning electrochemical microscope (SECM) response forifferent tip shapes seems possible.

cknowledgements

The author is very thankful to the referees for their valu-ble suggestions. The authors are pleased to acknowledge thenancial support of the Department of Science and TechnologyDST), Government of India to L. Rajendran, which has allowedhis work to be undertaken. The author also thanks Prof. A .K.hukla, Director, CECRI, Karaikudi, India and MR. Valliappan,eadmaster, SMSV Higher Secondary School for their encour-

gement. Useful discussions with Prof. P. N. Bartlett, Universityf Southampton, UK and Dr. C. A. Basha, Dy. Director, Pol-ution Control, CECRI, Karaikudi and Prof. N. Palaniappan,epartment of Mathematics, Alagappa University, Karaikudi

re gratefully acknowledged.

ppendix A. Nomenclature

Usual dimension Eq./Table involved

ideration cm (3)–(8)cm (15), Table 1 1st fig)cm (11), Table 1 (2nd fig)cm (12), Table 1 (3rd fig)

cm Table 1 (4th fig)cm Table 1 (5th fig)cm Table 1 (6th fig)

nd hemi- cm Table 1 (8th and 9th fig)

3 ctrochimica Acta 53 (2008) 3566–3578

A

S Usual dimension Eq./Table involved

b cm Table 1 (2nd fig)cm Table 1 (3rd fig)cm Table 1 (4th fig)cm Table 1 (6th fig)

ectrodes cm Table 1 (7th and 8th fig)cm Table 1 (9th fig)

c none (2)C mol cm−3 (3)D cm2 s1 (3)F C mol−1 (3)K none (4)–(8), (21)k s−1 (1)l none Table 1 (4th column), (22), (24), (32), (34) and (36)P cm Table 1 (3rd column)r none (2), (20)r cm (23)z none (2), (20)I none (8)I none (21)t none (21), (22)γ none (20)i A (21)I none (8)I none (21)a none (4)bAu

R

[[[

[

[[[[[[[

[[[[[25] O.K. Dudko, A. Szabo, J. Ketter, R.M. Wightman, J. Electroanal. Chem.

578 R. Senthamarai, L. Rajendran / Ele

ppendix A (Continued )

ymbol Meaning

Length of semi-minor axis of elliptical disc electrodeLength of inner radius of circular ring electrodeWidth of band electrodeRadius of hemicylinderLength of third semi-axis for hemi-oblate and hemi-prolate elLength of inner semi-minor axis of elliptical ring electrode

Dimensionless concentration

0 Bulk concentrationDiffusion coefficientFaraday constant 96487Dimensionless reaction rateReaction rate

0/a Steady-state diffusion-limited currentPerimeter of the electrodeDimensionless radical-co-ordinateWidth of the elliptical ringDimensionless axis co-ordinate

ss(K) Dimensionless steady-state current for EC′ reaction

T Dimensionless non-steady-state current for EC′ reactionDimensionless timeEuler’s constant (γ = 0.5772156. . .)

(t) Current density

ss Dimensionless steady-state current

T Dimensionless non-steady-state current

i coefficients in series expansion

i Coefficients in series expansion

1, B1, C1, D1 Coefficients of series, θ Integration dummy variable

eferences

[1] K. Stulik, C. Amatore, K. Holub, V. Marecek, W. Kutner, Pure Appl. Chem.72 (2000) 1483.

[2] M. Fleischmann, S. Pons, D. Rolison, P.P. Schmidt (Eds.), Ultramicroelec-trodes, Data Tech Systems, Morganton, NC, 1987.

[3] I. Rubinstein, in: C. Amatore (Ed.), Physical Electrochemistry, MarcelDekker, New York, 1995, p. 131.

[4] J.R. Delmastro, D.E. Smith, J. Phys. Chem. 71 (1967) 2138.[5] M. Fleischmann, F. Lasserre, J. Robinson, D. Swan, J. Electroanal. Chem.

177 (1984) 97.[6] M.A. Dayton, A.G. Ewing, R.M. Wightman, Anal. Chem. 52 (1980) 2392.[7] G. Denuault, M. Fleischmann, D. Pletcher, O.R. Tutty, J. Electroanal.

Chem. 280 (1990) 243.

[8] G. Denuault, D. Pletcher, J. Electroanal. Chem. 305 (1991) 131.[9] K.B. Oldham, J. Electroanal. Chem. 313 (1991) 3.10] M.A. Bender, H.A. Stone, J. Electroanal. Chem. 351 (1993) 29.11] Q. Zhuang, H. Chen, J. Electroanal. Chem. 346 (1993) 29.12] C.G. Phillips, J. Electroanal. Chem. 296 (1990) 255.

[[[

none (5)none (7)none (26), (30), (37), (23)

13] J. Galceran, S.L. Taylor, P.N. Bartlett, J. Electroanal. Chem. 476 (1999)132.

14] L. Rajendran, M.V. Sangaranarayanan, J. Phys. Chem. B 103 (1999) 1518.15] L.K. Bieniasz, D. Britz, Polish J. Chem. 78 (2004) 1195.16] D. Britz, Digital Simulation in Electrochemistry, Springer, 2005.17] I. Lavagnini, P. Pastore, F. Mango, J. Electroanal. Chem. 358 (1993) 193.18] L.K. Bieniasz, J. Electroanal. Chem. 360 (1993) 119.19] J.A. Alden, R.G. Compton, J. Phys. Chem. B 101 (1997) 9606.20] K. Hariman, D.J. Gavaghan, P. Houston, E. Suli, Electrochem. Commun.

2 (2000) 163.21] S.C.B. Abercrombie, G. Denuault, Electrochem. Commun. 5 (2003) 647.22] K. Gillow, D.J. Gavaghan, E. Suli, J. Electroanal. Chem. 587 (2006) 1.23] D.K. Cope, D.E. Tallman, J. Electroanal. Chem. 285 (1990) 85.24] S. Bruckenstein, J. Janiszewska, J. Electroanal. Chem. 538 (2002) 3.

586 (2006) 18.26] J. Galceran, S.L. Taylor, P.N. Bartlett, J. Electroanal. Chem. 466 (1999) 15.27] L. Rajendran, J. Theor. Comput. Chem. 5 (1) (2006) 11.28] L. Rajendran, Electrochem. Commun. 2 (2000) 679.