Influence of Uncompensated Solution Resistance on Diffusion Limited Chronocoulometric Response at Rough Electrode

Electrochimica Acta 180 (2015) 208–217

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Electrochimica Acta journal homepage: www.elsevier.com/locate/electacta

Influence of Uncompensated Solution Resistance on Diffusion Limited Chronocoulometric Response at Rough Electrode Shruti Srivastav, Rama Kant ∗ Complex Systems Group, Department of Chemistry, University of Delhi, Delhi-110007, India

a r t i c l e

i n f o

Article history: Received 22 July 2015 Received in revised form 10 August 2015 Accepted 10 August 2015 Available online 18 August 2015 Keywords: Reversible charge transfer Ohmic effects Double layer correction finite fractal electrode charge transient

a b s t r a c t Theory is developed for the charge transient under ohmic effect in reversible charge transfer process at disordered electrodes. Preliminary experimental results are also analyzed using this theory. Our results show that the deviation from Anson response is stemming from surface disorder and solution resistance. Detailed theoretical results are obtained and analyzed for fractal roughness with self-affine statistical properties. Expressions for concentration, charge density, and total charge have operator structure in Fourier transformed (deterministic) surface profile function. An elegant mathematical formula between the average charge transient and surface structure factor of roughness is obtained. Initial response is attributed to resistive effects of the electrolyte. The intermediate response is attributed to the interplay between various characteristics of the power spectrum and “ohmic equilibration layer thickness”. The theory marks that ohmic effects penetrate farther in time so much so to merge with time regimes which are outside the diffusion regimes giving nonideal behavior at electrode/electrolyte interface. Our results help with the quantitative understanding of generalized Anson response for Nernstian system affected by the uncompensated resistance at rough electrode/electrolyte interface. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Chronocoulometry has been used for kinetic studies [1–4], for rapid capacitance measurements at solid electrodes [5], studies on mass transport and enzyme kinetics [6], electrode mechanisms [7,8], conversion of quasi-reversible waves into reversible ones [9], heterogeneous kinetics of adsorption of organics with reversible charge transfer [10]. Chronocoulometry describes the amount of a species produced following a potential step as a function of time. The various contributing components to the charge such as double-layer, adsorption and diffusion can be separated easily in chronocoulometric experiments. This separation of components of the charge provides more quantitative information and the possible effects of the electrode surface configuration on the responses. It offers experimental advantages: (a) The measured signal often grows with time; hence the later parts of the transient, which are most accessible experimentally and are least distorted by nonideal potential rise, offer better signal to noise ratios. (b) The act of integration smoothens random noise on the current transients; hence the chronocoulometric records are inherently cleaner, (c)

∗ Corresponding author. E-mail address: [email protected] (R. Kant). http://dx.doi.org/10.1016/j.electacta.2015.08.035 0013-4686/© 2015 Elsevier Ltd. All rights reserved.

Contributions to the charge Q(t) from double layer charging and electrode reactions of adsorbed species can be distinguished from those due to diffusing species [11]. Chronocoulometry involves a rapid step in potential at a working electrode in an electrochemical cell containing the solution of interest. If one applies a sufficiently large step to the electrode, so that for time t > 0, the difference in surface concentration of the species undergoing reaction will be equal to 0, then the charge (QA (t)) behavior will follow the Anson behavior (integrated Cottrell response) [12–14]. √ 2nFA0 cO0 Dt QA (t) = (1) √  where n is the number of electron(s) involved in the reaction, F is the Faraday constant, cO0 is the concentration of the oxidized species, A0 is the area of the electrode and D is the diffusion coef√ ficient. Eq. (1) can be modified to QA (t) =

2nFA0 Cs Dt √ , 

with a more

general Nernstian electrode boundary constraint in the presence of both oxidized (cO0 ) and reduced (cR0 ) species, viz. ıcO (z = 0, t) = 0



−Cs = −(cO0 − cR0 )/(1 + ) [15], where  = e−nf (E−E ) , E 0 is the formal potential and E is the applied potential. If, however, the surface concentration does not go to zero instantly, in presence of uncompensated resistance, QA at t will be less than the value calculated from equation 1. One may thus consider these coulombs “lost”. If, as is the case, the actual potential rises slowly, the electron transfer

S. Srivastav, R. Kant / Electrochimica Acta 180 (2015) 208–217

List of symbols A ci0 ci Cs ıci D D √H Dt E  E0 F F J j(t) K K 

Geometrical or projected area of the surface Bulk concentration of the species, i=O, R Concentration of the species, i=O, R c 0 −c 0  O

R

1+

Change in concentration of species, i=O, R Diffusion coefficient Fractal dimension Diffusion length Applied potential Formal potential Faraday’s constant F/RT Flux Current density [Kx2 + Ky2 ]

1/2

p q

Vector (Kx , Ky ) Standard deviation of the surface height fluctuation Distance between the WE (reference plane z = 0) and the RE “diffusion-ohmic” coupling length Lower length scale cut-off Upper length scale cut-off Topothesy length Moments of power spectrum Number of electron transferred in redox reaction Normal vector drawn in the outward direction Average charge Charge at smooth electrode Charge in presence of uncompensated resistance at smooth electrode Laplace transform variable  p/D

q

2 [q2 + K ]

h e L  L  m2k n nˆ Q QA (t) QA (t)

1/2 2 1/2

 q, [q2 + |K − K | ] r Vector (x, y, z) r Vector (x, y) R Gas constant Area specific solution resistance R R* Roughness factor RF (t) Dynamic roughness factor for finite fractal Dynamic roughness Rm2 R(u) Dynamic roughness R† (t) Generalized time dependent roughness factor dS0 dx dy dS Area element (ˇdxdy) Inner crossover time ti to Outer crossover time W (|r − r |) Normalized correlation function z Coordinate representing distance away from electrode  0  e−nf (E−E ) √  −1 Dirac delta function in wave-vector K  ı(K  ) ı DH − 5/2 Resistivity of the electrolyte e ∇ Gradient (.) Incomplete Gamma function

Proportionality constant of power spectrum or strength of fractality

(r ) ˆ (K )

∇ ∂ ∂n

209

Arbitrary surface profile Fourier transform of the arbitrary surface profile Charge density ˆi∂x + ˆj∂y Partial derivative ˆ ∇ n.

rate will be determined by the actual potential during the rise, and thus charge lesser than the theoretical charge will have passed at any time, “t ; this will alter the concentration profile at the electrode surface. The impact of uncompensated resistance can be seen in most electrochemical data, regardless of the technique involved. Theoretically, not many avenues have been explored for the problem of ohmic contribution affecting the transient measurements at electrodes. Although this effect will be present in any potentiostatic method, chronocoulometry is most sensitive to this disturbance, since the integral of the current contains the history of the reaction from the time of initiation, which includes sensitivity to ohmic contribution too. √ According to equation 1, plots of QA (t) versus t should be linear with intercepts of q, and slopes proportional to the concentration of the reactant. However, in presence of uncompensated resistance the charge transient at the smooth electrode becomes mathematically isomorphic to a quasi-reversible process. It can thus be represented as [16],  2 √  √ L F( Dt/L ) L (2) +1− √ QA (t) = QA (t) 2Dt 2 Dt √ F(x) is given by x exp(x2 )erfc(x). Ohmic contribution is contained in the phenomenological length scale L . The phenomenological diffusion-resistance length (L ), is directly related to the experimental conditions such as resistivity of the solution (e ); distance between the working and reference electrode ( e ) and size of potential step contained in  [17]. L =

n2 F 2 D (cR0 + cO0 ) e e



RT

(1 + )

2

(3)

Refer to [17,18] for derivational details.  has the potential 0 dependence and is given as e−nf (E−E ) . Equation 2 is similar to the one obtained by integrating the chronoamperometric response for ohmic losses [17,19,20]. This is however true for a planar electrode. Kant and co-workers have established a series of relationship between the dynamic observables like current [21–26,28,27], admittance [29–32,34,33], absorbance [35,36], charge [15] and voltammetry [37,38] with the disordered topography and morphology of the electrodes via power spectrum of rough interface [39–42]. Recently, we adressed the problem of ohmic influence on potentiostatic current transients at randomly rough electrodes [17,28], double potentiostatic chronoamperometry [27] and impedance measurements [29]. We address the same phenomenon as the next relevant question during the chronocoulometric measurements. The results provide surprising insight into the question addressed and the techniques themselves. Osteryoung and co-workers has shown advantage of this transient technique over the other [1]. However, in presence of roughness and ohmic effects simultaneously, one might want to be sure of the methodology applied to one’s system. The important thing to realise is the length and time scale of the phenomenon one might be interested in [43]. The choice of a particular technique seems appropriate when it identifies the phenomenon with the given parameters and provides credible kinetic information reproducible over number of measurements.

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2. Experimental 2.1. Electrode Pretreatment Platinum wires of diameter 0.5 mm and 99.5 % purity purchased from “Arora Matthey and Company” was used as working electrode for electrochemical measurements and were labeled as Pt-1,Pt-2 and Pt-3. The electrodes were were mechanically roughened using emery paper of grit 300 for Pt-1,Pt-2 whereas Pt-3 was roughened using emery paper of grit 1500. After roughening, electrodes were kept in ethanol for 48 hours. Each platinum rough electrode was sonicated in ethanol for 10 minutes followed by sonication in tripled distilled water for 10 minutes. After that these platinum working electrodes were boiled in nitric acid for 10 minutes. Then these platinum electrodes were rinsed with triple distilled water. The electrodes were then electrochemically cleaned and activated in 0.5 M H2 SO4 between potential -0.2 V to 1.2 V Vs SCE at scan rate 100 mV/s for 100 cycles and is used as such for the electrochemical experiments. 2.2. Electrolyte Preparation Electrolyte solutions were prepared using “analytical grade reagents” in triply distilled water. All solutions were degassed with purified nitrogen. Electrochemical cleaning was done in 0.5 M Merck SuprapureTM sulphuric acid and electrochemical measurements were taken in mixture of 15 mM potassium ferrocyanide and 15 mM potassium ferricyanide prepared in the 3M NaNO3 supporting electrolyte.

observation with very high resolution and depth of field at very high scanning speeds. Before SEM imaging electrodes were stored in nitrogen environment. SEM images were recorded up to 100,000 magnification. Recorded images were coded in 1024 × 1024 pixels. 3. Formulation Fick’s first law relates the flux (J) to the gradient (∇ ) of concentration difference (ıc(r, t)) through a proportionality constant as diffusion coefficient (D): J = −D∇ ıc(r, t)

where ıc(r, t) is concentration difference defined as (ci (r, t) − ci0 ); ci0 is the bulk concentration (i = O, R); r is three dimensional position vector and t being time. Initial and bulk boundary conditions are ıci (r, t = 0) = ıci (r , z → ∞, t) = 0. For a single step charge transfer process, O + ne−  R, at a rough electrode (z = (r )) cathodic current density (j) in normal direction is given by j(z = (r ), t) = nFD ∂n ıc(r, t). The charge density is given by time integrating the above equation as



(t) =

t

 j(z = (r ), t  )) dt

(5)

0

and the total charge upto time t over the whole surface is given by the observable, i.e. average charge (Q), at the arbitrary interface is obtained by integrating the charge density ( ) over the whole surface (z = (r )).



Q (z = (r ), t) =

dx dy ˇ (z = (r ), t)

(6)

S0

2.3. Electrochemical Experiments Experiments were carried out using Autolab III. Prior to each experiment nitrogen purging had been done for about 15 minutes. The electrochemical experiments were conducted potentiostatically in a typical three electrode cell with a platinum sheet counter electrode and 3M SCE as reference electrode (E 0 = 0.242V vs standard hydrogen electrode (SHE)) at room temperature. The geometrical area of the working electrodes Pt-1, Pt-2 and Pt-3 exposed to electrolyte solution was 0.13, 0.11 and 0.07 cm2 , respectively. Pt1, Pt-2 and Pt-3 platinum electrodes were examine in mixture of 15 mM potassium ferrocyanide and 15 mM ferricyanide prepared in electrolyte medium in 3M NaNO3 . After this chronocoulometric measurements were carried out at various time durations at the potential bias of 0.335 V, 0.324V and 0.320 V vs SCE for Pt-1, Pt-2 and Pt-3 platinum electrode. These chronocoulograms were then superimposed to get the complete recording ranging from ms to tens of second. No iR compensation was applied while carrying out these measurements.

(4)

∂n = nˆ · ∇ signifies the outward drawn normal derivative at the electrode surface (z = (r )), nˆ = ˇ1 (−∇  (r ), 1), ∇  = (∂/∂x, ∂/∂y),



ˇ= 1 + (∇  )2 and ıc(r, t) refers to the difference in concentration at a given point and initial or bulk concentrations. The diffusive transfer at a random electrode for a Nernstian charge transfer reaction satisfies the diffusion equation for the concentration difference as

∂ ıci = D∇ 2 ıci ∂t The flux balance condition, initial and bulk boundary constraints along with simplifying assumption of DO = DR = D, the concentrations for O and R are related through ıcO (r, t) = −ıcR (r, t) = ıc(r, t). Hence, at the surface, this identity must be satisfied whatever be the surface profile or time. The surface constraint under reversible (Nernstian) charge transfer in the presence of ohmic contributions has the following form: cO0 − cR0  enfjR

2.4. Area Measurements

ıcO = −

Roughness factor (R* ) was calculated from Pt oxide region of cyclic voltammogram (CV) recorded in 0.5M H2 SO4 CV between -0.2V to 1.2V. The oxide reduction peaks were integrated using a charge density associated with reduction of oxide layer to be 420 C/cm2 at 25◦ C. Microscopic area calculated from 0.5M H2 SO4 cyclic voltammograms of Pt-1, Pt-2 and Pt-3 are 0.347, 0.345 and 0.252 cm2 , respectively.

R is the solution resistance, E 0 is the formal potential. In absence of solution resistance contribution, viz, R → 0, boundary constraint simplifies to ıcO (z = (r ), t) = −Cs = −(cO0 − cR0 )/(1 + ) [44]. Linearizing equation 7 and rearranging the resultant equation we obtain electrode surface boundary constraint [17,21] (Appendix A),

2.5. SEM Recording

To solve the diffusion equation with the given boundary constants, the concentration field is to be defined at the electrode/electrolyte interface. This concentration field is the function of boundary profile, therefore we can say that roughness is sensed by the concentration field in an electrochemical measurements. To write this concentration profile we use the perturbation of

SEM (scanning electron microscopy) characterization of electrodes were carried out using Quanta 200F SEM-FEI instrument used in secondary electron imaging or morphological mode. This mode provides an excellent topographic image of the surface under

1 +  enfjR

L ∂n ıcO (z = (r ), t) − ıcO (z = (r ), t) = Cs

(7)

(8)

S. Srivastav, R. Kant / Electrochimica Acta 180 (2015) 208–217

boundary profile. This is done by using Taylor expansion of surface boundary condition about a mean reference plane (z=0).

where braces  notation for   inverse Fourier transform is: f (K  ); K  → r ≡ (1/(2)2 ) d2 K exp(iK  .r )f (K  ). ˆ0 , ˆ1 are operators same as in equation 11 and ˆ2 is

3.1. Perturbation Solution in Surface Profile ˆ2

≡

The perturbative solution of concentration profile in LaplaceFourier domain (upto second order term is) [17,21,44]

 Cˆ0

≡

Cˆ2

≡

e−qz 1 + qL

 ; Cˆ1 ≡





e−qz (2)



d2 K 

2

−

q e−q z 1 + q L

 ;

qq, q2 + q L + 1 2(q L + 1)

K  · (K  − K  ) L K  · (K  − K  ) L q q − + (L q + 1) q, L + 1 2(q L + 1) q L + 1

q = (p/D)

where

1/2



2 1/2

; q = q2 + K



(10) ; q, =

1/2 K  − K  |2 ) .

(q2 +

| The expression for charge density for the arbitrary surface profile upto second order perturbation term is given by



1

d2 K 

2

(2) p −

−Cs ıc(K  , p) = [Cˆ0 (2)2 ı(K  ) + Cˆ1 ˆ (K  ) + Cˆ2 ˆ (K  ) ˆ (K  − K  )] p (9) ˆ (K  ) is Fourier transform of the surface profile ((r )). The operators Cˆ0 , Cˆ1 and Cˆ2 are defined below

211

−

 1 2

q q L + 1

−

q q L + 1



+

K  · (K  − K  ) q 1 − q, L + 1 L q(q L + 1) q, L + 1 | K  − K  |2 q(q,

L + 1)

−

q q, q (q L + 1)

(15)

q

K  · (K  − K  )

2q (q L + 1)

q L + 1



where the integral is performed over the surface S0 (i.e. z = 0 plane). It is necessary to mention here that ˆ2 and ˆ2 are similar except for numerical coefficient of the last term. These operators consist of diffusion characteristic (q) and ohmic loss characteristic (L ) while equations 9, 12 and 14 emphasize the roughness profile of the interface to highlight the mathematics involved in an amenable manner. Explicitly, the first operator defines the effect of diffusion and ohmic phenomenon for a smooth surface whereas the first order and the second order terms take into account the surface modulation or the fluctuations around the reference plane. It’s worth realizing that these expressions are valid for any arbitrary surface profile. These equations, i.e. 9, 11 and 14 are useful in predicting local concentration, current density profile and total current. This includes known deterministic profiles like sine, conical, triangular profiles etc. Surface profiles which can be represented in terms of cosine or sine functions yield easy to write final expressions for concentration and current as their Fourier representation have only Dirac delta functions. 3.2. Random Surface Electrode

(K  , p)

nFD Cs [ ˆ0 (2)2 ı(K  ) + ˆ1 ˆ (K  ) + ˆ2 ˆ (K  ) ˆ (K  − K  )]

=

(11) The operators ˆ0 , ˆ1 and ˆ 2 are defined below

ˆ0 ≡



1 q p (q L + 1)



 ; ˆ1 ≡

1 qp



q q − q L + 1 q L + 1

(r ) = 0



ˆ2

≡



1 p(2) + −

1 2

d2 K 

2

q q, q(q L + 1)

−

| K  − K  |2 q(q, L + 1)

q q − q L + 1 q L + 1

K  · (K  − K  ) 1 2 L q (q L + 1) q L + 1



(13) This charge density expression presents two structures in the equation; first term is the planar response to the ohmic contribution and subsequent terms represents the first and second perturbation corrections in surface profile. The expression for the total charge is given by taking the surface integral of the charge density and retaining terms upto second order in surface profile (ˆ (K  )).



Q (p, r )

=

d2 r

nFD Cs So



(r )(r ) = h2 W (|r − r |)

(17)

where  .  denotes ensemble average over various possible surface configurations. The normalized correlation function (W (|r − r |))

K  · (K  − K  ) q 1 − q, L +1 L q(q L + 1) q, L + 1 +

(16)

and the two point correlation function is (12)



For a random surface electrode, the surface profile is characterized by a centered Gaussian field with the statistical properties - the ensemble averaged value of the random surface and the two point correlation function [45]. The ensemble averaged value of the random surface is

ˆ0 + ˆ1 ˆ (K  ) + ˆ2 ˆ (K  )ˆ (K  − K  ); K  → r

 (14)

gives the measure of rapidity of variation of surface and h2 is mean square height fluctuations or width of interface. The correlation function vanishes on increasing the relative distance |r − r |. This correlation function contains information about the surface morphological characteristics as area, mean square height, slope, curvature, correlation length. Averages in the Fourier plane can be written as ˆ (K  ) = 0

(18)

ˆ (K  )ˆ (K  ) = (2)2 ı(K  + K  )|ˆ (K  )|2 

(19)

It is to be brought to notice that for a random surface it is not the local roughness quantities but rather average roughness measures represented by ensemble average over surface configuration space, which are quantities of interest. The average charge, in Laplace transform domain is, Q (p)

=

nFCs A0 p L





L 1 + q(qL + 1) (2)2



d2 K |ˆ (K  )|2 

(K L )2 1 1 − + qL + 1 q L + 1 2(qL + 1)2



(20)

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The above equation is the time integration of the average current in the Laplace domain related by 1/p. The inverse Laplace transform of the equation 20 gives the average charge transient equation. The average charge transient is given by,

input for equation 21. This power spectrum of a band limited fractal surface is given by [42]

ˆ |ˆ (K  )|2  Q (t) = QA (t) + Q

where DH is the Hausdorff dimension,  is the topothesy length,  and L are lower and upper length scale cut-offs of fractality, respectively. The moments of above power spectrum are related to various morphological features of rough surface, viz root mean square (rms) width (m0 ), rms gradient (m2 ), rms curvature (m4 ), etc. The general moments of power spectrum (i.e 2k-th moments, m2k ) are easily obtained for the above-mentioned power spectrum and amount to important morphological characteristic of surface roughness. m2k is given by the expression,

(21)

where, QA (t) is the integrated Cottrell response at a smooth elecˆ operator gives the trode with ohmic losses given by equation 2. Q roughness response for any arbitrarily rough electrode while operating upon the power spectrum. This relates the two important phenomenological length scales dynamically, predicting the various time regimes in the response. The roughness contribution to chronocoulometric response is

  |2DH −7 , H −3 |K |ˆ (K  )|2  = 2D 

1/L ≤ |K  | ≤ 1/

(25)

2(ı+1)



ˆ |ˆ (K  )|2  Q

t

=

dt 0



IC (t) (2)2

√ (1 − F +

Dt L

√

⎡ d2 K |ˆ (K  )|2 

 )

1 − DtK2

⎢ ⎢ ⎣ −e

Dt L

K2 F

−K2 Dt

 (−2(ı+k) − L−2(ı+k) ) 4(ı + k)

⎤ ⎥ ⎥ ⎦

Q (t)

= QA (t) +

(22) −

IC (t) is the Cottrell response at a smooth electrode without ohmic ˆ is defined as losses. The operator Q

ˆ ≡ Q

QA (t) (2)

2



 d2 K

ˆ 0 (t) + K2 M ˆ 2 (t)+ M

√

−

ˆ 0 (t) M

 √ 2 2L Dt(1 − (K L ) )

≡

1 2

≡

−K 2 Dt 

√ √ √ F( Dt/L ) + DtK erf(K Dt)



2

2Dt(1 − (K L ) )

(23)

 √

√ ˆ 2 (t) M

e

 √ F( Dt/L )  − ; √ Dt L Dt

 √ 2 √ L 1 L F( Dt/L ) − F( Dt/L ) − + √ 4Dt 2 2 4 Dt

(26)

ı = DH − 5/2 is the deviation from Brownian fractal dimension. Average charge at an isotropic self-affine fractal surface electrode is obtained using equation 21 and the power spectrum in equation 25. The average charge transient for random fractal electrodes is given by,

2

2 L

m2k =

(24)

Equation 21 is the generalization of Anson equation for a random electrode roughness under the influence of ohmic effects. Under limit L → 0, equation 21 reduces to Anson equation for a random electrode roughness [15]. 4. Result and Discussion 4.1. Random Fractal Electrodes In order to capture the complexity arising from the irregular interfaces (i.e. rough, porous and partially active interfaces) one often uses the concept of fractals [22–24,34,46,47]. The fractal irregularities are usually understood in particular in terms of selfsimilar [48,49] or in general as self-affine [25,26,31,39–41,49] fractals. These fractal boundaries exhibit statistical self resemblance over all length scales and can be described using power law power spectrum. For realistic electrode surface we use band limited power spectrum to circumvent the mathematical difficulty of nondifferentiability and non-stationarity. These band limited fractals have a lower and an upper length scale cut-off. The quantity we seek for an isotropic fractal surface is the power spectrum as the

QA (t) 2

(2) √ 



 dK 

e

−K 2 Dt 

√ √ √ F( Dt/L ) + DtK erf(K Dt) 2

2Dt(1 − (K L ) )



2DH −3

ˆ 2 (t)  ˆ 0 (t) + K2 M +M √ 2 2L Dt(1 − (K L ) )

|K  |2DH −6

(27)

This equation includes the fractal features dependent powerlaw as well as contribution from ohmic efffects at the electrode. The resulting charge-time behavior is an interplay between the interfacial potential, the resistance of the medium and the fractal roughness features of the interface. Limiting behavior for both phenomena can be achieved through this equation. In the absence of surface roughness, the smooth electrode response is observed with ohmic contributions and in absence of ohmic component the response is the generalized Anson equation for the rough electrode [15]. To elucidate the effect of ohmic losses on the charge transient, we consider the changes in the chronocoulograms with respect to the varying parameters for roughness and uncompensated resistance of the electrolyte. The resulting transient response show the variation of these parameters to provide a better understanding of both roughness and ohmic factors present together at the interface. The curves are generated using numerical evaluation for equation 27 using Mathematica software. Fig. 1(a) elucidates the effect of solution resistance on the chronocoulogram. We find that the chronocoulograms are affected significantly at short time by the ohmic loss thus delaying the onset of diffusion control. As the resistance of the solution increases the charge transient decreases. Higher the resistance of the electrolytic solution less prominent is the crossover between resistance controlled region and diffusion control region in the curves. One can segregate the effect on an approximate basis of the ohmic and microscopic roughness parameters, their interplay and then dependence on the gross macroscopic parameter of roughness factor. These plots then follow a power-law behavior in the intermediate time domain as the depletion layer near electrode starts forming. One can see the effect on dynamic response from morH −3 ) in phological features like DH ,  and strength of fractality(2D  the intermediate region. This behavior then merges with the diffusion controlled response. The response however never goes to planar response rather becomes parallel with the separation of a constant factor, viz. macroscopic observable quantity roughness factor. For a given system with some uncompensated resistance and

S. Srivastav, R. Kant / Electrochimica Acta 180 (2015) 208–217

213

Fig. 1. (a) Effect of electrolyte’s resistance (R ) on the chronocoulograms. R is varied as 50, 75, 100 cm2 , DH = 2.46, ␶ = 2.50 m and  = 40 nm. Dotted line is the planar Anson curve. (b) Effect of DH on the charge transient. DH is varied as 2.1, 2.2, 2.25, 2.3. (c) Effect of lower length scale of fractality () on the charge transient.  is varied as 25, 50, 75, 100 nm, DH = 2.3 (d) Effect of  on the charge transient.  is varied as 19.64, 43.01, 62.10, 107.55 m. The plots are generated by using fixed parameters: projected area (A0 = 0.164 cm2 ), distance between working and reference electrode e = 0.73 cm, diffusion coefficient (D = 5 ×10−6 cm2 /s) and concentration (cO = cR = 5 mM) are used in our calculations. Dotted curve in plots (b-d) are smooth electrode response in presence of uncompensated solution resistance.

roughness the slope varies dynamically. In principle it should transcend from higher value to ideal Anson slope of 0.5. This transition is strongly controlled by the uncompensated solution resistance present in electrochemical set-up. The transition is faster for the lower values of the resistive parameters and at high enough values of resistance, ideal Anson slope of 0.5 may never be realized. This is a very interesting and important situation to analyze. This indicates that whatever kinetic phenomenon is occurring at the electrode will be deferred and there sets in psuedo kinetics control in the presence of the ohmic equilibration layer. However, no significant effect of surface roughness is observed on the ideal Anson slope. The deviation is far less prominent than for resistive parameters for just roughness features. This also indicates that if the resistance is high enough, it restricts the electrochemical sensitivity towards dynamic roughness features and only the gross roughness factor can be realized through the measurements. This is in contrast with the purely diffusion controlled case where the dynamic roughness features contribute significantly towards the intermediate time response of the system. Fig. 1(b), shows the effect of fractal dimension DH on the charge transient in presence of sizeable resistance offered by the electrolyte at an isotropic self-affine fractal electrode. The quantity DH

is the Hausdorff dimension of the surface and is a direct measure of the surface irregularity. As the fractal dimension increases surface becomes more rough and the charge at the interface increases. This increase in the charge transient is however scaled down in presence of ohmic effect in short times and it strongly affects the early response. The ohmic effect is most pronounced at early time and carried till the onset of the diffusion control in the system. Fig. 1(c) shows the effect of varying the lower length scale cut off () on the charge transient. Lower length scale of fractality () can be identified as the extent of the coarseness of the surface. It is seen that increasing  decreases the charge transient. The effect of finer features of the surface on the response is however propagated to a larger time in the plots. It is evident that  not just effects the short time (ms to 0.1 s) but influences the intermediate time domain to a larger extent showing that the current transients are strongly dependent on the lower length scale cut off. The change in the slope of the curve is however enhanced in presence of the ohmic effect in the early time domain. Fig. 1(d) shows the effect of topothesy length ( ) or the width of the interface in presence of ohmic effect. The effect of  is similar to that of DH . It shows the same influence over the current transient as DH , i.e. when  increases the charge transient also increases and

214

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Fig. 2. Anson response of rough electrode (R* = 6.3) evaluated from classical Anson relation. Red and black solid lines represent current transient at rough and smooth electrode, respectively. While blue fitted curve. Red line is generated using DH = 2.25, =45 nm, cO = cR = 7.7 mM, D = 5 ×10−6 cm2 /s. Table 1 Error in estimating concentration or diffusion coefficient from rough electrode data from classical Anson equation. R*

RMSW h( m)

%error in (ci )

%error in diffusion coefficient (D)

2.78 6.34 15.48

0.63 1.40 2.25

1.14 3.59 18.72

1.42 51.65 350.52

response enhancement due to roughness is seen for a larger time. As one increases the strength of roughness, it increases the electrochemically active area. Hence, increasing  increases the charge transient which finally becomes parallel with the smooth electrode response when the time is sufficiently larger than  and all irregularities of the surface is smaller than the yardstick depletion layer thickness after its onset. The crossover time from intermediate anomalous region to diffusion controlled response increases with increase in the width of interface. An important feature common to all plots is the time delay. This time delay does not allow the system to sense the roughness features of the surface instantaneously. Only when the depletion layer grows larger than the diffusion-ohmic length, roughness features are sensed. Chronocoulometry being an integration techniques it is often assumed that the discontinuities even out after integration. However, the morphological roughness features also get integrated over time and the effect is resounded in the measure for a longer time affecting the response significantly. This means that the response may not ever merge with the smooth electrode response. So, the response over the whole time is actually the complex interplay of ohmic and roughness parameters making the

Fig. 3. (a), (b) and (c) shows reconstructed electrode surface of Pt-1, Pt-2 and Pt-3 electrodes, respectively, using CV-SEM Method [41].

accurate determination of kinetic parameters less certain if the morphology and resistance parameters are not accounted in the experiments. The estimation of parameters can be severely affected in presence of roughness features at the electrode, if one uses the classical Anson equation for calculating the unknown parameters

Table 2 Morphological parameters of Pt-1, Pt-2 and Pt-3 obtained using CV-SEM method and R from impedance data. CV-SEM Method

Pt-1 Pt-2

CA

EIS

A0 (cm2 )

R*

m2

h ( m)

DH

 ( m)

 (nm)

L ( m)

D cm2 /s × 10−6

R 

0.13 0.11

2.80 3.27

4.165 5.968

1.29 1.48

2.41 2.30

2.03 2.78

241.0 215.3

1.89 1.90

7.6 7.6

4.5 5.1

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as shown in Fig. 2. We examine quantitatively these affect the measurements if one ignores the ohmic effects and roughness on the estimation of concentration and diffusion coefficients as summarised in Table 1 This reflects the need to characterize morphology separately and then use the features thus obtained in calculating the parameters like diffusion coefficient and unknown concentrations. We elucidate such measurements in the experimental section. 4.2. Characterization of Surface: CV-SEM Method CV-SEM [41] method is employed here to characterize the electrode roughness. Elegance of this method lies in the statistical en route that it adopts for connecting the 3D morphology of the surface to the 2D SEM images through its PSD of roughness. This method will work for systems where length scale of random material heterogeneities are much smaller than the smallest roughness size. This method does not require prior assumption of fractal as nonfractal nature of roughness, for 3D surface reconstruction. SEM gives better representation of large surface area as scanning over surface is a fast process. Like AFM, SEM does not have mechanical contact between tip and surface, therefore, tip related problems are absent in CV-SEM method. CV-SEM method can be used to obtain fractal morphological parameters like fractal dimension (DH ), lower cutoff length (), upper cutoff length (L), topothesy length ( ). In this communication, we have used this method to quantitatively characterize roughness of platinum roughened electrodes (shown in Fig. 3 and their morphological parameters are listed in Table 2). Fig. 4 shows the comparison of theoretical curves obtained from equation 27 with the experimental data obtained for the roughened Pt electrodes in solution of 15 mM K4 [Fe(CN)6 ] and 15 mM K3 [Fe(CN)6 ] in 3M NaNO3 . The value of R was estimated from the impedance of the system carried out for the same experimental set up (see Table 2). The theoretical results are found to be in excellent agreement with the experimental recording. 5. Conclusions The paper establishes the effect of solution resistance on the chronocoulometric measurement. The chronocoulometric transients are shown to be significantly affected by the ohmic and roughness parameters. Plots show that the uncompensated resistance effect is greatest in the short times where the kinetic information in the data is buried. The combined effect of roughness and resistance loss renders a time delay to the system. Stochastic morphology offers realistic models of roughness which may be useful in predicting the non-ideal behavior in the charge measurement response at the solid electrode with geometric disorders. We have analyzed our general model for roughness with known statistical self-affine fractal properties and power-spectrum of such roughness model has four fractal characteristics fractal dimension (DH ), lower length scale cut off (), upper length scale cut off (L), H −3 ) which is proportional to mean and strength of fractality (2D  2 square height (h ). This paper establishes for such roughness the following points:

Fig. 4. (a), (b) and (c) Chronocoulometry at Pt-1, Pt-2 and Pt-3 electrode in solution of 15 mM K4 [Fe(CN)6 ] and 15 mM K3 [Fe(CN)6 ] in 3M NaNO3 . Solid line (Red) is the theoretical plot for rough electrode with ohmic contribution. Blue circles represent experimental data.

• We show that the chronocoulometric measurements distort under the conditions of Nernstian charge transfer (reversible) due to the ohmic drop across the solution and presence of the surface roughness. The formalism enables us to discern the effect of ohmic drop as well as morphology of an arbitrary rough surface. • The deviation from the classical Anson behavior in the short time domain is found to be dependent primarily on the resistance of the electrolytic solution and the real area of the surface. This short time domain is mainly influenced by the external experimental

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•

•

•

•

•

•

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conditions, resistivity of the solution (e ) and, the distance between the probe and the working electrode (e ). In the absence of surface roughness, the charge crosses over to classical Anson response as the diffusion length exceeds the diffusion-ohmic length, but in the presence of roughness, there is formation of anomalous intermediate region followed by roughness factor times the classical Anson response. The intermediate region is marked by the anomalous power-law behavior. The presence of an ohmic effect delays the onset of anomalous power-law region. The anomalous power-law region arises due to formation of a depletion layer which leads to the emergence of dynamic effects of roughness in the response. The 2 /D. depletion layer start forming as the time exceeds the value L The deviation from the Anson slope is governed primarily by the “ohmic equilibration layer” thickness giving rise to non-idealistic response. This intermediate time response contains several morphological details of the surface, viz. mean square width, mean square gradient, mean square curvature, etc. These finer features give shape to the anomalous intermediate region for a rough surface. For large resistive losses, the dynamic roughness response evens out in time due to large depletion layer thickness, and the roughness is sensed through the gross roughness factor of the surface. Estimation of diffusion coefficient, unknown concentration can be severely compromised in presence of roughness and ohmic losses, if classical Anson relation is used to fit the data. The criterion to assess the severity of ohmic and roughness effects is possible with the help of techniques like AFM and CV-SEM. These measurements will provide a fair idea about the roughness factor and width of interface. These information can be used to get an idea about the extent of deviation in chronocoulometric response and the longest time upto which deviation will be seen. If the longest crossover and ohmic-diffusion time scales lies between experimental time window roughness will influence the response. Very short time experimental data is affected due to capacitive charging and ohmic drop, and their contribution to charge transient (Qnf ) can be corrected in the experimental data. Good estimate for capacitive charging contribution can be obtained from equation: Qnf (t) = R∗ A0 (E − E 0 )cd exp(−t/R cd ) where cd is the area specific capacitance of the working electrode.

Finally, one can say that the theory presented here offers an understanding of more realistic picture of non ideal behavior in charge measurements for diffusion controlled processes under resistive experimental conditions at irregular interfaces. Acknowledgement RK thanks University of Delhi for financial support under Scheme to Strengthen R and D Doctoral Research Programme. References [1] J. Osteryoung, R.A. Osteryoung, The advantages of charge measurements for determining kinetic parameters, Electrochimica Acta 16 (1971) 525. [2] H. Randriamahazaka, C. Plesse, D. Teyssié, C. Chevrot, Relaxation kinetics of poly(3,4-ethylenedioxythiophene) in 1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)amide ionic liquid during potential step experiments, Electrochimica Acta 50 (2005) 1515. [3] M. Sharp, Determination of the charge-transfer kinetics of ferrocene at platinum and vitreous carbon electrodes by potential steps chronocoulometry, Electrochimica Acta 28 (1983) 301. [4] M. Orlik, G. Gritzner, Chronocoulometric studies of the kinetics of the electroreduction of Eu3+ in dimethylsulfoxide: Experiment and simulation, Journal of Electroanalytical Chemistry 582 (2005) 144. [5] J. Richer, J. Lipkowski, Measurement of physical adsorption of neutral organic species at solid electrodes, Journal of Electrochemical Society 133 (1986) 121.

[6] J. Rishpon, Chronocoulometric techniques in the study of computerized enzyme electrode systems, Journal of Biotechnology Bioengineering 29 (1987) 204. [7] J. Osteryoung, J.H. Christie, Alternative reaction pathways in potential-step chronocoulometry, Journal of Electroanalytical Chemistry 25 (1970) 157. [8] M.S. De Vicente, M.R. Moncelli, R. Guidelli, Chronocoulometric study of electrode kinetics in the case of an adsorbed redox couple: Progesterone electroreduction on mercury from aqueous solutions, Journal of Electroanalytical Chemistry 248 (1988) 55. [9] M.L. Foresti, R. Guidelli, L. Nyholm, Potential step chronocoulometric procedure for the conversion of quasireversible waves into reversible ones, Journal of Electroanalytical Chemistry 269 (1989) 27. ´ V. Svetliˇcic, ´ M. Lovric, ´ I. Ruˇzic, ´ J. Chevalet, Electron transfer kinet[10] V. Zˇ utic, ics of an adsorbed redox couple by double potential-step chronocoulometry: Methylene blue/leucomethylene blue, Journal of Electroanalytical Chemistry 177 (1984) 253. [11] A.J. Bard, L.R. Faulkner, Electrochemical methods: Fundamentals and applications, Wiley, NY, 1980. [12] F.C. Anson, Application of potentiostatic current integration to the study of the adsorption of Cobalt(III)-(Ethylenedinitrilo(tetraacetate) on mercury electrodes, Analytical Chemistry 36 (1964) 932. [13] F.C. Anson, Innovations in the study of adsorbed reactants by chronocoulometry, Analytical Chemistry 38 (1966) 54. [14] F.C. Anson, Charge-step chronocoulometry, Analytical Chemistry 38 (1966) 1924. [15] Md. M. Islam, R. Kant, Generalization of the Anson equation for fractal and nonfractal rough electrodes, Electrochimica Acta 56 (2011) 4467. [16] D.H. Evans, M.J. Kelly, Theory for double potential step chronoamperometry, chronocoulometry, and chronoabsorptometry with a quasi-reversible electrode reaction, Analytical Chemistry 54 (1982) 1727. [17] S. Srivastav, R. Kant, Theory of generalized Cottrellian current at rough electrode with solution resistance effects, Journal of Physical Chemistry C 114 (2010) 10066. [18] S. Srivastav, R. Kant, Theory of generalized Cottrellian current at rough electrode with solution resistance effects, Journal of Physical Chemistry C 114 (2010), 10066(Supporting Material). [19] P. Delahay, C.C. Mattax, T. Berzins, Theory of voltammetry at constant current. IV. Electron transfer followed by chemical reaction, Journal of the American Chemical Society 76 (1954) 5319. [20] O. Wein, Effect of ohmic losses on potentiostatic transient response of a reversible redox system, Journal of Applied Electrochemistry 21 (1991) 1091. [21] R. Kant, S.K. Rangarajan, Diffusion to rough interfaces: finite charge transfer rates, Journal Electroanalytical Chemistry 396 (1995) 285. [22] R. Kant, Diffusion-limited reaction rates on self-affine fractals, Journal of Physical Chemistry B 101 (1997) 3781. [23] S.K. Jha, R. Kant, Theory of partial diffusion-limited interfacial transfer/reaction on realistic fractals, Journal of Electroanalytical Chemistry 641 (2010) 78. [24] R. Kant, Can current transients be affected by the morphology of the nonfractal electrode? Physical Review Letters 70 (1993) 4094. [25] S.K. Jha, A. Sangal, R. Kant, Diffusion-controlled potentiostatic current transients on realistic fractal electrodes, Journal Electroanalytical Chemistry 615 (2008) 180. [26] R. Kant, S.K. Jha, Theory of anomalous diffusive reaction rates on realistic selfaffine fractals, Journal of Physical Chemistry C 111 (2007) 14040. [27] S. Dhillon, R. Kant, Theory of Double Potential Step Chronoamperometry at Rough Electrodes: Reversible Redox Reaction and Ohmic Effects, Electrochimica Acta 129 (2014) 245. [28] R. Kant, M. Sarathbabu, S. Srivastav, Effect of uncompensated solution resistance on quasireversible charge transfer at rough and finite fractal electrode, Electrochimica Acta 95 (2013) 237. [29] S. Srivastav, R. Kant, Anomalous Warburg impedance: Influence of uncompensated solution resistance, Journal of Physical Chemistry C 115 (2011) 12232. [30] R. Kumar, R. Kant, Generalized Warburg impedance on realistic self-affine fractals: Comparative study of statistically corrugated and isotropic roughness, Journal of chemical Sciences 121 (2009) 579. [31] R. Kant, R. Kumar, V.K. Yadav, Theory of anomalous diffusion impedance of realistic fractal electrode, Journal of Physical Chemistry C 112 (2008) 4019. [32] M.B. Singh, R. Kant, Debye-Falkenhagen dynamics of electric double layer in presence of electrode heterogeneities, Journal Electroanalytical Chemistry 704 (2013) 197. [33] M.B. Singh, R. Kant, Theory of Anomalous Dynamics of Electric Double Layer at Heterogeneous and Rough Electrodes, Journal of Physical Chemistry C 118 (2014) 5122. [34] R. Kumar, R. Kant, Theory of generalized Gerischer admittance of realistic fractal electrode, Journal of Physical Chemistry C 113 (2009) 19558. [35] R. Kant, Md. M. Islam, Theory of absorbance transients of an optically transparent rough electrode, Journal of Physical Chemistry C 114 (2010) 19357. [36] Md. M. Islam, R. Kant, Theory of single potential step absorbance transient at an optically transparent rough and finite fractal electrode: EC’ mechanism, J. Electroanalytical Chemistry 713 (2014) 82. [37] R. Kant, General theory of arbitrary potential sweep methods on an arbitrary topography electrode and its application to random surface roughness, Journal of Physical Chemistry C 114 (2010) 10894. [38] R.Kant Parveen, Theory for staircase voltammetry and linear scan voltammetry on fractal electrodes: Emergence of anomalous Randles-Sevˇcik behavior, Electrochimica Acta 111 (2013) 223.

S. Srivastav, R. Kant / Electrochimica Acta 180 (2015) 208–217 [39] J.M. William, T.P. Beebe Jr., Analysis of fractal surfaces using scanning probe microscopy and multiple-image variography. 1. Some general considerations, Journal of Physical Chemistry 97 (1993) 6249. [40] R. Kant, Statistics of approximately self-affine fractals: Random corrugated surface and time series, Physical Review E 53 (1996) 5749. [41] S. Dhillon, R. Kant, Quantitative roughness characterization and 3D reconstruction of electrode surface using cyclic voltammetry and SEM image, Applied Surface Science 282 (2013) 105. [42] G.A. Niklasson, D. Ronnow, M.S. Mattsson, L. Kullman, H. Nilsson, A. Roos, Surface roughness of pyrolytic tin dioxide films evaluated by different methods, Thin Solid Films 359 (2000) 203. [43] R. Kant, J. Kaur, M. B. Singh,;1; Nanoelectrochemistry in India, Chapter 10 in Specialist Periodical Reports: Electrochemistry, 12, 336-378, Royal Society of Chemistry, Jay D Wadhawan (Editor), Richard G Compton (Editor).

217

[44] R. Kant, S.K. Rangarajan, Effect of surface roughness on diffusion-limited charge transfer, Journal Electroanalytical Chemistry 368 (1994) 1. [45] R.J. Adler, The Geometry of Random Fields, John Wiley and Sons Ltd, 1981. [46] A.R. Osborne, A. Pastorello, Simultaneous occurence of low-dimensional chaos and colored random noise in nonlinear physical systems, Physics Letters A 181 (1993) 159. [47] P. Ocon, P. Herrasti, L. Vazquez, R.C. Salvarezza, J.M. Vara, A.J. Arvia, Fractal characterisation of electrodispersed gold electrodes, Journal of Electroanalytical Chemistry 319 (1991) 101. [48] O.I. Yordanov, N.I. Nickolaev, Self-affinity of time series with finite domain power-law power spectrum, Physical Review E 49 (1994) R2517. [49] J. Feder, Fractals, Plenum, New York, 1988.