Modeling of Catalytic Reaction in Protein-Film Linear Scan Voltammetry at Rotating Disk Electrode

Transcription

1 Portugaliae Electrochimica cta 9, 7(4), DOI: 1.415/pea.9455 PORTUGLIE ELECTROCHIMIC CT ISSN Modeling o Catalytic Reaction in Protein-Film Linear Scan Voltammetry at Rotating Disk Electrode Milivo Lovrić * Department o Marine and Environmental Research, Ruder ošković Institute, P.O. ox 18, HR-1 Zagreb, Hrvatska (Croatia) Received 9 March 9; accepted 1 pril 9 bstract The numerical method or the simulation o linear scan voltammetry on the rotating disk electrode is adusted to the problem o irreversible redox reaction between the adsorbed catalyst and the diolved reactant under transient conditions. The response consists o the wave and the imum. The peak current depends on the scan rate in linear scan voltammetry, while the limiting current o the wave depends on the rate o rotation o the working electrode. The rate constant o catalytic reaction is determined rom the kinetic current under steady-state conditions. Keywords: electro-catalytic reaction, rotating disk electrode, chemical reaction rate constant, linear scan voltammetry, protein-ilm voltammetry. Introduction The conversion o energy occurring in photosynthesis and respiration is realized through a complex sequence o electron transer reactions [1, ]. The rate constants o these transers are key parameters in the mentioned and several other biological procees ranging rom cell deense to gene control [3-5]. Voltammetry is a simple and powerul method o investigating biologically relevant redox-active compounds [6-8]. For instance, by adsorption o the redox protein onto the surace o some suitable lipophilic electrode, the kinetics o transer o electrons rom the immobilized enzyme to diolved substrate can be measured by cyclic voltammetry and chronoamperometry [9, 1]. For sensors based on electro-catalytic reactions, the application o rotating disk electrodes (RDE) was proposed [11-13]. The sensors usually work under theoretically well described steady-state conditions [1], but or kinetic studies o catalytic eects the transient conditions are equally important [14]. In this communication the * Corresponding author. addre: mlovric@irb.hr

2 M. Lovrić / Port. Electrochim. cta 7 (9) numerical method or the simulation o linear scan voltammetry on the RDE [15] is adusted to the problem o irreversible redox reaction between the adsorbed catalyst and the diolved reactant under transient conditions. This reaction scheme belongs to the catalytic EC mechanism which was investigated on stationary electrodes under the conditions o various voltammetric methods [16 - ]. The model It is aumed that a certain redox couple is strongly adsorbed on the surace o RDE: n + + ne ads ads (1) The electrode reaction (1) is ast and reversible: Γ Γ exp(ϕ) () where Γ and Γ are surace concentrations o the reactant and product, respectively, ϕ nf( E E / )/ RT is dimensionle potential and E / is the standard potential. During the experiment, the sum o surace concentrations o the reactant and product does not change: Γ + Γ Γ (3) It is urther aumed that in the solution there is a compound which can be oxidized by the product o the surace redox reaction: + + (4) n+ n+ ads x ads Z x In the investigated potential range the compound can not be electro-oxidized on the bare electrode surace due to very slow electron transer. The ma transport towards the surace o RDE is deined by the partial dierential equation: c c c D + κ x (5) t x x and the boundary conditions: c D k Γc, x (6) x dγ x I nfs k Γ c, x 3 1 where κ.51 ω ν, while ω is the angular rotation rate o the disk, ν is the kinematic viscosity o the solution, c is the concentration o the compound, D is the diusion coeicient, I is the current, S is the electrode surace area, n (7) 56

3 M. Lovrić / Port. Electrochim. cta 7 (9) is the number o electrons, F is Faraday constant and k is the rate constant o irreversible reaction (4). Equation (5) is solved by Galerkin method o variable diusion layer thickne [15]. It is aumed that the concentration proile is the linear unction o the space coordinate between the electrode surace ( x ) and a certain distance which is the unction o time: x ( c c ) c ( x, t) c, x +, x, or x (8) c ( x, t), or x > (9) c where c is the bulk concentration o. Equation (5) is integrated rom x to ininity: c c dx D t x and by using the aumed gradients or and or x > : c t c x x dc, x c x c + κ x dx (1) x x : ( c c ) 1, x c, x x d (11) (1) c c (13) t x it is transormed into the dierential equation on time: dy 4 3 y dc, x 4D κ y + (14) 3 c c, x where y. Under steady-state conditions: dy dc, x c, x the solution o equation (14) is: (15) 1 3 3D (16) κ which is the steady-state diusion layer thickne. general solution o eq. (14) can be obtained by the numerical integration. y combining equations (6) and (1) one obtains: c D c, x (17) D + k Γ 57

4 M. Lovrić / Port. Electrochim. cta 7 (9) lso, the combination o eqs. () and (3) gives the ollowing relationship: Γ Γ (18) 1+ In linear scan voltammetry the derivations o c, x and Γ on time are: dc, x c k D dγ d + Γ (19) ( D + k Γ ) dγ Using the approximation: nf RT de Γ [ 1+ ] () dy y t y +1 y t equation (14) is transormed into the system o recursive ormulae: y1 D t () (1) y + 1 y D + Dexp( ϕ ) + k Γ y exp( ϕ ) D t κ t y D Dexp( ϕ ) k y exp( ϕ ) + + Γ 3 nf RT y D t D + Dexp( ϕ ) + k Γ de y exp( ϕ ) (3) nf de ϕ Est + t E / (4) RT The current is deined by eq. (7): + y (5) I I di nf de Γ exp( ϕ ) RT k Γ exp( ϕ ) + (6) Dc [ 1+ exp( ϕ )] D[ 1+ exp( ϕ )] + k Γ exp( ϕ ) I di Dc nfs (7) Equations (6) and (7) are obtained by combining eq. (7) with eqs. (17), (18) and (). The irst term on the right-hand side o eq. (6) is peak-shaped. It tends to zero i both E << E / and E >> E /. Equation (18) shows that Γ Γ i E >> E /. So, i E >> E /, the limiting kinetic current appears: 58

5 M. Lovrić / Port. Electrochim. cta 7 (9) k Γ c D lim (8) I E >> E nfs D + k / Γ Equation (7) is the limiting value o eq. (8) i D << Γ. The simulation was perormed using constant values o diusion coeicient and kinematic viscosity, while the rotation rate o RDE and the scan rate in CV were varied. k Results and discuion Electro-catalytic oxidation o the compound on RDE depends on the product o the oxidation rate constant and the surace concentration o the catalyst ( k ) and on the ratio o the latter and the bulk concentration o the reactant ( Γ / c ). Fig. 1 shows simulated cyclic voltammograms o coupled reactions (1) and (4) on RDE, or various products k and the constant ratio Γ / c. The current is Γ Γ made dimensionle by dividing with the diusion steady-state current deined by eq. (7). Voltammograms consist o the wave and the peak. The latter originates rom the surace electrode reaction (1). The net peak current depends on the ratio Γ / c and does not change with the variation o the product k. Γ Figure 1. Dimensionle LSV o electro-catalytic oxidation o on RDE. D 1-5 cm /s, ν 1 - cm /s, n 1, ω 1 π s -1, de /.1 V/s, Γ / c.1 cm and k /cm s -1 (1),. (),.1 (3),.1 (4), 1 (5), 1 (6) and 1 (7). Γ Fig. shows the imum and steady-state currents o voltammograms as unctions o the logarithm o k product. The curve 1 in this igure shows that Γ I I I.5 V di, where I / Idi.9916 is the imum o the curve 1 in Fig. 1 and I / I.5 V di is the dimensionle current corresponding to the potential dierence E.5 V. The potential o imum is equal to the standard E / 59

6 M. Lovrić / Port. Electrochim. cta 7 (9) potential i either log( k Γ peak potential increases to ) < -3, or log( k E / Γ E.6 V at ) >. Within these boundaries, the k. cm/s. The steady-state currents are limiting currents o the waves. They change rom zero to one, as can be seen in Fig.. This component o the response originates rom the catalytic oxidation o the compound. Considering that in Fig. 1 D 1-5 cm /s and cm, the limiting value I lim Idi is achieved i k Γ > 1, which satisies the condition D << Γ (see eq. 8). k Γ Figure. Dependence o the imum ( ( I / I.5 V di I / Idi ; marked as 1) and the steady-state ; marked as ) dimensionle CV currents o oxidation on the logarithm o the product k. ll parameters are as in Fig. 1. Γ The hal-wave potentials o the wave-components o the voltammograms in Fig. 1 satisy the relationship: E 1/ E / -.59 log( k ) -.1 V, i k 1 and n 1. This potential can be calculated or steady-state conditions using equations (6) (8): Γ Γ D + k c Γ ( D + k Γ ) exp( ϕ ) ( D + k Γ ) Dexp( ϕ 1/ ) 1/ where ϕ nf( E E ) RT 1/ 1/ / / k c Γ D. The solution is: (9) RT D E1 / E / ln (3) nf D + k Γ I k, Γ E 1/ E / and i k Γ >> D the hal-potential is: E 1/ E / RT D ln nf RT nf ln ( k Γ ) (31) 51

7 M. Lovrić / Port. Electrochim. cta 7 (9) This shows that very ast chemical reaction consumes the product o electrode reaction and shits the oxidation wave towards lower potentials. The hal-wave potential o the curve 7 in Fig. 1 is E 1/ E / -.37 V. In this curve it can be noted that the peak component developes on the plateau o the wave component. The potential range within which this occurs can be calculated using equation (6): k Γ >.95 (3) D + D + k Γ ( ) For the curve 7 in Fig. 1 the solution is: E E / > V (33) The second condition is: nf RT de Γ Dc ( 1+ ) For the same curve the solutions are: <.5 (34) E E / < -.11 V (35) E / >.11 V (36) E So, between V and -.11 V the wave component stagnates and the peak component developes. ecause o constant ratio Γ / c, voltammograms shown in Fig. 1 correspond to various oxidation reactions (4), each with the dierent rate constant k. However, the variation o the product k can be achieved by the variation o the surace concentration o the catalyst, but in this case the ratio Γ / c must be also changed. n example is shown in Fig. 3. The main dierence between Figs. 1 and 3 is the change o the net peak current because o the change o the ratio Γ / c in Fig. 3. The curve 6 in Fig. 3 and the curve 4 in Fig. 1 are identical because all parameters are equal in both igures, but the curve 1 in Fig. 3 is signiicantly dierent rom the curve 3 in Fig. 1 because Γ / c is.1 cm in Fig. 3 and.1 cm in Fig. 1. The dimensionle steady-state current o the curve 1 in Fig. 3 is equal to the dimensionle steady-state current o the curve 3 in Fig. 1 because the second term on the right-hand side o eq. (6) does not depend on the ratio Γ / c. For analytical purposes, the relationships between either transient, or steady-state current and the bulk concentration o the compound are important. Fig. 4 shows that the latter relationship is linear: I /.5V nfs c (see curve 1). The slope o this straight line is deined by eq. (8). In Fig. 3 it can be seen that the ratio o the imum and steady-state currents decreases as the ratio Γ 511

8 M. Lovrić / Port. Electrochim. cta 7 (9) Γ / c is diminished. Consequently, the imum current depends on c nonlinearly, starting rom I nfs( nf / RT )( de / ) Γ / 4, or c, and approaching the linear relationship asymptotically (see curve in Fig. 4). Using the catalytic oxidation on RDE, the compound can be determined in the concentrations higher than 1-5 M. In the presence o several electroactive substances, the peak current rather than the limiting current can be used or the analysis o the compound. Figure 3. Dimensionle LSV o coupled reactions (1) and (4). ( Γ c ) (1),. (),.3 (3),.5 (4),.7 (5) and.1 (6); Γ / / cm.1 k / cm s -1.1 (1),. (),.3 (3),.5 (4),.7 (5) and.1 (6). ll other parameters are as in Fig. 1. Figure 4. Dependence o normalized steady-state (1) and imum () currents on the bulk concentration o compound. Γ 1-9 mol cm - and k 1 8 cm 3 s -1 mol -1. ll other parameters are as in Fig

9 M. Lovrić / Port. Electrochim. cta 7 (9) The investigated responses consist o the transient part, which is the rising portion o the wave and the whole imum, and o the steady-state part, which is the limiting current o the wave. The scan rate in LSV can inluence only the transient part, while the rotation rate o RDE inluences only the steady-state part. The dependence o dimensionle imum current on the scan rate is shown in Fig. 5. It is a straight line i de / > 3 mv/s. t the lowest scan rates the imum current tends to the steady-state current, which is.546 I di or the aumed experimental parameters. For this reason the net peak current ( I I. 5V ) is a linear unction o scan rate i de / > 15 mv/s, and tends to zero below this boundary. Figure 5. Dependence o I / Idi (1) and ( I I.5 V )/ I () on the scan rate. di k Γ.1 cm s -1 and all other parameters are as in Fig. 1. The relationship between the steady-state current and the rate o rotation o RDE is deined by eqs. (8) and (16). It can be transormed into linear dependence o reciprocal o current on the inverse value o square-root o the rotation rate: nfs I lim ν 1/ 1 + (37) / 3 k Γ c c D ω This dependence is shown in Fig. 6, or three dierent values o the product k. The intercepts o these straight lines serve or the determination o the Γ oxidation rate constants. The surace concentration o the catalyst Γ can be determined rom the peak current I in the absence o the compound ( c ; see the intercept o curve in Fig. 4). Hence, the advantage o RDE is in the establishment o steady-state conditions under which the kinetic current can be analyzed according to equation (37). 513

10 M. Lovrić / Port. Electrochim. cta 7 (9) Figure 6. Dependence o normalized reciprocal o steady-state current on the inverse value o square-root o RDE rotation rate. Γ 1-9 mol cm -, c 1-7 mol cm -3 and k / cm 3 s -1 mol (1), 1 7 () and (3). ll other parameters are as in Fig. 1. Conclusion The model described in this paper is an adequate representation o protein-ilm voltammetric experiments [6]. Some examples o electro-catalytic reactions considered here are the oxidation o glucose by glucose oxidase adsorbed on the electrode surace [1] and the oxidation o diolved succinate to umarate catalyzed by succinate dehydrogenase adsorbed on the rotating disk electrode []. In linear scan voltammetry on RDE the theoretical response o this type o reactions can be calculated by the proposed numerical method. It is developed or the irst, transient part o the response, in which the diusion layer thickne is time-depending and the concentration o the reactant at the working electrode surace depends on both the electrode potential and the oxidation rate constant. The steady-state part o the response appears as a special case o the general solution. Finally, it is shown that or both kinetic and analytical measurements the steady-state conditions are the most appropriate. cknowledgement Valuable contributions by the reerees and the inancial support by the Croatian Ministry o Science, Education and Sports are grateully acknowledged. Reerences 1. F.. rmstrong, ioelectrochemistry o iomacromolecules, irkhauser, asel, G. Volkov, M.I. Volkova-Gugeshashvili, C.L. rown-mcgauley,.j. Osei, Electrochim. cta 5 (7) K. Satpati, M. Kumbhakar, S. Nath, H. Pal, J. Photochem. Photobiol. : Chem. (8)

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