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ЭЛЕКТРОНИКА
UDC 517.958:536.71
SIMULATION OF THE DOUBLE HEMICYLINDER ELECTRODE SYSTEM THROUGH CONFORMAL MAPPING. APPLICATION TO STEADY-STATE ELECTROGENERATED CHEMILUMINESCENCE
OLEINICKA.I., AMATORE C.A., SVIR I.B.
The analytical expressions for steady state currents and electrogenerated chemiluminescence (ECL) intensities are derived for two-hemicylinder assemblies operating in the ECL generation mode based on a specific conformal mapping transform. This transform is also used for fast and accurate simulations of transient currents and ECL intensities leading to an excellent agreement at long times with the analytical steady state solutions. These results are compared to those relative to two-band assemblies to investigate the effect of hemicylindrical electrode protuberance on ECL efficiencies. This provides evidence that electrode protuberance results in a drastic increase of the ECL generation and shortening of commutation times when the gap is not too large as compared to the electrode radii.
1. Introduction
Using conformal mapping techniques for the simulation of two dimensional electrochemical problems at microelectrodes represents the most efficient and easiest way of obtaining an accurate numerical solution for diffusion at microelectrodes or microelectrode assemblies. Conformal transforms allow mapping of a twodimensional real space, where the flux lines may be extremely curved or present singularities, onto a space in which the flux lines become (almost) parallel [1,2]. It is probably the best numerical approach for the symmetrical electrode problems like diffusion at a disk [3], spherical/hemispherical [4,5] and cylindrical/ hemicylindrical [6-8] electrodes. The conformal mapping approach has been successfully used for solution ofband electrode and double band microelectrode problems [4,9-11]. These all transformations then were applied by many authors for the simulation of different electrochemical problems.
Furthermore, conformal mappings have the important own properties and often allow obtaining the exact solution of the problem. There are a few analytical solutions for the steady state current in double band and channel double band microelectrode generator-collector systems using the conformal mapping [4,11-13] and electrogenerated chemiluminescence (ECL) steady state intensity for the double band microelectrode system
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[11]. Recently we used the conformal transformation for the simulation of the double hemicylinder generator-collector microelectrode assembly [8], which to apply here for investigation of ECL in cell with two parallel-hemicylindrical microelectrodes. The analytical expressions for the steady state current and ECL intensity in cell with two microelectrode hemicylinders are presented here.
Using of the paired microelectrodes or arrays of such electrodes is popular in modern electrochemistry because it offers the possibility to generate different species (anion and cation radicals) at different electrodes which for one’s turn allows the ECL investigations. Our next aim is to compare the efficiency of ECL emission in two different microelectrode systems: with two bands and two hemicylinders, where the equal conditions have been created for: i) the ECL excitation (the same reaction scheme, eqs. (1) — (4)), ii) equal electrode squares (with the corresponding radii, widths and length of the electrodes) and iii) common numerical approach for simulations (i.e. the conformal mappings and the ADI method) in both cases [8, 14-15].
2. Theory
ECL model
We consider the simple mechanism of the ECL realization in cell with two parallel microhemicylinders
with equal radii, rhc, (the case when radii ofhemicylinders are different we consider in [8] for the generator-collector electrode system) and gap, g , between the
electrodes (Fig.1):
Ag - e ^ A+ (anode), (1)
Ag + e ^ A_ (cathode), (2)
A ++ A_—kbl >JA* + Ag (solution), (3) 1A* kf > уeci + Ag (solution), (4)
where Ag is the luminophor species in its basic state; A+ is the cation-radical species; A_ is the anion-
I *
radical species; A is the singlet species; уeci is the photon of ECL emission; kbl is the bimolecular rate constant; kf is the pseudo monomolecular rate constant (kf = фесі /x* where фесі is the quantum fluorescent output (in our calculations фесі = 1); т is the life time of a singlet species).
Fig. 1. Scheme of the orthogonal cross-section of an electrochemical cell with two hemicylinder electrodes in the real space with definitions of the geometrical parameters used here
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Mathematical model
The mathematical model corresponding to the kinetic equations (1) — (4) represents a system of non-linear partial differential equations with the initial and boundary conditions. Numerical simulation was carried out in the conformal space. The detailed description of this conformal transformation for case of the generator-collector system of two hemicylinders with different radii is represented in article [8]. As a result of these simulations we obtained the concentration distributions
і _ 1 *
of the all species (A , A , A and A g ) in solution. Conformal mapping
We used the following change of spatial variables (X and у are the Cartesian dimensionless coordinates defined as: X = x/rhc; Y = y/rhc ) [8]:
X =
ae (a-b)cosq + be^
2(cosh ^ - cos q)
Y =
(a - b) sin q 2(cosh £, - cos q) ’
(5a)
(5b)
where geometrical parameters a and b were defined as:
a - -^2 + G - VG2 + 4Gj ; b = — , where the dimensionless parameters corresponds to:
Rhc = ~ = 1;G = ~ (see fig.1, 2).
rhc rhc
Fig. 2. Representation of the two hemicylinder electrodes in the real (a) and conformal space (b) (solid lines are electrode surface); and forms of the computational grids in the real (a) and transformed coordinates (b)
The back transformations for conformal coordinates are given by
£ = ln
7
(X - b)(X - a) + Y2 + (a - b)2Y
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q = arctan
(X -b)2 + Y2
(a - b)Y
(X - b)(X - a) + Y2
,(6a)
(6b)
Transient current and ECL intensity
The expressions for transient current at the anode and cathode in conformal space are:
2л
Ia(x) = -FDLc0 J
1 О + a c “]
і Oj <Лт Oj cTG 1
dq,
2л
Ic(x) = FDLc0 J
1 n + a c ■]
і Oj cTG Oj cTG 1
dq.
(7a)
(7b)
where x = Dt/rhc is the dimensionless time; F is the Faraday constant; D is the diffusion coefficient, which is assumed equal for all four species; L is the length of the hemicylinders; c0 is the initial concentration of species Ag ; C+ = c+ /c0 and C_ = c_ /c0 are the normalized concentrations of the cation- and anion-radicals correspondingly.
The transient ECL intensity is defined as
Ф eclCO = Na
^2 2n *
= NA Фkf L rhcc0 J J C
І1 л
dn
Yecl
dt
a (X, Y)
a (£, q)
dq d^,
(8)
here Na is the Avogadro constant; dn y / dt is the ECL flux; ф is the ECL quantum efficiency (ф = 10_2 in our calculations); C* = c*/c0 is the normalized concentration ofthe singlet species; д(X, Y)/d (^, q) is the Jacobian of the transformation, where
a (X, Y) _ (a - b)2 a(^, q) 4(coshcosq)2 .
Steady-state current and ECL intensity
In the case of equal cylinders the analytical solution for the current and ECL intensity can be directly obtained. The ordinate axis (q) is a symmetry axis, when cylinders have the same radius, and species A+ and A_ coexist only in a thin region near the q -axis. When kbi ^ да and t ^ да the thickness of the coexistence region tends to zero. So it is necessary to solve the following problem to obtain the concentration distribution at steady-state:
a 2c+ a^2
a 2c+
aq2
= 0
(9)
with the boundary conditions:
s = ^; TC<q< 2% ; C+ = 1; (10a)
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The ECL dimensionless flux defined as
8 C+
t>1 <^< 0; Л = ^ or q = 2к; = 0, (10b)
Sq
^ = 0; л < q < 2к; C += 0 . (10c)
The analogous problem formulated for the species A _ in the region: 0 2, Л < q < 2к. The solution
of the system (9) — (10a), (10b), (10c) is a linear function:
C+ S
C (11)
If we incorporate this result into equations (7a) and (7b), and note that the flux of anion-radicals (8C~ /) at the anode (^ = E,i) is equal to zero, we get the analytical expression for the current:
Ia =-FDLc0 f = FDLC0 = Ic . (12)
Si S 2
Ф =
Ф ecl
Na фDLc0
(15)
In the assumptions given above, ECL flux must be equal to fluxes at each electrode. So the steady-state ECL flux for double hemicylinder assembly is:
фss
в
(16)
The analytical value of ECL intensity in photon/s units for specified parameters can be evaluated from Eq. (15).
Results and discussion
We consider the following typical experimental parameters: C0 = imM , D = 10 cm s ,
L = 0.2 cm , kf = 108 s-1. Other experimental parameters (rhc , g, kbi) are documented in each investigated situation.
Due to the symmetry of the system the fluxes at the anode and cathode will be equal, so the dimensionless current (with definition f = I/nFDLc0) is:
ss
-щ. ш>
It should be mentioned that the steady-state flux for the ECL is twice greater than for the same system operated in generator-collector mode [8]. This result is coherent with the same behavior of the double-band system [4,11].
The flux of photons is equal to flux of molecules [11], so that the expression for the ECL intensity is:
Ф eScl = NaDLc0 фА = NaDL^-^
M S 2'
(14)
The equation (12) is an exact solution for the current at steady-state, while (14) is only an asymptote to the exact solution. The steady-state ECL intensity tends to the value defined by equation (14), but never reaches it, because some part of anion- and cation-radicals becomes inaccessible for participation in reaction (3) through the diffusion towards infinity.
The computational results for the current are shown in Fig. 3a. It can be seen that when the system achieves steady-state behavior the current at both electrodes exactly reaches the value predicted by eq. (12). The same situation is with the ECL intensity (Fig. 3b), but it should be noted that ECL response needs much more time to achieve steady-state. Simulation parameters were: rbc = g = 10 mm and kbi = 109M_1s_1
(simulation grid: №, x Nq x Nt = 100 x 100 x 10000 ).
We compare the steady-state ECL intensity for both systems of electrodes in case when the squares of band and hemicylinder electrodes are equal. The width, w , and length, L , ofaband electrode were: w = 10pm and L = 0.2 cm correspondingly. The length of the hemicylinder electrode was the same (L = 0.2 cm ), the hemicylinder radius was defined as: rhc = w /к ~ 3.1831 pm . The square of any electrode (band and hemicylinder) is
S = wL = кrbcL = 2 x 10_4 cm2.
Fig. 4 shows the results of this comparison which are the analytically obtained steady-state responses of both systems (two bands and two hemicylinders) as a function of gap value. It can be seen that effect of electrode
Fig. 3. Comparison of the simulated results (solid lines) and analytical predicted solutions (dash dots) for the values of currents (a) and ECL intensities (b)
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protrudance for hemicylinders suppresses with the increasing of the distance between electrodes, but it strongly influence on the ECL intensity in case when gap is decreasing, because the effective electrode area of the hemicylinders significantly increases. Thus when gap is a big the ECL intensity of both electrode systems (two bands and two hemicylinders) practically equals.
Fig. 4. Dependence of the steady-state ECL intensity on values of the gap between electrodes for two hemicylinders (1) and two bands (2)
Conclusions
The analytical expressions for the current and ECL intensity for the double hemicylinder electrodes system are presented. The simulated results of the transient current and ECL intensity results have an excellent agreement with the analytical steady state solutions of the current and ECL intensity in cell two microelectrode hemicylinders.
The comparison of two different microelectrode systems — two bands and two hemicylinders, — with regard to the ECL efficiency of these systems gave the primary superiority of two hemicylinders. The ECL efficiency in cell with two microelectrode hemicylinders is higher than in similar cell with two microelectrode bands.
References: 1. Amatore C., in: I. Rubinstein (Ed.), Physical Electrochemistry: Principles, Methods and Applications (Chapter 4), M. Dekker, New York. 1995. 2. Lavrentiev M.A., Shabat B.V. Methods of the complex function theory. Moscow: Science. 1973, 736 p. 3. Amatore C.A., Fosset B., J. Electroanal. Chem. 328 (1992) 21. 4. Fosset B., Amatore C.A., Bartelt J.E., Michael A.C., Wightman R.M., Anal. Chem. V. 63, 1991, p. 306. 5. Amatore C, Fosset B., Anal. Chem. V. 68, 1996, p. 4377. 6. Amatore C.A., Fosset B, Deakin M.R., Wightman R.M., J. Electroanal . Chem. V. 225, 1987, p. 33. 7. Amatore C. A., Deakin M.R., Wightman R.M., J. Electroanal. Chem. V. 207, 1986, p. 23. 8. Amatore C, Oleinick A.I., Svir I.B., J. Electroanal. Chem. V. 553, 2003, p. 49. 9. DeakinM.R., Wightman R.M., Amatore C.A., J. Electroanal. Chem. V. 215 1986, p. 49. 10. Amatore C.A., Fosset B, Deakin M.R., Wightman R.M., J. Electroanal. Chem. V. 225, 1987, p. 33. 11. Amatore C.A., Fosset B, ManessK.M., WightmanR.M., Anal. Chem. V. 65, 1993, p. 2311. 12. Arkoub I.A., Amatore C, Sella C, Thouin L, Warkocz J.-S., J. Phys. Chem. B V. 105, 2001, p. 8694. 13. Amatore C, Sella C, Thouin L., J. Phys. Chem. B V. 106, 2002, p. 11565. 14. SvirI.B., OleinickA.I., J. Electroanal. Chem. V. 499, 2001, p. 30. 15. Svir I.B., Oleinick A.I, Klimenko A.V., J Electroanal. Chem. V.513, 2001, p. 119.
Поступила в редколлегию 09.10.2003 Рецензент: д-р техн. наук, проф. Стоян Ю.Г.
Олейник Александр Игоревич, аспирант кафедры БМЭ ХНУРЭ. Научные интересы: математическая физика и численное моделирование. Адрес: Украина, 61166, Харьков, пр. Ленина, 14, тел. 702-13-64.
Аматор Кристиан Андрэ, академик французской академии наук, профессор, зав. кафедрой химии. Адрес: Ecole Normale Superieure, Departement de Chimie, UMR CNRS-ENS-UPMC 8640 “PASTEUR”, 24 rue Lhomond, 75231 Paris Cedex 05, France.
Свирь Ирина Борисовна, д-р техн. наук, гл. науч. сотр., профессор кафедры БМЭ ХНУРЭ. Научные интересы: численное моделирование физико-химических процессов. Адрес: Украина, 61166, Харьков, пр. Ленина, 14, тел. 702-1364.
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