Abstract: The generation of light by chemical reactions between species generated by electrolysis at an electrode surface (electrogenerated chemiluminescence, ECL) is governed by the rate at which these species diffuse together and react. This diffusion controlled process may be modeled with the 1D diffusion PDE under semi-infinite boundary conditions. The analytic solution of the resulting IBVP's and integration of these solutions provides many opportunities for applying the computer algebra system Maple to model an interesting chemical system. A Levenburg-Marquardt procedure, as implemented in Mathcad or gnuplot may be used to estimate diffusion and reaction rate constants by fitting the resulting models to experimental measurements of the rate of light generation in an ECL system.

Introduction: In the ECL process, an electrochemical cell is used to generate light via a recombination reaction. A precursor species (M) is in solution (along with a supporting electrolyte) in the cell. In a typical ECL experiment the working electrode may be held at a potential at which the precursor will be reduced to the radical anion: M + e^- => M^-. The radical anions which form will then diffuse away from the electrode surface. The potential is then switched so that the precursor will be oxidized to the radical cation; this reaction is: M => M⁺ + e^-. The radical cation formed in this step will diffuse away from the surface as well. The radical ions are very reactive species, so it is possible (indeed very likely) that they will react with an impurity in the solution and so be destroyed. However, it is also possible (and does happen) that a radical cation will encounter a radical anion; in this case an electron is transferred between them and two precursor molecules are formed. The energy released in this reaction will leave one of the new precursor molecules in an excited state: M⁺ + M^- => M + M*. The excited molecule will emit a photon and return to its ground state. This process has many variations; they have been well reviewed¹. An elegant model for a related process (where one of the radical ions is at uniform concentration in the bulk of the solution and the other is generated as above) has been proposed by Akins and Snider². In our work, the precursor is perylene (C₂₀H₁₂) and the solvent is a low melting organic salt; the experimental work on this system has been described previously³.

Results and Discussion: The electrochemical cell used for these ECL measurements may be idealized as an infinite, smooth, planar electrode with a uniform current density across its surface. (The actual electrode used for the measurements was a rectangular mesh of very fine platinum wires.) The distance from this ideal surface may be represented as x, and the time after the experiment begins as t. The concentrations, C₁ and C₂, of the radical ions are then functions of x and t. These concentrations are solutions to the 1D diffusion equation: or the 1D reaction-diffusion equation where j is either 1 or 2 for the first or second species generated. The first of these equations applies when the model does not include losses of the radical ions to reactions with impurities. The second equation applies when such losses are included; is the rate constant for this loss reaction. In either case is the constant diffusion coefficient. The solutions to these equations depend quite strongly on the boundary and initial conditions that the solutions must satisfy. In this paper we focus on a case where the boundary conditions include the flux at the electrode surface. The flux is defined⁴ as: . Other boundary conditions are possible and have been used; for example, Faulkner⁵ has modeled the concentrations in ECL using concentration boundary conditions. Once the conditions have been chosen, the IBVP may be solved using Laplace transforms on the time variable, solving the resulting ODE with elementary methods, then using tables⁶ and convolution integrals to invert the Laplace transforms. This gives analytic expressions for . Evaluation of some of the convolution integrals provides an opportunity to use tools like Maple.

The rate of the light producing recombination reaction is proportional to the product of the concentrations of the two radical ions. The experimental geometry of the system is such that light is observed from all locations at once, rather than from any particular distance from the electrode surface. The electrochemical cell is 1cm wide; the magnitudes of the diffusion coefficients for the ions and the durations of the potential pulses is such that the electrochemically generated ions diffuse only a few hundred micrometers. The walls of the cell are effectively at infinity. Thus the experiment is the measurement of the light intensity; this is proportional to the intensity integral: . It is in the evaluation of this integral that Maple finds the greatest use. The experimental intensity curve (along with model curves) is shown in Figure 1.

The model studied for this paper has species 1 produced at constant flux at the electrode surface for the duration from t=0 to t=; its production ceases and species 2 is produced from t= to t= after which neither species is produced. The corresponding boundary conditions for the loss free model are: for ; for ; and for , where H(t) is the Heaviside unit step function, and F₁ is a positive constant. The first two boundary conditions satisfied by the second species are the same, but the third is for .

Once the solutions for this pair of closely related IBVP's has been found, the solutions for the corresponding problem where losses are included may be obtained via a trick due to Farlow⁷: if u_j(x,t) is a solution to the loss free problem, then C_j(x,t) = exp(-r_jt) u_j(x,t) is a solution to subject to slightly modified boundary conditions. The first two BC's are the same as above, but the last is: , and likewise for species 2.

The solution for the first species in the loss free case is known⁶ and is expressed in terms of the integrated complementary error function, ierfc(x). This function is defined as:
. The solution for species 1 is:
. The solution for species 2 may be found from this one by using the translation theorem for Laplace transforms. The boundary conditions for species 2 are the same as for species 1, except that the time origin has been translated from t=0 to t=. Thus, the solution to the loss free PDE for species 2 is:

. The solutions for the case where losses are included are obtained from these by multiplying by the appropriate exponential function.

Next the light intensity integral will be evaluated for the loss-free model. This requires integration of products of the form

.

With the assistance of Maple and use of parametric differentiation⁸, the following intermediate integrals are obtained:

,

,

and

.

Combining these results yields

.

As a partial check, all definite integral formulas were compared to results from Maple's numerical integration algorithms for various sample values of the parameters A and B. The results are in agreement to at least 7 or 8 digits for Maple's default setting of 10 digits. Increasing the digits setting produces higher precision agreement.

The light intensity integral can be written as a sum of four integrals of the last type listed above. Using Maple to combine, collect, and simplify terms yields:

The intensity integral for the model with loss terms is obtained by multiplying the above result by .

Conclusions: The loss free model fits the data quite poorly in that it suggests that the light pulse should be several seconds wide for an experiment where the species are generated for one second each. The model including losses is more successful at fitting the data. It fits most of the experimental data quite well except for a few points near the pulse maximum. The fit also suggests reasonable values for the model parameters: 6.65x10^-5 and 4.37x10^-6 cm²s^-1 for the diffusion coefficients for species 1 and 2 respectively, and 11.82 s^-1 for the sum of the rate constants for the loss reactions.

ECL Data and Model Results

Figure 1. Graph of results: intensity is plotted vertically in photons per second, and time is plotted horizontally in seconds. The traces are: experimental ECL light pulse (open circles), loss free model (dotted line), and best fit to model including loss term (solid line).

References:

1) Faulkner, L. R.; Glass, R. S.; Chemical and Biological Generation of Excited States; Adam, W.; Cilento, G. Editors, p. 191, Academic Press, New York, 1982; Feldberg, S. W.; J. Amer. Chem. Soc. 1966, 88, 390; J. Phys. Chem. 1966, 70, 3928.

2) Akins, D. L.; Snider, A. D.; J. Comp. Chem. 1981, 2, 368.

3) Coffield, J. E.; Zingg, S. P.; Sienerth, K. D.; Williams, S. D.; Lee, C.; Mamantov, G.; and Smith, G. P.; Materials Science Forum, 1991, 73-75, 595; Mamantov, G.; Sienerth, K. D.; Lee, C. W.; Coffield, J. E.; and Williams, S. D.; J. Electrochem. Soc. 1992, 139, L58.

4) Crank, J.; Mathematics of Diffusion, Second Ed., Clarendon Press, Oxford, 1989.

5) Faulkner, L. R. . J. Electrochem. Soc. 1977, 124, 1724.

6) Carslaw, H. S.; Jaeger, J. C. Conduction of Heat in Solids, Clarendon Press, Oxford, 1959.

7) Farlow, S. J.; Partial Differential Equations for Scientists and Engineers, Dover, New York, 1982.

8) Feynman, R. P. Surely You're Joking Mr. Feynman, W. W. Norton, New York, 1985; Squire, W. Integration for Engineers and Scientists, Elsevier, New York, 1970.