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This paper presents the description of triplet-triplet annihilation in the case of nondominant first-order decay of molecules in the triplet state. The kinetics of the statistical system is influenced by joined action of two processes: the first- and second-order decies. This kinetics can be described with analytical function if the rate parameter of second-order reaction is constant. The approach presented here combines the well-known Smoluchowski formula with the previously published intuitive and non-Fickian models of diffusion-controlled triplet-triplet annihilation. The kinetics of the delayed fluorescence of anthracene is used as a practical example of applicability of the model proposed. The advantages and limits of the proposed model are discussed.

The reaction where the diffusion plays an important role was studied for many years [

In the case of triplet-triplet annihilation (TTA) in viscous solvents where the process takes place on microsecond timescale the deviations from diffusional Smoluchowski model [^{2} s^{−1}) at room temperature. The Smoluchowski model can be applied to describe the kinetics of statistical system with partly reflecting boundaries. However, instead of the geometrical radius of interaction the effective one has to be used [

Also the reacting species can be treated as spherically symmetric. There are two reasons that make this approximation valuable. The first one is the very fast rotation of the molecules in comparison with their translational diffusion. This aspect was discussed in earlier papers [

The complicated spatial feature of Dexter mechanism of interaction [

To sum up, in the description of TTA kinetics of in viscous solutions the molecules can be treated as hard spheres which react with each other only when they are in the contact.

In order to make the discussion of the joined action of first-order decay and triplet-triplet annihilation the description is done within basic assumption of the classical Smoluchowki [

the Brownian motions of the reacting particles can be described by diffusion in continuous medium;

the molecules at distances larger than annihilation radius do not interact with each other;

the annihilation within contact pair takes place immediately, it means that the molecules in the T_{1} state cannot diffuse away if they are within an annihilation radius from each other;

the evolution of the local density

In this paper we extend the modifications of the Smoluchowki time-dependent rate parameter to the case of nondominant first-order decay. The modified models (two variations) are applied to the description of the TTA kinetics of anthracene in the mixture of

Let us come back to the analysis of the TTA presented in paper [

In order to extend the intuitive model presented in paper [

Let us take into account three molecules in the

the stationary part of the standard Smoluchowski rate parameter

the time-dependent part

Now one has to find an expression for

The expression for the time-dependent intensity of the delayed fluorescence has the form [

The purification of anthracene and the components of mixture solvent, as well as the preparation of the sample were described in previous papers [

The experimental setup used for these experiments was similar to that described previously [^{+} laser and two slow choppers pulsed laser was used. The excitation wavelength was equal to 363.3 nm. Two configurations of excitation source were used:

dye laser pumped N_{2} laser;

dye laser (Lambda Physik LPD 3000) pumped excimer laser (Lambda Physik LPX 100).

Modification of the previously described [

The decays were measured in the temperature range from 132 K to 152 K in the case of middle excitation pulse and in the range 132 K–150 K for the short pulse. The increment between the temperature values was equal to 1 K.

In order to make our model mathematically as simple as possible we use here similar procedure as in the previous papers [

The delayed fluorescence was analyzed with the following functions:

The parameters obtained from nonexponential decay were compared with the parameters obtained from model described in [

The functions used to compare the nonexponential versions of the intuitive (

As in the previous papers [

Phosphorescence decays were analyzed with the following functions:

In Figure

squares—calculated from the delayed fluorescence with exponential model (

circles—obtained from the delayed fluorescence within nonexponential approach (

solid line—calculated from phosphorescence with exponential model (

dot-dashed line—calculated from phosphorescence with application of nonexponential model (

^{−1}and 30 s

^{−1}. In the case of the exponential fit of the delayed fluorescence and for the delay time larger than about 1 ms the values of the first-order rate constant

The fitted parameters as dependent on the delay (the starting point of evaluation—

The parameters obtained for the excitation with the “short pulse” have very similar dependence on the delay

it increases with the increase of

in the case of

Since the kinetics of delayed fluorescence depends on the mobility of molecules in

in the first one the exponential model gives reasonable description of the kinetics of TTA;

in the other one the application of nonexponential model is necessary in order to obtain reasonable values of the kinetic parameters.

In Figure ^{−1} and 30.8 s^{−1} is shown. This is done in order to present small changes of the first-order rate constant obtained from different approaches. The behavior of the parameter in both parts of Figure

First-order rate constant as a function of temperature. Part (a): measurements performed with middle excitation pulse (dye laser pumped excimer laser), part (b) measurements in the case of excitation with short pulse (dye laser pumped nitrogen laser). Results of following measurements and fits are compared: exponential fit of the delayed fluorescence (squares), nonexponential fit of the delayed fluorescence (circles), exponential fit of the phosphorescence (crosses), and nonexponential fit of the phosphorescence (triangles). All fits were performed for intuitively modified model.

The values of the first-order rate constant obtained from phosphorescence with application of both models are not very different from each other. Generally, the values of

Indicator of nonexponential behavior as dependent on the temperature measured for excitation with the middle pulse—part (a) and short pulse—part (b). Comparison of the data obtained from the fit of the delayed fluorescence (circles) and phosphorescence (triangles).

Since the middle excitation pulse has significantly larger energy than the short one the

138 K–142 K for middle excitation pulse;

138 K–146 K in the case of the short excitation pulse.

The temperature dependence of the

The values of

In Figure

Comparison of the kinetic parameters obtained from intuitive and non-Fickian models is presented in Figures

Comparison of the values of the first-order rate constant calculated with intuitive (a) and non-Fickian (b) models. The values of the parameter are presented as dependent on temperature. The measurements were performed for so-called middle excitation pulse.

Comparison of the values of the indicator of nonexponential behavior calculated with intuitive (a) and non-Fickian (b) models. The values of the parameter are presented as dependent on temperature. The measurements were performed for socalled middle excitation pulse.

In Figure

The behavior of the first-order rate constant for short excitation pulse is similar to that presented for middle excitation pulse. The differences between exponential and nonexponential models are smaller because the energy of excitation pulse was smaller in comparison with the energy of middle excitation pulse. The lower energy of excitation pulse results in lower concentration of molecules in the triplet state.

The temperature dependence of the indicator of nonexponential behavior (

To sum up, the introducing of the nonexponential model offers better description of the kinetics within this temperature range in which the combination of the energy of excitation pulse and the mobility of molecules results in significant deviation from the exponential model. Both models: the intuitive and non-Fickian has their limits. The intuitive model has the tendency to overestimate the first-order rate constant calculated from the delayed fluorescence in comparison with the value obtained from the phosphorescence. The course of the first-order rate constant calculated from the delayed fluorescence with non-Fickian model does not show this systematic difference. At the other side the non-Fickian model gives a good fit to the experimental decay within the smaller range of the temperature.

The indicator of nonexponential behavior obtained from intuitive model equals to zero within the temperature range where the exponential model offers good description. In the case of non-Fickian model this parameter tends to 0.01 instead of zero with the standard deviation significantly smaller than 100% in the temperature range where the exponential approach offers good description of the kinetics.

The most important conclusion coming out from the paper is that the nonexponential approach significantly enhanced the range of experimental conditions where the kinetics of diffusion-controlled TTA has reasonable description. The indicator of nonexponential behavior—parameter

The annihilation radius can be calculated from equation for

The authors thank Professor Jürgen Troe (Max-Planck Institute, Göttingen, Germany) for generous support of the research project. The discussion with Professor K. Rotkiewicz (Institute of Physical Chemistry PAS, Warsaw, Poland) was very fruitful for preparation of this manuscript. The support of stay in Göttingen (Germany) of one of the authors (P. B.) by Max-Planck-Gesellschaft (from 1.10.2000 to 30.09.2001) and by Alexander von Humboldt-Foundation (from 1.10.2001 to 31.12.2002) is gratefully acknowledged. Dr B. Nickel passed away 27/01/2002.

_{2}-thioketones studied by the Smoluchowski-Collins-Kimball model: standard systems