The Kinetics of Joined Action of Triplet-Triplet Annihilation and First-Order Decay of Molecules in T1 State in the Case of Nondominant First-Order Process: The Kinetic Model in the Case of Spatially Periodic Excitation

Paweł Borowicz 2, 3 and Bernhard Nickel 1 Spectroscopy and Photochemical Kinetics Department, Max Planck Institute for Biophysical Chemistry, Am Fassberg 11, 37077 Göttingen, Germany Department of Photochemistry and Spectroscopy, Institute of Physical Chemistry, Polish Academy of Sciences, Kasprzaka 44/52, 01-224 Warsaw, Poland Department of Characterisation of Nanoelectronic Structures, Institute of Electron Technology, Al. Lotników 32/46, 02-668 Warsaw, Poland


Introduction
e technique of measuring the diffusion coefficient using spatially periodic excitation was �rst developed by Avakian and �erri�eld [1], Ern et al. [2], and Ern [3].In those papers the authors used a pattern that consists of parallel slits to create the spatially inhomogeneous distribution of the measured species, namely, triplet excitons in anthracene crystals.e technique was modi�ed by Nickel [4], instead of the pattern with parallel slits the interference of two laser beams was used to create spatially periodic distribution of the density of the molecules in the T 1 state.In this way the drawback of original Avakian's method, the diffraction of the light on the slits was avoided.is method was applied in measurements of the diffusion coefficient of different organic molecules [5,6].
e main aim of this work is to take into account the effect of the joined action of the �rst-order decay of the molecules in the triplet state and diffusion-controlled triplettriplet annihilation (TTA) in the kinetic model that takes into account the effects caused by spatially periodic excitation.e starting points are the intuitive treatment of the spatially homogeneous excitation [7] and the Nickel's model of TTA with spatially periodic excitation [6].e main aim of the present approach is the introducing of the joined action of TTA and �rst-order decay to the term describing the kinetics of the TTA for inhomogeneous excitation-in the diffusional relaxation term.e derived model was applied for the evaluation of the diffusion coefficient of anthracene in the T 1 state.e values obtained with the introduced model are compared with the data obtained from the Nickel's model developed in the framework of standard Smoluchowski approach.e temperature and viscosity dependence of diffusion coefficient is another problem discussed in this paper.e hydrodynamic theory of diffusion predicts the linear dependence between the diffusion coefficient and the ratio -Stokes-Einstein (SE) model [8].At the other side the hole theory of diffusion [9] predicts that viscosity and diffusion constant vary exponentially with the temperature.In this model diffusion is described as a process that has an activation barrier.To allow the molecule to diffuse a free space (hole) must be created in the solute.e energy that is necessary for the creation of this hole is described in terms of the activation barrier.In the hole model the diffusion coefficient tends to the limited value when temperature tends to in�nity, whereas in the SE model the diffusion coefficient tends to in�nity with the increase of temperature.e Stokes theory of diffusion was modi�ed by �ierer and �irtz [8,10].Instead of the continuous medium they took into account the �nite dimensions of the solvent and solute molecules.ey introduced to the SE model a correction which is a function of the radii of solvent and solute species.
In this work we postulate the equation for the diffusion coefficient as dependent on the  which combines the �nite limit of the diffusion constant for large values of  as in the hole theory of diffusion and the linear dependence of the diffusion coefficient for small  as predicted by SE equation.

Description of the Kinetic Model
2.1.Temporal Behavior of Delayed Fluorescence aer Spatially Periodic Excitation.e kinetic equation derived for spatially periodic excitation within framework of standard Smoluchowski model is the start position of the discussion.In the case of strongly dominant �rst-order decay: (4     0   ) ≪ 1% and for the spatial period     the equation for the delayed �uorescence intensity can be written in the following way [6]: where  0 is the constant,   is annihilation radius,   is mutual diffusion coefficient,   is the �rst-order rate constant,  0 is the density of the molecules in T 1 state at the time  = 0, and  is the normalized parameter specifying the degree of interference.Its value varies from 0-no interference to 1-complete interference.e parameter  (diffusional relaxation constant) describes the diffusional relaxation resulting from inhomogeneous (here spatially periodic) distribution of the molecules in the T 1 state at  = 0 and is a function of the spatial period of excitation  and the absolute diffusion coefficient  0 (in the case of TTA  0 =   2): Equation ( 1) can be divided into two parts: DF () =  homo DF ()  period DF

(𝑡𝑡) . 󶀡󶀡1′󶀱󶀱
e �rst one describes the decay of the molecules in the T 1 state aer spatially homogeneous excitation in standard Smoluchowski approach:  homo DF () =  0 4     2 0 1 +        −2   =   2 ()  2 () , where the convention of the indications (  2 ()) is the same as in the paper [11] it means   2 () is the time-dependent rate parameters in the Smoluchowski original model.e modi�cation of the �homogeneous� kinetics was described previously [7] for the case of strongly dominant �rst-order decay and extended to nondominant �rst-order decay in the �rst paper of this series [12].
e other term introduces the changes in the kinetics caused by spatially periodic excitation, namely, describes the diffusional relaxation of the inhomogeneous initial distribution of the molecules in the triplet state: is relaxation modi�es the kinetics of homogeneous distribution of the molecules in the triplet state.Since TTA do not contribute directly to this term the changes in timedependent intensity of the delayed �uorescence are caused only by the average decay of the concentration of the molecules in T 1 state in the sample.e changes in the local concentration involve the changes in the efficiency of the TTA in different areas of the sample.e lack of the direct contribution of the TTA to the diffusional distribution manifest itself by the absolute not relative diffusion coefficient appearing in the term  period DF .e diffusional relaxation will be in�uenced by the �rstorder decay which decrease homogeneously the concentration of molecules in the triplet state.e diffusional relaxation constant depends on the gradient in the distribution of molecules in the T 1 state between areas of the highest and the lowest concentration of the reacting species.is gradient is built by spatially periodic excitation, so one of the factors in�uencing the parameter  is the spatial period .e mobility of the molecules is represented by absolute diffusion coefficient  0 .However, due to homogeneous decay of the molecules in the T 1 state via �rst-order decay the initially built gradient will change with the time by joined action of the redistribution and the �rst-order decay.�e assume that the �rst-order decay can be treated as dominant and the diffusional relaxation is the function of the mobility of molecules.at means that homogeneous decay do not in�uence the redistribution process in other way as changing the gradient via homogeneous decrease of the concentration of the molecules in the T 1 state.In such a case the parameter  introduced as diffusional relaxation constant should be corrected by time-dependent factor.Taking into account the independence of �rst-order decay and the redistribution of the molecules one can represent the diffusional relaxation in terms of joined action of the �rstorder decay and the diffusional redistribution by introducing the time-dependent parameter  � (   (   instead of  in the periodic term: So intuitively the modi�ed equation for the time-dependent intensity of delayed �uorescence should have the form in the case of dominant �rst-order decay: ere is also one more point in the model that should be discussed.Since, as mentioned before, in different places in the sample one has different intensity of the exciting laser light it cannot be excluded that in the areas with higher exciting intensity small components of nonexponential behavior of the sample exist.Due to irradiation with so-called long excitation pulse, where the intensity is so small that by homogeneous excitation the nonexponential behavior of the kinetics can be generally avoided, we expect to have the nonexponential behavior only in the part directly related to the TTA, it means in homogeneous term of  DF (.e most sensitive part in  homo DF ( in the context of nonexponential behavior is  2 (, because in  2 ( the changes of ( contribute linearly to the time-dependent rate parameter.So the most simple way to test the nonexponential behavior of the spatially periodic decay is to use the following modi�cation of the model: (1) the time-dependent rate parameter has the form as in the case of the intuitively modi�ed model with �rstorder decay to be dominant: (2) the density of the molecules the T 1 state has the nonexponential form: where  donate (1 +   /      , (3) the periodic part of  DF ( is the same as in (5).
e density described in point 2 has the form as in (4) in the �rst paper of this series �12], but with  TT as timedependent rate parameter de�ned by (7).
In the temperature above 150 K the contribution of the short time effect to the intensity of the delayed �uorescence decreases very fast.If the spatial period of the irradiation is enough large the diffusion relaxation resulting from inhomogeneous excitation takes place mainly for so large values of the delay  0 that the contribution of the nonstationary part can be neglected.Since it takes place generally at the temperatures, where the diffusion coefficient   is large the nonexponential approximation equation (9) seems to be the proper choice in this case.However due to negligible contribution of the short-time effect the equation can be simpli-�ed by neglecting the contribution of the nonstationary term.Neglecting the ratio: where  is the Boltzmann constant and  is the radius of the particle.Viscosity  is a function of temperature and it decreases with the increase of .In SE model  0 → 0 if  → 0 and  0 → +∞ if  → +∞.e lower limit of  0 is reasonable: with  → 0 the viscosity  → ∞, so the mobility of the molecules should be very small.e other limit:  0 → +∞ seems to be arti�cial.is limit suggests that at very high temperatures the process of diffusion should be in�nitively fast, it means the in�nitively large mass should diffuse through �nite area within the �nite time period.At the other side the hole model of the diffusion postulate the diffusion coefficient to tend to �nite limit  ∞ and to be exponentially dependent on the temperature: In this model the process of diffusion of solute molecule is described as series of jumps between equilibrium positions of the solute molecule.e equilibrium positions are separated by barriers.is equilibriums result from the �nite (nonzero) dimensions of the solvent and solute molecules.e origin of the barrier is to create of the "hole" among the solvent molecules.is "hole" is the free space where the solute molecule can migrate.e energy  is the free energy of the barrier which corresponds to the process of creating a hole in the solvent.Since the energy  depends on viscosity, the diffusion coefficient also depends on  even if it is not written explicit in (12).e hole model of diffusion takes into account the corpuscular nature of the solvent.
Since measured values of diffusion coefficient show nonlinear dependence as a function of  and the course of the functions seems to tend to �nite limit (see Section 4) it would be sensible to postulate an equation for  0 satisfying following conditions: (1)  0 → 0 for  → 0; (2) for small  the new equation should be approximated by SE model; (3)  0 →  ∞ for  → ∞.
We postulate the equation in the form: e model de�ned by (13) will be named as combined model because it has  as an argument of the function as in the SE model and has the exponential dependence of the diffusion coefficient from the independent variable and �nite limit of the diffusion constant for high temperatures as in the hole model.
Since the difference (1 − exp(−())) equals to 0 for  = 0,  0 ( = 0) = 0; for  → ∞  0 →  =  ∞ and the expansion of (13) in Taylor series around  = 0 gives From (11) and ( 14) one can obtain e �nite limit of the diffusion coefficient when the value of the argument of function tending to in�nity is the great advantage of the combined one.is kind of the dependence of the diffusion coefficient shows that the mass transport tends to saturation at very high temperatures.is kind of behavior, namely, the saturation of the efficiency of the process for conditions tending to the limit is observed in the case of other transport processes.e example of the setup where the saturation is observed is charge transport in structures like Junction Field-Effect Transistors (JFET) or Metal Semiconductor Field-Effect Transistors (MESFET) [13].e saturation effect in the electronic structures is observed for drain current for JEFT and MESFET structures [13] and as well for accumulation-mode (AM) -channel siliconon-insulator (SOI) metal-oxide-semiconductor-�els-effecttransistor (MOSFET) or enhancement-mode (EM) -channel SOI MOSFET [14].e saturation effect in the case of electronic devices is so important that the models of charge transport developed for structures like SOI -type MOSFET [15] or high electron mobility transistors (HEMTs) based on AlGaN/GaN [16] structures have this effect as an necessary condition of correctness of the model.e sublinear behavior of charge transfer with the tendency to saturation is also observed for organic crystals that is for perylene or deuterated naphtalene [17].

3.2.
Apparatus.e apparatus was described in details also in previous papers [7,11,18] where so-called side excitation was used.Here we will concentrate on the parts of the setup that make possible to realize two way of excitation: the side one and the bottom one.In this paper we will use the names: illumination/excitation from the side (side excitation) or from the bottom (bottom excitation).Special attention will be focused on the elements necessary to excite the sample in spatially periodic way.
In order to create the spatially periodic distribution of the molecules in the T 1 state in the sample in wide range of viscosity two options of the excitation were used.e details of the part of the apparatus generating spatially inhomogeneous distribution of the molecules in the T 1 state are presented in Figure 1.Starting from side illumination: e laser beam prepared by the fast chopper [7,11] was re�ected in the vertical direction towards mirror M8 (see scheme of the setup in [7,11,12]).M8 was used to re�ect the laser beam horizontally to generate the homogeneous or spatially periodic distribution of the molecules in the T 1 state.One of the wall of the sample cell was partly covered with the dielectric mirror re�ecting back the laser beam from Ar  laser.Since the coherence length of the laser beam was 50 mm and the pathway in the cuvette was equal to 10 mm the re�ection of the 30 s long pulse generated the standing wave in the sample.In the case of homogeneous excitation the cell was moved down and the excitation beam was passing the sample over the top edge of the dielectric mirror [7,11,18].is option of spatially periodic excitation is presented in Figure 1 F 1: Two methods of creation of spatially periodic pattern of the exciting light: (a) excitation from the side: illumination with standing wave, (b) excitation from the bottom: illumination with two crossing beams.  is the refractive index of the cis-/trans-DMCH at the wavelength equal to ,  0 : the spatial period in the case of side excitation, : spatial period for excitation from the bottom, -the angle of the prism, and   is the incident angle of the laser beam in the case of the bottom excitation.
other possibility of inhomogeneous excitation: the illumination from bottom is presented in Figure 1(b).e mirror signed as M7 in [7,11] was replaced by M7 � .Mirror M7 � is adjusted in such a way that it directs the laser beam to the mirror M8 � and then to the beamsplitter BS instead of re�ecting it in the vertical direction to the M8.en the laser beam was divided by beam splitter BS.Both beams were re�ected by mirror M0 and entered the cell through the prism placed on the bottom wall of the cuvette.e prism angle was equal to 60 ∘ .Due to refraction of both beams on the prism walls the directions of both laser beams changed in such a way that the beams crossed in the sample and generate the spatially periodic distribution of the exciting light.In order to measure the decay in the case of homogeneous excitation from the bottom one of the crossing beam was cut-off.In the case of interference of two laser beams having equal intensities the light intensity in the maximum should be four times larger than the intensity of each interfering beam.To have the same intensity of the exciting light for homogeneous excitation and in places of positive interference of laser beams different energy of laser pulses were used for homogeneous and spatially periodic excitation.e energy of the pulse in the case of homogeneous excitation was 4 times larger in comparison with the energy of the laser pulses used for spatially periodic excitation.
e spatial period of the distribution of the exciting light can be calculated from the following equation [4]: where  is the wavelength of the exciting laser light,   the refraction coefficient of the solvent for the wavelength , and (2) is the angle between two crossing beams in the solvent.e angle   2 corresponds to the standing wave (the side illumination).e idea of spatially periodic excitation is presented in Figure 2.

Evaluation
Procedure.e data were �tted using the following procedure.First the decies obtained for spatially homogeneous excitation were �tted with intuitively modi�ed model [7]: where  0 is a dark current,  1 is the amplitude,  2 will be called Smoluchowski parameter ( 2    (  ) 12 , and  3 is the �rst-order decay time ( 3    −1 ).Obtained from the �ts of homogeneously excited decies the parameters  2 and  3 were used in the �tting of decies measured for spatially periodic excited samples.To �t the periodically excited decies the following functions were used: where  4 describes the degree of the interference,  5 is the diffusion relaxation time ( 5   −1 ), and  6 plays the role of the indicator of nonexponential behavior and equals e function de�ned by (18) describes the model developed by Nickel [4][5][6] within standard Smoluchowski approach.It was used here as a reference.e function de�ned by ( 19) can be treated as intuitively modi�ed (18) and describes the condition of strongly dominant �rst-order decay.e equation (20) introduces to (19) the nonexponential approximation on the lowest possible level (see Section 2).e functions given by ( 19) and ( 20) take into account both components of the decay in intuitively modi�ed model: the short time part and the stationary part.e diffusional relaxation has to play important role in the kinetics on different time scale in comparison with short time effect.Otherwise it will be impossible to differentiate between these two components of the model in the case of analysis of the data measured with spatially periodic excitation of the sample.
e measurements of the diffusion coefficient were performed in the temperature range from 135 K to 155 K. e viscosity of the mixture solvent changes over four orders of magnitude.In the vicinity of the ends of the temperature range the differences in time scales of the short time effect and diffusional relaxation may be so large that the simpli�ed equation for the  DF ( may offer more stable �t than (19) or (20).Taking the diffusion coefficients used in previous papers [7,11] and the approximate value of annihilation radius equal to 1 nm and the spatial period of excitation as used in the experiment one can estimate the contributions of the short time term and the diffusional relaxation term to the whole decay of the delayed �uorescence.e measurements were done for two values of spatial radius: 119 nm and 305 nm.In the next paragraph the estimation of the short time term and the diffusional relaxation term will be presented.
In the case of low viscosity where the diffusion is fast the short time effect decay also very fast.In order to have reasonable time for the diffusional relaxation one should excite the sample with enough large spatial period.For example, in the case of the decay measured at    K aer the delay equal to about 2 ms the term describing short time effect equals to about 0.06 whereas the diffusional relaxation term: exp(− exp(−   equals to ∼0.60 if the spatial period of excitation equals to about 305 nm.Aer the delay equal to 20 ms the short time term equals to 0.02 and the diffusional relaxation equals to about 0.28.So, if the spatial period of excitation is enough large the diffusional relaxation mainly takes place in the delay range where �ow can be approximated with stationary part of the intuitively modi�ed Smoluchowski equation.Since the bottom excitation gives small signal, because only small part of the illuminated part of sample contributes to the measured signal the number of the �tted parameter should be limited in order to make the �tting procedure as stable as possible.In such a case (20) should be simpli�ed:

󶀡󶀡2𝑇′󶀱󶀱
At the other side in high viscosities the diffusional relaxation can also be important for longer delay than the short time effect.For example, in the case of the decay measured at 135 K and aer the delay equal to about 3 ms the short time term decreases to the value equal about 0.09 whereas the diffusional relaxation term equals to about 0.95 if the spatial period of excitation equals to 119 nm.Aer 20 ms the short time term decreases to 0.06 and diffusional relaxation term to about 0.80.So, as in the case of high temperature the main part of diffusional relaxation takes place when the process can be described with stationary part of the intuitively modi�ed Smoluchowski equation.�ere the intensity of the delayed �uorescence is low due to low mobility of the molecules.In such conditions the �t of the function with minimum number of important parameters can give more stable results than the �t of the whole function de�ned by (19).e simpli�ed equation (19) has the following form:  DF (    +    +  4  −( − 2     −2 3 .9′ e other approximation used here is the same as in previous papers [7,11]: it means that modi�ed initial condition introduced by Nickel et al. [19] have signi�cant in�uence on the initial part of the decay.In those papers the decay was divided into anti-Smoluchowski and Smoluchowski time ranges.Within the anti-Smoluchowski time range the modi�ed initial conditions introduce to the temporal behavior of the decay signi�cant changes in comparison with standard Smoluchowski behavior.In the Smoluchowski time range the course of the decay of the delayed �uorescence is the same for both types of initial conditions.e points are systematically cut off from the beginning of the decay of spatially homogeneous excited samples.In this way the part of the decay where the modi�ed initial condition introduce the important changes to the kinetics is removed.e parameters  2 and  3 were obtained from the evaluation of the decies measured for the spatially homogeneous excited samples.ey are kept constant in the case of the �tting of the spatially periodic excited samples.e values of the parameters  4 ,   and in the case of nonexponential behavior  6 are obtained from Smoluchowski time range of spatially periodic excited decay.e anti-Smoluchowski and Smoluchowski time ranges are assumed to be the same in the case of homogeneously and spatially periodic excited samples.e reason to avoid the anti-Smoluchowski time range in the �tting procedure is the same as in previous papers: (i) the modi�ed initial conditions do not affect the general idea of the joined action of two processes: the �rst-order decay and ��A� (ii) the mathematical description of the kinetics is much simpler for the standard Smoluchowski initial conditions.
e Smoluchowski, anti-Smoluchowski time range are presented in Figure 3 together with the delay range of particular interest.e values of the parameters  4 ,  5 , and eventually  6 were obtained as follows.e functions de�ned by ( 18) ÷ (20) were �tted to the measured decies with the systematic cutting off the points from the beginning.Using this procedure one obtains the parameters as functions of the delay,  0 .In the case of (19) for exponential behavior or (20) in the case of nonexponential behavior one should obtain the constant values (within the accuracy of the measurement and the �tting procedure) of parameters  4 ,  5 , and eventually  6 in the time range of particular interest (see Figure 3).In the case of the decies measured near the limits of the temperature range the simpli�ed functions: equation 19′ for exponential behavior and 20′ for nonexponential case were �tted and the results were compared with the �ts performed for the whole functions (see (19) and ( 20)).e option giving more stable �t and precise results were taken to the further analysis.e values of parameters  4 and  5 obtained from the �tting of (18) were taken as reference.e values of the parameters were averaged over the range of particular interest.e mutual diffusion coefficient was calculated from the parameter  5 : where the spatial period  was calculated for the wavelength of the Ar + laser  = 6 nm and the refractive index of cis/trans-DMCH.e refractive index   of the mixture solvent was measure and published previously [20].e temperature and viscosity dependence of the diffusion coefficient was examined with the following procedure.First the linear functions was �tted to the values of the absolute diffusion coefficient calculated from (22): Since in the SE model  0 (( = 0 = 0 the coefficients  0 and  1 obtained from �ts of (23a) and (23b) should be equal within experimental error.e coefficient  1 can be treated as an additional component of the standard deviation of  0 .e contribution of this component should be important in high viscosities.Also the other model of temperature/viscosity dependence of the diffusion constant was �tted to the experimental points: F 3: e in�uence of the modi�ed initial conditions and the spatially periodic excitation.e main picture: the intensity of the delayed �uorescence of anthracene measured at 143 �. e upper curve presents spatially periodic excitation, lower-homogeneous.e ordinate is presented in logarithmic scale in order to expand the initial period where the differences between the temporal behavior of the homogeneously and spatially periodic excited sample are signi�cant.e anti-Smoluchowski time range the extent of the delay where modi�ed (�ickel�s) initial conditions introduce signi�cant changes to the kinetics of diffusion controlled ��A, the Smoluchowski time range the extent of the delay where the temporal behavior of the sample is the same for both type of initial conditions.e time range of particular interest the extent of the Smoluchowski time range where the temporal behavior of homogeneously and spatially periodic excited sample are signi�cantly different.Insert: the initial part of the main picture with the ordinate in linear scale.
e values of diffusion coefficient were calculated from both models: SE model (see (23a), (23b), and combined model (24)).e annihilation radius   was calculated from  2 (Smoluchowski) parameter obtained from the �ts of the decies of homogeneously excited sample: using values of   calculated from the both models of temperature/viscosity dependence of the diffusion coefficient.e values of   calculated for the experimental data resulting from measurements for short and middle excitation pulses are compared with those obtained for the long excitation pulse [7,11].Also the values of the annihilation radius calculated with the diffusion coefficient obtained from SE and combined models are compared with each other.

�.�. �omparison o� the �ickel�s an� �o�i�e� �pproaches�
Exponential Decay.In Figure 4 there is presented the comparison of the �t of two models: original �ickel�s approach (18) and intuitively modi�ed one (19).Since the parameters  2 and  3 were taken from the �t of homogeneously excited decay they are not presented here.e comparison is reduced to the diffusional relaxation time  5 , -Figure 4(a) and the degree of interference  4 -Figure 4(b).Both parameters were obtained from the �tting procedure performed for the decay measured at 142 K. e parameters  4 and  5 are presented as dependent on the delay,  0 .e course of the parameters are similar for both models.In the case of  5 (Figure 4(a)) aer the decrease for  0 < ∼500 s the values reach shallow minimum which can be treated constant within the accuracy of the measurement and �tting procedure.For  0 > ∼2500 s the diffusional relaxation time has clear tendency to grow up with the increase of the delay.e values of  5 obtained from the intuitive model are about 500 s smaller than those from Nickel's original approach in the range of  0 , where the plateau is reached.e course of the parameter  4 (Figure 4(b)) as a function of the delay corresponds to the course of  5 .For  0 < 500 s the values of the degree of interference increase with the increase of the delay.In the range 500 s <  0 < 2500 s there is a shallow maximum where the values can be treated as constant.For  0 > 2500 s the values of  4 decrease with the increase of the delay.e insert in Figure 4(b) presents  2 as a function of  0 .For the delay larger than 500 s the values of the  2 can be treated as constant and equal to about 1.2 for both models.
In Figure 5 there are presented the values of diffusion coefficient   (average values) calculated from original Nickel's and intuitively modi�ed model as a function of  in the range below 0.12 K/m Pa s (what corresponds to the temperature below 149 K). e experimental data are presented together with the �t of SE model.e measure� ments performed in the range  < 0055 K/m Pa s (the temperature below 145 K) were done for the side excitation.In the case of higher temperatures the larger spatial period of excitation was necessary.ese measurements were done  Nickel's formula for the temporal dependence of the intensity of delayed �uorescence.e correction gives the results consistent with SE model in the range 0.055 K/m Pa s <  < 0.12 K/m Pa s. e difference between experimental diffusion coefficient and SE model in the range ∼0.015 K/m Pa s <  < ∼0.055 K/m Pa s is not fully corrected with the intuitive modi�cation of the Nickel's equation.e additional difference between the experimental values and SE model results from nonexponential decay of the spatially periodic excited sample.is is due to the side excitation of the sample within this range of  where the energy of the exciting pulse was signi�cantly larger than in the case of bottom excitation (see Section 3).

�.�. �onexponential Contrib�tion to the �odi�ed �odel in the
Case of Spatially Periodic Excitation.e parameters  5 and  4 as a function of the delay are presented in Figure 6, where the values obtained from exponential and nonexponential models (see Description of the model, part Temporal behavior of delayed ��orescence a�er spatially periodic excitation) are compared.In Figure 6(a) there is presented diffusional relaxation time  5 as a function of the delay.For exponential model the values of  5 decrease with the increase of the delay for  0 < ∼400 s.For ∼400 s <  0 < ∼1300 s the values obtained from exponential model show shallow minimum.e values within this range of  0 can be treated as constant within experimental error.For  0 > ∼1300 s the rapid increase of  5 with the delay is observed.In the case of nonexponential the course of  5 is similar to the exponential case: for  0 < ∼400 s the decrease of  5 with the increase of the delay is observed.For ∼400 s <  0 < ∼1400 s the values of  5 can be treated as scattered around the constant.For  0 > ∼1400 s the parameter  5 has tendency to decrease with the increase of  0 .Also the increase of the standard deviation Δ 5 is observed for  0 > ∼2200 s.e behavior of degree of interference-Figure 6(b) corresponds to that of diffusional relaxation time.For  0 < ∼400 s the values of  4 obtained from both types of modi�ed model (exponential and nonexponential) increase with the increase of the delay.In the case of exponential �t parameter  4 reaches the shallow maximum for ∼400 s <  0 < ∼600 s and for larger  0 the decrease of the value of  4 with the increase of the delay is observed.e values of  4 resulting from nonexponential �t can be treated as constant within the accuracy of the measurement and �tting procedure for ∼400 s <  0 < ∼1000 s.For  0 > ∼1000 s the values of the parameter  4 are scattered and the standard deviation Δ 4 is much larger than for the delay below 1 ms.e insert in Figure 6(b) presents the indicator of nonexponential behavior- 6 .Although the values are very small (not larger than 0.003) they can be treated as constant within the presented accuracy for the delay between ∼400 s and a few milliseconds.
In Figure 7 there is presented diffusion coefficient obtained from kinetic model with taking into account the nonexponential character of the decay in the range ∼0.015 K/ m Pa s <  < ∼0.055 K/m Pa s. e presented values were calculated from exponential:  < ∼0.015 K/m Pa s and nonexponential: ∼0.015 K/m Pa s <  < ∼0.055 K/m Pa s �ts.e �experimental� values of the diffusion constant are presented together with the SE model.e range of  from 0.003 to 0.052 K/m Pa s corresponds to the temperature range 135 K ÷ 145 K. e difference between SE model and the result of the �tting of exponential modi�ed model of the periodically excited delayed �uorescence increases with the increase of .e nonexponential model of the decay seems to reproduce better the SE dependence of the diffusion coefficient than the exponential approximation of the decay.e effect is small but the tendency seems to be unequivocal.

Negligible Short Time Effect.
As mentioned in Evaluation procedure in some cases the short time effect can be neglected and the simpli�ed functions of  DF ( can be used.In Figure 8 there is presented an example of evaluation of the decay measured at 155 K.Here the results of the �tting procedure obtained from whole and simpli�ed kinetic models are compared.e course of the parameters obtained with or without the contribution of the short term are almost the same.e diffusional relaxation time (Figure 8(a)) shows the increase of the value of the parameter  5 for  0 < ∼250 s.
For the delay from the range ∼250 s to ∼1 ms the values are scattered around the constant.For  0 > 1 ms the increase of the standard deviation Δ 5 and the value of  5 is observed.e increase of Δ 5 re�ects the decrease of the contribution of the diffusional relaxation to the total decay.e values obtained for the  4 parameter (Figure 8(b)) show the decrease as a function of the delay in the range:  0 < ∼250 s.For ∼250 s <  0 < ∼700 s the values of  4 can be treated as scattered around the constant.e dispersion of the values obtained with the neglecting of the short time effect 20′ is signi�cantly smaller in comparison with the data resulting from the �t of (20).For  0 >∼700 s the the values of  4 decrease with the increase of the delay.e parameter  4 tends asymptotically to zero.e values of  2 are presented in the insert in Figure 8(b).e values of  2 are constant and equal to about 1.05 for both version of the kinetic model.
In Figure 9 there is presented similar comparison as in Figure 8.Here however, the evaluation was performed for the decay measured at 136 K. e complete function of the kinetic model was de�ned by (19), the simpli�ed one by 19′.In Figure 9(a) the diffusional relaxation time is presented as dependent on the delay.In the case of the �t with short time term included only �rst two points are presented in the plot.e other values obtained from the �t of ( 19) are much larger than the scale and their standard deviation is larger than the value of the parameter.So they are omitted in the plot.e results of the �t of 19′ show the decrease of  5 with the increase of the delay for  0 < ∼800 s.For ∼ 800 s <  0 < ∼8000 s the values of the diffusional relaxation time can be treated as scattered around the constant.e standard deviation Δ 5 grow up with the delay.is behavior re�ects the decrease of the contribution of the diffusional relaxation to the whole decay with the increase of the delay.For  0 > ∼8 ms the decrease of the  5 with the increase of the delay is observed.e degree of interference shows similar behavior-Figure 9(b).e data obtained from �t of (19) (with short-time term included) are not presented due to the standard deviation overcoming 100% of the value of the parameter.In the case of the analysis neglecting short time effect the values of  4 are scattered around the constant for  0 < 8 ms.e exception is the �rst point which standard deviations is larger than the value.e increase of the standard deviation Δ 4 is observed as in the case of the diffusional relaxation time.is increase is very clear for the delay above ∼5.5 ms.For  0 > ∼8 ms the increase of the  4 with the increase of the delay is observed.Also the increase of Δ 4 for the delay above 8 ms is much more signi�cant.From the course of  4 and the ratio Δ 4  4 one can deduce that the contribution of the diffusional relaxation to the whole decay decreases signi�cantly within the delay range ∼5.5 ms <  0 < 8 ms.For the delay larger than 8 ms this contribution seems to be negligible.e insert in the Figure 9(b) presents the  2 as dependent on the delay.e value obtained for the �t of (19) are spread between 1 and 2.4, whereas the values of  2 resulting from �t of 19′ are constant and equal to 1 for  0 > ∼500 s.
It seems that in the case of periodically excited samples in some cases the simpli�ed functions with reduced number of parameters give better results than the complete one.is is due to the complicated temporal dependence of the delayed �uoresce which is described with function having up to 7 parameters.In some cases simpli�cation of �tted function results in avoiding the parameters that values are negligible.is gives in turn more stable �t and more reasonable results.Let us compare the diffusion coefficient measured for organic molecules (having the size similar to anthracene) in organic solvents at room temperature with the data extrapolated with both models.e collection of the anthracene absolute diffusion coefficients ( 0 ) in different solvents can be found in [6].e values at 25 ∘ C are in hexane (3.15÷3.18)×10−5 cm 2 s −1 , in octane 1.97÷2.04×10−5 cm 2 s −1 , in hexadecane 0.536 ÷ 0.545 × 10 −5 cm 2 s −1 , in per�uororohexane (1.80 ± 0.03) × 10 methylcyclo-hexane (1.61 ± 0.03) × 10 −5 cm 2 s −1 .e diffusion constant of pyrene and 9,10-diphenylanthracene in hexane equal to (2.93 ± 0.04) × 10 −5 cm 2 s −1 and (1.80 ± 0.02) × 10 −5 cm 2 s −1 , respectively.Taking into account that   = 2 0 the cited data can be summed up: the absolute diffusion coefficient at room temperature in the range between 10 −5 and 10 −4 cm 2 s −1 .e value of  0 extrapolated with the combined model ∼0.7 × 10 −4 cm 2 s −1 can be treated as reasonable, whereas the values calculated with SE model overestimate the experimental day by a few orders of magnitude.e expansion in the Taylor series of the combined model gives the product of the coefficients  3 and  3 (see (24)) equal to (5.3 ± 1.1) × 10 6 nm 2 K (m Pa s 2 ) −1 , where the standard deviation Δ( 3  3 ) was calculated from e slope obtained from the �t of SE model equals to (4.51 ± 0.05) × 10 6 nm 2 K (m Pa s 2 ) −1 .Taking into account standard deviations of the above discussed quantities both presented above values can be treated as equal.

Annihilation Radii.
Using the diffusion coefficients   obtained from both discussed above models the annihilation radii were calculated.In Figure 11 there are presented annihilation radii obtained for the measurements with short excitation pulse and spatially homogeneous excitation.e data presented in Figure 11 consist of values calculated from exponential and nonexponential approaches of the temporal dependence of the delayed �uorescence intensity.
In Figure 11  shows that the standard deviation of   does not increase with the drop of the temperature as in the case of SE model.In Figure 12 there is presented the annihilation radius   as a function of temperature for the short excitation pulse and the model based on the non-Fickian treatment of TTA [11].e partition of the temperature range between exponential and nonexponential part is the same as in First the annihilation radii calculated from the non-Fickian treatment are larger in comparison with that from the intuitive modi�cation of Smoluchowski equation.e average value for the "intuitive"   equals to about 0.6 nm, whereas the non-Fickian approach gives the value of annihilation radius equal to about 1 nm.is discrepancy was already discussed in previous paper [11].It results from the screening effect of the �rst coordination shell which is reproduced in the non-Fickian approach.e other difference is the scattering of the values of the annihilation radius in the range 132 K ÷ 134 K which is not observed in the case of intuitively modi�ed model.�t the other side, the values of   calculated for  2 parameter obtained for intuitively modi�ed kinetic model has the tendency to decrease with the decrease of the temperature in the range 132 K ÷ 134 K.In the case of non-Fickian kinetic model this tendency is not observed.
In Figure 13 there is presented the annihilation radius   as dependent on the temperature.Here the values were calculated for middle excitation pulse and intuitively modi-�ed model of temporal behavior of the intensity of delayed �uorescence.e convention is the same as in the case of Figures 11 and 12.It means: in Figure 13 Figure 11.e changes of the standard deviation Δ  shows the same behavior as a function of temperature for short and middle excitation pulses.It means that the main factor contributing to the accuracy of   is the standard deviation of the mutual diffusion coefficient-Δ  .Also the average value of   in the temperature range where annihilation radius can be treated as constant is for both kind of excitation pulse equal within the accuracy of the measurement and the accuracy of the evaluation procedure.
In Figure 14 there is presented the temperature dependence of   in the case of middle excitation pulse and the non-Fickian model of the temporal behavior of the intensity of delayed �uorescence.e convention is the same as in Figures 11,12, and 13: part (a) presents values of   calculated with   from SE model and part (b)-with   from combined model.e behavior presented in Figure 14 is similar to that one presented in Figure 12 for short excitation pulse.ere is only one difference: the average value of   in the temperature range where the annihilation radius can be treated as constant equals to about 0.7 nm for middle excitation pulse.is value is slightly smaller than average   for short excitation pulse and the same model of temporal behavior of  DF (.

Concluding Remarks
e introduction of the nonexponential modi�cation to the model describing the temporal behavior of the intensity of delayed �uorescence allowed to enlarge the range of the applicability of the model.In the case exponential model to obtain a good �t one has to perform the experiment in such condition that the measurement of single decay takes very long time (several hours).e measurement of the kinetics of TTA taking long time is extremely difficult because the temperature of the sample must be stabilized within the range of ±0.1 K. Otherwise the �uctuations of viscosity are too large to get reasonable values of parameters.In the case of application of exponential model the mathematical condition:  2 (    < 0.01 must be satis�ed.In the case of nonexponential approach one can obtain good �t for the same parameter equal up to about 0.06.e values of the annihilation radius obtained from nonexponential model are in a good agreement with that from literature and from modi�ed exponential models [7,11]. It seems that the combined model reproduces quite well the dependence between the diffusion coefficient and the ratio temperature to viscosity over several orders of magnitude.is treatment also removes the arti�cial upper limit of the diffusion coefficient in SE model:  0   when   .In the case of very high viscosities application combined or SE model to calculate   from intuitively modi�ed Smoluchowski equation results in the decrease of annihilation radius with the decrease of temperature.is kind of decrease of   is not observed for the data calculated from non-Fickian kinetic approach.In the case of SE model the increase of Δ  with the decrease of the temperature for   1 K is observed.e reason for this behavior is increasing contribution of the uncertainty concerning with the constant factor from (23b).
ere are two limits of the discussed model of the kinetics of TTA.First, the nonexponential modi�cations are based on time-dependent rate parameter  2 ( from Smoluchowski equation.So in the matter of fact it contains the traces of the time-dependent rate parameter applied for the case of dominant �rst-order decay [7,11,18].e other limit is concerned with the case of spatially periodic excitation.e evaluation of spatially periodic excited samples is performed as for the spherically symmetric problem, although this irradiation introduces the cylindrical symmetry to the sample which axis is perpendicular to direction of the changes of the intensity of exciting light.

F 2 :
e idea of the creation of the spatially periodic excitation in the sample.Two laser beams having the wavelength equal to  in air cross each other in the sample and create the spatial pattern of the light with period equal to .e angle between crossing beams equal 2, the refractive index of the sample equals to   .

F 4 :
e diffusional relaxation time (part (a)) and the degree of interference (part (b)) as dependent on the delay.e comparison of two models: circles: Nickel's approach, squares: intuitively modi�ed model.Both models are applied in the form of dominant �rst�order decay.e insert presents  2 function as dependent on  0 .

F 6 :
e diffusional relaxation time (part (a)) and the degree of interference (part (b)) as dependent on the delay,  0 , measured at 145 K. e comparison of two versions of the modi�ed model for spatially periodic excitation: squares-exponential �t, circles-nonexponential �t.e insert presents the indicator of nonexponential behavior as a function of the delay.

4. 4 .F 8 :F 9 :
Viscosity and Temperature Dependence of the Diffusion Coefficient.In Figure 10 there is presented the diffusion coefficient   as dependent on .Together with experimental points �tted models are presented: SE model and a combined model.e course of experimental points shows that the both models give the similar values of the diffusion coefficient up to   ∼0.15 K/m Pa s.For larger values of  the diffusion coefficient shows the deviation from SE model towards lower values.e other model seems to Diffusional relaxation time (a) and the degree of interference (b) as dependent on the delay.e comparison of the exponential model of temporal behavior of the intensity of delayed �uorescence in the case of included short time effect (squares) and neglected short time effect (circles).Fits were done for the decay measured at 155 K with bottom excitation.Both variations of the model show almost the same course of the presented parameters.e insert presents the  2 as dependent on  0 .e values of are equal to about 1.05.e same comparison as presented in Figure 8 in the case of measurement at 136 K. e values of diffusional relaxation time and degree of interference obtained for included short time term are omitted, because they are pathologically large and their standard deviations overcome 100% of the values.e function  2 (insert in the part (b)) shows the quality of both �ts.�n the case of neglected short time term the values of  2 equal to about 1.05 for  0 > ∼500 s, whereas for the other case the values of  2 are spread between 1 and 2.5.reproduce better than the SE model the course of the experimental points.e values of   extrapolated to the room temperature with different models are as follows, (1) SE model: (1.50 ± 0.02) × 10 9 nm 2 s −1 = (1.50 ± 0.02) × 10 −1 cm 2 s −1 , (2) combined model: (1.43 ± 0.16)×10 6 nm 2 s −1 = (1.43±0.16) × 10 −4 cm 2 s −1 .

Figure 11 (
b) there is presented annihilation radius calculated from the combined model.e values of   are presented as dependent on the temperature.e behavior of   as a function of the temperature is practically the same in the case of Figure 11(b) as described above for Figure 11(a).e only difference between Figures 11(a) and 11(b) is the behavior of standard deviation Δ  in the temperature range 132 K ÷ 138 K.In the case of Figure 11(b) it is constant what

Figure 11 .
Parts (a) and (b) present the data calculated for the same model of the viscosity and/or temperature dependence of   as in the case of Figure 11: (i) part (a):   calculated from SE model; (ii) part (b):   calculated from combined model.In the case of Figures 12(a) and 12(b) the course of   as a function of the temperature is similar to that presented in Figures 11(a) and 11(b).e are two differences between the behavior presented in Figures 12(a) and 12(b), and Figures 11(a) and 11(b).

F 11 :F 12 :
e annihilation radius calculated using the values of mutual diffusion coefficient obtained from spatially periodic excitation and different models of temperature and/or viscosity dependence of   : (a) SE model and (b) combined model.e  2 parameter was calculated from the intuitively modi�ed Smoluchowski model [7� of temporal behavior of the intensity of delayed �uorescence.e plots present the combination of exponential and nonexponential versions of the  DF (.e experimental data were measured with the short excitation pulse.e annihilation radius calculated using the values of mutual diffusion coefficient obtained from spatially periodic excitation and different models of temperature and/or viscosity dependence of   : (a)-SE model and (b)-combined model.e  2 parameter was calculated from the non-Fickian model [11� of temporal behavior of the intensity of delayed �uorescence.e plots present the combination of exponential and nonexponential versions of the  DF (.e experimental data were measured with the short excitation pulse.

F 13 :F 14 :
(a) the values of   are calculated using the diffusion coefficients from SE model and in Figure 13(b)-from combined model.e course of the annihilation radius as a function of temperature presented in Figure 13 is very similar to that shown in e annihilation radius calculated using the values of mutual diffusion coefficient obtained from spatially periodic excitation and different models of temperature and/or viscosity dependence of   : (a) SE model and (b) combined model.e  2 parameter was calculated from the intuitively modi�ed Smoluchowski model [7� of temporal behavior of the intensity of delayed �uorescence.e plots present the combination of exponential and nonexponential versions of the  DF (.e experimental data were measured with the middle excitation pulse.e annihilation radius calculated using the values of mutual diffusion coefficient obtained from spatially periodic excitation and different models of temperature and/or viscosity dependence of   : (a) SE model and (b) combined model.e  2 parameter was calculated from the non-Fickian model [11� of temporal behavior of the intensity of delayed �uorescence.e plots present the combination of exponential and nonexponential versions of the  DF (.e experimental data were measured with the middle excitation pulse.
To sum up: the time-dependent intensity of the delayed �uorescence for nonexponential decay with spatially periodic excitation should have the form: −5 cm 2 s −1 , and inF 10: e comparison of experimental data (squares) of the mutual diffusion coefficient   as dependent on temperature and/or viscosity.Presented data are calculated with application of two models: SE model (straight line) and combined model (dotted line).
(a) the values of annihilation radii calculated with   from SE model are presented.For the temperature range 132 K ÷ 146 K the values of   are calculated from parameter  2 obtained from exponential �t of  DF () (squares).In the case of the temperature range 142 K ÷ 150 K the values are taken from nonexponential �t (circles).e values calculated from exponential model of the kinetics of TTA can be treated as constant for the temperature range 138 K ÷ 144 K.For the temperatures below 138 K the the values of   decrease with the decrease of the temperature.e increase of the standard deviation Δ  with the decrease of the temperature in the range 132 K ÷ 138 K re�ects the growing up uncertainty of the diffusion coefficient   with the decrease of the temperature.For the temperatures 145 K and 146 K the   obtained from exponential kinetics has the tendency to grow up with the temperature.In the case of the temperature above 146 K the values of the annihilation radii were not average, because there was no range of the delay  0 (for  0 < ∼10 ms) where  2 parameter can be treated as constant in the case of exponential model.In the case of nonexponential �t the values of   were calculated for the decies measured in the temperature range 142 K ÷ 150 K. e values obtained from nonexponential �t are constant within this temperature range.For the temperature between 142 K and 144 K the values of   resulting from exponential and nonexponential models of the decay are equal.In