7.1 TADF Kinetics
The basic model for thermally activated delayed fluorescence (TADF) kinetics in the condensed phases is a three‐state scheme, involving excited‐state interconversion between S1 and T1, with the ground state S0 as the initial and final state (Scheme 7.1):where kF and kP are the radiative rate constants for fluorescence and phosphorescence, respectively, and are the internal conversion rate constant for S1 → S0 deactivation and the intersystem crossing (ISC) rate constant for T1 → S0 deactivation, and and are the direct (ISC) and reverse intersystem crossing (rISC) rate constants for transitions between S1 and T1. and are also denoted in the literature as kISC and krISC, respectively. The rISC rate constant is temperature dependent and is given by [1–3]
where kv is the rISC rate constant of the vth vibrational level of T1 (v representing the full set of vibrational quantum numbers) and Ev is the respective vibrational energy. Assuming that kv is a step function, equal to a constant A for Ev ≥ ΔEST, where ΔEST is the S1–T1 energy gap, and zero otherwise, and further assuming that the energy difference between consecutive vibronic levels is much smaller than kBT and that the density of states is approximately constant, Eq. (7.1) becomes the simple Arrhenius equation [2, 3]:
which, owing to the absence of detailed information on kv and on the density of vibrational states, is the commonly used form, empirically validated, for the rISC rate constant. The approximate nature of Eq. (7.2) may explain, in part, why the recovered ΔEST does not always exactly match the spectroscopic value (when available). For this and other reasons, ΔEST should, in fact, be regarded as an activation energy for rISC that has a value close – but not necessarily identical – to the S1–T1 energy gap.
In the above analysis, it is assumed that a relatively fast equilibration exists among the triplet sublevels that can therefore be treated as a single entity whose intrinsic decay rate is the Boltzmann‐weighted average of the sublevel decay rates . This is valid for all temperatures of interest where TADF is operative, as the zero‐field splitting in aromatic organic molecules (typically tenths of cm−1) and in organometallic complexes (at most a few cm−1) is much smaller than kBT when T exceeds a few kelvin, and the kinetics of triplet sublevel equilibration are usually fast when compared with radiative and nonradiative triplet relaxation processes [4a, 4b]. It is also assumed that the upper triplet states either do not contribute significantly to the TADF process or can be grouped together with T1 for TADF analysis purposes, although in at least one case experimental results are compatible with a temperature‐dependent T2 contribution [4c].
At sufficiently high temperatures Eq. 7.2 gives . Assuming that the molecule is stable under such conditions and that S1 and T1 are in fast equilibrium,
Furthermore, at these temperatures and in the absence of significant structural differences between S1 and T1 molecules, the relative populations follow the respective spin statistical weights [5, 6]:
where ΦT is the quantum yield of triplet formation (also called ΦISC),
and is the singlet state lifetime in the absence of TADF (see Section 22.214.171.124).
Owing to the typical values of ΔEST for TADF molecules, the rISC rate constant is in most cases strongly temperature dependent.
7.1.2 Excitation Types
In the case of optical excitation (photoluminescence), the T1←S0 (radiative) transition is forbidden, and only S1 is generated by photon absorption (Scheme 7.2):
The function Iexc may correspond to pulsed, modulated, or continuous excitation.
On the other hand, in the case of excitation by electrical current (electroluminescence, usually continuous excitation), both S1 and T1 are produced (Scheme 7.3), in a statistical ratio of 1 : 3, according to the respective spin multiplicities [7, 8], where the quantity Iexc is defined with respect to excitation by electron‐hole recombination.
126.96.36.199 Rate Equations
The rate equations for weak photoexcitation (i.e. nonsaturating and avoiding triplet–triplet interaction) are
where is called here the phosphorescence lifetime but, owing to TADF, does not correspond to a real decay time (see Section 188.8.131.52). This system of coupled equations can be solved exactly by a number of methods. The solution is well known (see e.g. ) as the set of differential equations is mathematically identical to that of monomer–excimer kinetics [10, 11].
184.108.40.206 Fluorescence and Phosphorescence Decays
where is formally identical to the low‐temperature phosphorescence lifetime but refers to the temperature of the system.
In both TADF kinetics and monomer–excimer kinetics, the intensity of the higher‐energy emitter (excited singlet state and monomer, respectively) increases with temperature, owing to an increase of the rate of the back step (reverse intersystem crossing, excimer dissociation). There is nevertheless one important difference between the two kinetics (apart from the molecularity of the direct step): Monomer and excimer intrinsic lifetimes are usually not very different, whereas singlet and triplet excited‐state intrinsic lifetimes differ by several orders of magnitude. For this reason, the decay constants given by Eq. 7.12 can be simplified in the TADF case to 
It is seen that the fluorescence decay has a short component with a lifetime 1/λ2, that is, smaller than the fluorescence lifetime τF and a long component (delayed fluorescence (DF) lifetime) with a lifetime τDF = 1/λ1, that is, smaller than the low‐temperature phosphorescence lifetime. The higher the temperature, the shorter these two lifetimes are.
where τDF is the DF lifetime, associated with the slow component of both fluorescence and phosphorescence and which does not coincide with the phosphorescence lifetime τP defined above. For a simple derivation of Eq. 7.17, see ref. .
Equation 7.16, in turn, becomes
and defines the prompt fluorescence (PF) lifetime.
and it is thus always very small as .
220.127.116.11 Steady‐state Fluorescence and Phosphorescence Intensities
In the case of continuous excitation (steady‐state experiment, denoted as ss), writing Iexc = I0, where I0 is the number of moles of photons (einstein) absorbed per unit time and unit volume, and setting the time derivatives in Eqs. 7.8) and 7.9) equal to zero, gives
is defined as the quantum yield of singlet formation by rISC  (also called ΦrISC in the more recent literature), compare Eq. 7.6). The corresponding fluorescence and phosphorescence intensities (wavelength integrated) are
where the PF quantum yield ΦPF is the fluorescence yield in the absence of TADF, the phosphorescence quantum yield is ΦP = ΦTθP, and θP is the phosphorescence quantum efficiency, given by .
In the absence of reverse intersystem crossing (ΦS = 0, e.g. owing to triplet quenching by oxygen along with negligible singlet quenching), all the fluorescence is PF:
where ΦDF in Eqs. 7.28 and 7.29 is the DF quantum yield. Equation 7.28 can be used to compute the rISC rate constant (or at least ΔEST) from experimental data [5, 13, 16, 20], e.g. in the form (see also Section 18.104.22.168). Equation 7.29, derived and used by Parker [18, 19], was previously obtained by Rosenberg and Shombert .
The maximum possible fluorescence yield corresponds to ΦS = 1, and Eq. 7.26 gives
In this way, strong TADF effectively eliminates the ISC nonradiative channel by always returning the excited molecule to S1 (see in the next section the TADF cycle perspective).
22.214.171.124 Excited‐state Cycles
In the customary description of the TADF mechanism, it is said that after photoexcitation to Sn (n ≥ 1) and once attained S1, ISC to the triplet manifold occurs, followed by rISC from T1 back to S1 and then by fluorescence emission. However, this description of TADF is incomplete. It was shown that the excited molecule may go through several S1–T1–S1 cycles before fluorescence finally takes place [9, 21], as exemplified in Figure 7.1 for a single molecule undergoing three excited‐state cycles.
In Figure 7.1 both ISC and rISC are for simplicity depicted as vertical lines. However, intrinsic intersystem crossing steps connect isoenergetic levels. In the S1 → T1 case, direct ISC is quickly followed by vibrational relaxation, whereas in the S1←T1 case, thermal activation (according to the Boltzmann distribution) precedes the rISC step.
The existence of excited‐state cycles is compatible with the kinetic results already derived. This can be explicitly shown using a convolution approach , where the evolution equations are directly written in integral form. The S1 and T1 populations are given by the following coupled equations :
where ⊗ stands for the convolution between two functions, , and τF and τP were previously defined (both lifetimes only have direct experimental meaning in the absence of reversibility).
Hence the first term for the singlet decay can be associated with PF (zero S1 → T1 → S1 cycles) and the remaining terms with DF, the nth term resulting from n − 1 S1 → T1 → S1 cycles. Summation of the terms of Eq. (7.34) leads to Eq. 7.10 (convolved with Iexc). Analogous results are obtained for the triplet decay.
We now turn to the steady‐state situation. Again, for strong TADF to occur, the following inequalities must hold: . In most cases it is also observed that and . Interconversion of the singlet and triplet emissive states then occurs several times before photon emission or nonradiative decay can take place. In this way, a pre‐equilibrium between S1 and T1 exists, and the cycle S1 → T1 → S1 repeats a number of times before fluorescence emission occurs. It is interesting to consider the following question: For a given set of rate constants, how many times is the cycle S1 → T1 → S1 completed on the average, before return to the ground state occurs? Clearly, for a pre‐equilibrium to exist, this cycling must occur many times. In order to quantitatively answer the above question, and related aspects, it is convenient to view TADF as the sequential process depicted in Scheme 7.4.
One then has
compare Eq. 7.26. The first term corresponds to PF (0 cycles), and the remaining terms correspond to DF, the nth term resulting in general from n − 1 S1 → T1 → S1 cycles. Equation (7.35) can also be derived from Eq. (7.34).
and, using Eq. 7.27,
Hence the increase in fluorescence intensity owing to TADF is a direct measure of the average number of S1 → T1 → S1 cycles performed . This result follows from the fact that each return to S1 brings a new opportunity for fluorescence emission.
In the absence of reversibility, . On the other hand, for the fastest possible excited‐state equilibration , one has
and the maximum possible fluorescence intensification factor is 1/(1 − ΦT), as already discussed, cf. Eq. 7.30.
Using the photophysical parameters for fullerene 13C70 in a polymer matrix (Zeonex) , ΦT = 0.997, τF = 700 ps, = 96 ms, and ΔEST = 33 kJ mol−1 (340 meV), the maximum average number of cycles is estimated to be 332 and the maximum fluorescence intensification factor to be 333. The average number of cycles as a function of temperature, computed using Eq. 7.27, is displayed in Figure 7.2 (for simplicity, the temperature dependence of  is neglected). It is seen that TADF sets in at about 225 K and a large number of excited‐state cycles are already effected at room temperature. The maximum value is expected to be attained at about 500 K; however this can happen only with a suitable matrix (not Zeonex) and in the absence of triplet quenching and thermal reactions.
126.96.36.199 TADF Onset Temperature
Efficient TADF can occur only in the absence of quenching by molecular oxygen (or other triplet state quencher); otherwise the excited‐state loop will be broken by T1 deactivation. Given that back intersystem crossing is always thermally activated, degassing or oxygen diffusion blocking is not enough to set in TADF: A minimum temperature is also required. Indeed, rISC (S1←T1) competes effectively with T1 → S0 deactivation channels only above a certain temperature, characteristic of each molecule (in a given medium), T0. This may be expressed quantitatively by imposing a certain IDF/IPF ratio, e.g. IDF/IPF = 1, meaning a doubling of the total fluorescence (that is given by IF = IDF + IPF) owing to TADF. According to Eq. (7.38), this also means that the average number of excited‐state cycles is equal to 1 at such temperature.
where Tg is the singlet–triplet gap characteristic temperature:
The TADF onset temperature T0 (for which ) can thus be defined as
It is seen that T0 is controlled by three parameters: ΦT, ΔEST, and the ratio . For the example given in Figure 7.2, T0 = 225 K (−48 °C).
Equation (7.44) also shows that T0 as defined does not exist for ΦT ≤ 0.5. There cannot be efficient TADF for ΦT ≤ 0.5, in the sense that fluorescence cannot be doubled, whatever the temperature (see Eq. (7.40)), given that only a minor fraction of S1 goes to T1. The same holds if . Nevertheless, usually .
188.8.131.52 Conditions for Efficient TADF
As discussed in the previous sections, fluorescence enhancement by TADF (efficient TADF) means that the PF quantum yield is moderate in its absence, implying a high ΦT, that is, . The absence of TADF can also be due to low temperature or to triplet quenching. Suppression of the direct ISC by TADF raises the fluorescence quantum yield ΦF up to ; hence ideally . In order to have fast reversibility, ΦS must be close to 1; hence or, equivalently, , as expressed by Eq. (7.41).
The discussion of TADF efficiency can also be based on Eq. 7.26, rewritten as
This function is plotted in Figure 7.3. It is seen that ΦF approaches the ceiling value not only when ΦT is small but also when ΦT is significant, if ΦS is also important (significant rISC).
For a given fluorescence yield ratio , the value of ΦS – hence the average number of cycles – is defined by the value of ΦT:
For instance, if ΦT = 0.90 and the ratio target value is R = 0.90, then ΦS = 0.988, cf. Figure 7.3, for which .
Equation (7.44) can be used to estimate a maximum permissible value for ΔEST, given a certain T0:
Let us take ΦT = 0.90, τF = 5 ns, and . Considering that, for practical purposes, TADF should be effective above 0 °C (T0 = 273 K), one gets .
7.1.4 Electrical Excitation
184.108.40.206 Steady State
Under steady‐state conditions, the rate equations for weak excitation (i.e. nonsaturating and avoiding triplet–triplet interaction) are
It follows that
confer Eqs. 7.20–7.29. Equations (7.56) and (7.57) are especially noteworthy. Compared with their photostationary counterparts, Eqs. 7.28 and 7.29, it is seen that IDF/IPF is always higher in the case of electroluminescence, by a minimum factor of 4 (when ΦT is close to 1), and is much higher when ΦT is close to 0. This is understandable, as in electroluminescence S1 is (formally) obtained from T1 even when ΦT = 0, which does not happen under photoexcitation conditions. On the other hand, IDF/IP is identical for both excitation mechanisms.
220.127.116.11 Conditions for Efficient Electroluminescence
compare Eq. 7.26. Here, the maximum hypothetical intensification factor (ΦS = 1) is 4/(1 − ΦT), four times the photostationary one, owing to the contribution of directly excited triplets. The effective fluorescence quantum yield may attain 4 (if ), as three emitting singlets (out of four) originate in the triplet manifold and direct ISC is effectively suppressed. When considering the overall electroluminescence efficiency ΦEL and assuming negligible phosphorescence, Eq. (7.58) must be divided by 4:
Equation (7.59) can be rearranged to give
as given by Adachi [24, 25]. The first term corresponds to the contribution of directly excited singlets (containing prompt and delayed components), whereas the second is the (delayed) contribution from directly excited triplets, :
Note that Eq. (7.61) can be written as , with and .
Equation (7.61) allows concluding that there are two extreme cases for which the electroluminescence efficiency can be high. In both, and ΦS must be close to 1, implying that and that the rISC rate constant must dominate over the other triplet decay channels whose rate is and in particular .
In the first case, ΦPF is low owing to a high ISC rate constant, ; hence ΦT is close to 1. However, efficient rISC (ΦS close to 1 assumed) effectively eliminates the ISC channel, making ΦDF approach 1; thus ΦEL is also close to 1. In this case both ISC and rISC are fully operative, and the average number of S1 → T1 → S1 cycles is high. This is the situation observed in efficient TADF under photoexcitation, and it can occur in electroluminescence as well (see Figure 7.4). Most TADF emitters intended for organic light‐emitting diode (OLED) applications fall in this case .
However, the second term in the r.h.s. of Eq. (7.61), specific of electrical excitation, allows a second solution: If ΦPF is close to 1, implying , then ΦT is low, and the average number of cycles is small to negligible. PF dominates the contribution from directly excited singlets and accounts for ¼ of the yield (see Figure 7.4). For very low ΦT, Eq. (7.62) reduces to ΦSΦPF, meaning that a single T1 → S1 step occurs before emission. The dependence on ΦS is linear (see Figure 7.4). For ΦS close to 1, the triplet contribution approaches ¾ (see Figure 7.4).
This function is plotted in Figure 7.4. For very low ΦS, only the PF contribution exists, and the ratio is, at most, 1/4 (for ΦT = 0), as mentioned.
However, for increasing values of ΦS, the relative luminescence yield increases, owing to DF coming from both singlet and triplet. The important point is that this increase occurs for all values of ΦT and is fast for small values of ΦT (see Figure 7.4).
Equation (7.63) gives the value of ΦS for a given ratio and a given ΦT:
The required value of the rISC rate constant can next be computed:
Equation 7.7 finally relates this value with the remaining parameters (fluorescence lifetime, singlet–triplet gap, temperature). Several combinations of these three parameters correspond to a given value of the rISC rate constant.
It is thus mathematically viable to attain high values of , even in the near absence of cycles (ΦT = 0), that is, of significant TADF.
At this point, the question arises: Is it possible to reconcile a significant rISC (leading to a high ΦS) with a low ΦT, knowing that direct and reverse ISC rate constants are proportional?
The following example shows that this is indeed feasible: Let us assume a potentially highly fluorescent molecule, with . Using a typical radiative lifetime for an allowed transition, 5 ns, one gets kF = 2.0 × 108 s−1 and . Let us also impose ΦS = 0.90. In order to proceed, we consider a phosphorescence lifetime . This value gives . Assuming further that T = 300 K and that ΔEST = 100 meV, the pre‐exponential factor A in Eq. 7.2 is 4.3 × 106 s−1, and . The computed values for the direct ISC and rISC rate constants are not unrealistic and give ΦPF = 0.85, ΦDF = 0.044, and ΦT = 0.055. The quantum yield of triplet formation is thus quite small. From these parameters, the DF lifetime is τDF = 11 μs and ΦEL = 0.83; hence . It is thus in principle possible, in electroluminescence, to have efficient triplet harvesting with a low ΦT (implying at the same time that the molecule will display very weak TADF under photoexcitation in the example ΦDF/ΦPF = 0.052).
A similar plot can be drawn for any desired REL value.
Figure 7.5 clearly shows that the demands on rISC increase with the value of ΦT and are also more stringent for short . Molecules with low ΦT and long require lower for the same efficiency REL. In addition, should be as high as possible, in order to have a high ΦEL. These seem to be important guidelines in the design of third‐generation OLED molecules.
7.1.5 More Complex Schemes
The three‐state kinetic scheme implies that the S1 Franck–Condon and emissive states are relatively similar. This is not always the case, especially with donor–acceptor molecules with strong charge transfer (CT) in the excited state [30–33]. In such situations, a distinction must be made between the photoexcited state and the relaxed CT state. The latter is the one participating in the photophysical mechanism represented by Scheme 7.1, although in many cases the full picture is still wanting.
If the photoexcited state to CT state conversion is fast enough, then the three‐state kinetic scheme still holds, otherwise more complex kinetics ensues, with at least an additional rate constant corresponding to the conversion.
7.2 TADF Data Analysis
Analysis of photophysical observables, by themselves or combined, like fluorescence and phosphorescence intensities (steady‐state measurements) and fluorescence and phosphorescence decay times (time‐resolved measurements) allows determining all kinetic parameters of Scheme 7.1. From the temperature dependence of some of the observables, the TADF activation energy ΔEST can also be estimated. There are several possible methods, depending on the quantities for which data was measured (owing not only to experimental techniques available but also to system's properties, e.g. phosphorescence can be essentially undetectable in some cases). These methods are described in the next three sections, with examples of application to two different systems, both degassed, eosin in glycerol and fullerene C70 in a cycloalkane polymer, Zeonex.
7.2.2 Steady‐state Data
18.104.22.168 Delayed Fluorescence and Phosphorescence Intensities as a Function of Temperature: Rosenberg–Parker Method
where C is a constant. As an example, a plot for eosin in glycerol is shown in Figure 7.6 and gives a very good straight line, in support of the form of Eq. 7.2). The recovered ΔEST is 40 kJ mol−1 (0.41 eV). The original measurements by Parker and Hatchard  for the same system gave 42 kJ mol−1 (0.43 eV). The spectroscopic value is 43 kJ mol−1 (0.45 eV). The method was applied to a number of molecules, including xanthene dyes [18, 34, 35], ketones [6, 36], thiones , polycyclic aromatic hydrocarbons , and fullerenes .
22.214.171.124 Prompt and Delayed Fluorescence Intensities as a Function of Temperature
In several cases, the phosphorescence is undetectable in the temperature range of interest, and the previous method cannot be used. Furthermore, in such cases the spectroscopic estimation of ΔEST is also not possible. To deal with this situation, a method using only prompt and DF steady‐state intensities at several temperatures was devised . This method is based on Eq. 7.28, rewritten as [13, 39]
It is therefore possible to obtain ΔEST from the temperature dependence of the ratio IPF/IDF. However, the correct value of ΦT (assumed temperature independent) is required for a linear least‐squares fit. The shape of the plot is a very sensitive function of ΦT, not being, in general, a straight line. Variation of this parameter in the search for maximum linearity yields its best value and, simultaneously, ΔEST. Application of this method to fullerene C70 in Zeonex  (Figure 7.7) gives , ΦT = 0.995, and ΔEST = 34 kJ mol−1 (0.35 eV). This type of plot has been frequently used in the OLED field .
and because ΦS is smaller than or equal to unity, a lower bound for ΦT is
Z is closest to the fitted value of ΦT, the closest ΦS is to one, and the highest is the temperature. In the above example, taking the IDF/IPF ratio for the highest temperature measured (95 °C), is obtained, already very close to the fitted value.
126.96.36.199 Delayed Fluorescence Intensity as a Function of Temperature
In case PF is too weak but assumed to be independent of temperature, Eq. (7.70) can be slightly modified to read
where β/α = b/a and ΔEST can still be obtained from parameter c. An example, eosin in glycerol, is shown in Figure 7.9. The fitting gives 40 kJ mol−1 (0.41 eV), in agreement with the value obtained using the Rosenberg–Parker method (see Section 188.8.131.52).
7.2.3 Decay Data
Measurement of the PF decay (fast decay) gives τF. Measurement of the DF decay (slow decay) provides the DF lifetime, τDF (Eq. 7.17), rewritten as
where B = (1 − ΦT)A. In the case of TADF, this is also the phosphorescence decay time, as discussed in Section 184.108.40.206.
Determination of τDF versus temperature allows obtaining ΔEST and also to estimate (if assumed to be temperature independent in a relatively narrow range). This was the first method used in TADF analysis, going back to Lewis, Lipkin, and Magel , in the approximate form
where C is a constant.
Determination of all three parameters in Eq. (7.75) for a limited temperature range is difficult, owing to parameter correlation. It is preferable to set ΔEST at the steady‐state value (Eq. (7.67) or (7.70)) and then carry out the fitting for the remaining two parameters. An example of this procedure is shown in Figure 7.10. It refers to C70 in Zeonex in the temperature range 30–95 °C. Using the steady‐state values (see Section 220.127.116.11) ΔEST = 34 kJ mol−1 (0.35 eV) and ΦT = 0.995, the following values are obtained: A = 8 × 108 s−1 and = 30 ms. In studies covering more extended temperature ranges, parameter correlation is no longer a problem .An approximate equation that has also been used to analyze TADF decay data [4c, 33] is based on the assumption of fast thermal equilibrium [14, 33] between S1 and T1 and can be written as
7.2.4 Combined Steady‐state and Decay Data
18.104.22.168 Linear Relation Between Delayed Fluorescence Lifetime and Intensity Ratio
This relation allows the determination of and of ΦT from a linear plot of τDF versus IDF/IPF, assuming that is constant in the temperature range in question. An example is shown in Figure 7.11, C70 in Zeonex. The obtained photophysical parameters are = 32 ms and ΦT = 0.995.
22.214.171.124 Linearized Relation for the Determination of ΔEST
Equation 7.28, written as
Again, if IPF is not available but can be assumed to be constant, Eq. (7.80) gives
where C is a constant. An example of this plot, for eosin in glycerol, is shown in Figure 7.12, for which ΔEST = 39 kJ mol−1 (0.40 eV) is obtained.
The study of TADF started almost a century ago, when the fluorescence and phosphorescence mechanisms were still unclear . After significant fundamental work, establishing the nature and relevance of TADF (see refs. , , and references therein), applications in the field of temperature  and trace oxygen [43, 44] sensing appeared. More recently, third‐generation OLEDs relying on TADF were proposed [15, 45] and are under active development, with an already vast and rapidly growing literature.
In this chapter, a general view of the kinetics of TADF was presented, stressing the difference between photoluminescence and electroluminescence and discussing optimal conditions for both situations, from the point of view of photophysics. The methods of TADF analysis used for the determination of several photophysical parameters were also described, with examples given for each case.
This project was carried out within projects RECI/CTM‐POL/0342/2012 (FCT, Portugal) and FAPESP 2017/2014 (FCT, Portugal).
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