# 7

TADF Kinetics and Data Analysis in Photoluminescence and in Electroluminescence

*
Tiago Palmeira and
Mário N. Berberan‐Santos*

*Universidade de Lisboa, CQFM‐IN and IBB – Institute of Bioengineering and Biosciences, Instituto Superior Técnico, 1049‐001 Lisboa, Portugal*

## 7.1 TADF Kinetics

### 7.1.1 Introduction

The basic model for thermally activated delayed fluorescence (TADF) kinetics in the condensed phases is a three‐state scheme, involving excited‐state interconversion between S_{1} and T_{1}, with the ground state S_{0} as the initial and final state (Scheme 7.1):where *k*_{F} and *k*_{P} are the radiative rate constants for fluorescence and phosphorescence, respectively, and are the internal conversion rate constant for S_{1} → S_{0} deactivation and the intersystem crossing (ISC) rate constant for T_{1} → S_{0} deactivation, and and are the direct (ISC) and reverse intersystem crossing (rISC) rate constants for transitions between S_{1} and T_{1}. and are also denoted in the literature as *k*_{ISC} and *k*_{rISC}, respectively. The rISC rate constant is temperature dependent and is given by [1–3]

where *k*_{v} is the rISC rate constant of the *v*th vibrational level of T_{1} (*v* representing the full set of vibrational quantum numbers) and *E*_{v} is the respective vibrational energy. Assuming that *k*_{v} is a step function, equal to a constant *A* for *E*_{v} ≥ Δ*E*_{ST}, where Δ*E*_{ST} is the S_{1}–T_{1} energy gap, and zero otherwise, and further assuming that the energy difference between consecutive vibronic levels is much smaller than *k*_{B}*T* 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 *k*_{v} 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 Δ*E*_{ST} does not always exactly match the spectroscopic value (when available). For this and other reasons, Δ*E*_{ST} should, in fact, be regarded as an activation energy for rISC that has a value close – but not necessarily identical – to the S_{1}–T_{1} 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 [4]. 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 *k*_{B}*T* 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 T_{1} for TADF analysis purposes, although in at least one case experimental results are compatible with a temperature‐dependent T_{2} contribution [4c].

At sufficiently high temperatures Eq. 7.2 gives . Assuming that the molecule is stable under such conditions and that S_{1} and T_{1} are in fast equilibrium,

Furthermore, at these temperatures and in the absence of significant structural differences between S_{1} and T_{1} molecules, the relative populations follow the respective spin statistical weights [5, 6]:

hence

where Φ_{T} is the quantum yield of triplet formation (also called Φ_{ISC}),

and is the singlet state lifetime in the absence of TADF (see Section 7.1.3.2).

If is essentially temperature independent, then Eq. (7.5) holds for all temperatures; hence an approximate relation for the rISC rate constant is [5, 6]

Owing to the typical values of Δ*E*_{ST} 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 T_{1}←S_{0} (radiative) transition is forbidden, and only S_{1} is generated by photon absorption (Scheme 7.2):

The function *I*_{exc} 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 S_{1} and T_{1} are produced (Scheme 7.3), in a statistical ratio of 1 : 3, according to the respective spin multiplicities [7, 8], where the quantity *I*_{exc} is defined with respect to excitation by electron‐hole recombination.

### 7.1.3 Photoexcitation

#### 7.1.3.1 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 7.1.3.2). This system of coupled equations can be solved exactly by a number of methods. The solution is well known (see e.g. [9]) as the set of differential equations is mathematically identical to that of monomer–excimer kinetics [10, 11].

#### 7.1.3.2 Fluorescence and Phosphorescence Decays

In the case of delta‐pulse excitation, *I*_{exc}(*t*) = *I*_{0}*δ*(*t*), the singlet decay is given by a sum of two exponentials of time and the triplet decay as a difference of the same two exponentials [9–11]:

where

with

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 [9]

It is seen that the fluorescence decay has a short component with a lifetime 1/*λ*_{2}, that is, smaller than the fluorescence lifetime *τ*_{F} and a long component (delayed fluorescence (DF) lifetime) with a lifetime *τ*_{DF} *=* 1/*λ*_{1}, that is, smaller than the low‐temperature phosphorescence lifetime. The higher the temperature, the shorter these two lifetimes are.

For , as is usually the case, Eq. 7.15 further reduces to [5, 6, 9, 12–14]

where τ_{DF} is the DF lifetime, associated with the slow component of both fluorescence and phosphorescence and which does not coincide with the phosphorescence lifetime τ_{P} defined above. For a simple derivation of Eq. 7.17, see ref. [14].

Equation 7.16, in turn, becomes

and defines the prompt fluorescence (PF) lifetime.

It also follows from Eq. 7.10 that the relative amplitude of the fluorescence slow component (DF) is [9]

and it is thus always very small as .

#### 7.1.3.3 Steady‐state Fluorescence and Phosphorescence Intensities

In the case of continuous excitation (steady‐state experiment, denoted as ss), writing *I*_{exc} = *I*_{0}, where *I*_{0} 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

where

is defined as the quantum yield of singlet formation by rISC [13] (also called Φ_{rISC} in the more recent literature), compare Eq. 7.6). The corresponding fluorescence and phosphorescence intensities (wavelength integrated) are

where the PF quantum yield Φ_{PF} is the fluorescence yield in the absence of TADF, the phosphorescence quantum yield is Φ_{P} = Φ_{T}*θ*_{P}, and *θ*_{P} is the phosphorescence quantum efficiency, given by .

In the absence of reverse intersystem crossing (Φ_{S} = 0, e.g. owing to triplet quenching by oxygen along with negligible singlet quenching), all the fluorescence is PF:

hence

and therefore [5, 6, 9, 13, 15, 16]

where Φ_{DF} 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 Δ*E*_{ST}) from experimental data [5, 13, 16, 20], e.g. in the form (see also Section 7.2.4.2). Equation 7.29, derived and used by Parker [18, 19], was previously obtained by Rosenberg and Shombert [17].

The maximum possible fluorescence yield corresponds to Φ_{S} = 1, and Eq. 7.26 gives

In this way, strong TADF effectively eliminates the ISC nonradiative channel by always returning the excited molecule to S_{1} (see in the next section the TADF cycle perspective).

#### 7.1.3.4 Excited‐state Cycles

In the customary description of the TADF mechanism, it is said that after photoexcitation to S_{n} (*n* ≥ 1) and once attained S_{1}, ISC to the triplet manifold occurs, followed by rISC from T_{1} back to S_{1} and then by fluorescence emission. However, this description of TADF is incomplete. It was shown that the excited molecule may go through several S_{1}–T_{1}–S_{1} 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 S_{1} → T_{1} case, direct ISC is quickly followed by vibrational relaxation, whereas in the S_{1}←T_{1} 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 [22], where the evolution equations are directly written in integral form. The S_{1} and T_{1} populations are given by the following coupled equations [9]:

where ⊗ stands for the convolution between two functions, , and *τ*_{F} and *τ*_{P} were previously defined (both lifetimes only have direct experimental meaning in the absence of reversibility).

The general solution can be obtained either by Laplace transforms or by insertion of Eq. 7.32 into Eq. 7.31 and then by repeated substitution of the left‐hand side on the right‐hand side [9]:

Hence the first term for the singlet decay can be associated with PF (zero S_{1} → T_{1} → S_{1} cycles) and the remaining terms with DF, the *n*th term resulting from *n* − 1 S_{1} → T_{1} → S_{1} cycles. Summation of the terms of Eq. (7.34) leads to Eq. 7.10 (convolved with *I*_{exc}). 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 S_{1} and T_{1} exists, and the cycle S_{1} → T_{1} → S_{1} 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 S_{1} → T_{1} → S_{1} 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 *n*th term resulting in general from *n* − 1 S_{1} → T_{1} → S_{1} cycles. Equation (7.35) can also be derived from Eq. (7.34).

The probability for fluorescence emission to occur after exactly *n* S_{1} → T_{1} → S_{1} cycles obeys a geometric probability distribution [9, 21]:

The average number of cycles is thus given by [9, 21]

Comparison of Eq. (7.37) with Eq. 7.28 gives immediately

and, using Eq. 7.27,

Hence the increase in fluorescence intensity owing to TADF is a direct measure of the average number of S_{1} → T_{1} → S_{1} cycles performed [9]. This result follows from the fact that each return to S_{1} 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 *−* Φ_{T}), as already discussed, cf. Eq. 7.30.

Using the photophysical parameters for fullerene ^{13}C_{70} in a polymer matrix (Zeonex) [4], Φ_{T} = 0.997, τ_{F} = 700 ps, = 96 ms, and *ΔE*_{ST} = 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 [4] 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.

#### 7.1.3.5 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 T_{1} 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 (S_{1}←T_{1}) competes effectively with T_{1} → S_{0} deactivation channels only above a certain temperature, characteristic of each molecule (in a given medium), *T*_{0}. This may be expressed quantitatively by imposing a certain *I*_{DF}/*I*_{PF} ratio, e.g. *I*_{DF}/*I*_{PF} = 1, meaning a doubling of the total fluorescence (that is given by I_{F} = I_{DF} + I_{PF}) 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.

Using Eq. 7.7 and solving for *T*, Eq. (7.41) becomes

where *T*_{g} is the singlet–triplet gap characteristic temperature:

The TADF onset temperature *T*_{0} (for which ) can thus be defined as

It is seen that *T*_{0} is controlled by three parameters: Φ_{T}, Δ*E*_{ST}, and the ratio . For the example given in Figure 7.2, *T*_{0} = 225 K (−48 °C).

Equation (7.44) also shows that *T*_{0} as defined does not exist for Φ_{T} ≤ 0.5. There cannot be efficient TADF for Φ_{T} ≤ 0.5, in the sense that fluorescence cannot be doubled, whatever the temperature (see Eq. (7.40)), given that only a minor fraction of S_{1} goes to T_{1}. The same holds if . Nevertheless, usually .

#### 7.1.3.6 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 Φ_{T}, 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 Φ_{F} up to ; hence ideally . In order to have fast reversibility, Φ_{S} 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 Φ_{F} approaches the ceiling value not only when Φ_{T} is small but also when Φ_{T} is significant, if Φ_{S} is also important (significant rISC).

For a given fluorescence yield ratio , the value of Φ_{S} – hence the average number of cycles – is defined by the value of Φ_{T}:

For instance, if Φ_{T} = 0.90 and the ratio target value is *R* = 0.90, then Φ_{S} = 0.988, cf. Figure 7.3, for which .

Equation (7.44) can be used to estimate a maximum permissible value for Δ*E*_{ST}, given a certain *T*_{0}:

Let us take Φ_{T} = 0.90, τ_{F} = 5 ns, and . Considering that, for practical purposes, TADF should be effective above 0 °C (*T*_{0} = 273 K), one gets .

### 7.1.4 Electrical Excitation

#### 7.1.4.1 Steady State

In the case of electroluminescence (Scheme 7.3), S_{1} and T_{1} are produced in a 1 : 3 ratio [7, 8].

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 *I*_{DF}/*I*_{PF} is always higher in the case of electroluminescence, by a minimum factor of 4 (when Φ_{T} is close to 1), and is much higher when Φ_{T} is close to 0. This is understandable, as in electroluminescence S_{1} is (formally) obtained from T_{1} even when Φ_{T} = 0, which does not happen under photoexcitation conditions. On the other hand, *I*_{DF}/*I*_{P} is identical for both excitation mechanisms.

#### 7.1.4.2 Conditions for Efficient Electroluminescence

The intensification of fluorescence owing to rISC is obtained from Eq. (7.53) [23]:

compare Eq. 7.26. Here, the maximum hypothetical intensification factor (Φ_{S} = 1) is 4/(1 − Φ_{T}), 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 Φ_{EL} and assuming negligible phosphorescence, Eq. (7.58) must be divided by 4:

Equation (7.59) can be rearranged to give

and, using also the *photoluminescence* DF quantum yield, Eq. 7.27, Eq. (7.60) becomes

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 Φ_{S} 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, Φ_{PF} is low owing to a high ISC rate constant, ; hence Φ_{T} is close to 1. However, efficient rISC (Φ_{S} close to 1 assumed) effectively eliminates the ISC channel, making Φ_{DF} approach 1; thus Φ_{EL} is also close to 1. In this case both ISC and rISC are fully operative, and the average number of S_{1} → T_{1} → S_{1} 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 [26].

However, the second term in the r.h.s. of Eq. (7.61), specific of electrical excitation, allows a second solution: If Φ_{PF} is close to 1, implying , then Φ_{T} 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 Φ_{T}, Eq. (7.62) reduces to Φ_{S}Φ_{PF}, meaning that a single T_{1} → S_{1} step occurs before emission. The dependence on Φ_{S} is linear (see Figure 7.4). For Φ_{S} close to 1, the triplet contribution approaches ¾ (see Figure 7.4).

Equation (7.59) can be rewritten as (compare Eq. (7.45))

This function is plotted in Figure 7.4. For very low Φ_{S}, only the PF contribution exists, and the ratio is, at most, 1/4 (for Φ_{T} = 0), as mentioned.

However, for increasing values of Φ_{S}, 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 Φ_{T} and is fast for small values of Φ_{T} (see Figure 7.4).

Equation (7.63) gives the value of Φ_{S} for a given ratio and a given Φ_{T}:

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 (Φ_{T} = 0), that is, of significant TADF.

At this point, the question arises: Is it possible to reconcile a significant rISC (leading to a high Φ_{S}) with a low Φ_{T}, 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 *k*_{F} = 2.0 × 10^{8} s^{−1} and . Let us also impose Φ_{S} = 0.90. In order to proceed, we consider a phosphorescence lifetime . This value gives . Assuming further that *T* = 300 K and that Δ*E*_{ST} = 100 meV, the pre‐exponential factor *A* in Eq. 7.2 is 4.3 × 10^{6} s^{−1}, and . The computed values for the direct ISC and rISC rate constants are not unrealistic and give Φ_{PF} = 0.85, Φ_{DF} = 0.044, and Φ_{T} = 0.055. The quantum yield of triplet formation is thus quite small. From these parameters, the DF lifetime is τ_{DF} = 11 μs and Φ_{EL} = 0.83; hence . It is thus in principle possible, in electroluminescence, to have efficient triplet harvesting with a low Φ_{T} (implying at the same time that the molecule will display very weak TADF under photoexcitation in the example Φ_{DF}/Φ_{PF} = 0.052).

Efficient emitters with photophysical parameters matching this situation (low Φ_{T}) were indeed recently reported [27–29].

As a more systematic approach, let us now proceed according to Eqs. (7.64) and (7.65). Imposing *R*_{EL} = 0.90, a plot of versus Φ_{T} is obtained (Figure 7.5).

A similar plot can be drawn for any desired *R*_{EL} value.

Figure 7.5 clearly shows that the demands on rISC increase with the value of Φ_{T} and are also more stringent for short . Molecules with low Φ_{T} and long require lower for the same efficiency *R*_{EL}. In addition, should be as high as possible, in order to have a high Φ_{EL}. 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 S_{1} 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

### 7.2.1 Introduction

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 Δ*E*_{ST} 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 C_{70} in a cycloalkane polymer, Zeonex.

### 7.2.2 Steady‐state Data

#### 7.2.2.1 Delayed Fluorescence and Phosphorescence Intensities as a Function of Temperature: Rosenberg–Parker Method

This method, first used by Rosenberg and Shombert [17] and shortly afterward by Parker and Hatchard [18, 19], relies on Eqs. 7.29 and 7.2) combined, in a linearized form:

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 Δ*E*_{ST} is 40 kJ mol^{−1} (0.41 eV). The original measurements by Parker and Hatchard [18] 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 [37], polycyclic aromatic hydrocarbons [38], and fullerenes [13].

#### 7.2.2.2 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 Δ*E*_{ST} is also not possible. To deal with this situation, a method using only prompt and DF steady‐state intensities at several temperatures was devised [13]. This method is based on Eq. 7.28, rewritten as [13, 39]

It is therefore possible to obtain Δ*E*_{ST} from the temperature dependence of the ratio *I*_{PF}/*I*_{DF}. However, the correct value of Φ_{T} (assumed temperature independent) is required for a linear least‐squares fit. The shape of the plot is a very sensitive function of Φ_{T}, not being, in general, a straight line. Variation of this parameter in the search for maximum linearity yields its best value and, simultaneously, Δ*E*_{ST}. Application of this method to fullerene C_{70} in Zeonex [4] (Figure 7.7) gives , Φ_{T} = 0.995, and Δ*E*_{ST} = 34 kJ mol^{−1} (0.35 eV). This type of plot has been frequently used in the OLED field [24].

To estimate a minimum value for Φ_{T} in a simple way, Eq. 7.28 is rewritten as [13]

and because Φ_{S} is smaller than or equal to unity, a lower bound for Φ_{T} is

Z is closest to the fitted value of Φ_{T}, the closest Φ_{S} is to one, and the highest is the temperature. In the above example, taking the *I*_{DF}/*I*_{PF} ratio for the highest temperature measured (95 °C), is obtained, already very close to the fitted value.

Alternatively (or subsequently), a nonlinear fitting can be performed [40]. Rewriting again Eq. 7.28:

with

A fitting to the data shown in Figure 7.7 is displayed in Figure 7.8 and gives the same photophysical parameters.

#### 7.2.2.3 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 Δ*E*_{ST} 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 7.2.2.1).

### 7.2.3 Decay Data

Measurement of the PF decay (fast decay) gives τ_{F}. Measurement of the DF decay (slow decay) provides the DF lifetime, τ_{DF} (Eq. 7.17), rewritten as

where *B* = (1 − Φ_{T})*A*. In the case of TADF, this is also the phosphorescence decay time, as discussed in Section 7.1.3.2.

Determination of τ_{DF} versus temperature allows obtaining Δ*E*_{ST} 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 [41], 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 Δ*E*_{ST} 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 C_{70} in Zeonex in the temperature range 30–95 °C. Using the steady‐state values (see Section 7.2.2.2) Δ*E*_{ST} = 34 kJ mol^{−1} (0.35 eV) and Φ_{T} = 0.995, the following values are obtained: *A* = 8 × 10^{8} s^{−1} and = 30 ms. In studies covering more extended temperature ranges, parameter correlation is no longer a problem [4].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 S_{1} and T_{1} and can be written as

This equation reduces to Eq. 7.15 when Φ_{T} is close to 1 and to Eq. 7.17 when, additionally, Δ*E*_{ST} ≫ *k*_{B}*T* (as is usually the case).

### 7.2.4 Combined Steady‐state and Decay Data

#### 7.2.4.1 Linear Relation Between Delayed Fluorescence Lifetime and Intensity Ratio

Elimination of from Eqs. 7.17 and 7.28 leads to [9]

This relation allows the determination of and of Φ_{T} from a linear plot of τ_{DF} versus *I*_{DF}/*I*_{PF}, assuming that is constant in the temperature range in question. An example is shown in Figure 7.11, C_{70} in Zeonex. The obtained photophysical parameters are = 32 ms and Φ_{T} = 0.995.

#### 7.2.4.2 Linearized Relation for the Determination of Δ*E*_{ST}

Equation 7.28, written as

was probably first obtained by Callis et al. [5]. It has been frequently used (including in the OLED field, e.g. [20]) to obtain from the remaining parameters:

With data obtained at several temperatures, Eq. (7.79) allows the determination of Δ*E*_{ST} [16]. Using Eq. 7.2, Eq. (7.80) becomes

Again, if *I*_{PF} 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 Δ*E*_{ST} = 39 kJ mol^{−1} (0.40 eV) is obtained.

## 7.3 Conclusion

The study of TADF started almost a century ago, when the fluorescence and phosphorescence mechanisms were still unclear [42]. After significant fundamental work, establishing the nature and relevance of TADF (see refs. [19], [41], and references therein), applications in the field of temperature [40] 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.

## Acknowledgment

This project was carried out within projects RECI/CTM‐POL/0342/2012 (FCT, Portugal) and FAPESP 2017/2014 (FCT, Portugal).

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