I W. J. McCarthy and J. D. Winefordner University of Florida Gainesville, 32601
The Selection of Optimum Conditions for Spectrochemical Methods
I
11.
Quantum efficiency a n d decay
time o f luminescent molecules
It is well known that there are many factors which influence the measured signal in analytical techniques based upon the measurement of lurninescence (fluorescence, phosphorescence, and delayed fluorescence) of molecules in solution (condensed phase). So far no detailed treatment of the experimental factors and the extent of these factors affecting the measured detector signal has appeared in the literature. Therefore in an experimental analysis, the investigator is usually forced to rely on a trial and error approach to obtain the l'optimum" results. This generally requires a long time and even then is suhject to considerable error. Perhaps the most significant parameter in determining the signal is the quantum efficiencv of the luminescence Drocess. In this manuscript, general approach to-quantum efficiencies is presented. Using these equations the influence of experimental parameters on the quantum efficiency and ultimately the measured signal can he predicted. The kinetic approach of Fonter is used to derive equations for the quantum efficiencies and decay times of molecules (in the condensed phase) which exhibit fluorescence, phosphorescence, and delayed fluorescence. A number of authors have derived equations for specific cases of luminescence measurements; however, the general treatment has never been p r e sented nor have the implications of the general equa-
a
This research was carried out as a part of a study on the phoe phorimetric analysis of drugs in blood and urine, supported by a. grant from the U. S. Public Health Service (GM 11373-03).
Figure 2. Schematic diagram of activation and deactivation processes for molecules; 1 = activation step$ 2 = radiationless deastivotion of S*; 3 = radiational (fluorescence) deactivation of S*; 4 = intersystem crossing S* + To; 5 = intersystem crossing To -+ S*; 6 = rodiotionlesr ded i v o t i o n of To; 7 = radiational (phosphorewence) deactivation of To.
136
/
Journal of Chemical Education
tions been discussed. The analyst, in particular, will be concerned with the quantum efficiency behavior of numerous luminescing compounds. Rather than accurately determining the absolute value of the quantum efficiency, the analyst will he more concerned with simply increasing the quantum efficiency of the luminescence process. The general equations presented herein will provide the analyst with sufficient information to investigate a particular system and decide whether appropriate changes in the system should he made to increase the quantum efficiency and ultimately the sensitivity of a particular analysis. Quantum Efficiency and Decay Time Fluorescence and Phos~horescence
In this section, the quantum efficiency, 41,and the decay time, T,, which is often called lifetime (8) will he evaluated for fluorescence and phosphorescence by considering recent concepts of the kinetic processes occurring in solution when the molecule is excited by radiational means. When a molecule is excited by the absorption of ultraviolet or visible radiation, the molecule may then undergo a number of possible deactivational steps, each of which has a specific rate constant as designated in Figure 2. The energy levels r e p resented in Figure 2 are simplified to demonstrate only the eround sinelet electronic state., So, -.the fint excited singyet state, i*, and the lowest energy triplet electronic state, TO. Transitions to states of higher energy than the first excited singlet state and the lowest energy triplet state are omitted because radiationless deactivation to the excited states shown would generally occur rapidly compared to the transitions from the higher excited states to the ground states. No a& tempt will he made to classify the electronic states or the transitions involved other than with respect to multiplicity because further classification would require a detailed knowledge of the structure of the specific molecule being studied. Hence, only the states generally involved in the processes of fluorescence, and
Part 1 of this paper, discussing the use of signal-to-noise ratio theory, appeared in the February issue of T H I S JOURNAL (44, 80, 1967). The numbers of the figures, tables, and literature cited in this paper follow consecutively those in Part l . The third part of this paper will appear in the April issue of T H I S JOURNAL.It will discuss the sensitivity of atomic fluorescence. ahsomtion. and emkion flame wectrometni.
Table 2. Type Repmentation of process tmns1rat* tion (see Reactants Lproducts constant Fig. 2)
+ hv* S* + So
So
ha
kaa
ks r
S*
Transition rateb
Reference (8-1 6)
Excitation step
2So
Quenching due to collision with So
Rse = ksa[S*I [Sol
(8,16-19)
Quenching due to collision with Y
Rsu = ksu[S*l [Yl
(8,18-87)
RSP = k 9 ~ [S*l RSD = kso[S*l [Sol
(9)
-
+ SO
kw
Description of trrtnsition
S*
sv+y-S"+Y S*
Activation and Deactivation Processes for Molecules.
RA = krPsb~[sal
Products Photochemical reaction
kso
ksn
(So),
s*+y-SO+Y*
Dimerimtian
(18, 27)
FBrster-type intermolecular energy transfer
Rsa = ksa[S*l[Yl Ra = ka[S*] Intersystem crossing (spin forbidden) Rsr = ksr[S*l Intersystem crossing(spin forbidden) RTI = krr[Tol exp (-AEIkT)
Fluorescence step
Quenching due to collision with So Quenching due to collision Y To To + So
To+ Y
k?~ kro
krr
Products Photochemical reaction (So)%
Dimerization
So + Y
FBrster-type intermolecular energy transfer
TodSo+ hvp Phosphorcecence step k~
(18, dl, 88, 29) (8,11,14,16,18) (8,9,18, 18,80,88,SO, 91) (8,9,18, do, $8, 31,SS) (418, SO, 93)
RTQ = krp[Tol [Sol RTY = ~TY[ToIIYI
(8,18, SO, SO, 33-36)
RTP = kw[ToI RTD = ~TD[ToI [Sol
(18)
(9)
RTF = k~~[TallYl RP = ICPITOI
(18, 88, $9) (8, IS, 15, 18, SS, 36)
Symbols used in the above table are defined as: So= ground singlet st& of molecule; S* = excited singlet state of molecule; To = lowest tnplet state of molecule; PSb. = radiant power absorbed by molecule; h = Plenck's constant; va = frequency of absorpt~ontransltlon; = frequency of fluorescence transition; v p = frequency of phosphorescence transition; Y = solvent or impur~tymolecule; Y* = excited non-luminescing solvent or impurity molecule; (So)% = non-luminescing dimer; AE = energy dserence between lowest vibrational level of S* and lowest vibrational level of To. b [I designates concentration of designated species: R = rste of transition in mole I-' see-'.
phosphorescence are indicated in Figure 2. Also the singlet-triplet absorption step is not shown because luminescence processes are generally a result of the singlet-singlet absorption step (.9). Also, because of the multiplicity change, the singlet-triplet absorption is forbidden and so would occur to a much smaller extent than the singlet-singlet absorption transition. No attempt has been made in Figure 2 to represent internuclear distance along the abscissa because it is unimportant in this discussion. Vibrational and rotational energy levels are excluded in Figure 2 for simplicity. The processes shown schematically in Figure 2 may be described by rate equations for the specific types of transitions or reactions occurring (see Table 2). Each process is assigned a rate constant designated by the symbol k with a suitable subscript to indicate the appropriate process. From each rate constant, a rate equation can be written, and the resulting rate, R, is indicated in Table 2 with a suitable subscript to represent the process involved. The references listed in Table 2 give a thorough discussion of the particular process described. Using the processes described in Table 2 and using the general approach of Forster, the rate equations for the luminescing system are readily obtained. The rate of change of S* concentration with time, i.e., d[S*]/dt is given by:
defined in Table 2 or previously in the text and K r is given by: Ka = ksp[Sal
+ ksu[Yl + ksp + ka~[Sal+ ks~[Yl+ ksr
+ k~
(2)
The rate of change of To concentration with time, i.e., d[To]/dt is given by:
assuming the direct excitation of state TOfrom state So is negligible. The symbols in eqn. (3) have been previously defined in the text or in Table 2, and K , is given by: K, = k ~ ~ [ S a l km[YI ~ T P km[Sd krr[YI
+
+
+
k~
+ + + k n exp ( - U l k T )
(4)
Because all measurements are assumed to be made under steady state conditions, d[S*]/dt = 0 and d[Tol/dt = 0, and therefore: 0 = krP.ba[Sol
+ krrITo1 exp (-AEIkT)
- KFWI
(5)
-
If the quantum efficiency is defined as the number of emission transitions per absorption transition, the +L for the fluorescence, i.e., all S* Sotransitions, and phosphorescence processes is defined as:
where PIb is the power (in watts) of incident radiation absorbed by the sample, all other symbols have been Volume 44, Number 3, Morch 1967
/
137
where the subscripts of +, F and P represent, respectively, fluorescence and phosphorescence. Combining eqns. (5) and (6) with eqns. (7) and (S), the quantum efficiencies for fluorescence and phosphorescence can be evaluated in terms of the rate constants given in Table 2, i.e.:
and
The decay times (lifetime), 8 s of the luminescence process are given, according to the general approach given by Forster (IS), by:
and
It is interesting to note that under the assumptions listed for this special case which is an expression often given in the literature concerning luminescence. Special (High Sample Temperature) Case III. If the sample is at a high temperature, e.g., 273"K, or above, then the expressions for the 4's and .r's can be simplified as shown below.
and TF-'
S K P - k m exp ( - A E / k T )
+
Ka - ksi esp ( - A E / k T ) (25) exp ( - A E / ~ T ) ] - ' ( 1 2 )
Some special (limiting) cases can be given, and these cases result in considerably simpler equations for the 4's and 7's than are given above for the general case. Several useful special cases are listed below. Special (Low Sample Temperature-Some Quaching) Case I. If the sample is at a very low temperature, e.g., 77'K or lower, and if all quenching processes are considered, then exp ( - A E I k T ) E 0, and therefore:
and
The above special cases correspond to the most common experimental conditions used for observation of fluorescence and phosphorescence. It should be noted that the above equations for the &'s and TL'S apply only to measurements of the luminescence of molecules in solution (condensed phase) and do not strictly apply to luminescent molecules in the gaseous state unless additional factors such as the sensitized (energy transfer) steps are considered. Also the above equations do not apply directly to molecules in inorganic matricies unless additional factors are considered. However, in analytical applications of luminescence measurements, solutions are generally used. Delayed Fluorescence
Also and rp
Kp-'
(16)
Special (Low Sample Temperature-No Quenching) Case II. I n many cases, quenching due to foreign species or self-quenching (concentration quenching) will be negligible if the sample temperature is very low, e.g., 77'K or lower. Self-quenching will generally be negligible if the sample is kept at low temperatures and if the sample is dilute. Also, in many cases the efficiency for photochemical decomposition will be negligible due to lack of sufiicient thermal activation energy. I n this case, it will be assumed that because the sample is dilute and at a low temperature, the above quenching processes are negligible. It will also be assumed that photochemical decomposition is negligible. Therefore, for this case, the expressions for +L and rr, can be further simplified to:
I n this section, the quantum efficiencyfor two types of delayed fluorescence will be evaluated using the results of recent observations for the phenomena. Four distinct types of delayed fluorescence can be identified; the mechanism of production of each of the types of delayed fluorescence has been thoroughly discussed by Parker (S7), and so only a summary of the mechanism will be presented here. The nomenclature and s.ymbols will be consistent with the previous section where applicable. E-type Delayed Fluorescence. The simplest mechanistic delaved fluorescence scheme is observed when a molecule in the triplet state, TO,crosses over by thermal activation to the excited singlet state, S*, with a rate of kTIITo]exp (-AElkT) (see Table 2). The molecule then decays with the normal fluorescence efficiency, +F, as given in eqn. (9). The quantum efficiency for E-type delayed fluorescence, + ", is thus defined as:
By using the equations for the steady state population of the triplet state which are obtained using eqns. (5) and (6) and substituting into eqn. (26), +,,E is then given by: 138 / Journal o f Chemical Education
k d m k a exp (- AE/kT) ksrkrz exp (- AE/kT) IKPKF - A-.Frkr~a p t - b E l k T ) I { K -~KP (27)
1
The lifetime of E-type delayed fluorescence, +D,E is thus seen to be the sum of the lifetimes of the triplet state, Kp-l, and the singlet state, Kp-', i.e., QD#
=
+
KP-~ Kpd
Because, in general Kp-' as:
of populating S*; therefore, the term k ~ , [To] exp (-hE/kT) may be neglected in eqn. (31). Also Roo should generally be much less than the rate due to direct absorption of radiation, i.e., kaP.b,[So] >> KDD[Sz*]. With these simplifying assumptions, the steady state populations of the states involved will be given by:
(28)
>> Kp-', TDP may be given and
Thus E-type delayed fluorescence will have a lifetime approximately equal to the phosphorescence lifetime and a spectrum which is identical with fluorescence. There should be considerable temperature dependence observed for the quantum efficiency. Considerable efforthas been expended on the E-type delayed fluorescence of several molecules, namely eosin and proflavine hydrochloride among others. All of this work has been reviewed by Parker (37). P-type Delayed Fluorescence. The mechanism of Ptype delayed fluorescence appears to be well established, although there is some discrepancy over actual individual steps in the process (37-39). The method proposed by Parker, et al. (59), appears to be the most general and will be extended here to a general treatment (however, this approach will still encompass the specific mechanisms suggested by others (20, 40-43)). Consider that under steady state illumination an encounter occurs between two triplet state molecules. The following possible consequences have been suggested on the basis of experimental observations; namely Triplet-triplet annihilation
Fluoreseenee
The rate equation for the population of the state S2* can be obtained by making the usual steady state approximation, namely,
+
where KD = kDQ[Y] koD. It has also been aasumed that k,[ToIZ ~ - ..- - ,. (36) KEIRS,R. J., BRILITT, R. D., AND WENTWORTH, W. E., Anal. Chem. 29, 202 (1957). C. A., "Phosphores~enceand Delayed Fluorescence (37) PARKER, in Solution" in "Advances in Photochemistry," Vol. 2, (Edilors: NOYES,HAMMOND, and P ~ r r s ) , Interscience Publishers (division of John Wiley & Sons, Ine.), New York, 1964. C. A,, Tram. Farday Soe. 60, 1998 (1964). (38) PARKER, C. G., Trans. Farday Soe. (39) PARKER,C. A., AND HATCHARD, 59, 284 (1963). S. P., J . Chem. Phys. 38, 2773 (40) AZUMI,T., AND MCGLYNN, (1963). (41) Ibid., 3533 (1963). A. T., AND MCGLYNN, 8. P.,J. (42) ~MITH,F. J., ARMSTRONG, C h . Phys. 44, 442 (1966). S. P., J . C h . Phgs. 42,4308 (43) SMITH, F. J., AND MCGLYNN, (1965). ~ ~ ~ ~ , (44) LIM, E. C., AND SWENSON, G.W., J . Chem. Phys. 36, 118 (1962). (45) LIM, E. C., AND WEN,W., J . C h . Phys. 39, 849 (1963).
~~
\
Volume 44, Number 3, March 1967
/
141
