7488
J . Phys. Chem. 1990, 94, 7488-7494
Strong Transient Effects of Nonstationary Diffuston on Excimer Formation. Test of the Concept of Convolution Kinetics J. Vogelsang and M. Hauser* Institute of Physical Chemistry, University of Stuttgart, Pfaflenwaldring 55, D - 7000 Stuttgart 80, FRG (Received: October 1 1 , 1989; In Final Form: May 8 , 1990) The concept of convolution kinetics (CK) remains valid if rate coefficients depend on time, where rate equations (RE) fail, Excimer kinetics are without strong spurious influences, thus being convenient for a crucial test of CK. With strong excimer dissociation, the expectedly small effect of nonstationary diffusion becomes a large one, thus making the solution of integral equations in CK necessary. The rules of CK are given, and the most novel and uncommon part of the procedure is rederived. With time-independent coefficients, CK give the same result as RE. Some simpler applications, where CK are straightforward, are sketched. The problem as given in the title is initially solved correctly with CK. The integral equations can be solved by Laplace transformation and the back transformation is accomplished via series expansion. As far as convolutions cannot be performed in closed form, a Simpson procedure maintaining the general dependence on kinetic parameters is applied. On the basis of Smoluchowski's approach, and assuming reasonable values of disposable experimental parameters, considerable deviations from results of the RE procedure are predicted by CK with high viscosity and reasonably small reaction radius even at low dissociation rate; with strong dissociation, however, such deviations happen even at low viscosity. Both these influences mutually enhance when combined. As a first experimental test, the time dependence of pyrene excimer on flash excitation was measured with 7 = 0.152 P, T = 334 K, and 0.05 mol/L (about the half value concentration). The curve shape, readily explained by CK with a reaction radius 2.5 X IO-'' m, is in disagreement with the RE calculation. Smaller deviations between the CK curve and the experimental one may be explained by insufficient validity of Smoluchowski's approach, but mistakes from not applying CK are much more significant and must not be attributed to the former.
I. Introduction If processes obeying master equations participate in a reaction mechanism, the kinetic equations become very complicated in general. Certain important types of such processes with distributed typical rate coefficients may alternatively be described, however, by one time dependent rate coefficient (TDRC), but then the usual concept of rate equations (RE) can be shown to fail.lJ Convolution kinetics (CK) are a general calculus for treating reaction mechanisms consisting of first-order processes the rate coefficients of which may depend on time. The main field of application of CK has been kinetics of first excited singlet (FES) states, where the populations and fluorescence intensities additionally depend on excitation intensity, which also is a function of time. The foundation and procedure of CK has been published together with some simpler tests of more complicated applications of CK were sugge~ted.~TDRC's are well known in FES kinetics; the most important case is Forster type energy transfer? and another is the influence of nonstationary diffusion in Smoluchowski's approachS which may often be considered a smaller effect, but in contrast to energy transfer it may be investigated without spurious influences. In energy-transfer experiments, trivial absorption of donor fluorescence by the acceptor is almost inevitable thus making the process less convenient for a crucial test of any kinetic calculus. The system to be investigated here was chosen because the TDRC (of presumably Smoluchowski type) of excimer formation is combined with the strong influence of dissociation. It will be shown by CK that the former "small" effect amounts to a considerable one. Many papers deal with sophisticated features in connection with diffusion6*'but as long as no backward reaction is included, such ( I ) Hauser, M. Presented a t the XIIIth Conference on Photochemistry Budapest, 1987. (2) Hauser, M.; Wagenblast, G.In Time-Resolved Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E. Eds.; NATO Series A, Life Science; Academic: New York, 1983; Vol. 69, p 463. Wagenblast, G. Dissertation, University of Stuttgart, 1982. (3) Hauser, M. Acta Phys. Chem. Szeged 1984, 30, I . (4) Fcrster, Th. Natunvissenschaften 1946, 33, 166. Forster, Th. Z. Naturjorsch. 1949,4a, 321. Hauser, M.; Klein, U. K. A.; Gbele, U. Z. Phys. Chem. (Munich) 1976, 101, 255. ( 5 ) Smoluchowski, M. v. Z . Phys. Chem. 1917, 92, 129. (6) Ware, W. R.; Andre, J. C. In Time-Resolved Fluorescence Spectroscopy in Biochemistry and Biology; Cundall, R. B., Dale, R. E., Eds.; NATO Series A, Life Sciences; Academic: New York, 1983; Vol. 69, p 363. Andre, J. C.; et al. in a series about Kinetics of partly diJfusion controlled reactions, parts 1-20 in different scientific journals in the years since 1978.
0022-3654/90/2094-7488$02.50/0
one-step "mechanisms", as for instance quenching, need not be discussed here extensively, though TDRC's or master equations may be concerned. Joshi, Gryczinski, and Lakowicz* could not fit well their quenching results to a Smoluchowski decay function and interpreted them in favor of the radiation boundary condition. Andre, Baros, and Reis e Sousag proposed a model based upon Poisson distribution of quenchers neighboring the excited species, which correctly describes the steady state but fails with short pulse excitation. A great deal of such work, namely all cases obeying a TDRC, could be treated properly by CK in its simplest form, which is still within the scope of linear response theory.I0 Among the reactions with many steps, especially with backward step, excimer kinetics have been the most investigated ones. A particular situation is given with excimer kinetics coupled with energy migration and transfer in polychromophoric systems dealt with by Kauffmann.Il Irrespective of strongly nonexponential decays, weak intramolecular coupling (and weak excimer dissociation) is operative there. We are dealing with intermolecular processes in solution, where the single step is nonexponential (namely with TDRC), and where the influence of the inverse process (e.g., excimer dissociation) may be strong. The treatment in terms of RE as, e.g., by Donner, Andre, and BouchyI*is justified only if any TDRC is negligible. Birch et a1.I3 pointed out that a RE treatment followed by reconvolution reveals additional deviations of the measured time dependences from the calculated ones when the influence of nonstationary diffusion characterized by the t'/* term is stronger, namely at the beginning of the decay and with higher viscosities. The authors' conclusions on the limited validity of the linear Smoluchowski approach are premature before treating the problem correctly by CK. Inde(7) Stevens, B. Chem. Phys. Lett. 1987,134, 519. Stevens, B.; McKeithan, D. N. J . Pholochem. Photobiol. A 1987, 40, I . (8) Joshi, N.; Gryczynski, 1.; Lakowicz, J. R. Chem. Phys. Lett. 1987,135, 200.
(9) Andre, J. C.; Baros, F.; Reis e Sousa, A. T.J. Photochem. Photobiol. A 1988, 42, 233. (10) Kubo, R. J. Phys. SOC.Jpn. 1957, 12, 570.
( I I ) Mollay, B.; Landl, G.; Kauffmann, H. F., to be published in J. Chem. Phys. Landl, G.; Langthaler, T.; Engl, H. W.; Kauffmann, H . F., to be published in J . Chem. Phys. (12) Donner, M.; Andre, J . C.; Bouchy, M. Biochem. Biophys. Res. Commun. 1980, 97, 1183. (13) Birch, D. S.J.; Dutch, A. D.; Imhof, R. E.; Nadolski, B. J . Photochem. 1987, 38, 239.
0 1990 American Chemical Society
Effects of Diffusion on Excimer Formation pendently, allowance has been made for time dependence of excitation and/or linear distortions of the apparatus by convolution for many years. Under the aspects of CK, this may lead to a correct description in special simple cases, but in general a more or less inconsistent mixture of RE and linear response theory will result. In the absence of TDRC such convolution may be applied or not, as time-dependent excitation may be treated as inhomogeneity in the RE as well. But even a single-step reaction with TDRC in principle is outside the scope of RE, which becomes obvious when treating the steady state. A correct treatment of such case is by convolution of the 6-response with the excitation function.' But this follows from linear response theory2*l0or from a master equation description, not to be confused with the CK calculus, as being merely a part of it. CK include all simpler cases but become important with more than one step of energy transfer,I4 A iL B iL C (even though A may be &excited), and in energy migration* where A = B = C ..., or if A is an excimer followed by one-step energy transfer.' The two-step transfer should be expected to be appropriate for a crucial test of kinetic calculus, but it turned out that undesired processes obscure the significant result^.'^ The systems just mentioned have in common that the application of CK is straightforward, while the excimer system requires the solution of an integral equation. In two papers quoting CK in that case, this calculus was not applied correctly. The authors of ref 16, though making use of some elements of CK, applied unnecessary approximations. In ref 17, a rederivation of CK in that special case may be recognized, but the final results differ from concise application of that calculus, mainly because the authors did not construct the production term. Before the rules of CK are given once again, some general remarks on the present situation with respect to fluorescence and FES kinetics are necessary. There is some stagnancy in development and proper understanding since multiparametric curve fitting without quantitative kinetic understanding has become widespread. It was shown by Ware, Doemeny, and Nemzek18 long ago that multiexponential fitting has hardly to do with physical or mechanistic sense. Grinvald and Steinberg19demonstrated that, in a weighted sum of two exponentials, with weight factor ratio 3, this ratio can be inversed and time constants can be altered 20%without changing the root mean square (rms) of difference more than 2 X lo4. The same rms is obtained when best-fitting the sum of three exponentials by two only. Moreover, curve fitting is little helpful for distinction between reasonable and wrong theories or models, not to speak about artifacts and spurious influences. Otherwise it could hardly be understood that crude misinterpretations of experiments and even wrong formulas persist in the literature and have entered textbooks." We accept curve fitting to find an algebraic presentation of experimental data. A minimum requirement for kinetic significance of fitting parameters is that the influence of changing experimental parameters such as concentration and temperature can be calculated to be checked by experiment. Even if erroneous influences are absent, the number of parameters to be determined from one curve should be kept as small as possible, which may often be achieved by proper choice of experimental conditions. If experimental findings turn out not to be understood on the basis of actual state of knowledge, the possibility of misinterpretation and other errors should be carefully investigated. If discrepancies persist, one should try to establish another theoretical approach instead of hindering the progress of understanding by mundane curve fitting. Under such aspects we consider the procedures practised by Kauffmann et al." as exemplary in his field. Concluding Remark. The common RE procedure, where optical excitation is allowed for as an inhomogeneity, leads to un(14) Wahl, C. R. M. Dissertation, University of Stuttgart, 1987. (15) Wahl, C. R. M.; Hauser, M., unpublished results. (16) Martinho, J. M. G.; Winnik, K. A. J. Phys. Chem. 1987, 91, 3640. (17) Sienicki, K.; Winnik, M.A. J . Chem. Phys. 1987, 87, 2766. (18) Ware, W. R.;Doemeny, L. J.; Nemzek, T. L. J. Phys. Chem. 1973, 77, 2038. (19) Grinvald, A.; Steinberg, 1. Z . Anal. Biochem. 1974, 59, 583. (20) Cf.: Hauser, M. Eer. Bunsen-Ges. Phys. Chem. 1984, 88, 1244.
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 1489 reasonable results if rate coefficients depend on time. Linear response theory can properly treat only one-step reactions of that type but has no algorithm for more s t e p including backward step. This is provided by CK, the rules of which are given below, thus making CK a generalization of linear response theory. 11. The Concept of Convolution Kinetics The mechanism (viz., a reaction scheme) may be given in the usual manner, where the processes increasing or decreasing the populations of excited species Xi are symbolized by arrows, above which the rate coefficients (with or without time dependence) are noted. All are of the first order or pseudo first order, as the concentrations or populations of unexcited partners are constant in FES kinetics. At least one arrow, denoted by Za(t), symbolizes the optical excitation, which in the RE procedure (not in CK) would be allowed for as an inhomogeneity. Instead of writing down R E S , in CK the following rules are to be applied: 1. An excited-state population ( X j * ) ( t )is given by convolution of its formal &response with its production term.
(Xj* ) ( t ) = Px,**~~x,*
(1)
2. The formal &response follows immediately from the reaction scheme
where the k'k are the rate coefficients of all processes depopulating ( X j * ) . Eack process increasing ( X j * )is described by a production term (which may be a sum of production subterms) Px.. The construction of the Px,, is the most important part of CK, because it represents the fundamental difference between linear response theory and CK and is given below as the third rule. In the simplest case, namely production by direct light absorption, we have Px,. = I,(?), where I,(?) is the quantum intensity absorbed by X j . If this is the only production path of Xi.,we get according to linear response theory: ( X j * ) = Ia(t)*fax,.. In this case, may be measured, provided that I,(?) corresponds to a sufficiently short light pulse. But if one of the processes of the reaction scheme, contained infax., produces xk*, where k = j f 1 say, we have to deal with the formal &responses only, which are not measurable in general, and the corresponding production terms. What may be called the measurable responses, including &excitation, are to be calculated by CK. Only in very special cases the RE procedure may also give correct results.' 3. The production term Px*is given by convolution of the production term Px,. of species ki*,from which the process arrow starts, with the negative time derivative ofhX,., but omitting the terms of that derivative not producing Xj*. This definition may be written
&.
Px,* = -Px,.*(
-1
)
dt - dfax,. %XI*
dt = PXI**(k,jhX,*)
kU=O
(3)
where kij is the corresponding rate coefficient. By application of this recursion procedure one finally arrives at production term(s) given by light absorption. As the construction of the production term is somewhat unusual, we give the following explanation, starting with a few words about the kinetic meaning of convolution. At time 9 an incremental amount of ( Xj*) is produced d(Xj*)(8) = PX,.(r9)dr9 (if the production is by light absorption d(Xj*)(r9) = I,(r9) do). At time t the fraction d(X,*)(t) = Px,*(8)fax,*(t-r9) dr9 is still excited and the total amount at t is found by integration, giving eq 1 (Chart I), viz.
7490 The Journal oJPkysical Chemistry. Vol. 94. No. 19. I990
CHART I
Vogelsang and Hauser shown the recursion procedure of construction of production terms. Equation 3a is equivalent to eq 3 referring to the production of Xi* from Xi*. So far the procedure for calculating the FES populations is complete. 4. The connection between population (Xi*) and its total fluorescence intensity F,is given by the fourth rule, being generally valid in FES kinetics.
t
P(
Fi = k,(Xi*)
(5)
Simply speaking, Fi is the number of fluorescence quanta, generated by the process Xj*
The decay rate of increments forming the integrand in eq l a may be written
kV
Usually one of the rate constants of processes depopulating Xi* will populate Xk*. where k = j I , with a probability
*
-w.I k = w kj
kjk --
xkjk k
hX,.
It is evident that the rate coefficient -kik depopulating Xi: is the same as the rate cocfficient kki populating Xk*. Performing the inner integration we get
The terms in the square brackets may be called the increment of (Xk*) produced at time ( r - 8 ) . Finally including the decay of (Xk*) we get
where obviously
namely the production term of Xk* from Xi*. Thus we have also
(6a)
From the rate equation of spontaneous emission de -(Xi*) dr
= kej(Xj*)
describing that contribution to the depopulation of (Xi*), and with the volume integral dV
we get eq 5 when applying Gauss' formula. But the concentration [Xj*] depends on space coordinates (cf. Beer's law) and may change with time (e.g., by diffusion), in addition to the processes influencing the population (Xi*) dealt with above. For kinetic purposes it is usually sufficient to measure a spatial and/or spectral fraction of F,,while for the fluorescence quantum yield the total fluorescence intensity Fi needs to be considered.
Ill. Application of CK 1. The case without backward reactions, where the general reaction scheme is
r.(o
= [Pxy*(kk~f&x,*)] do'
(6)
d i v j . = k,[Xj*]
(Xj*) = $[Xi*]
d 8 dfi'
d5
A.
which also may be written
= Px,.(t-O)k&,.
$$
is the surface integral of the closed shell and is the fluorescence quanta flux density (=product of the spatial number density of quanta and light velocity). and 5 is the surface element vector. Eauation 5 is strictlv valid if reabsorotion of fluorescence is negligible. Anisotropy of emission may or may not be discounted on the basis of Brownian rotation. In the former case with the concentration [Xi*] we have
giving an incremental production of Xk*, d(d(Xk*)) = -wk; d((Xj*)), cf. eq 4 d(d(Xk*)) = - P x l * ( f - 8 ) w y z d 8 d o '
+ kv
and passing an arbitrary closed shell per unit time surrounding the population (Xi*),, and kei is the spontaneous emission coefficient of Xi*. For simple understanding of eq 5 we may refer to radioactivity, where the decay rate measures the amount or population of active material (and not its "concentration"). This rate is independent of the other processes which may influence (Xi*) simultaneously. More accurately Fi =
(as the production PXi(t-8) d9 is independent of the decay). From the definition, eq 2, we get
Xi
A*
- -k srl4
B* kcal4 C*
can be treated by straightforward application ofeqs 1-3. Here
and corresponding expressions forjsB. etc. are obtained. The production terms are PA. = Ia(r); P,. = kBJaA.*PA.; Pc. = kca/ie.*Ps. Examples are given in refs 1 and 2. Only convolution integrals need to be evaluated. The two-step energy transfer, where JE, = exp[-kct], turned out to be difficult to realize without spurious influences." By
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7491
Effects of Diffusion on Excimer Formation
-
proper choice of A, B, and C direct excitation can be limited to A only and (more important) direct transfer A C can be prevented. Translation diffusion, which is a strongly cooperative effect with energy transfer?* can be kept negligibly small by high enough solvent viscosity. But it is practically impassible to prevent reabsorption of donor fluorescence by the acceptor(s). If the product of concentration and extinction coefficient of an acceptor is, e.g., 300 cm-l (a typical value in energy-transfer experiments) one should apply I 1 O-" cm sample thickness to keep such reabsorption below 10% but then various surface effects become significant. In the first approximation reabsorption can be allowed for by treating gikei(Xi*)as additive excitation light; the factor gi depending on cuvette thickness and extinction coefficient(s) can be estimated and checked by application of quenchers as additional acceptors.I4 In general, quantitative interpretations of energytransfer experiments in the literature often do not withstand appropriate criticism, except perhaps the one-step case with donor fluorescence only. 2. In the general case, backward steps in the mechanism must be considered. The main feature can be seen from the following simple scheme with practical importance
Id0
:
10-1
0
a
E m
-a d
D
k E
6
10-2
X
a I
I
I
I
100
200
300
400
I 500
time in Cnsl
Figure 1.
First we show that without the time-dependent rate constant (b = 0) we get the same result as with the RE procedure. From eq 8
kw
U* ==e V*
lku ikv kw
Applying rule 3 we get Pu. = Za(f)
As (A*) = FA*fA* and (.&A? = FAA*fu*, and the transforms of the formal 6-responses arehA*= (s + kA + kJ' andhAA*= (S+ kAA+ kD)-l, We get
+ Pv.*kuvfv.
Pv* = Pu**kvUf6~*
(7)
The solution of these coupled integral equations can be given after Laplace transformation (denoted by a tilde) i(t) PU. = 1 - (kUVhU*)(kVufW*)
-
ja(r)kr
-
(AA*)= (S + kA + k,)(s + kAA
kD) - k,kD
(1 1)
Back transformation by usual methods (e.g., via partial fractions) gives with I a ( t ) = 6 ( t - 0) the real (measurable) &responses: The case where only kVu is a function of time and kuv is constant will be completely solved below (excimer kinetics). The other case with kVUis constant and kuv(t) may be important in excited-state protolysis ROH* .s RO-* + H+.
IV. Excimer Kinetics with Nonstationary Diffusion Assuming the Smoluchowski case with k, = ka[A], where k, is constant and [A] is constant (=concentration of unexcited A), we may write
-
k,(l + b/f")
A*
J"
AA*
A
(9)
ikA4
A + A
under the conditions of FES kinetics discussed, we get from eqs 1-3 ( A * ) = P A *faA*
with A1 A2 = kA + kAA + k, + kD and h1A2 = kA(kAA + kD) kAAk or equivalently 2Al/2 = -kll - k22 f [(kZ2- kll).Z+ 4k,kD]'j2, where kll = kA k, and k22 = kAA + kD. Equations 11 are well-known from the RE procedure,2 and in the general case of la(?) we get (A*) = Z,(t)*(A*)* and (AA*) = Z,(t)*(AA*),
+
+
Another procedure of back transformation we shall make use of in the case b # 0 starts from eqs 1 and 10, keeping in mind that krk&A*fAA* =
1
By expansion of 1/( 1 - q ) in a geometric series, which can be back transformed, we get the 6-responses m
(AA*) = P A A * * ~ A A * and cf. eqs 7 we obtain P A * = Ia(t) PAA.
= k,(l
+ b/t1/2)fsA**PA*
with k, = k,[A]
and for the 6-responses we get &A*
= exp(-(k, hAA*
m
+ knfsAA**PAA* + kA)t - 2k,bt1l2)
= exp[-(kD + kAA)tl
(21) Hauser, M.; Heidt, G.2.Phys. Chem. Munich 1970, 69, 201. (22) Ghele, U.; Klein, U. K.A.; Hauser, M.; Frey, R.Chem. Phys. Lett. 1975, 34, 519. GBsele, U.Prog. Reoct. Kinet. 1984, 13, 63.
(AA*)* = 1 /kD
k-0
L-'(qk+']
(12)
where L-I means the inverse Laplace transformation. Assuming the following typical rate coefficients (of pyrene at room temperature) kA = 2 X lo6 s-', ku = 17 X lo6 s-I, and kD = 7 X IO6 s-l and a concentration [A] = 2[A], 2, where the half-value concentration is [AIll2 = kA(kAA+kD)/duk,, we get from the closed form eq 1la and from eq 12 the (AA*)aas shown in Figure 1. As in eq 12 only terms up to k = 3 were taken into account, the corresponding curve is negligibly lower a t t > 400 ns.
7492 The Journal of Physical Chemistry, Vol. 94, No. 19, 1990
After having demonstrated that the back transform by series expansion, which is more general, functions reasonably, we treat the interesting case 6 > 0 of nonstationary diffusion. Decoupling of the production terms via Laplace transform gives the same eqs 12 as above, but the q expression is now more complicated: 4 =
krkDLI(1
+ b/t1/2)faA*lf8AA*
Q 0,
E
(14)
entering eq 14 were given in the context of eq 9; thus with eqs 12 and 14 the treatment is complete, merely needing various convolutions to be performed. We start with L-'(q) = k,kD(Il + b12) containing two integrals
=
10-1
0
b
hA*and faAA*
12
%
(13)
After back transformation of eq 11 we get
L-Ik?) = k $ ~ C f 6 ~+* b f A * / t 1 / 2 ) * f a A A *
Vogelsang and Hauser
0
:
10-2
6
200
100
300
400
500
time in C n s l
Figure 2.
j i A * ( r ) / r l / ~d r
x>*AA*(t-T)
which can be calculated in the closed form. We must distinguish the case k, + k A kD k A A ; using the abbreviations p1= k, + k A and p2 = k D + k A A We get for P I > P2 with
+
z = (pl
- p 2 ) 1 / 2 t 1 /+2 k r b / ( p l - P ~ ) ~ / ~ ( 1 5 )
4:0-3!.
" " "
' ' 1 , ' , ,
100
k&T1/tf6~~*(?)82(0) (PI
I2 =
[
- P2)II2
z ( t ) - erf z(0))
"
200
'
"
1 '
" "
300
' ' ' , " ,
400
500
time in C n s l
(erf z ( t ) - erf z(0))
zPI] 1-i 2P2~ 2 ( o ~ A A . ( t ) ( e r f
-
Figure 3.
(17)
we get
powers of 10. kA is usually not highly dependent on temperature but may be influenced by proper choice of excimer forming fluorescer rather than by addition of a quencher, which would increase the complexity of the kinetics. kAA is almost independent of the special excimer considered (1 / k A A = 40-60 ns from experience) and is also weakly dependent on temperature. The interesting parameter b = 2[ 3?7/ k T ] ' / 2
2krbfA*(t) (P2
-
erf x = 2 / x 1 / = Jxe-"
(Daw z ( t ) - Daw z(0))
du; Daw x = e-x2 x'eu' du
We now know, cf. eq 12, the first-order approximation (AA*)al = k r ( l l k12) where the zero approximation (AA*)& = 0, while the corresponding (A*)M,=fsA.. All further convolutions, e&, (A*)8l =hA.+ f a A * * L - ' ( q ) and the repeated self-convolutions of L-I{q),entering the higher approximations can be performed by a Simpson formula, eq 18, with variable upper integration limit, maintaining the general dependence on the five parameters of the reaction scheme. F ( t ) * G ( t ) = r/6[F(O) G ( t ) + 4F(t/2) G ( t / 2 ) + F ( t ) G ( 0 ) ]
+
(18)
Such simple approximation turns out to be useful for our purposes; it can be improved or tested by decreasing the distance of base points, of course. The influence of parameter variations on the measurable time dependences can be overlooked or even mathematically discussed. The main variable parameter of excimer kinetics are kD, since its activation energy is about 240 kJ/mol (e.g., with pyrene); k, = k,[A] = (8RT/3000v)[A], as [A] and viscosity may vary some
allows one to determine the radius r; the other parameters should be chosen so as to get a significant influence of b on the measuring curves. This is crucial for testing of C K and of validity of Smoluchowski's theory. Next we assume again k A = 2 x IO6 s-I, k A A = 17 x IO6 s-l, and kD = 7 x lo6 s-I (values of pyrene at room temperature of 293.1 K),2' but a rather high viscosity of 7 = 1.34 P (e.g., paraffin oil at room temperature). A concentration [A] = [A],p (with b=O) of [A] = 0.0582 mol/L should give a typical excimer time dependence, where 6 excitation may be fairly well approximated with an excitation flash of about 2 ns half-width. In curve 3 of Figure 2, got by C K and the Simpson procedure, a reasonable r=5X m was chosen. The shape deviates strongly from curve 1, which was calculated with RE, assuming b = 0. This is justified in RE23at least for t 1 lO-'s (=lo0 ns), as b I (23) From the reaction scheme eq 9 we get the RE d(A*) - - [ k ~ -+ k,(l + b/f'12)](A*) + k,(AA*) + I&) dt
d(AA*) -dt
-
- k,(l + k/t'12)(A*) - [ k , + ~ D ] ( A A * )
The unreasonable nature of the treatment is evident in the steady state I, = constant I a,giving (A*),t and (AA.),,, which, surprisingly, are independent of b. Generally it is the decisive fault of RE that the strong influence of b / t 1 l 2is allowed for at t = 0 only, thus justifying the above approximation. Really excited species produced at arbitrary time must be treated the same way, as is the case in CK; cf. section 11.2, to be realized with pyrene by moderately raising T o r alternatively by the choice of, e.& naphthalene at room temperature.
Effects of Diffusion on Excimer Formation
The Journal of Physical Chemistry, Vol. 94, No. 19, 1990 7493
in
103
=
102
;
101
-
c
C J
8
100
100
200
300
time in Cnsl
Figure 5. Excimer time dependence on flash excitation.
+
s 1 l 2 appears only in the combination (1 b/t'I2). In Figure 2, the deviation of curve 1 from the correct curve 3 is most stringent for the above time domain. From CK one can get curve 2 (almost m. equal to curve 1) only by choice of an unreasonable r = We see that the RE are acceptable only for b = 0 (namely if the rate coefficient does not depend on time). In Figure 3 it is demonstrated that strong excimer dissociation (k,) drastically influences the shape of (AA*)&,which is reflected only by CK. Here kA, k A A , and T are the same as in Figure 2 throughout, but with 7 = 0.01 P we have to use a low-viscosity solvent. In curve 3 (by CK) and curve 4 (by RE) kD = 7 X IO6 s-l as above, but [A] = 4.345 X IO-" mol/L = [A],/, (resulting from 7). No significant deviation between RE and CK is observed with low values of kD and 7. With an increased2 k D = 70 X lo6 s-l and correspondingly higher [A] = 1.575 X lo-' mol/L no significant change of shape results from RE, curve 2, which is surprising; but CK, curve 1, indicate a strong influence of large k,; a qualitative explanation will be given below. From the reasons given above, the comparison of CK and RE curves should be restricted to t 2 100 ns. Only the CK curves m. are valid at all t : here again r = 5 X Figure 3 would give essentially the same features if all curves had been calculated with the same [A]. Now the explanation of the necessarily strong kD influence: If k D = 0, an individual monomer underlies the nonstationary influence at one time, giving the well-known "small" effect. But if kDis large, the k, (coupled with b and t1I2)must also be large to give monomer and excimer amounts corresponding [A] In other words, the individual species associates and dissociates many times, undergoing the nonstationary effect many times. This is the meaning of our sentence "the small effect powers up to a large one" in the Introduction. With a reasonable reaction radius it was shown in Figure 2 that RE fail at high viscosity though excimer dissociation is weak; in Figure 3, at low viscosity, R E also fail with strong excimer dissociation. In Figure 4 both effects causing deviations from RE independently were combined. kA, kAA,and T were as before; kD = 70 X IO6 s-', 7 = 1.34 P,24(both high) and [A] = 7.5 X IO-' mol/L. The CK curve no longer resembles a usual (AA*)*, as expected: it looks rather like a (A*)&,which could never result from R E additionally it could be reasonably described not even by three exponentials.
V. A First Experimental Test of CK Literature data on excimer kinetics are most extensive for pyrene and have been determined by various authors. But we must keep in mind that these data originate from evaluations based upon RE, so to speak from fits to an imperfect theory. A reevaluation with CK is impossible unless the complete measuring data were available. Instead of carrying out all measurements once more for the purpose of a proper evaluation (such extended work is in (24) Strong excimer dissociation at room temperature (which may be convenient for a high viscosity) could be realized with pyrene bearing bulky substituents.
progress) we next performed only as much as needed for a test of CK. As far as literature data were used, uncertainties do not influence the quantitative conclusions to be drawn. From an Arrhenius plot of excimer dissociation rate constant k D in ref 10 we took kD = 70 X IO6 s-I at T = 334.1 K. At this temperature we measured the viscosity of our solvent paraffin oil 7 = 0.1525 P. We chose a concentration [A] = 0.050 mol/L where the uncertainties of kA = 2 X IO6 s-I and kAA= 17 X IO6 S-I may amount f30% without markedly influencing the shape of excimer time dependence. In Figure 5, curve 1 was calculated with R E as above, where the neglect of b is justified for t > 100 ns, and curve 3 with CK using a reaction radiusZSof r = 2.5 X m, which is then the only disposable parameter in the Smoluchowski approach. Both curves were convoluted with the time dependence of the excitation flash and normalized to comparable height of maximum with the measured curve 2. The latter was obtained with an Ortec single photon counting system (1024 channels, about 0.1 ns time resolution, 2 ns half-width of flash). The measured excimer time dependence, curve 2, is explained by CK only as being influenced strongly by nonstationary diffusion. This influence, however, is at the moderately high viscosity chosen and rather small r (with twice or half that value the curve shape changes drastically) brought about by the large k D , which is characteristic for CK. The real case considered here is of the general ("combined") type as discussed in the context of Figure 4, but there we had much higher viscosity and r = 5 X m. The deviation of the CK curve (3) from the measured one at t > 200 ns diminishes, if a more accurate Simpson approximation is applied instead of eq 16, which must not be confused, however, with the neglect of higher terms in Smoluchowski's approach. Small but perhaps noticeable is the deviation of curve 3 over curve 2 in the maximum region; seemingly the RE curve 1 fits better there. But in that region, curve 1 is dubious, being calculated with r = 0. (For comparison with RE at small f we had to solve the coupled differential equations given in ref 23. Taking the overshot to be significant would indicate slower association than follows from Smoluchowski's approach. We are inclined to assume that a process of proper orientation of excimer partners may participate, but cf. ref 25.
VI. Final Remarks We demonstrated various kinetic features of a TDRC not properly understood otherwise, especially the strong enhancement of transient effects with participating back reaction. The influence of reaction radius, which may reach more than 20 X m (e.g., with strong attractive forces between partners) may be much more drastic than in our example; in subnanosecond kinetics such should be the case with small radii already. Also the influences of shape of reaction surface and of nearest-neighbor molecular dynamics should be revealed in this time region. The special system investigated here, which was chosen so as to exclude spurious influences, perhaps does not belong to the most ( 2 5 ) This value is influenced by neglecting higher powers of q in eq 12. Not doing this would rquire extended numerical calculations of repeated convolutions; the simple survey of parameter changes would be lost.
J . Phys. Chem. 1990, 94, 7494-7500
7494
important applications of CK. In other cases with TDRC(s) such as energy transfer, mistakes from not applying CK may be even more strikingz6and the CK calculations simpler. Many important features of energy transfer, such as the influence of dimensionality, mutual orientation of partners and its dynamics, are far from being disclosed at present. As long as CK is not applied, misinterpretations will persist and spurious or trivial influences will not be recognized, especially if plain multiparametric curve fitting is practiced. (26) According to the RE treatment, no energy transfer takes place in the steady state.',2 In the three-dimensional homogeneous system without cooperative diffusion, the energy transfer appears only in t1I2terms of RE similar to ref 23.
CK is restricted to first- and zero-order processes, being a generalization of linear response theory. This limitation is unimportant in excited-state kinetics, except certain laser applications and triplet-triplet second-order processes. TDRCs may be operative in second-order elementary reactions too, of course. But the development of a procedure with performance comparable to that of CK is very difficult for such systems: in the simplest case, the shape of the 6-response depends on its amount, which could be. allowed, if the temporally generated local reaction centers would develop in time without interaction. Proper linearization for application of CK is recommended rather than approximating with a description by RE, which are inapplicable anyway. We are hopeful that our contribution will help progress in excited-state kinetics.
Products and Mechankms of the Gas-Phase Reactions between NOs and a Series of Alkenes J. Hjorth,* C. Lohse,+ C. J. Nielsen,f H. Skov, and G. Restelli Commission of the European Communities Joint Research Centre-Ispra (Received: November 6, 1989; In Final Form: March 27, 1990)
Site, 21020 Ispra (VA), Italy
Products and mechanisms for the gas-phase reactions of NO3 radicals with a series of alkenes in air have been studied. The experiments with propene, isobutene, trans- and cis-2-butene, 2-methyl-2-butene, and 2,3-dimethyl-2-butene were carried out at 295 f 2 K and 740 5 Torr in a 480-L Teflon-coated reaction chamber where NO3 was generated by the thermal dissociation of N2O5. Reactants and products were observed to produce nitroxy-nitroperoxy intermediates that decayed with formation of carbonyl, nitroxycarbonyl, nitroxy alcohol, and dinitrate species. The product distribution was found to be dependent on the alkyl substitution pattern around the double bond. Evidence was also found of a reaction between NO3 radical and organic peroxy radicals with a rate constant measured in the case of the 2,3-dimethyl-2-nitroxy-3-peroxybutane cm3 molecule-' s-l. to be of the order of 5 X
*
Introduction The nitrate radical, NO3,is formed in the atmosphere by the reaction between 0, and NOz: NO2 + 03 --* NO3 + 0 2 (1) During the day NO3 is rapidly photolyzed but at night it builds up and establishes the fast equilibrium
NO2 + NO3 & N 2 0 5
(2)
In recent years, the chemistry of NO3 and NzOsin air has been studied extensively and found to play an important role in the troposphere.' Kinetic studies have shown that the reactions between the nitrate radical and alkenes are relatively fast with reaction rate constants increasing with increasing alkyl substitution at the double bond.z However, investigations of mechanisms and products are still limited to the case of p r ~ p e n e . ~ -The ~ first attempts to identify reaction products in air were reported in refs 3 and 4 where an absorption band a t 1670 cm-l was tentatively attributed to a nitrite. This work was followed by a series of investigation^^-^ of the NO3-propene-air reaction performed in smog chambers and specifically aiming at the understanding of the reaction mechanism. Hoshino et al.5 found 1,2-dinitroxypropane to be a major product together with acetaldehyde, propene oxide, NO2, and H N 0 3 as minor products. FT-IR and GC-MS were used as analytical techniques. 'On leave from the Chemistry Department, University of Odense, Campusvej 55, 5230 Odense M,Denmark. *On leave from the Chemistry Department, University of Oslo, P.O. Box 1033, Blindern, 0315 Oslo 3 , Norway.
0022-3654/90/2094-7494$02.50/0
Bandow et a1.8 tentatively identified, using FT-IR,&nitroperoxy- 1-nitroxypropane (and/or 1-nitroperoxy-2-nitroxypropane) as intermediates that subsequently reacted to form 1,2-dinitroxypropane, which was identified by comparison with the spectrum of the synthesized compound. Formaldehyde and acetaldehyde were found as minor products; evidence was also found for the formation of 2-nitroxypropanal. Shepson et ale9employed a NOz to O2ratio about a 100 times lower than that used in the experiments of Bandow et ale8 Applying GC-MS and HPLC as analytical techniques they found 1-nitroxy-2-propanone and nitroxypropanols as the dominant products, concluding that under tropospheric conditions the formation of the 1,2-dinitroxypropane was unimportant. Acetaldehyde and formaldehyde were observed together with minor amounts of 1,2-dinitroxypropane. The results above could be accounted for with a reaction scheme that proceeds after the initial addition of NO3 to the double bond, e.g., at the 1-position, ( I ) Finlayson-Pitts, B. J.; Pitts, Jr., J. N. Atmospheric Chemistry; Wiley: New York, 1986. ( 2 ) Atkinson, R.; Aschmann, S. M.; Pitts, Jr., J. N. J . Phys. Chem. 1988, 92, 3454. ( 3 ) Morris, Jr., E. D.; Niki, H.J. Phys. Chem. 1974, 13, 1337. ( 4 ) Japar, S. M.; Niki, H. J . Phys. Chem. 1975, 16, 1629. ( 5 ) Hoshino, M.; Ogata, T.; Akimoto, H.; Inoue, G.; Sakamaki, F.; Okuda, M. Chem. h i t . 1978, 1367. ( 6 ) Akimoto, H.; Hoshino, M.;Inoue, G.; Sakamaki, F.; Bandow, H.; Okuda, M. J. Enuiron. Sci. Health. 1978, A13(9), 611.
(7) Akimoto, H.; Bandow, H.; Sakamaki, F.; Inoue, G.; Hoshino, M.; Okuda, M. Enuiron. Sci. Technol. 1980, 14(2), 172. (8) Bandow, H.;Okuda, M.; Akimoto, H.J. Phys. Chem. 1980,84,3604. ( 9 ) Shepson, P. B.; Edney, E.0.; Kleindienst, T. E.; Pittman, J. H.; Namie, G . R.; Cupitt, L. T. Enuiron. Sci. Technol. 1985, 19(9), 1985.
0 1990 American Chemical Society
