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Reversible fluorescence quenching: Generalized Stern–Volmer equationson the basis of self-consistent quenching constant relationsWolfgang Naumann Citation: J. Chem. Phys. 112, 7152 (2000); doi: 10.1063/1.481325 View online: http://dx.doi.org/10.1063/1.481325 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i16 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors
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JOURNAL OF CHEMICAL PHYSICS VOLUME 112, NUMBER 16 22 APRIL 2000
Reversible fluorescence quenching: Generalized Stern–Volmer equationson the basis of self-consistent quenching constant relations
Wolfgang Naumanna)
Institut fur Oberflachenmodifizierung, Permoserstraße 15, 04318 Leipzig, Germany
~Received 7 December 1999; accepted 31 January 2000!
For irreversible fluorescence quenching in solution, it is shown that the kinetic prediction of theSmoluchowski approach, which is exact under target model conditions, can also be alternativelyformulated in terms of well-defined non-Markovian rate equations. For the well-knownsuperposition approximation, it is demonstrated that the definition of an approximate quenchingconstant by a self-consistent relation can also be formally transferred to the reversible quenchingprocesses if only the low-density limits of thenet forward rate kernels in the generalized rateequations are known. Fluorescence quenching by reversible excimer formation and by reversibleexcitation transfer meet this requirement due to the recent findings of several authors. It isdemonstrated that the proposed quenching constant approximation procedure leads to nonlinearplots with positive curvature which correct the zeroth-order linear plots in the higher quencherconcentration region. The influence of the yield-reducing back reaction effect is discussed. ©2000American Institute of Physics.@S0021-9606~00!50116-0#
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I. INTRODUCTION
Recently, much theoretical work has been devotedimproving the traditional diffusion-kinetic approaches in slution kinetics in order to extend their range of applicationmore complex kinetic situations. This renewed interestdiffusion kinetics has also been advanced by the experimtal observation of new, nonclassical phenomena in the kiics of reversible solution reactions.1 Hence, some theoreticastudies have tackled the highly interesting relaxation behior of association–dissociation-type reactions, which at lotimes follow a power law instead of the formal-kinetic expnential one~see, e.g., Refs. 2–5!. Another line of recent the-oretical work studied back reaction effects if a reversireaction is part of a complex fluorescence quenchmechanism.6–17
In Refs. 7 and 8, Molski and Naumann investigatedfluorescence quenching by reversible excimer formatwithin the framework of hierarchy-type diffusion approachpreviously applied to the kinetic study oassociation–dissociation.18,19 The quenching kinetics waanalyzed under the conditions of low excitation intensity alow excimer concentration, leading for time-scale separabetween reaction and diffusion to approximate rate equatwith so-called phenomenological rate coefficients. In Reit was shown that by dropping the time-scale separationgument, the quenching kinetics has to be more generallymulated in terms of non-Markovian rate equations. Exanalytical expressions were derived for the rate kernels inlow-density limit.
Apart from these studies of unimolecular back reacteffects, the influence of a bimolecular back reaction onquenching kinetics has also been investigated by sevauthors.10–16In Refs. 10 and 12, Burshtein and Lukzen stu
a!Electronic mail: [email protected]
7150021-9606/2000/112(16)/7152/6/$17.00
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ied the kinetics of fluorescence quenching by reversiblecitation transfer in the framework of a so-calledintegral en-counter approach.20 In a more recent paper,16 Naumanntackled the same problem by using a truncated hierarchyproach in the sense of Lee and Karplus21 ~superposition ap-proximation!. For a pseudo-first-order model, allowing ongeminate-type back transfers, it was demonstrated that inlow-reactant density limit the kinetic prediction of this cocept is equivalent to that of the integral encounter approaleading to similar generalized, non-Markovian rate equtions. When modeling the transfer events as contact retions, the rate kernel limits derived in Ref. 16 exactly agrwith those obtained in Refs. 10 and 12. In a very recpaper,17 Sung et al. studied the reversible energy transfproblem with a hierarchy-type approach truncated athigher three-particle equation level. For the special casepseudo-first-order reaction, on which the treatment wasstricted in Ref. 16, these authors arrived at similar geneized rate equations as found with the two other approacHowever, the rate kernels could be given in higher ordcorrected for many-particle effects.
The aim of this paper is to suggest a procedure forextension of linear Stern–Volmer plots to the nonlineargion of higher quencher concentrations. The proposedproximation method is applicable if analytical expressionsthe net forward kernels in the generalized rate equationsknown, at least in the low-density limit. The paper is orgnized as follows: Sec. II deals with irreversible fluorescenquenching under the conditions of the target model~immo-bile fluorophore, independently moving quenchers!. It willbe shown that the exact kinetic prediction of the Smochowski approach can also be equivalently formulatedterms of generalized, non-Markovian rate equations. In SIII it is shown that the well-known superposition approximtion of the irreversible quenching constant, defined by a sconsistent relation, can be cast into a form which also s
2 © 2000 American Institute of Physics
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7153J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Reversible fluorescence quenching
gests a formally similar quenching constant approximatfor reversible fluorescence quenching, provided thatzeroth-ordernet forward kernel is known. The approximation procedure is demonstrated for fluorescence quencby reversible excimer formation and by reversible excitattransfer, which just meet this requirement. The back reaceffect on the obtainednonlinear Stern–Volmer plots is ex-plicitly discussed. Finally, in Sec. IV we give a summarythe obtained diffusion-kinetic results.
II. IRREVERSIBLE QUENCHING
The kinetics of an irreversible fluorescence quenchreactionA* 1Q→productscan be deduced from the averafate of a single excited fluorophore if theA* concentrationrA* is small enough for the competition of the fluorophorfor the surrounding quencher moleculesQ to be neglected.Moreover, if the reactive system additionally fulfills the coditions of atarget model~static fluorophores, independentdiffusing quenchers!, an exact analytical expression for thexcited fluorophore concentration can be derived. For a lintensity excitation, the formula
rA* ~ t !5E0
t
dt I ~t!rA0e2(t2t)/tAS0~ t2t!, ~1!
was derived, where the excitation profileI (t)rA0 is convo-
luted with the survival probability of an excited fluorophoinitially surrounded by an equilibrium distribution oquenchers,e2t/tAS0(t), with
S0~ t !5expH 2rQE0
t
dt kir~t!J . ~2!
In Eq. ~1!, rA0 is the initial fluorophore concentration andtA
the fluorescence lifetime. The functionS0(t) depends onkir(t), the time-dependent rate coefficient for an irreversireaction ofA* with Q,22 and on the quencher concentratiorQ .
The treatment of fluorescence quenching on the basiEqs. ~1! and ~2! is the conventional Smoluchowskapproach.23 It was recently shown by Naumann and Szathat it can be generalized to handle excitation pulses of atrary intensity.24
By differentiation and partial integration of Eq.~1!, itcan easily be demonstrated that the Smoluchowski approcan also be formulated in terms of the generalized, nMarkovian rate equation
drA*dt
5I ~ t !rA02
rA*tA
2rQH kir~0!rA* ~ t !
1E0
t
dtdkir~ t2t!
dte2(t2t)/tAS0~ t2t!rA* ~t!J .
~3!
By defining the rate kernel~memory function!
F~ t !5kir~0!d~ t !1dkir~ t !
dte2t/tAS0~ t !, ~4!
this equation can be written in the more compact form
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drA*dt
5I ~ t !rA02
rA*tA
2rQE0
t
dt F~ t2t!rA* ~t!. ~5!
Equations~18! and ~5! qualify the Smoluchowski approacof the target model as a particular case of an integral-ttheory,25,26 for which an exact formula of the memory kerncan be given.
Since Eq.~5! is a linear equation forrA* (t), Laplacetransform can be applied (f (s)5*0
`dt f(t)e2st) to obtain theformula
rA* ~s!5I ~s!rA
0
s11/tA1F~s!rQ
, ~6!
with
F~s!5kir~ t50!1E0
`
dtdkir
dte2(s1tA)tS0~ t !. ~7!
As a rule, the evolution ofrA* (t) during the quenching process can only be obtained from Eq.~6! numerically, providedthat the rate coefficientkir(t) is known. On the other handhowever, Eqs.~6! and ~7! immediately allow a formula forthe total fluorescence yieldFA* . For a finite pulse, withFA*defined by
FA* 5 limt→`
tA21E
0
t
dt rA* ~t!Y E0
t
dt I ~t!rA0
5 rA* ~0!/ I ~0!rA0tA , ~8!
the Stern–Volmer relation
FA* 51
11kQrQtA~9!
is obtained, with the quenching constant
kQ5 lims→0
F~s!5kir~ t50!1E0
`
dtdkir
dte2t/tAS0~ t !. ~10!
It can easily be shown that Eqs.~9! and ~10! also give thesteady-state yield in a continuous irradiation experim@ I (t)5constant#, as expected for a linear response.
An example of an analytically possiblekQ calculation isirreversible fluorescence quenching by a diffusion-controlcontact reaction. In this case, we havekir(t)5kD(11As2/pDt) ~with s as the contact radius,D as the relativediffusion coefficient, and the diffusion-controlled rate costantkD54pDs), andkQ can easily be obtained by comparing the Stern–Volmer relation~9! with the exact excitedfluorophore decay formula calculated by Szabo@see ~Eq.~5.1! in Ref. 23#. This leads to the complex expression
kQ53F~DtA /s2!1A3C exp~3C/p!erfc~A3C/p!
rQtA@12A3C exp~3C/p!erfc~A3C/p!#,
~11!
with the density parameterF54ps3rQ /3 and whereC53F2/(3F1s2/DtA), which is valid for neglected inter-action forces between quencher and fluorophore.
In a recent comparative study, it was demonstratedNaumann and Szabo24 how a usefulkQ approximation can be
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7154 J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Wolfgang Naumann
obtained by applying a hierarchy approach21 to the infinitecoupled set of equations satisfied by the reduced maparticle functions of the reactive problem. For the targmodel it was shown that by truncation at the pair equatlevel ~superposition!, a closed set of coupled reactive equtions is obtained@Eqs.~28!, ~29! and~35! in Ref. 24#, leadingfor the special case of continuous excitation to the steastate quenching coefficientkQ
ss determined by the ‘‘self-consistent’’ relation
kQss5$1/tA1kQ
ssrQ%kir$1/tA1kQssrQ%. ~12!
This equation was previously derived by Szabo23 using aheuristic ‘‘mean-field’’ approach. Subsequently, it was otained within the framework of an ‘‘extended’’ Smoluchowski theory of Molski27 and the statistical thermodynamics of Keizer.28 More recently, Eq.~12! was obtained byYang, Lee, and Shin25 as a diffusion level result in a kinetistudy of the stationary fluorescence quenching problem.extension of Eq.~12! for arbitrary light intensity was firstfound by Lee and co-workers.29
For diffusion control, Eq.~12! has the solution23
kQss5kD~113/2F1As2/DtA13F19/4F2!. ~13!
III. REVERSIBLE FLUORESCENCE QUENCHING
In the following it is shown that the quenching constaapproximation by a self-consistent relation, as given by~12! for irreversible quenching, can also be formally etended to the treatment of a reversible fluorescence queing process where theA* fluorescence yield is reduced byunimolecular or bimolecular back reaction~reversiblequenching!.
It can easily be seen that relation~12! for kQssalso follows
if S0(t) in kQ definition ~10! is approximated byexp(2rQkQ
sst). On the other hand, by using the rate kernelthe low quencher density approximation of the exact rate~5!,
drA*dt
'I ~ t !rA02
rA*tA
2rQE0
t
dt F0~ t2t!rA* ~t!, ~14!
which according to Eq.~7! has the Laplace transformF0(s)5$s11/tA%kir$s11/tA%, relation ~12! can also bewritten as
kQss5E
0
`
dt F0~ t !e2rQkQsst5F0~rQkQ
ss!. ~15!
Equation ~15! demonstrates thatkQss is a well-defined
functional of the zeroth-order kernelF0(t). Its structure sug-gests a similar quenching constant approximation for revible fluorescence quenching, by using instead ofF0(t) thelow-density limits of the correspondingnet forward rate ker-
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nels which, for example, for quenching by reversible exmer formation and by reversible energy transfer werecently obtained by several authors.7–10,12,16,17 In thefollowing we will demonstrate how the proposed approximtion procedure can be realized for these two reaction typ
A. Quenching by reversible excimer formation
In a very recent study, we investigated fluorescenquenching by reversible excimer formation9 underlying thereaction scheme
A→I ~ t !
A* , ~16a!
A* →tA
21
A, ~16b!
A* 1A→C* , ~16c!
C* →A1A* , ~16d!
C* →tC
21
A1A. ~16e!
This includes the photoexcitation of monomers (A) with thetime profile I (t) @reaction~16a!#, the decay of the excitedmonomers (A* ) by fluorescence@reaction ~16b!; omittingany nonradiatiative decay for mathematical convenien#and the A* quenching by formation of excimersC*[(AA)* @reaction~16c!#, which together represent the patial reactions of an irreversible quenching problem. Adtionally, it is assumed that the excimers are able to dissoc@reaction ~16d!# or decompose radiatively into unexcitemonomers@reaction ~16e!#. It was shown that in the low-density limit the quenching kinetics follows the generalizrate equations
drA*dt
5I ~ t !rA02
rA*tA
2rA0E
0
t
dt F0ex~ t2t!rA* ~t!, ~17!
drC*dt
5rA0E
0
t
dt F0ex~ t2t!rA* ~t!2
rC*tC
, ~18!
with the net forward rate kernelF0ex(t) defined by the
Laplace transform
F0ex~s!
5~s11/tA!kir~s11/tA!
11@kd /~s11/tC!#@~s11/tA!kir~s11/tA!/keq#. ~19!
In these equations,tA and tC denote the excited monomeand excimer radiative lifetime, respectively, andkd is theintrinsic excimer dissociation constant.
Taking the pattern of Eq.~15!, we assume that a finitedensity approximation of theA* quenching constantkQ
en canbe defined via the self-consistent relation
kQex5F0
ex~rA0kQ
ex
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ms
7155J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Reversible fluorescence quenching
Equation ~20! can be expected to be useful for bocontact-type and longer-range reactivities since it was shin Ref. 9 that the kernel formula~19! results for all physi-cally reasonable encounter models under detailed balaconditions. Comparing the self-consistent relations~12! and~20!, it can be seen that by the reverse reaction effectkQ
ex isgenerally smaller thankQ
ss.Figure 1 illustrates how the Stern–Volmer curv
change if excimer formation and dissociation are appromated as contact events, with
skir~s!5kakD~11As2s/D !/$ka1kD~11As2s/D !%,
by the Smoluchowski–Collins–Kimball theory.22 In this spe-cial case, Eq.~20! takes the form
kQex5
kakD
ka /@11A~s2/D !~1/tA1rA0kQ
ex!#1kD~11kdtC!,
~21!and forx5kQ
exrA0tA we obtain the transcendental equation
x5$11A~s2/DtA!~11x!%H 3FDtA
s22
kD~11kdtC!
kaxJ ,
~22!
with the density parameterF54ps3rA0/3.
From Eq.~22! it can be seen that the excimer dissoction effect enters the Stern–Volmer relations via the paraeter kD(11kdtC)/ka . Since this term disappears for diffusion control (ka→`), we can conclude that under contaencounter conditions a significant back reaction effect onexcited monomer fluorescence yield only occurs for a suciently slow excimer formation rate. In this case, a finiteka
value enables a partialA* escape from the monomer aftethe excimer dissociation.22 From Fig. 1 we can also see thfor sufficiently low ‘‘effective quencher reactivities’ka /kD(11kdtC), the Stern–Volmer curves are acceptabapproximated by linear plots with the low-density limkQ
ex(rA0[0) of the quenching constant~21!.
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B. Quenching by reversible energy transfer
The effect of a bimolecular back reaction was recenstudied by us16 for reversible energy transfer under pseudfirst-order kinetic conditions allowing only a geminate-typreverse reaction. We underlied the treatment the followreaction scheme:
A→I ~ t !
A* , ~23a!
A* →tA
21
A, ~23b!
A* 1B→$A1B* %, ~23c!
$A1B* %→A* 1B, ~23d!
B* →tB
21
B, ~23e!
which in addition to the excited fluorophore (A* ) generationby photoexcitation@reaction~23a!#, theA* fluorescence@re-action ~23b!# and theA* quenching by reversible energtransfer to an acceptorB @reactions~23c! and ~23d!# alsoinvolves the possible first-order deactivation of excitedBmolecules@reaction~23e!#.
It was shown16 that in the low concentration limit thekinetics of this complex reaction can be formulated in terof generalized, non-Markovian rate equations
drA*dt
5I ~ t !rA02
rA*tA
2rB0E
0
t
dt F0en~ t2t!rA* ~t!, ~24!
drB*dt
5rB0E
0
t
dt F0en~ t2t!rA* ~t!2
rB*tB
, ~25!
with a net forward rate kernelF0en(t) defined by the Laplace
transform
errward
asng the
F0en~s!5
~s11/tA!kfir~s11/tA!
11@kreq/~s11/tB!kr
ir~s11/tB!21#@~s11/tA!kfir~s11/tA!/kf
eq#. ~26!
In these equations,tA andtB denote theA* andB* fluorescence lifetime, respectively.rB0 is the initial concentration of the
acceptorB. kfir and kr
ir are the Laplace transforms of the irreversible rate coefficients of the partial reactions~23c! and ~23d!,respectively, with the initial valueskf
ir(0)5kfeq and kr
ir(0)5kreq.
The F0en expression~26!, which was first obtained by Burshteinet al. by applying a so-called integral encount
approach,10,12 is only valid for contact transfer events. It was shown in Ref. 16 how it can be generalized for modeling foand back energy transfer with more realistic reactivity ranges.
Following the pattern of the self-consistent relations~15! and~20!, we postulate forkQen, theA* quenching constant in the
complex mechanism~23!, the self-consistent relation
kQen5F0
en~rB0kQ
en
7156 J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Wolfgang Naumann
skfir~s!5
kakD~11As2s/D !
$ka1kD~11As2s/D !%, skr
ir~s!5kbkD~11As2s/D !
$kb1kD~11As2s/D !%,
with ka[kfeq andkb[kr
eq as the intrinsic forward and reverse transfer constants, respectively.By means of these relations, Eq.~27! takes the special form
kQen5
kakD
ka /@11A~s2/D !~tA211rB
0kQen!#1kb /@11A~s2/D !~tB
211rB0kQ
en!1kD
, ~28!
-ib
tl-thte
du
inhiea
getetic
ed,
ledbe
iontheingldion
tly
the
forhatell-
hinhin
ingcon-
and forx5kQenrB
0tA we get the transcendental equation
x5$11A~s2/DtA!~11x!%H 3FDtA
s22
kD
ka
3S 11kb
kD~11A~s2/DtB!~11x!D xJ , ~29!
with the density parameterF54ps3rB0/3.
From Eq.~29! it follows that, like quenching by the reversible excimer formation and also quenching by reversenergy transfer, the reverse reaction effect on theA* quench-ing yield disappears in the limit of diffusion control (ka
→`). It can also be seen that the back transfer effecsuppressed fortB→0, since then all excited acceptor moeculesB* are rapidly deactivated by luminescence beforereverse reaction~23d! becomes effective. In the opposicase,tB→`, by comparing Eqs.~22! and~29! it can be seenthat the back transfer from the acceptor has the same reing effect on kQ
en as the excimer dissociation onkQex, if
kb /kD5kdtC . Therefore, the Stern–Volmer curves givenFig. 1 are also representative of the fluorescence quencby reversible energy transfer on these conditions. Figurdemonstrates how the Stern–Volmer curves change by btransfer for a finite lifetimetC of the excited acceptorsB* .
FIG. 1. Comparison of the Stern–Volmer plots for fluorescence quencby reversible excimer formation predicted by the zeroth-order quencconstant approximationkQ
ex(F50) ~broken lines! and by the completekQex
formula ~21! ~full lines!.
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IV. SUMMARY
For irreversible fluorescence quenching under tarmodel conditions, it has been shown that the exact kinprediction of the Smoluchowski approach@Eqs.~1! and ~2!#can equivalently be formulated in terms of a generaliznon-Markovian rate equation@Eq. ~3!#. For the quenchingconstant approximation by a self-consistent relation@Eq.~12!#, which, e.g., can be obtained by applying a so-calsuperposition approach, it was shown that it can alsotaken as a functional of the low-density limit rate kernel@Eq.~15!#. This suggests a similar higher density approximatfor reversible fluorescence quenching by postulatingquenching constants as functionals of the correspondzeroth-ordernet forward rate kernels. This procedure coube realized for quenching by reversible excimer formatand by reversible excitation transfer@Eqs. ~20! and ~27!#,where the low-density limits of the rate kernels were recenobtained by several authors.10,12,16,17At higher quencher con-centrations, the Stern–Volmer curves obtained show thepositive curvature caused by the short-time behavior ofrate kernels@see Eq.~15!#, which is well-known from theirreversible fluorescence quenching.23 It was demonstratedthat the reducing effect of the back reaction disappearsdiffusion control on contact encounter conditions and tthe quenching constants expressions coincide with the w
ggFIG. 2. Comparison of the Stern–Volmer plots for fluorescence quenchby reversible energy transfer predicted by the zeroth-order quenchingstant approximationkQ
en(F50) ~broken lines! and by the completekQen for-
mula ~28! ~full lines!.
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7157J. Chem. Phys., Vol. 112, No. 16, 22 April 2000 Reversible fluorescence quenching
known irreversible result@kQss in Eq. ~13!# in this limit.
Hence, a significant back reaction effect can only bepected for sufficiently slow, diffusion-influenced forward ractions. In this case, the finite reactivity of the quenchprocess means that not all newly formed back reaction pucts are necessarily immediately retransformed and, thfore, can reduce the quenching yield.
ACKNOWLEDGMENTS
This work was kindly supported by the Deutsche Fochungsgemeinschaft, by the Sa¨chsische Ministerium fu¨r Wis-senschaft und Kunst, and by the Fonds der Chemischendustrie.
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