Impurity quenching of fluorescence in intense light. Violation of the Stern–Volmer law

Impurity quenching of fluorescence in intense light. Violation of theStern–Volmer lawO. A. Igoshin and A. I. Burshtein Citation: J. Chem. Phys. 112, 10930 (2000); doi: 10.1063/1.481732 View online: http://dx.doi.org/10.1063/1.481732 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v112/i24 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 24 22 JUNE 2000

Impurity quenching of fluorescence in intense light. Violationof the Stern–Volmer law

O. A. Igoshin and A. I. BurshteinChemical Physics Department, Weizmann Institute, Rehovot, 76100, Israel

~Received 1 February 2000; accepted 31 March 2000!

The original integral encounter theory~IET! was used to describe the kinetics and quantum yield offluorescence in the presence of intense pumping light. We compare the energy quenching throughan impurity induced interconversion, with energy quenching produced by bimolecular chargetransfer from the excited donor to the acceptor of the electron. In the former case, the convolutionrecipe which expresses the quantum yield via the system response tod-pulse excitation is confirmedunder special conditions, while in the latter case it was shown to not be applicable at all. By meansof IET we found the stationary concentrations of excitations and ions and demonstrated thequalitative violation of the classical Stern–Volmer law at high intensity of pumping light. Themodified form of this law was proposed instead and the light dependence of its constant wasdetermined in the contact approximation. ©2000 American Institute of Physics.@S0021-9606~00!51624-9#

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I. INTRODUCTION

Conventional chemical kinetics of bimolecular reactioin liquids is based on Markovian rate equations of the maction law bilinear in reactant concentrations. This approcompletely ignores the nonstationary evolution of phochemical reactions initiated by light excitation of the fluorecent molecules. The non-Markovian generalization ofrate approach known as differential encounter theory~DET!is necessary to account for the nonstationary energy tranoriginated from a distortion of the uniform pair distributioof reacting particles.1–6 The time evolution of this distribu-tion, accompanied by the decay of excited reactants, affnot only the energy quenching kinetics but also the fluorcence quantum yield. The latter depends on the rates ointramolecular processes accompanied energy transfecluding excitation rate~light strength! whose increase turnthe fluorescence signal into a nonlinear system responsecontemporary non-Markovian theories account for arbitrlight strength only in case of great excess of energy queners when the rate equations are linear in variable concention of reactants. Here we will use for the same goalalternative but also non-Markovian integral encountheory7–13 ~IET! which is beyond the rate concept. It is freof the above-mentioned limitation and suitable not onlyelementary energy quenching but also for the multistageergy or electron transfer processes of arbitrary complexi

Let us illustrate the problems of the rate theories byexample of elementary quenching reaction representedthe following scheme:

D* 1Q⇒D1Q. ~1.1!

The non-Markovian rate equation for survival probabilityexcitationP(t)5@D* #/@D* # t50 is very simple:

P52k~ t !cP . ~1.2!

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Here c5@Q#5const andk(t) is the time dependent ‘‘rateconstant.’’ The energy dissipation of this sort is usually acompanied by mono-molecular decay of the excitation wthe lifetimet:

D* →D . ~1.3!

The term representing decay rate,2P/t, can be added to theright-hand side of Eq.~1.2!. Unfortunately, the excitation ofluorescent molecules with a rateI cannot be accounted for inthe same way. It has been stressed many times that thepumping cannot be incorporated into non-Markovian eqtions in the same fashion as in their Markovian analog~wherek5const).6,14–16Therefore, the stationary concentrtion of excited and non-excited molecules,Ns* and Ns , aswell as the relative fluorescent quantum yieldh, cannot befound by the same means as in conventional chemical kiics. In the extreme case of static energy quenching, the nMarkovian rate constantk(t)→0 with t→`, so that inser-tion of the pumping term in an equation like~1.2! leads toconfusion: the stationary solution does not exist. Instead,simple convolution recipe is in common use forN* (t)5@D* # calculation at arbitrary light excitationI (t). At I5I 05const it provides the well-known formula for the quatum yield calculation:6,17

h5E0

`

P~ t !e2t/t dt/t. ~1.4!

This formula valid for only infinitely weak stationary pumping (I 0→0) was later on extended to arbitrary strong fielby appropriate generalization of the convolution recipe.18,19

However, the validity of the latter is strongly restricted bthe linear kinetic equations for excited particle concenttions N* .

The kinetic equation like~1.2! is evidently linear be-causec5const, but this is not a case when the quenchersexpended in course of reaction. One has either request t

0 © 2000 American Institute of Physics

cense or copyright; see http://jcp.aip.org/about/rights_and_permissions

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10931J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 Impurity quenching of fluorescence

to be in great excess and neglect the expenditure or lookalternative theory. For instance, energy quenching cancarried out by charge transfer from excited electron doD* to electron acceptorA according to the multistage reaction scheme:

D* 1A⇒@D1•••A2#↘↗D11A2⇒D1A

@D•••A#.~1.5!

The forward electron transfer must end by backward tranin the geminate ion pair@D1•••A2# followed by ion sepa-ration and subsequent bimolecular recombination in the bThe recombination completes the cycle with nonexciproductD, making possible the stationary regime under pmanent excitationD→D* .

The convolution recipe is applicable if not only the linearity of rate equations but also an equality of encoundiffusion coefficients of reactants and reaction productsguaranteed. Assuming these conditions are fitted, the colution recipe can be confirmed as well as the conventioStern–Volmer law, representing the quantum yield conctration dependence:

1/h511kct. ~1.6!

However, in the case of reaction~1.5! both conditions arenever fit simultaneously and consequently there is no pfor the convolution recipe. The way out is provided by toriginal integral encounter theory~IET!,7 which is free ofboth limitations.

The IET uses memory function formalism insteadconventional rate equations and allows additive inclusionlight pumping into integro-differential kinetic equations. Whave already taken advantage of this possibility calculathe stationary photoconductivity and other parallel proceslinear in I 0.12,13 Here we will solve the similar problem ostationary fluorescence but more generally, keeping all nlinear effects in field strengthI 0. These results of IET will beconfirmed by comparing them to the results obtained eawith the generalized convolution recipe and many-partiapproach to the simplest quenching mechanism~1.1! allow-ing an exact solution. However, extension of the theorymore complex reactions like~1.5! is only possible withinIET and leads to essential correction of the Stern–Volmlaw or at least its constant. We will propose the generaliform of this law which relatesh in a usual way not to totaconcentration of electron acceptors but to their neutral frtion reducing in the high fields. We will also show that fboth quenching mechanismsk5k(I 0 ,t) is field dependentand this dependence in the case of ionization is ratherusual. The latter accounts for the new effect of non-uniforegeneration ofD in the bulk due to charge recombinatioSuch an effect is trapped only by IET memory function fomalism.

The outline of this paper is as follows. In the next setion we will discuss the different recipes to the calculationrelative fluorescence quantum yield at strong illumination.Sec. III general IET formalism will be set out and appliedthe simplest energy quenching reaction~1.1!. There the va-lidity limits of the convolution recipe and original SternVolmer law will be established and the field dependence


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the Stern–Volmer constant disclosed using the contactproximation or exponential quenching rate. In Sec. IV ttheory will be extended to a multistage electron transferaction ~1.5! and the generalized Stern–Volmer law will bestablished. Using contact approximation we will study tfield dependence of its constant as well as the nonlinearlationship between concentrations of charged and neutraceptors responsible for the nonlinearity ofh21(c) depen-dence. In the conclusions we will summarize the resultsthe present study of fluorescence quenching.

II. RELATIVE FLUORESCENCE QUANTUM YIELDAND GENERALIZED CONVOLUTION RECIPE

Under stationary excitation with a rateI 5I 0, one usu-ally studies the fluorescence quantum yieldx defined viastationary concentrations of excited and ground states offluorophor,Ns* 5@D* # andNs5@D#:

x5ANs*

BI0Ns, ~2.1!

whereA andB are Einstein coefficients. The quantum yiein absence of quenchers is

x~0!5A

Bt , ~2.2!

where we used the relationNs* /Ns5I 0t and the definition oft established in Appendix A.

The bimolecular impurity quenching parallel to monmolecular decay makes the quantum yield of fluorescedependent on the quencher concentrationc5@Q#. Thehigher isc, the smaller isx(c). To study this dependence amerely an impurity effect, the relative quantum yieldh isused:

h~c!5x~c!/x~0!5Ns*

I 0tNs. ~2.3!

HereNs* andNs are the stationary populations of excited aground states in the presence of quenchers in solution.

An important question arises: how to relate these polations to the survival probability of excitations generatedt50 by a d pulse. The conventional inter-relationship btweenP(t) and the stationary state populations under permnent illumination, is given by the simplest convolutiorecipe:6

Ns* 5I 0NsE0

`

P~ t !e2t/t dt, ~2.4!

whereP(t)exp(2t/t) is the kinetics of a decay of excitationssubjected to simultaneous monomolecular decay and imrity quenching of the sort~1.1!. The latter is actually a catalysis of fluorophor deactivation by impurities whose nuber does not change in the course of reaction~1.1!: c5const. This reaction named~i! in Ref. 18 was studied thereafter d pulse excitation and under the action of a light puof arbitrary shape. The interrelationship between the twoproaches is represented by a generalized convolurecipe:19


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10932 J. Chem. Phys., Vol. 112, No. 24, 22 June 2000 O. A. Igoshin and A. I. Burshtein

N* ~ t !5I 0N0E0

t

e2(I 011/t)t8P~ t8!dt8, ~2.5!

which implies that the total number of fluorescent molecu

N* ~ t !1N~ t !5N0 ~2.6!

remains constant. This recipe obtained by NaumannSzabo,19 within the many-particle formalism, is valid at abitrary c, provided thatD* is immobile andQ’s move inde-pendently. According to Eqs.~2.5! and ~2.6! the stationarydensities of fluorophors in the excited and ground states

Ns* 5I 0N0P~ I 011/t!,

Ns5N0@12I 0P~ I 011/t!#, ~2.7!

where

P~s!5E0

`

e2stP~ t !dt.

Substituting Eq.~2.7! into Eq. ~2.3!, we obtain the mostgeneral, contemporary definition of the relative quantyield resulting from the convolution recipe:

1

h5

t

P~@11I 0t#/t!2I 0t. ~2.8!

This is an extension to arbitraryI 0 of the low field definition~1.4!, following from Eqs.~2.4! and~2.3!. If the terms of theorderI 0t are small and can be neglected, the general form~2.8! reduces to~1.4!: h5 P(1/t)/t. The latter can bebrought into the form of the Stern–Volmer law~1.6! if onekeeps only a linear part of the concentration expansionP(c,1/t). Only this linear part is reproduced in IET whilalternative approaches, like superposition approximatio20

modified encounter theory,21 and DET, keep also the nexterm quadratic inc. In fact, DET keeps even all higher termthough in a single particular case whenD* is immobile andQs move independently.22,23 The second order terms arslightly different in all the theories while the linear one, reresented byk, is the same in all of them.24

The complex dependencek(1/t) was the subject of numerous investigations~see for instance Refs. 6 and 24 areferences herein!. However, the results of them should brevised if the excitation rate is strong. This revision may sfrom Eq.~2.8! and result in the specification of thek(I 0,1/t)dependence. However, this is enough if and only if the cvolution recipe~2.5! is actually valid. Otherwise, one shoufirst revise the Stern–Volmer law itself starting from thmost general definition of the quantum yield given in E~2.3!. Unfortunately, for complex mechanisms of enerquenching this law need not hold true when the illuminatis so strong thatI 0t@1. At least nothing similar to a convolution recipe~2.5! was established in Ref. 18 for reactiontype ~ii ! to which the electron transfer~1.5! should be as-signed. Fortunately, IET provides an alternative way to fiNs* due to the radical change of the mathematical formalisfrom differential to integral.


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III. ENERGY QUENCHING BY INERT IMPURITIES

A. General IET formalism

According to reaction scheme~1.1! Q’s are inert par-ticles whose concentrationc remains unchanged. But thfluorophor has at least two electronic states, the excitedground one, whose populations compose the vector

DW 5S N*

N D . ~3.1!

The components of this vector obey the particle conservalaw ~2.6!.

Equations~A4! and ~2.6! for monomolecular processeswritten for DW , take in IET the following matrix form:9

d

dtDW ~ t !5QDW ~ t !, where Q5S 21/t I 0

1/t 2I 0D . ~3.2!

Taking into account bimolecular quenching, IET providthe following generalization of this equation:9

d

dtDW ~ t !5QDW ~ t !1cE

0

t

dt8 M ~ t8!DW ~ t2t8!. ~3.3!

From the two matrices denoted by the bold font,Q repre-sents the monomolecular transitions either light inducedspontaneous, whileM (t) is the matrix memory function representing the bimolecular quenching of the excitation. Tlatter is given by its Laplace transformationM (s), denotedwith a tilde throughout this paper. It is defined by the folowing integral over the space:

M ~s!5E d3r W~r !F~r ,s!@sI2Q#. ~3.4!

HereI is an identity matrix, whileW is a reactivity operatorof the same rank:

W5S 2Wq~r ! 0

Wq~r ! 0D . ~3.5!

The matrix elementsWq(r ) represent the distance dependerate of quenching borrowed from quantum mechanics, wha pair density matrixF(r ,t) obeys the equation

] tF5LF1WF1QF. ~3.6!

There is initial condition to it,F(r ,0)5I , and reflectingboundary condition at contact. The elements of the oper

L5S L1 0

0 L2D ~3.7!

run the relative motion in pairsDB andD* B. It is usuallyassumed to be diffusional, with or without force interactibetween the particles.

RepresentingF(r ,t) andM (r ,t) as follows:

F5S n1 m1

m2 n2D , M5S 2S1 2S2

S1 S2D , ~3.8!

and writing matrix equations~3.3! in components, we havethe following set of kinetic equations for the state densitunder permanent illumination:


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dt8 S1~ t8!N* ~ t2t8!

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t

dt8 S2~ t8!N~ t2t8!52d

dtN. ~3.9!

The kernels,

S1~s!5E d3r Wq~r !@ n1~r ,s!$s11/t%2m1~r ,s!/t#,

~3.10!

S2~s!5E d3r Wq~r !@m1~r ,s!$s1I 0%2 n1~r ,s!I 0#,

~3.11!

are expressed through the pair densities. They obey thelowing set of equations:

~] t2L11Wq~r !11/t!n1~r ,t !5I 0m2~r ,t !, ~3.12!

~] t2L11Wq~r !11/t!m1~r ,t !5I 0n2~r ,t !, ~3.13!

~] t2L21I 0!n2~r ,t !5~Wq~r !11/t!m1 , ~3.14!

~] t2L21I 0!m2~r ,t !5~Wq~r !11/t!n1 , ~3.15!

with initial conditions

n1~r ,0!5n2~r ,0!51, m1~r ,0!5m2~r ,0!50.

To simplify the above system let us introduce the flowing auxiliary density:

m~r ,s!5 n1~r ,s!I 02m1~r ,s!~s1I 0!. ~3.16!

Performing the Laplace transformation of Eqs.~3.12!–~3.15!we find the integral equation form(r ,s):

@s2L11Wq~r !11/t1 P#m~r ,s!50. ~3.17!

Here P is an integral operator in coordinate space withkernel

P~r ,r 0 ,s!5I 0g2~r ,r 0 ,s!,

where g2 is the Green function of Eqs.~3.14! and ~3.15!.Since there is no source term in Eq.~3.17! its solution istrivial: m(r ,t)[0. Using this result in Eq.~3.16! we obtainan important relationship:

m1~r ,s!5I 0

s1I 0n1~r ,s!. ~3.18!

Making use of it in Eqs.~3.10! and ~3.11! we can reducethem to the following set:

S1~s!5sH s1I 011/t

s1I 0J E d3rWq~r !n1~r ,s!, S250.

~3.19!

Using Eqs.~2.6! and ~3.18! in Eq. ~3.9! we obtain thefinal equation for the kinetics of excitation accumulation uder permanent illumination:

d

dtN* 52

N*

t2I 0N* 1I 0N02cE

0

t

dt8S1~ t8!N* ~ t2t8!,

~3.20!


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whereS1(t) is defined through its Laplace transformatio~3.19!. The solution of equation~3.20! also can be found byits Laplace transformation:

N* ~s!5I 0N0 /s1N* ~0!

s11/t1I 01cS1~s!. ~3.21!

This is a good starting point to check the validity of thconvolution recipe.

B. Confirmation of the convolution recipe

Here we will derive the IET equivalent of the convolution recipe and demonstrate its consistency with the relatship ~2.5! obtained previously in Refs. 18 and 19. In theworks it was essentially simplified assuming the motionpairsD* B andDB is the same as well as the correspondioperatorsL15L2. The diffusion coefficients and force interactions are really the same in the excited and ground stprovided their charge states do not change. Under theseditions we can get an important relationship by summingEqs. ~3.12! and ~3.15!:

n1~r ,t !1m2~r ,t !51. ~3.22!

Using this relationship in Eq.~3.12! and performing itsLaplace transformation we obtain the following equationn1:

@s2L11Wq~r !11/t1I 0#n1~r ,s!511I 0 /s. ~3.23!

Let us introduce now a new densityn(r ,t) that obeys thekinetic equation

@] t2L11Wq~r !11/t1I 0#n~r ,t !50, ~3.24!

with the initial conditionn(r ,0)51. The Laplace transfor-mation ofn simply relates to that given by Eq.~3.23!:

n~r ,s!5sn1~r ,s!/~s1I 0!.

In the new terms the Laplace transformation of the kernelS1

becomes much simpler:

S1~s!5$s1I 011/t%E d3r Wq~r !n~r ,s!. ~3.25!

It is of interest that the same equality may be represenas

S1~s!5S0~s1I 011/t!, ~3.26!

where the newly introduced kernel,

S0~s!5sE d3r Wq~r !n~r ,s!, ~3.27!

is expressed through the conventional pair densityn(r ,t)commonly used in DET.4,5 The Laplace transformation othe latter simply relates to that ofn:

n~r ,s1I 011/t!5 n~r ,s!,

and its original,n(r ,t), obeys Eq.~3.24! but with I 051/t50. Therefore, whenS0(t8) is substituted forS1(t8) in Eq.


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~3.20!, the latter continues to describe the quenching kinebut in the absence of light pumping and monomolecularcay.

Let us now consider the survival probabilityP(t) in thelong-lived excited state (t5`), after instantaneous excitation at t50. It can be obtained from Eq.~3.20!, keeping inmind thatP(t)[N* (t), if N* (0)51 and I 051/t50. Theresult is given by its Laplace transformation,

P~s!51

s1cS0~s!. ~3.28!

The relaxation of the excitation density under permanenlumination, N* (t), from N* (0)50 to its stationary valueNs* , simply relates to this result. Substituting Eq.~3.25! intoEq.~3.21!, we come to the following relationship betweethese two quantities:

N* ~s!5I 0N0P~s1I 011/t!/s. ~3.29!

Making the inverse Laplace transformation of this relatioship we come to the convolution recipe~2.5! obtained in Ref.19. Hence, for equal diffusion coefficients, the convolutiformula can be derived from integral encounter theory,well as from the many-particle~essentially differential!theory used in Ref. 19. Though at long timesP(t) obtainedwith these theories is different,21 as well asN* (t), the rela-tionship betweenN* (t) andP(t) remains the same.

Generally speaking, this relationship is nonlinear in tfield intensityI 0. However, if the applied light is weak oncan neglectI 0 in the integrand of Eq.~2.5!, thus reducing itto the simplest convolution formula~2.4! proposed earlier.25

However, the validity criterion of such a simplification is nobvious. To make the situation clearer one must investigthe stationary solutions of Eq.~3.20! with the kernel definedin Eq. ~3.25! through the pair correlation function that obeEq. ~3.24!. In the contact approximation the required statioary solution will be obtained analytically in the next subsetion and used for a quantum yield analysis.

C. Stern–Volmer constant field dependence

The stationary concentration of excited molecules canobtained either as a solution of Eq.~3.20! with the left-handside equal to zero, or by passing the limits→0 in Eq.~3.21!multiplied by the Laplace variables. In any case the result ithe following:

Ns* 5N0

I 0t

11I 0t1cS1~0!,

Ns5N02Ns* 5N0

11cS1~0!

11I 0t1cS1~0!. ~3.30!

Using this result in Eq.~2.3! we confirm the Stern–Volmelaw ~1.6! with a constant

k~ I 0!5S1~0!5~ I 011/t!E Wq~r !n~r ,0!d3r . ~3.31!

The analytical evaluation of thek(I 0) dependence iseasily available in the contact approximation of the quen


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ing rate, relatingWq(r ) to the kinetic rate constantkq

5*Wq(r )d3r . Coming to the extreme, one can replace tsharp exponential decrease of the ‘‘exchange’’ quenchrate,

Wq5W0 exp@22~r 2s!/L#, ~3.32!

by the following ‘‘contact’’ approximation:

Wq~r !5kq

d~r 2s!

4pr 2, ~3.33!

wheres is a distance of the closest approach. In the abseof a force interaction between reactants, the diffusionalerator L15DD. Substituting this operator and contact ra~3.33! into Eq. ~3.24!, one can easily solve it and using thresult in Eq.~3.31!, find the contact solution of the problem

k5kqkD

kD1akq. ~3.34!

HerekD54pDs is the diffusional rate constant, and impotant correction factor,

a51/~11Atd~ I 011/t!!, ~3.35!

includes the dependence onI 0 , t and the diffusional en-counter timetd5s2/D. This is the straightforward extension for I 0Þ0 of the well-known result9,8

k~0!5kqkD

kD1kq /~11Atd /t!, ~3.36!

wheret[tD .In the kinetic control limit (kD→`) there is no depen-

dence of the Stern–Volmer constantk5kq on either 1/t orI 0. However, it builds up with both of them in the alterntive, diffusion control limit (kq→`):

k~ I 0!5kDAtd~ I 011/t! 1kD . ~3.37!

This field dependence is actually the qualitative testwhether the quenching is under kinetic or diffusional contrThese mechanisms can be discriminated experimenwithout heavy measurements of diffusion coefficients:there is no field dependence, this is kinetic quenching, ifdependence is pronounced the quenching is diffusioMore quantitative statements are not possible in the conapproximation.

To make them possible one should return to Eqs.~3.12!and ~3.15!, resolve them with respect ton1 and using the

result in Eq.~3.19!, find k5S1(0). We areable to performthese numerical calculations having at our disposal a speprogram developed in Ref. 26. The results shown in Figdemonstrate that the difference between the real spacepersion of the quenching rate~3.32! and its contact mode~3.33! is essential. It significantly affects the field depedence of the Stern–Volmer constant: compare the scurve~b! and its contact analogue~dashed line! in Fig. 1. Inthe same figure we also demonstrate the sensitivity ofresults to the difference between the diffusion coefficientsthe stable and excited pairs,D•••Q andD* •••Q. This dif-ference ignored in the convolution recipe affects the fi


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n

ta


dependence no less than the contact approximation. Thisunique advantage of IET, that it is free of any limitationsthis sort and can be used for quantitative investigationsfield effects, at arbitrary diffusion coefficients and at aspace dispersion of quenching rate.

IV. QUENCHING BY ELECTRON TRANSFER

A. General IET formalism

According to the reaction scheme given by Eqs.~A1!and ~1.5! there are three electronic states of the electronnor, but only two of the acceptor which are involved in telectron transfer and the interaction with light. Namely,

DW 5S N*

N1

ND and AW 5S A

A2D . ~4.1!

A few conservation laws should be added to these detions:

N1N* 1N15N0 , A1A25c, N15A2. ~4.2!

Vectors ~4.1! obey the following general equations ointegral encounter theory:

d

dtDW 5TrAE

0

`

M ~ t8!@DW 3AW #~ t2t8!dt81QDDW , ~4.3!

d

dtAW 5TrDE

0

`

M ~ t8!@DW 3AW #~ t2t8!dt8. ~4.4!

HereDW 3AW is a direct vector product in the basis of colletive states of the donor–acceptor pair, which is a direct pruct of theD basis with theA basis. The relaxation matrixQD

represents in this basis the light induced and radiationtransitionsD*�D corresponding to Eq.~A1!:

FIG. 1. The field dependence of the Stern–Volmer constant for the disquenching rateWq5103 exp@22(r2s)/L# ns21 at different diffusion coef-ficients in pairs involving the excited or ground state molecule,DD* Q andDDQ : ~a! DD* Q50.1DDQ51025 cm2/s, ~b! DD* Q5DDQ51025 cm2/s@dashed line, the same, but in contact approximation withkq5*Wq(r )d3r51.93105Å3/ns#, ~c! DD* Q510DDQ51025 cm2/s. Other parameters,s55 Å, L51.0 Å, t510 ns.


s aff

-

i-

-

ss

QD5S 21/t 0 I 0

0 0 0

1/t 0 2I 0

D . ~4.5!

To account for the bimolecular reaction of electron transone must define the reactivity and relaxation matrices insame basis:

W5S 2WI 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 WR 0

0 0 0 0 0 0

WI 0 0 0 2WR 0

0 0 0 0 0 0

D , Q5I23QD .

~4.6!

HereI2 denotes the 232 identity matrix. Equation~3.4! willcontinue to be used for the Laplace transformation ofkernel,M (s), which is expressed through the pair distribtion matrix F. The latter obeys Eq.~3.6! with W and Qdefined by Eq.~4.6!.

Proceeding as before and making analytical matrix coputations by means ofMathematica, we wrote the matrixkinetic equations~4.3! and ~4.4! in components which obeythe following set of equations:

] tN* ~ t !52E0

t

S1~ t2t8!N* ~ t8!A~ t8!dt8

1E0

t

S2~ t2t8!N~ t8!A~ t8!dt8

2E0

t

S3~ t2t8!N1~ t8!A2~ t8!dt8

2N*

t1I 0N, ~4.7a!

] tN~ t !5E0

t

S4~ t2t8!N* ~ t8!A~ t8!dt8

2E0

t

S5~ t2t8!N~ t8!A~ t8!dt8

1E0

t

S6~ t2t8!N1~ t8!A2~ t8!dt81N*

t2I 0N,

~4.7b!

whereS i are elements of the matrix kernelM which is simi-lar to that of Eq.~3.8! but of higher rank (636). The par-ticle balance provides all other equations:

] tN15] tA

252] tA52] t~N1N* !. ~4.8!

Expressions for kernels, following from the matrix definitioof M in Eq. ~3.4!, take the form

S1~s!5E d3r WI~r !~@s11/t#n1~r ,s!2m1~r ,s!/t!,

~4.9a!

nt


al

rpwila

nd-

-ec-

e

er-

n athetheous

ll

the

de-ofaltainac-


S2~s!5E d3r WI~r !~ I 0n1~r ,s!2@s1I 0#m1~r ,s!!,

~4.9b!

S3~s!5sE d3r WI~r !m2~r ,s!, ~4.9c!

S4~s!5E d3r Wr~r !~@s11/t#m3~r ,s!2m4~r ,s!/t!,

~4.9d!

S5~s!5E d3r Wr~r !~ I 0m3~r ,s!2@s1I 0#m4~r ,s!!,

~4.9e!

S6~s!5sE d3r Wr~r !n2~r ,s!. ~4.9f!

Herenk (k51,2) are diagonal matrix elements ofF, equalinitially to unity (nk(r ,0)51), while m j ~j51,2,3,4! are off-diagonal elements of the same operator, which are initizeros. Here are the equations for all of them:

@] t2L11WI~r !11/t#n1~r ,t !5I 0m5 , ~4.10a!

@] t2L21WR~r !#n2~r ,t !5WIm2 , ~4.10b!

@] t2L31I 0#n3~r ,t !5m1 /t1WRm4 , ~4.10c!

@] t2L11WI~r !11/t#m1~r ,t !5I 0n3 , ~4.10d!

@] t2L11WI~r !11/t#m2~r ,t !5I 0m6 , ~4.10e!

@] t2L21WR~r !#m3~r ,t !5WIn1 , ~4.10f!

@] t2L21WR~r !#m4~r ,t !5WIm1 , ~4.10g!

@] t2L31I 0#m5~r ,t !5WRm31n1 /t, ~4.10h!

@] t2L31I 0#m6~r ,t !5WRn21m2 /t. ~4.10i!

Fortunately, this set of equations can be essentiallyduced because some of the pair densities are not indedent, as well as in the previous section. In Appendix Bestablish the following relationships between them, simto Eq. ~3.18!:

I 0n1~r ,s!2~s1I 0!m1~r ,s!5S250, ~4.11!

I 0m3~r ,s!2~s1I 0!m4~r ,s!5S550. ~4.12!

Equations~4.9a! and ~4.9d! for S1 andS4 can also be sim-plified using Eqs.~4.11! and ~4.12!:

S1~s!5s~s1I 011/t!

s1I 0E d3r WI~r !n1~r ,s!, ~4.13!

S4~s!5s~s1I 011/t!

s1I 0E d3r Wr~r !m3~r ,s!. ~4.14!

After reduction the kinetic equations~4.7a!–~4.7b! take theform


ly

e-en-er

] tN* 52E0

t

S1~ t2t8!N* ~ t8!A~ t8!dt8

2E0

t

S3~ t2t8!N1~ t8!A2~ t8!dt82N*

t1I 0N,

~4.15a!

] tN5E0

t

S4~ t2t8!N* ~ t8!A~ t8!dt8

1E0

t

S6~ t2t8!N1~ t8!A2~ t8!dt81N*

t2I 0N.

~4.15b!

Substituting these equations into Eq.~4.8!, we obtain anequation for ion concentration:

] tN15E

0

t

@S1~ t2t8!2S4~ t2t8!#N* ~ t8!A~ t8!dt8

2E0

t

@S6~ t2t8!2S3~ t2t8!#N1~ t8!A2~ t8!dt8.

~4.16!

In previous integral theoriesN* was linear inI 0 andthere was no need in the equation forN, becauseN5N0. Asa result, there were only three integral terms and correspoing kernels: one,R* [S1, for N* and two others,R†[S1

2S45R* 2S4 and Rf[S6, for N1.10 All kernels had theclear physical sense.S1 andS6, described by the bimolecular quenching of excitation and ion recombination, resptively. The kernelR†[S12S4 was responsible for primaryionization (S1) followed by geminate recombination of thion pair (S4).10,13

Only S3 is new and rather unusual. It describes the ovpopulation of ion pairs@D1•••A2#, produced by electrontransfer in closely situated products of ion recombination ibulk, excited before their separation. Since the density ofcorrelated pairs is enhanced due to bulk recombination,ionization becomes faster than at the entirely homogenedistribution of the reactants. The terms involvingS3 accountfor this effect. However, unlike other kernelsS3 vanishes inthe lowest order approximation with respect toI 0. Therefore,the integral terms involving this kernel were negligible in aprevious theories, developed in this approximation.

B. Stationary concentrations and fluorescencequantum yield

Let us introduce the stationary constants related toabove discussed kernels:

k I5S1~0!, kg5S1~0!2S4~0!, ~4.17!

k r5S6~0!, k85S3~0!. ~4.18!

Unlike standard Markovian rate constants these constantspend on the excitation lifetime and light intensity and twothem,kg andk8, have no analogies in conventional chemickinetics. However, only using these constants we can oba set of equations for the stationary concentrations of re


a

fro

t

e

re

x–-thr-

ono

-

te

nceem

le

e

e

tw

eec-

iftact:

isck-ns.ac-ame

in

ion

mthe-


tants and reaction products. For this purpose one has to ma change of variables (t95t2t8) in integrals of Eqs.~4.15b!and ~4.15a!, ~4.17! and pass to the limitt→`. As a result,

052k INs* As2k8@Ns1#22

Ns*

t1I 0Ns , ~4.19a!

05kgNs* As1~k82k r !@Ns1#2, ~4.19b!

As1As25As1Ns

15c, Ns1Ns11Ns* 5N0 . ~4.19c!

From Eqs.~4.19a! and~4.19b! one can easily find the ratio ostationary concentrations of excited and nonexcited fluophors:

Ns*

Ns5

I 0

1/t1As~k I1bk8!, ~4.20!

whereb5kg /(k r2k8). Substituting this ratio into the mosgeneral definition of the fluorescence quantum yield~2.3!,we can adjust it to a form similar, but not identical to thStern–Volmer law:

1/h511kAst , ~4.21!

with the ab initio defined constant:

k5k I1bk8. ~4.22!

The main difference lies in the fact thatAs is not a totalconcentration of electron acceptorsc, but a fraction of themwhich remain neutral in the stationary regime. Since theof them become anions the relationship betweenAs and c5@A#1@A2#, as well ash21(c) dependence, are compleand nonlinear at largeI 0. In this sense the original SternVolmer law ~1.6! breaks down with an increase in light intensity. Of no less importance also is the difference infield dependence ofk which is changed due to the appeaance of a completely new second term in Eq.~4.22!.

The general relationship betweenc andAs hardly can beobtained analytically, but is open to numerical inspectiHere we preface this investigation with a short analysisimportant particular cases. AssumingNs* is known, we useEqs.~4.19b! and ~4.19c! for establishing the following relationship betweenAs andc:

As5c2Ns15c2

AbNs* ~bNs* 14c!2bNs*

2. ~4.23!

There are two opposite cases, that should be discrimina

As;c2

bNs*!c, Ns

1;c at c!bNs* , ~4.24!

As;c,Ns1!c at c@bNs* . ~4.25!

In the latter case, the inverse quantum yield is linear icbecause there is practically no expenditure of electron actors. Contrary to that, in the former case almost all of thare charged and the number of the rest is quadratic inc, as inEq. ~4.24!.

As a matter of fact,Ns* is not a parameter but a variabwhich depends onAs in its turn. Therefore Eq.~4.23! cannotbe used for the elucidation of the true concentration dep


ke

-

st

e

.f

d:

p-

n-

dence ofAs or Ns1[x. The latter only can be found from th

cubic equation following from the set~4.19a!, ~4.19b!, and~4.19c!:

x2

b~c2x!5

I 0~N02x!2k8x2

I 011/t1k I~c2x!. ~4.26!

It is clear thatx(c) as well asAs(c) dependencies are nolinear in c. Therefore, the generalized Stern–Volmer la~4.21! is also nonlinear unlike the classical one~1.6!. Theextent to which these two differ from each other will bdemonstrated in the particular example given in the next stion.

C. Contact approximation

The simplest solution of the problem can be obtainedboth ionization and recombination are assumed to be con

WI~r !5ki

d~r 2s!

4pr 2, WR~r !5kr

d~r 2s!

4pr 2. ~4.27!

According to the analysis made in Refs. 27 and 28 thisonly possible in the NN case, when the forward and baward electron transfers occur in the normal Marcus’ regioFor the sake of simplicity we also neglect the force intertion between reactants and assume diffusion to be the sin all pairs:

L15L25L35DD.

Then it follows from Eqs.~4.10a!–~4.10i! that

n11m51m351, n21m21m651, n31m11m451.~4.28!

After such simplifications the analytic solution presentedAppendix C provides us with the following results:

k I5kikD~kr1kD!~11I 0t!

kD@kr1kD1I 0t~ki1kr1kD!#1ki~kr1kD1I 0tkr !a,

~4.29!

k85kikrkDI 0t~12a!

kD@kr1kD1I 0t~ki1kr1kD!t#1ki~kr1kD1I 0kr !a.

~4.30!

In the same fashion, we obtain that the free ion productrate constant is

kg5k Iw, ~4.31!

and the recombination rate constant

k r5krkD~kD~11I 0t!1ki~ I 0t1a!!

kD@kr1kD1I 0t~ki1kr1kD!#1ki~kr1kD1I 0krt!a.

~4.32!

Here w5 1/(11kr /kD) is the charge separation quantuyield in the contact approximation, a particular case ofmore complexw which is averaged over the initial distribution of photogenerated ions, either contact or noncontact.27,28

With k I and k8 from ~4.29!–~4.30! and b5kg /(k r

2k8) expressed viakg and k r from ~4.31!–~4.32!, we cansolve numerically Eq. ~4.26!. The results for x5As

2

5Ns1(c) and As(c)5c2As

2(c) are shown in Fig. 2. The


p

the

th

isre

ritohe

o

estbe-uldce.

d

on

ionn-

rms

atrixgey

inonle

hethenlyin

. ItT in

is

lin-the

ityc-

his

.


density of free ions is limited by the total number of accetors. At small c or high light intensity all of them arecharged,Ns

15As25c and this limit cannot be exceeded wi

a further increase inI 0. This is the saturation region near thorigin of the coordinates in Fig. 2. At higherc the saturationis removed and the charge concentration is controlled bylight intensity. The whole curveNs

1(c)5As2(c) can be stud-

ied experimentally by measuring photoconductivity, whichproportional to the density of free carriers whoever they aanions or cations.

Another side of the same phenomenon is the nonlineaof As(c) dependence at high fields. After substitution inEq. ~4.21! this dependence breaks the original form of tStern–Volmer law~1.6!. The resulting curvesh21(c) areshown in Fig. 3, taking into account the field dependence

FIG. 3. The bent Stern–Volmer dependence of 1/h on electron acceptorconcentrationc at the same light strengths as in the previous figure. Tparabolic distortion of high field curves in the low concentration regionshown in the insert. All the parameters are the same as in Fig. 3.

FIG. 2. The stationary concentrations of neutral acceptorsAs ~dashed lines!,and anionsAs

2 ~solid lines!, as functions ofc5As1As2 at different light

strengthI 051023 ns21 (A);1022 ns21 (B);1021 ns21 ~C!. The rest ofparameters are:ki5kr5104 Å3/ns, kD56000 Å3/ns, tD50.25 ns, t510 ns.


-

e

,

ty

f

the Stern–Volmer constantk(I 0). Except for the lowest fieldcurveA which reproduces the classical linear law, all the rare nonlinear at low concentrations and even intersecttween themselves. To avoid this complication one shomeasure the photoconductivity in line with the fluorescenIf the former allows findingAs

2 and henceAs5c2As2 , then

the linearity of Eq.~4.21! can be experimentally confirmeand used to findk defined in Eq.~4.22!. In Fig. 4 we showthat the latter is essentially affected by the second termthe right-hand side of Eq.~4.22! which is proportional tok8.It keeps the memory of correlated pairs generated by therecombination, which contribute to the Stern–Volmer costant in line with conventional bimolecular ionization.

V. CONCLUSIONS

Unlike more sophisticated binary theories~DET, super-position approximation20 or modified encounter theory! theintegral encounter theory keeps only the lowest order tein the concentration expansion ofh21 and similar quantities.On the other hand, this is an advanced method whose mformulation provides us with universal key to any multista~sequential and parallel! photochemical reaction of arbitrarcomplexity and under arbitrary strong illumination.

The alternative way to study stationary fluorescencestrong light is restricted to a validity range of the convolutirecipe. We confirmed the applicability of the latter to a singstage but non-Markovian impurity quenching, provided tencounter diffusion of reactants and reaction products issame. Otherwise, the quantitative study is available owithin IET and the results are sensitive to the differencediffusion and to space dispersion of the quenching rateshould also be stressed that there is no alternative to IEstudying the biexcitonic quenching of fluorescence whichnonlinear in the excitation concentration.11,12

The quenching through electron transfer is also nonear because of the expenditure of neutral reactants andbimolecular recombination of ions. This breaks the linearof the Stern–Volmer law in the total concentration of ele

e

FIG. 4. The field dependence of the Stern–Volmer constantk with andwithout accounting fork8, shown by solid and dashed lines, respectively


mabia

inhe

ed

th. Eote

lae

hv

t

ife

th

su

iti

so-the

ce

te

e-

-

n

.

ar


tron acceptors, keeping it only for a neutral fraction of theThe Stern–Volmer constant is also corrected taking intocount the spatial inhomogeneity resulting from ion recomnation. This is a unique capacity of IET to trap suchmemory effect, as well as geminate recombination, followthe photo-generation of ion pairs. The former is built into tmemory functionS3, while the latter is incorporated inS4

and allows discriminating betweenk I and kg . All nonzeroelements of the matrix kernelM play a definite role in thekinetics of the multistage reaction and must not be ignor

ACKNOWLEDGMENTS

Some numerical calculations were performed withhelp of the SSDP-2 software package developed by DrKrissinel and Professor Agmon.26 The authors are grateful tK. Ivanov for useful discussions. This work was supporby the Israeli Science Foundation.

APPENDIX A

If there are fluorescent moleculesD but no quenchers inthe system, one have to account for only monomolecuprocesses activating and deactivating the fluorophors. Thare light induced transitions with a rateI as well as a decayof the excited state with lifetimetD :

D↔I

D* , D* →1/tD

D. ~A1!

The molecular excitation is usually carried out by a liginduced transition from the ground state to the resonantbronic subleveli of D* . Due to fast vibrational relaxation irelaxes to the lowest vibrational leveli 50, in time tv ,which is usually 3 orders of magnitude shorter than the ltime of the electronic excitation,tD . This scheme shown inFig. 5~A! is represented by the set of rate equations forpopulations of corresponding vibrational states. IfI 0tv!1the quasistationary solution of these equations may be uto get a reduced description of the fluorescent state poption:

N* 5I 0

11I 0tvN2

N*

tD. ~A! ~A2!

In another scheme shown in Fig. 5~B!, the pumping fre-quency is in exact resonance with the fluorescence trans

FIG. 5. The simplified schemes of~A! off-resonant@through intermediatevibronic sublevel~i!# and ~B! resonant excitations of fluorescent moleculstateD* .


.c--

g

.

e.

d

rse

ti-

-

e

edla-

on

so there is no place for vibrational relaxation and the renant excitation is described by a single equation fromvery beginning:

N* 5I 0N2F I 011

tDGN* . ~B! ~A3!

In this particular situation, Eq.~A3! can be obtained fromEq. ~A2! neglecting I 0tv and substituting@ I 011/tD# for1/tD . With this reservation both cases~A! and ~B! can beconsidered represented by the same equation

N* 5I 0N2N*

t, ~A4!

wheret5tD for nonresonant excitation of the fluorescen@Fig. 5~A!# and 1/t5I 011/tD for the resonant [email protected]~B!#. It follows from this equation that the stationary stadensities relate to each other asNs* /Ns5I 0t.

APPENDIX B

In this appendix we derive an important relationship btween the pair densities. Let us introduce a new one,m(r ,t),whose Laplace transformation is

m~r ,s!5I 0n1~r ,s!2~s1I 0!m1~r ,s!. ~B1!

The latter obeys the following equation:

@s2L11WI~r !11/t#m~r ,s!

5I 0@11I 0m5~r ,s!#2~s1I 0!I 0n3~r ,s!. ~B2!

Using the Green functiong3(r ,r 0 ,t) of Eqs. ~4.10h! and~4.10c!, we can get from them the following integral relationships:

m5~r ,s!5E d3r 0 g3~r ,r 0 ,s!

3FWR~r 0!m3~r 0 ,s!1n1~r 0 ,s!

tG , ~B3!

n3~r ,s!51

s1I 01E d3r 0 g3~r ,r 0 ,s!

3FWR~r 0!m4~r 0 ,s!1m1~r 0 ,s!

tG . ~B4!

The other densities,m3 and m4, can also be expressed iterms ofn1 andm1 by means of the Green functiong2(r ,0 ,t)of Eqs. ~4.10f!, and ~4.10g!. By substituting them into Eqs~B3! and ~B4! and using the result in~B2!, we obtain thehomogeneous equation form(r ,s),


cu-

n

he-ae

ir

a

to

.

,

d

-

tt.


@s2L11WI~r !11/t#m~r ,s!

5I 0E d3r 0g3~r ,r 0 ,s!S m~r 0 ,s!

t

1E d3r 1 g2~r 0 ,r 1 ,s!WI~r 1!m~r 1 ,s! D . ~B5!

Since there is no source termm(r ,t)50 and Eq.~4.11! fol-lows from Eq. ~B1!. Then from Eqs.~4.10f!, ~4.10g! and~4.11! we obtain Eq.~4.12!.

APPENDIX C

In order to evaluatek I5S1(0) from Eq. ~4.13! oneneeds the pair densityn(s,s). Equations ~4.10a! and~4.10h!, with allowance for Eq.~4.28!, form the closed set

@~s2DD!I1Q0#S n1

m5D 52S WI 0

Wr WrD S n1

m5D

1S 1

Wr /sD , ~C1!

whereI denotes the identity matrix of rank 2. In the contaapproximation this set of equations can be easily solveding the Green functionG(r ur 0 ,s), which obeys the homogeneous diffusional equation

@~s2DD!I1Q0#G5Id~r 2r 0!

4prr 0, ~C2!

with a reflecting boundary condition atr 5s. By means ofthis Green function the solution of Eq.~C1! is straightfor-ward:

S n1~s,s!

m5~s,s!D 5H I1G~sus,s!S ki 0

kr krD J 21

S¢ , ~C3!

where the source is

S¢5H G~sus,s!S 0

kr /sD 1E d3r G~r ur 0 ,s!S 1

0D J . ~C4!

The integral term in this expression can be easily fouby a straightforward integration of Eq.~C2!:

E d3r G~r ur 0 ,s!5~sI1Q0!21. ~C5!

To find the contact Green function which determines anotterm one has to solve Eq.~C2! after diagonalization of matrix Q0. In fact, the solution for elements of the diagonmatrix is well known29,30 and we need only transform thlatter back to the initial basis:

G~sus,s!51

kD~t11/I 0! S a/I 01bt ~b2a!t

~b2a!/I 0 b/I 01at D ,

~C6!


ts-

d

r

l

wherea21511Atd(s1I 011/t) andb21511Astd. Sub-stituting this result into Eq.~C3! and solving it forn1, wecan use the result in Eq.~4.13! and passing the limits→0arrive at Eq.~4.29!.

To calculatek8 we need the contact value of the padensitym2. Since Eqs.~4.10e! and~4.10i! are very similar toEqs. ~4.10a! and ~4.10h!, the solution can be obtained inform of Eq. ~C3!, but with a different source term,

S8W5G~sus,s!S 0

kr /sD . ~C7!

A little cumbersome but straightforward calculation leadsEq. ~4.30!.

1M. V. Smoluchowski, Z. Phys. Chem., Stoechiom. Verwandtschaftsl.92,129 ~1917!.

2T. R. Waite, Phys. Rev.107, 463 ~1957!.3N. N. Tunitskii and Kh. S. Bagdasar’yan, Opt. Spectrosc.15, 303 ~1963!;S. F. Kilin, M. S. Mikhelashvili, and I. M. Rozman,ibid. 16, 576 ~1964!;I. I. Vasil’ev, B. P. Kirsanov, and V. A. Krongaus, Kinet. Katal.5, 792~1964!.

4I. Z. Steinberg and E. Katchalsky, J. Chem. Phys.48, 2404~1968!.5A. B. Doktorov and A. I. Burshtein, Sov. Phys. JETP41, 671 ~1975!.6V. M. Agranovich and M. D. Galanin,Electron Excitation Energy Trans-fer in Condensed Matter~North-Holland, Amsterdam, 1982!.

7V. P. Sakun, Physica A80, 128 ~1975!; A. B. Doktorov, ibid. 90, 109~1978!; A. A. Kipriyanov, A. B. Doktorov, and A. I. Burshtein, ChemPhys.76, 149 ~1983!.

8N. N. Lukzen, A. B. Doktorov, and A. I. Burshtein, J. Chem. Phys.102,289 ~1986!.

9A. I. Burshtein and N. N. Lukzen, J. Chem. Phys.103, 9631~1995!; 105,9588 ~1996!.

10A. I. Burshtein and P. A. Frantsuzov, J. Chem. Phys.106, 3948 ~1997!;107, 2872~1997!; 109, 5957~1998!.

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Impurity quenching of fluorescence in intense light. Violation of the Stern–Volmer law