Theory of diffusion-influenced fluorescence quenching: Dependence of the Stern-Volmer curve on light intensity

Chemical Physics 156 ( 199 1) 339-357 North-Holland

Theory of diffusion-influenced fluorescence quenching: dependence of the Stern-Volmer curve on light intensity Sangyoub Lee, Mino Yang, Kook Joe Shin, Kwang Yul Choo + Department of Chemistry, Seoul National University, Seoul 151-742, South Korea

and Duckhwan Lee Department of Chemistv, Sogang University, Seoul 121-742, South Korea Received 1 May 1991

A theory for describing the effects of quencher molecules on the intensity of fluorescence stimulated by steady-state illumination is presented, which is based on a hierarchical system of many-body Smoluchowski equations for the reactant molecule distribution functions. An important aspect of the present theory is that it describes the dependence of fluorescence quenching dynamics on the intensity of external illumination. In the conventional Smoluchowski approach it has been simply assumed that the intensity of illumination should be quite weak. Hence the validity of the present theory may be tested by a fluorescence quenching experiment under intense illumination.

1. Introduction

There have been many studies of the quenching of fluorescence from both experimental [ 1- 10 ] and theoretical perspectives [ 1 l-2 11. In some experiments [ 2-8 ] fluorescent molecules are produced by a light pulse of short duration and the decay of the fluorescence intensity from the sample is followed as a function of time. In other experiments [ 1,2,9,10] the effects of quencher molecules on the steady-state intensity of fluorescence stimulated by an illumination of constant intensity are examined. The present theory addresses the latter class of experiments although the basic framework is also relevant to the former. When the concentration of fluorescent molecules is much smaller than that of quencher molecules, a quencher molecule may be able to dispose of the energy it received from a fluorescent molecule before it has an opportunity to quench again. If this is the case, the fluorescence quenching kinetics can be described by the following reaction scheme [ 111: (excitation of fluorophors ) ,

D*

D + h ve z

(1.1)

kF

D* -

D + h r$

( flUOt%SCtXXX),

(1.2)

kNR

D* -

D

(nonradiative unimolecular decay) ,

(1.3)

kQ D*+A-

D +A

(bimolecular quenching) .

t Deceased January 24,199 1. 0301-0104/91/$03.50

0 1991 Elsevier Science Publishen B.V. All rights reserved.

(1.4)

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340

In these equations F, kR kNR,and k. represent the rate constants of the respective processes. k, and kNRmay be considered to be independent of time, and their sum, denoted hereafter by k,, can be determined experimentally as the inverse of the fluorescent life time 7, in the absence of quencher; that is, k,+kNR=ks=z,y’.

(1.5)

Fin eq. ( 1. 1 ), denoting the transition probability per unit time, depends on the intensity of radiation and thus varies with time in general. The bimolecular quenching rate coefficient kQ in eq. ( 1.4) depends on the distribution of the quencher molecules A around D*. When the relative diffusion between D* and A is slow, this distribution deviates from the equilibrium one and varies with time, and so does the bimolecular quenching coefftcient ko. An important aspect of fluorescence quenching that is neglected in the above reaction scheme is the static quenching mechanism [ 1,2,14], which involves a formation of complex:

D+A#DA.

(1.6)

Although in some cases this mechanism can play an essential part in determining the dependence of fluorescence quenching kinetics on the quencher concentration, we will neglect it for simplicity in the present work. According to the reaction scheme given by eqs. ( 1. 1)- ( 1.4)) the time-dependence of the concentration (number density) of D* molecules will obey the following phenomenological rate law: [D*]+F(t)

$[D*]=-ks

[D]--kq(t)

[D*] Ci,

(1.7)

where [D] and [D* ] denote the number densities of D and D* molecules at time t, respectively, and Ci is the number density of A molecules that is essentially constant in time. By solving this differential equation, we obtain [D*]=Cgid7F(7)exp[-ks(t-7)-

jdr,

k,(r,)],

F(7,)-Cijd7,

r

0

(1.8)

7

where C”, is the total number density of D molecules. That is, C”, = [D*] + [D 1. The above expression for [D*] is, however, only formal. We will see below that ko( t), being a functional of the nonequilibrium pair distribution of A molecules around D*, depends on the history of [D* ] variation. The timedependence of [D*] may also be expressed as [D*]= 1 drF(7)

[D]G(r,

(1.9)

t).

0

Here, F( ;> [D] d7 gives the number of D* molecules generated in a unit volume between times 7 and r+d7, and G( 7, t) denotes the probability that a D* molecule excited at time 7 remains in the excited state at time t. This is an exact expression. In conventional theories [ 111, however, it is assumed that each D* molecule which has been just excited is surrounded by an equilibrium distribution of A molecules and that the A molecule distribution around each D* follows the same time evolution thereafter. This means that the decay probability of D* depends only on the time elapsed after excitation regardless of when the excitation occurred. That is, G( 7, t) in eq. ( 1.9) is approximated as t--T

G(7, t)zG(t-r)=exp

-k,(t-7)-C:

I 0

dT, k:‘(r,)

1 ,

(1.10)

where kGs(7, ) is the quenching rate coefficient obtained for the initial condition that each D* molecule created

S. Lee et al. /Diffirrion-injluencedfruorescence quenching

341

at 7, = 0 is surrounded by an equilibrium distribution of A molecules; see eq. (3.26) below. Consequently, it is differentfromko(7,) ineq. (1.8). In most analyses of experimental data [ l-8 1, one goes one step further by assuming that [D] B [D*] and thus [D] remains nearly constant in time; that is, [D] = C”,. Then eq. ( 1.9) gives [D*]=Cb

jdrF(7)

ex[-k,(r-7)-Ci’Jrd7,

0

(1.11)

kss(7,)], 0

and for the relative intensity of unquenched to quenched fluorescence under an illumination of constant intensity (i.e., F(7) =const.) IO/Z= lim { ( [D*] with C’z --O)/( I-CO

[D*] with C~+O))=kr’{qdlexp[-~~-C~ 0

jdrlig(r)l)-’

.

(1.12)

0

Comparison of eq. ( 1.11) with eq. ( 1.8) shows clearly that the validity of eq. ( 1.11) is limited by the condition that F( 7) 2 0. That is, the intensity of illumination which excites D molecules ought to be weak to ensure that [D*] K [D]. One may also question the validity of the approximation made in eq. ( 1.10). The assumption that each D* molecule which has been just excited is surrounded by an equilibrium distribution of A molecules can be true if the D* molecule has been excited for the first time or re-excited after unimolecular radiative or nonradiative decay as represented by eq. ( 1.2) or ( 1.3). However, if the re-excited D* molecule was quenched recently by the bimolecular process [ eq. ( 1.4) 1, it will still see, on the average, more A molecules in its vicinity than the equilibrium distribution. For such D* molecules, the decay law would be different from that given by eq. ( 1.10). The opportunity of repeated excitations grows when in the intensity and time width of the light pulse increases, and the fraction of D* molecules re-excited after bimolecular quenching rather than after unimolecular decay increases as the quencher concentration increases. Hence, the conventional theories of fluorescence quenching [ 111 are expected to break down in such situations. Wilemski and Fixman [ 121 have proposed a general formalism which takes into account the re-excitation problem as described above, but they introduced several coarse approximations to obtain an explicit solution. Their final result implies a linear variation of the Stem-Volmer curve (lo/Z versus Cz ) for the whole range of Ci in contradiction to many experimental results [ 2, lo]. The purpose of the present work is to provide a more rigorous and computationally viable approach to this complicated problem, which is based on a hierarchical system of many-body Smoluchowski equations for the reactant molecules [ 221. The outline of this paper is as follows. In section 2 we present a general theoretical framework for describing the fluorescence quenching dynamics. The formalism is applicable not only to steady-state quenching experiments but also to the analysis of quenched fluorescence decay data obtained by using an initial excitation pulse of finite time width [ 231. In section 3 we then restrict ourselves to treating the steady-state fluorescence quenching kinetics. For the simplest case where the bimolecular quenching occurs only upon contact of D* and A molecules and the potential of the mean force between them has a negligible effect, exact expressions are obtained for the steady-state quenching rate coefficient and for the ratio lo/Z. For the general cases where the bimolecular quenching may also occur via long-range energy transfer processes and the potential of the mean force is not negligible, only approximate analytic solutions have been obtained [ 241. The present theory predicts a complicated nonlinear dependence of IO/Zon the quencher concentration Ci. The Stem-Volmer coefftcient KS”. which is defined by [ 2,141 KS, = (lo/Z- 1J/C: ,

(1.13)

depends on Ci and the intensity of light that excites the fluorophor as well as on other parameters that characterize the dynamics of the fluorophor and quencher in a given solution. In section 4 the results of model calculations based on the present theory are compared with those obtained from the conventional Smoluchowski

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342

theory [ 11,201, the mean field theory of Cukier [ 17 1, and the statistical nonequilibrium thermodynamic theory of Keizer [ 15,161. Section 5 concludes the present work.

2. The many-body Smoluchowski equation approach Consider a solution containing NO, molecules of species D, including both D* and D, and N”Amolecules of species A,.including both A* and A. We then introduce the following probability density functions: P,,:( r, t) = probability density that the ith D molecule is in the excited state and located at r at time t, Ppi( r, t) = probability density that the ith D molecule is in the ground state and located at r at time t, Pp:*,( r, r’, t) sprobability density that at time t (i) the ith D molecule is in the excited state and located at r and (ii) thejth A molecule is in the ground state and located at r’. P,,,Aj(r, r’ , t ) = probability density that at time t (i) the ith D molecule is in the ground state and located at r and (ii) the jth A molecule is in the ground state and located at r’ , PDp& (r, r' , r”, t) = probability density that at time t (i) the ith D molecule is in the excited state and located at r and (ii) thejth and the kth A molecules are in the ground state and located at r’ and r”, respectively,

and so on. The two one-particle probability density functions P,:( r, t) and PDi(r, t) satisfy the conservation condition dr [P,:(r,

t)+P,,(r,

f)l=l

,

(2.1)

and their evolutions are governed by the following kinetic equations [ 22 ] : %PD:(r,t)=L,,.P,:(r,f)-ksPo:(r,l)+FP,,(r,

O-,2,!

dr’ S(ryr’)P~:A,(ryr’yO

(2.2)

and (2.3)

Each term on the right-hand side of eq. (2.2) has the following physical meaning. The first term L,*P,z( r, t) represents the evolution of P,r(r, t) due to the thermal motion of Dt in the absence of the decay processes ( 1.2)-( 1.4). In the present work, we approximate the evolution operator L De by the Smoluchowski operator [ 221. The second term denotes the depopulation rate of Dt due to the unimolecular decay processes ( 1.2) and ( 1.3), while the third term is the rate of excitation Di-+D? due to the external radiation. When the intensity of the external radiation varies with time, so does the rate coefftcient F. But under an illumination of constant intensity it may be considered to be independent of time. .The fourth term, involving the sink function S( r, r’ ), describes the quenching rate via the bimolecular encounter with A molecules. Possible forms of the energytransfer sink function S( r, r’ ) have been discussed in ref. [ 111. The various terms in eq. (2.3) have the corresponding meanings. To solve eqs. (2.2) and (2.3), we should have the expressions for the two-particle probability density functions which in turn satisfy the following kinetic equations [ 221:

S. Lee et al. / Di&ion-injluencedjluorescence

r’, t)=LD’APD:A,(ry

$pD:A,tc

r’, t)-ksPD:Aj(r,r’,

quenching

t)-S(r,r’)PD~A,(c

r’,

t)+FPDi~j(r,r’,

343

t)

(2.4)

dr” S(r, r”)PDrAiAl,(rp r’, r”, t)

-kz,J k#i

and % PD,A,(r, +

r’?

t)

=LDApDiAj(ry

r’,

kz, ,fdr" S( r, r”)PD~,+.(r,

t)

+kspDTA,(rT

r’, t)+S(r,

r’)PDrA,(r, r’, t)-FPDiAj(r,

r’, r”, t) .

r’, 1)

(2.5)

k#i

Again, each term in these equations has a similar meaning as the corresponding term in eqs. (2.2) and (2.3). In eq. (2.4), for example, LpA is the Smoluchowski operator governing the thermal motion of a D*-A pair; the second and the third terms on the right-hand side represent the disappearance rates of the Dt-Aj pair due to.the unimolecular decay of Dr and the energy transfer from D: to A,, respectively; the fourth term represents the creation rate of the pair due to the excitation Di+Df by the external light source; and finally the fifth term, involving the three-particle probability density functions, describes the rupture of the DI-A, pair due to competitive takeover of the excitation energy of Df at r by A molecules at r” other than A, In writing eqs. (2.2)-(2.5), we have neglected the possibility of donor-to-donor energy transfers, Dt + D/PDi+ Dj’ [ 3 1. In addition, we have taken no account of the fate of any excited A molecule. For example, we have neglected the possibilities of the reversible energy transfer, Df +AjpDi+AT, and the rupture of the D:-A, pair due to the excitation OfAj by donor molecules other than Df. Hence the validity of eqs. (2.2)-( 2.5) is limited to a situation where Ni S+Ng and an A molecule can rapidly dispose of the energy it received from a D* molecule before it has an opportunity to quench again. Indeed this is the condition required for the validity of the reaction scheme represented by eq. ( 1.4). More general kinetic equations are discussed in appendix A. Inclusion of higher-order equations is straightforward, but the solution to the whole hierarchy of kinetic equations is difficult to obtain. To truncate the hierarchy at the level of two-particle kinetic equations we need to approximate the three-particle probability density functions contained in the competition terms in eqs. (2.4) and (2.5 ) in terms of one-particle and two-particle probability density functions. In the superposition approximation [ 22,25,26], we may write (2.6)

Here PA,<,:, (r’, t 1r) represents the conditional probability density that Aj is at r’ at time t given that w is at r. P+(Dz, (r”, t )r) is defined similarly. In passing, we note that in terms of this conditional probability density the two-particle probability density function can be expressed exactly as PD&(&

r’,

t)=PD:(rT

~)PA,(D~)(r’,

tir)

(2.7)

.

We now introduce various concentration (number density) fields that are related to the particle distribution functions as follows: CD*(r, t ) = the number density of D* molecules at r at time t = 1 I”; Pb:( r, t), CD(r, 2) = the number density of D molecules at r at time t= cfle PDi (r, t), CAcDtl (r’, t(r) =the number density of A r’IIOkCUkS

at r’

at ti!IIC

t giVeII

that Df is at r= ~~z’lPA,(D:)(rp, t(r).

Kinetic equations governing the evolution of these concentration fields may be obtained by summing eqs. (2.2)(2.5 ) for all reactant molecules. We make the usual assumptions that (i) the volume Vof the reaction vessel is large enough and the shape is such that surface effects may be neglected, and (ii) the initial distribution of the reactant molecules is an equilibrium one. Hence, C, (r, t ) and CD(r, t) are uniform over r and can be equated

S. Lee et al. / Dijkion-injluencedfluorescence quenching

344

to the bulk concentrations, namely, C&r, t)= [D*]

and

C,(r, t)= [D] .

(2.8)

The conditional concentration fields, CA(o: ) ‘s, are statistically equivalent for all i and depend only on the relative separation I r’ -rl ; that is, G(o:)(r’,

Ir’ -rl,t)=[Al~~~~(Ir’-~l,f),

tlr)=G~~~(

(2.9)

where we have also introduced the nonequilibrium pairdistribution function PAD*( I r’ - r I, t ) . With these assumptions, summing eq. (2.2) over i= 1,2, .... NODyields $[D*]=L,.[D*]-k,[D*]+F[D]-

Sdr’S(Ir’-rl)p,,.(Ir’-rl,t)[D*][A].

(2.10)

Since LD.= D,.Vz, where DD. is the diffusion coefficient of D*, and [D*] is independent of r, the first term on the right-hand side drops out and eq. (2.10) reduces to the familiar rate law, $ [D*]=-ks

[D*]+F[D]-b(t)

[D*] [A].

(2.11)

Here, the time-dependent quenching rate coefficient k. ( t ) is given by j dr’ s( jr’-r])PA&

ko(t)=

(r’-r],

t)= 1 dr47Cr2S(r)/.&,.(r, t) .

(2.12)

In the second line, the integration variable r denotes the relative separation I r’ -r I. Similar manipulation of eq. (2.3) yields the same rate law as eq. (2.11):

$ tD1 =ks P*l

-FtDl

+kdO P*l [Al .

To evaluate /co( t ), we need an explicit expression for the nonequilibrium pair-distribution function PAD*( r, t ). The kinetic equation governing the evolution of p Ap is in turn obtained from eq. (2.4). Summing eq. (2.4) first overj= 1,2, ... , Ni and then over i= 1,2, ... , NO, yields ; {P*l tAlPAD*(I;r’,

[D*l [AIPAF(~~r’, t)

~))=LD*A

-Its tD*l [A]PAD*(~,r’, t) -Wr, r’ 1 ID*1 [A]PALF(~,r’, t) +FPl

[AlPA&, r’, t) -b(t)

P*l tA12Pm*(“, r’, f) .

(2.13)

In obtaining this equation, we have applied the superposition approximation, eq. (2.6):

2jz,?, ‘DTAjAk(

r9 “?

“‘, f)z

N$

PDr(r, t) jf PAj(D:)(r’, tlr)

<=I

j=l

k#j

= 2,

rf,

PAk(DT)(f*, tlr)

k#j

pD~~r~ l)$!,

PA,CD:j(r’9

tlr){C~(~:)(f’,

=cD*(c

t)CA(D*)(r’,

flr)CA(D’)(

z&dry

t)CA(D*)(r’y

tlr)CA(D’)(r”,

tlr)--PA,cDt,(r”,

r“, tlr)-

tlr)

N3

i=l .

tlr)}

PDr(r, t) Nj PAjcDr,(r’, t)r)PAjcDt,(f”, j=l

tlr)

(2.14)

One can note that the term kept in the last line of this equation has a magnitude comparable to [D*] [A] 2, while the term neglected has a magnitude comparable to [D*] [A] / I/. Since the volume Vof the solution is a macro-

S. Lee et al. / Dl~ion-inJ]uenece

quenching

345

scopic quantity, that neglect practically does not amount to any further approximation. Since we are considering the situation where NP, w NO,, the time-dependence of [A] may be neglected so that eq. (2.13) reduces to [D*l

[Al

+FPl

a

-PAD*(~, at [Al

r’, f)=

b,4D(r,

[D*l

[A~J%=APAD*(~~

r’, t)--Pm(r,

r’, t)

r’, f)-

[D*l

[AlS(r,

f)p,,(r,

f,

t)

(2.15)

1.

Inderivingthisequation,weusedeq. (2.11). When [D*]=O,eq. (2.15)saysthatp,,.(r,r’,t)=p,,n(r,r’,t). Physically, this means that the separate evolutions of p AD.and &D start upon formation of D* mOk%XleS. For most experimental situations both p AD.and PADremain identical with the equilibrium radial distribution function gag ( )r’ - t 1) between D and A molecules for t Q 0 when [ D* ] = 0: PArr(r, r’, t)=l?m(r,

r’, O=&d(

If-4

1 9 (t
(2.16)

3

and a $?A”*(6

r’, t) = $4,@,

f’,

t)

=o,

(t
(2.17)

For t > 0, [D*] # 0 so that we can divide both sides of eq. (2.15) by [D*] [A] to obtain ip~D*(r,

r’, t)

i-F [D*l [D1

we

=LD*ApAD*(r,

[PAD(I;

I’,

r’,

f)

t)-s(r7

-P*D*(~,

r’,

r’

t)l

)pAD*trp

r’,

9 (t>o)

-

t,

(2.18)

Noting that PAD9(r, r’ , t) depends only on the relative separation, r= Ir’ -rip can reduce eq. (2.18 ) to [ 22 ]

%p,~(r, 0 =J%AP~D~~, t) -S(r)pAdr,

t) +F #

bAD(r,

t)

-pAD=(r,

t)

[i.e.,pAD*(r,

1,

f9

(b0).

t)=pAD*(6

t)l

(2.19)

Here, the reduced Smoluchowski operator L.“,, for the relative motion of D* and A is given by (2.20) dAb.( r) denotes the diffusion coefficient, which depends on r if the hydrodynamic interaction between D* and A is to be included, and UAD*(r) is the potential of the mean force. j?= 1/k,T with the Boltzmann constant kB and the absolute temperature T. If U,,(r) has a very steep potential wall at r=a, PAD-(r, t) must satisfy the reflecting boundary condition, (2.21) By the definition given by eq. (2.9 ) PAD*(r, t ) approaches unity as r goes to infinity, limpAD.(r,t)=l.

(2.22)

,-UZ

The initial condition for&D. ( r, t) is that given by eq. (2.16); that is, PAD*(~,

~=O)=PAD(~,

t=o)=exp[-BUAD(r)l

,

(2.23)

S. Lee et al. / Di@sion-influencedfluorescencequenching

346

where U,,(r) is the potential of the mean force between A and D molecules. Similar manipulation of eq. ( 2.5 ) gives a kinetic equation governing the evolution of PAD( r, 1) $*&

P*l

1) + g

t) =J%A/%&r, t) +S(r) [DIPA&

If%+k#c1

Pm*(r, t) -PAD(C t) 1 > (t>o) 3 (2.24)

where CP,=Ni/ Y= [A] + [A*] z [A] and L”DAhas the same structure as L& in eq. (2.20). The boundary and initial conditions for pAD(r, t ) are also similar to those for PAD*( r, t ) in eqs. (2.2 1 )- (2.23 ). Eqs. (2.11), (2.12), (2.19) and (2.24) together with the associated initial and boundary conditions constitute the set of integro-differential equations to be solved for the four unknowns, [ D* 1, k~ ( t ) , PAW and PAi,. We assume that k,, F and S(r) are known from quantum-mechanical calculations or independent experiments. However, if the physical characteristics of the excited molecule D* do not differ significantly from those of the ground state molecule so that the difference in the potentials of the mean force, UAn*(r) and UAu(r), is negligible, the problem can be simplified greatly. In such cases, we have [D*]PAD*(r, ~)+]D]&D(&

t)=CD*(A)(ry

t)+CD(A)(r,

(2.25)

t)=%d,ZL)(r).

Here, CD.(A)(r, t) represents the number density of D* molecules at the separation r, given that an A molecule is located at the origin. CD(A)(r, t) has the corresponding meaning. C”, denotes the bulk concentration of D molecules including both the excited and the ground state molecules, namely, (2.26)

C;=[D*]+[D]=N;/V. gJ.2 (r) is the equilibrium radial distribution function between D and A given by

(2.27)

g~~(r)=exp[-_~AD(r)l~exp[-~~AD*(r)l.

With the relations, eqs. (2.25) and (2.26), eq. (2.19) becomes decoupled from eq. (2.24) to give %PaD*(‘.t)=LO,*ApAD*(r,

t)--S(r)PAD*(r,t)+(FCOD/[D*l)[ga~(r)-pAD*(r,t)l,

(f>o).

(2.28)

Hence, only this equation together with eqs. (2.11) and (2.12) has to be solved to obtain the reaction kinetic information. Hereafter we will restrict our discussion to this situation. In manipulating eq. (2.28) it is convenient to introduce a function which measures the deviation of PAD*(r, t) fromga2d(r): f(rY t)

=pAD*(r?

t)-d%t)(r)

(2.29)

.

Noting L”D&& (r) = 0 if UAu.( r) z uAn( r), we can easily obtain the evolution equation forf( r, t): (2.30) where y(t) z FC’O,/[D*] . Boundary conditions forf( r, t) are limf(r, t)=O r-m

(2.31)

and

>

f(r, 0

1

=o. I=0

For the initial condition, eq. (2.23), we have

(2.32)

341

S. Lee et al. / Difiion-influenced fluorescence quenching

f(r, t=o)

=o .

3. Steady-state fluorescence quenching kinetics

When the external radiation which excites the D molecules has constant intensity, a steady state is attained at long times. For this steady-state, d[ D*] ldt =O and af( r, t)/at=O. Therefore, the equations we have to solve, i.e. eqs. (2.1 l), (2.12) and (2.30), become respectively ~~==FCO,/[D*],=ks+F+khC~, k&= j drS(r)[gc2)

(r)+.L(~)l=keqg+

(3.1)

Jdr~(r)_~(r),

(3.2)

and [&-LO+S(r)]f,(r)=-S(r)gC2’(r).

(3.3)

Here, Y,, [D*],, kb andf,(r) denote the steady-state values of y(t), [D*], b(t) andf(r, t), respectively, and the subscripts D*A of L&* and AD of gi*J (r) have been omitted for the brevity of notation. Note also that in eq. (3.2) we introduced the equilibrium quenching rate constant kg, which is given by k$=

s drS(r)g(*)(r)

(3.4)

.

If the redistribution of reactant molecules by diffusion occurred rapidly so that the pair-distribution between D* and A was given by g(*) (r), k$ would be the rate constant for the bimolecular quenching process. Once yS,kb and_&(r) are determined from eqs. (3.1)~( 3.3), the ratio of the steady-state fluorescence intensity in the absence (Zo) and in the presence (I) of quencher can be calculated from the following equation: ZolZ=[D*],(C~=O)/[D*],(C~#O)=y,(C~#O)/~,(C~=O)=l+k~C~/(k,+F)

.

(3.5)

A formal solution to eq. (3.3) may be written as L(r)=

-

J dro Wr,ro, y,)~(ro)g(*)(ro) ,

(3.6)

where the Green’s function Gs( r, r,, y,) satisfies the differential equation,

[h-L”(r)+S(r)lGs(r,ro,~,)=S(r-ro)/4nr~,

(3.7)

and the same boundary conditions as those forf,( r) in eqs. (2.31) and (2.32), i.e., lim G,(r, ro, y,) =O T-rcx)

(3.8)

and (3.9) The notation Lo(r) in eq. (3.7) denotes that it operates on the variable r. Gs( r, r,, ys) can be, in turn, expressed in terms of a simpler Green’s function: (3.10) where the reaction-free Green’s function G ( r, rl , ys) governs the D*-A pair dynamics in the absence of the energy transfer reactions. It satisfies the differential equation,

’

S. Lee et al. / Difiion-influencedfluorescence quenching

348

(3.11)

and the same boundary conditions as Gs( r, r,, y,). Now suppose that the quenching reaction can occur only when the D* and A molecules are brought into contact. In this case the sink function S(r) may be modeled as a delta-function: S(r)=

--

kg

&r-a)

g’*‘(a)

=

4x02

K8(

r-

a)

/4m*

(3.12)

,

where rc=ke$ /g’*’ (a). From eq. (3.10) we then have Gs(r, ro, y,)=G(r,

ro, Y,)-kG(r,

0, QG(o,

ro, M/[l+kG(o,

a7 r,)l

(3.13)

and from eq. (3.6)

fS(r)=-k~G(r,a,y,)/[l+~G(a,o,y,)l.

(3.14)

Substituting this expression forf,( r) in eq. (3.2), we obtain the steady-state quenching rate constant, kb=k$/[l+kG((&

%y,)l,

(3.15)

which is obviously smaller than the equilibrium rate constant k FJ. Finally, putting this expression for kh into eq. ( 3.1)) we obtain an algebraic equation for yS: y,~=k,+F+C:k~/[

l+rcG(a, c, y,)] .

(3.16)

By solving this equation for yS,we can calculate the steady-state quenching rate constant from eq. (3.15) and in turn the ratio Z,/Zfrom eq. (3.5). For simplicity, we further assume that the potential of the mean force U,,(r) vanishes for r2 Q but goes to infinity for r-c a We also assume that the hydrodynamic interaction between D* and A is negligible so that &n.(r) = DA+ Do. = D = constant. In this simplest case the Green’s function G( r, rl, y,) can be readily obtained: (3.17) where (Y= (7,/D) ‘I*. The equation for ySis then given by yS=kS+F+CO,k~/[l+k~/k,(l+ay:‘*/D1’*)],

(3.18)

where kD = 47toD. In appendix B is shown that the root of eq. ( 3.18 ) is given by r,=X,(k,+F+ke$C:). The expression for the appendix B. It depends the dynamics of D and With the expression explicitly given by

(3.19) coefftcient X2, which is less than unity, is rather complicated and given by eq. (B.4) in on Fas well as on the quencher concentration CP, and other parameters that characterize A in a given solution, and its value approaches unity as Ci -+O. for ySin eq. (3.19) the steady-state quenching rate constant kb in eq. (3.15) is now

k&=k”$/{l+(k’,q/k,)[l+(y,/D)“*a]-I}.

(3.20)

When this expression for kh is used in eq. (3.5), the ratio lo/Z can also be determined. Another expression for lo/Z can be obtained directly from eq. (3.19 ) by noting that the value of X2 becomes unity as c”, -+O: Zo/Z=yS(C~#O)/yS(C~=O)=X2[l+k$C0,/(ks+F)].

(3.21)

In the low quencher concentration limit X2, given by eq. (B.4) in appendix B, can be approximated as in eq. (B.6).Hence,eq. (3.21) becomes

S. Lee et al. / Dij7itsiominjluencedfluorescence quenching

4,/Z= 1+K’&Ci

(when CO,+O),

349

(3.22)

where the infinite dilution Stern-Volmer coefficient &

is given by k$/kD

l+(k$/kD)+[(ks+F)a2/D]“*

(3.23)

When F/k, is very small and k$ /kD becomes very large, the expression for Z& reduces further to

Kg”Z &F{l+[(kS+F)~2/D]‘/2}

(whenkv%k,),

s

z

2 [1+ (ksa2/D)“*]

(when Fe

k+) .

(3.24)

On the other hand, when kg /kD B= 1 and F/k, -=x1, Kg, has the following limiting expression: K&zkfJ/(k,+F) z kT/k,

(whenk$ek,), (when Fe

ks ) .

(3.25)

3. I. Relation to the conventional Smoluchowski theory

In most analyses of experimental data [ l-8,1 1] the validity of eqs. ( 1.9)-( 1.12) has been taken for granted, and the time-dependent quenching rate coefficient kgs( T) appearing in eq. ( 1.12 ) is evaluated by kG’(t)=

j drS(r)pcs(r,

(3.26)

1) .

Here, the nonequilibrium pairdistribution

&s~r, t) =LO(r)pcs(r,t) -S(r)Pcs(c

function pcs(r, t) is the solution of (3.27)

t)

with the same boundary and initial conditions as given by eqs. (2.2 1 )-( 2.23). For the simple case in which S(r) is given by eq. (3.12), U,,.(r)=0 for r>u, and &,.(r)=DA+DD.=D=constant, one can find an explicit expression for k?(t) [ 11,201: kGs(t)=k,[

1+ (kz/k,)

exp(X*t) erfc($“*)]

,

(3.28)

where k, is the value of kg( t) in the long-time limit, k,=

lim kGs(t)=kZJkD/(kg+kD),

,-a,

(3.29)

and x= [ 1 + (kg/k,) ]D’/*/a. With the expression for kGs( t) in eq. (3.28), the expression for lo/Z in eq. ( 1.12) can be evaluated only numerically. For the infinite dilution Stem-Volmer coefficient K&, however, an analytic expression can be derived [ 201, and is found to be identical with eq. (3.23) if (k,fF) is replaced by kP Deviation of the conventional Smoluchowski theory, eq. ( 1.12) along with eq. (3.28), from the present theory rep resented by eq. (3.5 ) or eq. (3.2 1) at higher quencher concentrations will be discussed below through numerical calculation. Nemzek and Ware [2] derived an approximate expression for lo/Z using a long-time approximation for kg(t) in evaluating the integral in eq. ( 1.12), namely,

1.

(3.30)

S. Lee et al. / Di~ion-injluencedfluorescence

350

quenching

The resulting expression for lo/Z is

10 -= Z

1+C:k,/ks 1- rc”29exp(522) erfc(Q) ’

(3.31)

‘I2 . As k3/k, goes to infinity eq. (3.30) for where Q=C0,k,[k$/(k$+k,)](a2/xD)“2(ks+C~k,)kzs( t) becomes exact and so does eq. (3.3 1) for IO/Z. Here, the word “exact” is used only in the scope of the

conventional Smoluchowski theory. For eq. (3.3 1) to be really exact the requirement of low light intensity and low quencher concentration should also be met. Nevertheless, eq. (3.3 1) is most frequently used by experimentalists. The expression for Z& corresponding to the approximation given by eq. (3.3 1) is found to be Kg,=

1 k$k, kSQkeq+k D

--

(3.32)

As mentioned above, when F-=x ks and k$ > kD eq. (3.32) for K& reduces to the correct expression given by eq. (3.24). Also in the opposite limit, i.e. when Fe h and k$ cx kD eq. (3.32) becomes exact; that is, it reduces to the expression for K& given in eq. (3.25). In fact, for this case of fast diffusion limit the phenomenological rate law of chemical kinetics [i.e., eq. (2.11) with ko( t) replaced by the time independent rate constant kq ] is applicable, and we simply have lo/Z= 1+ (k$/k,)Co,

(3.33)

for all quencher concentrations. This linear relationship between Z,-,/Zversus C”, is the Stem-Volmer law. 3.2. Relation to the mean$eld theory When the radiation intensity becomes very weak so that F+O, our expressions for kh and lo/Z become identical with the corresponding expressions obtained by Szabo [ 201 via the mean field approach of Cukier [ 17 1. Therefore, the present theory may be considered to be a generalization of the mean field theory of Szabo for the case of arbitrary light intensity, although the basic formalisms of the two theories appear to be different from each other. It is known that the mean field theory gives a correct estimation of the concentration effect on the bimolecular rate coefficient only to the lowest order [ 15 1. The validity of the mean field theory at higher quencher concentrations is uncertain. In the context of the present theory, it is expected that the superposition approximation [ eq. (2.6) 1, neglecting the direct correlation between the quencher molecules, may break down at higher quencher concentrations. In the present theory, the correlation between quencher molecules is taken into account indirectly via the competition for the same fluorophor molecule. Hence, as a concentration-dependent theory, the present theory may be just slightly better than the conventional Smoluchowski theory. 3.3. Relation to the statistical nonequilibrium thermodynamic theory Keizer [ 15 ] has formulated a general theory of diffusion-influenced bimolecular reactions based on statistical nonequilibrium thermodynamics [ 271, and applied the theory to treat the steady-state fluorescence quenching kinetics [ 15,161. It has been shown [ 201 that at low quencher concentrations Keizer’s theory predicts almost the same curvature in the Stem-Volmer plot as the mean field theory. However, as Ci increases, Keizer’s theory predicts a more drastic positive curvature. In the following section, we compare numerically the predictions of the various theories for a model fluorescence quenching reaction system.

S. Lee et al. /Di&.v’on-injluencedjluorescence quenching

351

4. Model calculations

Eq. (3.5) shows that the relative intensity of quenched to unquenched fluorescence depends on the intensity Cz and other parameters that characterize the diffusive dynamics of the fluorophor and quencher in a given solution. This is an important result since the conventional Smoluchowski approach [ 111 makes no reference to the probable dependence of lo/Z on F. Fig. 1 shows the F dependence of Stem-Volmer plot. The model parameters used are those for the fluorescence of 1,Zbenzanthracene quenched by CBr., in 1,Zpropanediol at 25 oC [ 2 1; r. = kg ’ = 38.5 ns, rr= 9.1 A, D=5.Ox10-7cmZs-‘andk~=6x10’0Lmol-’s-’=1x10-’0cm3s-‘.Inref. [2],NemzekandWareevaluated these parameters from an analysis of quenched fluorescence decay data based on the conventional Smoluchowski theory. However, the light pulse used by them to excite fluorophors had a finite width. Thus, their analysis may be erroneous due to the failure of the assumption, made in the conventional theory, that D* molecules just excited are surrounded by an equilibrium distribution of quencher molecules. In fact, they found [ 21 that the above parameters deduced from time-resolved fluorescence decay data are inconsistent with the curvature observed in the Stem-Volmer plots for the same system. Again, their analysis of Stem-Volmer plots was based on the conventional Smoluchowski theory. Although we feel that a consistent analysis of the time-dependent and steady-state fluorescence quenching data within the framework of the present theory is desirable, we are not able to carry out the analysis since no information on the intensity of illumination is given in any experimental paper on fluorescence quenching kinetics. For the present, we just take the above values for motional and reaction parameters as a reasonable choice for the input to model calculations to illustrate the implications of the present theory. As the light intensity decreases (i.e., F/&--+0), the Stem-Volmer curve approaches an asymptote. In fig. 1 this asymptotic curve is almost identical with the dashed curve for the case with F= 0.001 b. On the other hand, as F increases, the Stem-Volmer curve approaches the horizontal line lo/Z= 1. This behaviour can be under.stood by considering eqs. (3.5) and (3.20). As F increases k& increases only slightly, as displayed in fig. 2. Hence, the slope of the Stem-Volmer curve given by [ kb + C$ ( ilk& /Xi ) ] / ( ks + F) decreases as F increases. The F dependence of kh shown in fig. 2 is by itself of interest. As mentioned in the introductory section, increasing F results in the increase of the fraction of D* molecules re-excited shortly after the bimolecular F of steady illumination, as well as on the quencher concentration

40.

/

I

--_.-._ -..-. 30. --...-

I

I,

F F F F

= 3 =

I

II

13

I

I

I

---

0.001kg Olk, k, 10 k,

-'-'- -..-.

/

I

I

I

I

I

I

I

q=o.olN 7

q=o.o5u

i

q-OIN

..f$

,'/,

c,o

!!k /'!/ /..!I 1.' /I /" ,', _ -..-.._.._.._.._.._..-.' /' / /' 1 H' / -.-.-._______._.C.-'

20. -

I

I 0.1

I

I

I1

h

0.2

0.3

Ci

in M

Fig. 1. F dependence of the Stem-Volmer rameters used are described in the text.

1 0.4

I

_ 0.5

5’ -4

’

’ -3

’

’

’

-2

’ -1

’

’ 0

’

J 1

Wzto(F/ks) plot. The model Pa-

Fig. 2. F dependence of the steady-state quenching rate coeffk cient k&_The model parameters used are the same as in fig. 1.

352

S. Lee et al. / Di$%on-injluencedjluorescence quenching

/’

5

40. /

/’

/’ 1’ .’

1’ .’

.’

.’

.’

./

,’ .’

.’

/ .’ / ./ / / /.’ ,__.-.. ’ 0’ /..C.._. / /’ _..-. /.A ._..-.. ,_.. _... ... ...- ..-.. , 7 -I , 0. fi
20. -

W&/k,) Fig. 3. Dependence of the infinite dilution Stem-Volmer coeff~cient K$, on the ratio of F to ks.

C; in

M

Fig. 4. Variation of the Stem-Volmer plot with the values of kD changing from O.OOlk~ to 0.1 kg. The value of kg is fixed at 6x 1O”‘Lmol-’ s-‘.

quenching. Since such D* molecules will find a quencher molecule still hanging around, they can be quenched quickly. This explains the increase in the value of k& with increasing F. Note, however, that this enhancement in the quenching rate is appreciable only for very large value of F. In fig. 3 we display the dependence of the infinite dilution Stem-Volmer coefficient Z%, as given by eq. (3.23) on the ratio F/k,. We have used the same values for the parameters 70, o, D, and kg as in fig. 1. As F goes to zero the value of KO,vapproaches a limiting value which coincides with the prediction of the conventional Smoluchowski theory. On the other hand, as F increases K& decreases and eventually becomes unity for very large values of F. Fig. 4 shows that the quenching effect becomes more pronounced as the diffusive encounter rate of D* and A increases. We used the same values for the parameters 70, o, and kq as in fig. 1, but changed the value of D such that kD has a value ranging from 0.00 1kg to 0.1 k$. We fixed the value of F at 0.00 1ks. It is of interest to compare the Stem-Volmer plot calculated from the present theory when Fe ks with that calculated from other theories. In fig. 5 we present the results for the same reaction system as in fig. 1. One can see that within the concentration range shown in the figure the differences among the present theory, the conventional Smoluchowski theory and Cukier’s theory are quite small. However, the statistical nonequilibrium thermodynamic theory of Keizer predicts a more drastic positive curvature of the Stem-Volmer curve. When compared to the experimental Stem-Volmer plot reported in ref. [ 21, it is seen that none of the theories are satisfactory. This may be due to the inaccuracy in the parameter values deduced from time-resolved fluorescence decay data, as mentioned above. Perhaps the more positive curvature observed in the experimental data may be due to the static quenching [ 2 1. However, for the present, there is no spectroscopic evidence of a ground state complex [see eq. ( 1.6) ] between carbon tetrabromide and l,Zbenzanthracene, and the analysis of time-resolved fluorescence decay data would also be modified by the presence of such complex. It may be desirable to derive a more satisfactory theory to describe the concentration effect on the quenching rate coefficient *‘. Fig. 5 shows the case when the equilibrium quenching rate constant k3 is much larger than the diffusion-

*’ For the concentration effect on the bimolecular rate coefficient of diffusion-influenced reactions involving infinite life-time reactants, see ref. [28].

S. Lee et al. / DiQurion-influenced fluorescence quenching 12. .

I -

10.

_ ---.-.- - .. -’

8.-

0. 0.00

,

,

,

I

,

353

,

Present Theory Conventional Theory (Ref. 20) Cukier’s Theory (Ref. 17) Keizer’s Theory (Ref. 16) Experiment (Ret. 2) .

_

---.-.-..-.

Present Theory Conventional Theory (Ref. 20) Cukier’a Theory (Ref. 17) Keizer’s Theory (Ref. 16)

l

I

I 0.05

I

I

I

0.10

Ci

I 0.15

1 0.20

in M

Fig. 5. Comparison of various theories of fluorescence quenching when k$ s kD The model parameters used are the same as in fig. 1.

-0.00

0.05

0.10

0.15

0.20.

Cl in M Fig. 6. Comparison of various theories of fluorescence quenching when k? = kD Except for the value of k$, the model parameters used are the same as in fig. 1.

limited rate constant kD. When k$ is comparable to kD we find that the discrepancy among various theories may not be discernible for the whole range of Ci as long as the light intensity is kept low. Fig. 6 shows this trend. Here the value of kg has been set equal to kD but other parameters are the same as in fig. 1.

5. Concluding remarks

We have formulated a theory for describing the effects of quencher molecules on the intensity of fluorescence stimulated by steady-state illumination. The present theory, which is based on a hierarchical system of manybody Smoluchowski equations for the reactant molecule distribution functions, predicts a rather complicated nonlinear dependence of lo/l (the relative intensity of unquenched to quenched fluorescence) on the quencher concentration Ci. Nevertheless, an exact analytic solution has been obtained for the simplest case where bimolecular quenching occurs upon contact of donor and acceptor molecules and the potential of the mean force between them has a negligible effect; see eqs. (3.2 1) and (B.4). Effects of the potential of the mean force on the quenching dynamics can be evaluated with good accuracy by an approximate analytic equation, which will be presented elsewhere [ 241. However, even the full numerical calculation that solves eqs. (3.1)-( 3.3) directly is found to be not too demanding [ 241. An important aspect of the present theory is that it describes the dependence of fluorescence quenching dynamics on the intensity of the external illumination. The conventional Smoluchowski approaches [ 111 are mute in this respect; it has been simply assumed that the intensity of illumination should be quite weak. In this regard it should be mentioned that for the present experimentalists check for the effect of the light intensity F by just repeating the fluorescence quenching experiment with another F to make sure that nothing has changed. Hence the validity or utility of the present theory may be tested by a fluorescence quenching experiment under intense illumination. In appendix C, for the convenience of experimentalists, we describe a method to determine the value of F from time-resolved fluorescence data obtained in the absence of the quencher.

5. Lee et al. / Dijksion-injluencedfruorescencequenching

354

Acknowledgement

We wish to thank Dr. A. Szabo and an anonymous referee for a critical reading of the manuscript. This work was supported by a grant from the Basic Science Research Program, Ministry of Education of Korea, 1989.

Appendix A. General kinetic equations for fluorescence quenching reactions

Although the kinetic equations [eqs. (2.2)-(2.5)] described in section 2 are valid in many experimental situations, they neglect several important aspects of fluorescence quenching reactions such as the effects due to donor-to-donor energy transfer, reversible energy transfer, and slow decay rates of excited quencher molecules. When these events are taken into account, eq. (2.2) should be replaced by

+r$, jdr’

S,,(r,r’)P,,(r,

r’, t)-

ri j dr’ S,,(r,r’)Po:4(r,r’,

t)+ r< j dr’

sDD(r,r’)pD,D;(r,r’,t).

(A-1)

In this equation, the fifth term on the right-hand side involving the sink function SADdescribes the rate of reversible energy transfer from quencher molecules to the Dj molecule, while the last two terms involving the sink function SD, represent the donor-to-donor energy transfer events. A corresponding kinetic equation for PDi(r, t) can be readily given. The kinetic equation (2.4) for the two-particle probability density function PDrA,(r, r' , t ) should be in turn replaced by

dr ” SD, ( r, r “ ) pD:Apk (r, r’, r”, t)-

k#i

$: j dr” &&(r’, r” )PDpjDj(r, r’, r”, t) I#1

+sAD(r,

r’)PD,A;(ry

r’, f)+

I$,j dr”

&D(r?

r”)PDiA&(r9

r’?

r”y

t)

k#j

+

i:

dr” SAD(r), r”)PDf+,(r,

j

r’, r”, t)-

I#i

+

2

2

I dr”

sDD(r,

r”

)PD:A,D,tr9

r’y

r”y

l)

Izi

j

dr” &D(r,

r”)PDiAjDf(rv

r’,

r”,

t)+k$(f)PD:A;(ry

r’v

l)

Izi

-

kz, k#j

j

dr” SAA(r), r”)PDrApf(r,

r’, r”, t) + $,

j dr” &A(r’, r”

)PD:AfAk(r,

r’,

r”,

t)

.

(A-2)

k+J

The physical meaning of each term on the right-hand side of this equation should be self-evident. For example, the sixth term describes the rupture of the D:-A, pair due to excitation of the Aj molecule by the competitive energy transfer from D molecules other than D,. This term, as well as the tenth arid the eleventh terms describing

S. Lue et al. /Di&ion-influenced fluorescence quenching

355

the rupture and regeneration of the D:-A, pair due to donor-to-donor energy transfers, is negligible if the concentration of donor molecules is small (i.e., if Cg -S 1). The seventh to the ninth terms involving the sink function S,, describe the reversible energy transfers. The twelfth term represents the rate of regeneration of the D:-A, pair due to unimolecular de-excitation of A:. Finally, the last two terms involving the sink function S,, describe the acceptor-to-acceptor energy transfers. In addition to eq. (A.2) for P pAj we need to consider kinetic equations for PDiA,, PDiAp P,y.+ P,,p,, and PDi,,r. Detailed treatment of this extended set of kinetic equations will be presented in a forthcoming paper.

Appendix B. Determination of ys from eq. (3.18)

Eq. ( 3.18) can be rearranged to give -

A

1-x

CX”2 ,

-B=

(B.1)

where XS y,/ yeqwith yesz kS +F+ kg Ci. The three constants A, B and C, are all positive:

(B-2) Fig. 7 shows that the root for eq. (B. 1) is given by X,. To evaluate its value analytically each side of eq. (B. 1) is squared to yield a cubic equation for X X3+b,X2+b2X+b3=0,

(B-3)

withb,~-(2+B2/C2)Oandb~~-[(A-B)/C]2
cos[ (&2rt)/3]

-b,/3,

(E.4)

wherecos0=R(-Q3)-“2withQ=(3b2-b:)/9andR=(9b,b,-27b,-2b3)/54. An approximate expression for X, that is valid in the limit of low quencher concentration [i.e., when Cz +O ] can be obtained as follows. We first note that in this limit X, = 1 - E with e being a very small positive number. An algebraic equation for e derived from eq. (B. 1) is then solved approximately to obtain e=A/(B+C).

(B.5)

With this expression for E,we finally have X,=1-

(&lkD) [@‘l(ks +F) 1 { l+(k&‘/kD)+[(k,+~)a2/Dl”Z

c”, ,

(when p+o)

.

(B.6)

S. L.ee et al. / Di$ksion-infltrencedjluorescence quenching

356

Fig. 7. Locationofthe

root X2 foreq. (B.1).

Appendix C. Experimental determination of F One can determine the value of F from time-resolved fluorescence data obtained in the absence of quencher as follows. When Cz = 0 eq. (2.11) becomes $ [D*]=-ks[D*]+F(t)[D]

.

(C.1)

Solving this equation for [D* 1, we obtain [D*]=CO,exp[-&-If F(

7)

=

Fzconstant,

/drF(r)]jdrF(r)exp[ksr+

jdr,F(r,)].

0

0

0

(C-2)

we have

[D*]=[FCO,/(ks+F)]{l-exp[-(ks+F)t]}.

(C.3)

If one can measure the absolute value of [D*], k and F can be determined separately from this equation; the transient portion of the data gives the value of ( ks+ F) and then the steady-state value of [D*] gives the value of F separately. Else, if only the relative magnitude of [D*] can be measured, one needs to perform a second experiment using a light pulse of short duration. When F(t) vanishes for

F) has been determined from the transient data obtained with constant illumination, one can then obtain the value of F.

References [ I] J.B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970);

S. L.ee et al. /Dimion-injluencedJluorescence quenching

357

J.B. Birks, in: Organic Molecular Photophysics, ed. J.B. Birks (Wiley, New York, 1975) p. 409. [2] T.L. Nemzek and W.R. Ware, J. Chem. Phys. 62 ( 1975) 477. [ 3 ] D.P. Millar, R.J. Robbins and A.H. Zewail, J. Chem. Phys. 75 ( 198 1) 3649. [4] R.W. Wijnaendts van Resandt, Chem. Phys. Letters 95 (1983) 205. [ 51 N. Tamai, T. Yamazaki, I. Yamazaki and N. Mataga, Chem. Phys. Letters 120 ( 1985) 24. [6] J.R. Lakowicz, M.L. Johnson, I. Grynynski, N. Joshi and G. Laczko, J. Phys. Chem. 91 ( 1987) 3277. [ 71 N. Periasamy, S. Doraiswamy, B. Venkataraman and G.R. FIeming, J. Chem. Phys. 89 ( 1988) 4799. [ 81 G.C. Joshi, R. Bhatnagar, S. Doraiswamy and N. Periasamy, J. Phys. Chem. 94 ( 1990) 2908. [9] F. Heisel and J.A. Miehe, J. Chem. Phys. 77 ( 1982) 2558. [ IO] J.K. Baird and S.P. Escott, J. Chem. Phys. 74 ( 1981) 6993. [ 111 S.A. Rice, Diffusion-limited Reactions in Comprehensive Chemical Kinetics, Vol. 25, eds. C.H. Bamford, C.F.H. Tipper and R.G. Compton (Elsevier, Amsterdam, 1985). [ 121 G. Wilemski and M. Fixman, J. Chem. Phys. 58 (1973) 4009. [ 131 U. Giisele, M. Hauser, U.K.A. Klein and R. Frey, Chem. Phys. Letters 34 ( 1975) 5 19. [ 141 J.K. Baird, J.S. McCaskill and N.H. March, J. Chem. Phys. 74 (1981) 6812; 78 (1983) 6598. [ 151 J. Keizer, J. Phys. Chem. 86 (1982) 5052; Chem. Rev. 87 (1987) 167. [16]J.Keizer,J.Am.Chem.Soc.105(1983)1494;107(1985)5319. [ 171 RI. Cukier, J. Chem. Phys. 82 (1985) 5457. [ 181 B. Stevens, Chem. Phys. Letters 134 ( 1987) 5 19. [ 19) J. Najbar, Chem. Phys. 120 (1988) 367. [20] A. Szabo, J. Phys. Chem. 93 (1989) 6929. [ 2 1 ] H. Zhou and A. Szabo, J. Chem. Phys. 92 ( 1990) 3874. [22] S. Leeand M. Karplus, J. Chem. Phys. 86 (1987) 1883. [ 231 M. Yang, S. Lee, K.J. Shin, KY. Choo and D. Lee, to be submitted for publication. [24] M. Yang, S. Lee, K.J. Shin, K.Y. Choo and D. Lee, Bull. Korean Chem. Sot. 12 (1991) 414. [25] L. Monchick, J.L. Magee and A.H. Samuel, J. Chem. Phys. 26 (1957) 935. [26] T.R. Waite, Phys. Rev. 107 (1957) 463. [27] J. Keizer, J. Chem. Phys. 63 (1975) 398; 64 (1976) 1679; 65 (1976) 4431. [28] B.U. Felderhof and J.M. Deutch, J. Chem. Phys. 64 ( 1976) 4551; D.F. Calef and J.M. Deutch, Ann. Rev. Phys. Chem. 34 ( 1983) 493.