Theory of Diffusion- Influenced Fluorescence Quenching - American

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6929

J. Phys. Chem. 1989, 93, 6929-6939

Theory of Diffusion-Influenced Fluorescence Quenching Attila Szabo Laboratory of Chemical Physics, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20892 (Received: February 16, 1989; In Final Form: May 9, 1989)

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A unified and comprehensive analysis is presented of various (including Smoluchowski, mean field, statistical nonequilibrium thermodynamic) approaches to the description of the kinetics of the reaction A* + B A + B within the framework of the diffusive model of microscopic dynamics. (A* is an excited state, and B is a quencher or “trap”.) The Smoluchowski formalism is derived in a way that clearly shows its limitations as well as its strengths (Le., it is in fact an exact treatment of many-particle diffusive dynamics under certain conditions). When the calculation of the Smoluchowski steady-state rate constant is based on the time-dependent rather than the steady-state diffusion equation, the result is well-defined and quencher concentration dependent in all dimensions. The formal relationship between the above approaches is clarified; their predictions are compared as a function of quencher concentration, of the ratio of A* to B mobilities, and of spatial dimensionality; and their range of validity is assessed. Within the Smoluchowski framework, models where the nonradiative lifetime of an A*-B pair depends on interparticle distance are considered, and their relevance to finding a unified description of both the transient and steady-state fluorescence intensity is discussed. The simplest model of this class, in which the nonradiative lifetime is constant over a finite range, is shown to predict, on a time scale longer than the nonradiative lifetime, the same transient behavior as the so-called radiation boundary condition model. This analysis not only provides a justification for the somewhat physically artificial “radiation” model but also gives a more microscopic interpretation of the intrinsic rate constant which shows that this constant can be viscosity dependent.

I. Introduction From both experimental and theoretical perspectives, fluorescence quenching is one of the simplest chemical reactions in solution. This reaction is characterized by the kinetic scheme A* B A B where A* is an excited state of A and B is a quencher. This mechanism is pseudo first order since the concentration of B does not depend on time. Since the B’s act like indestructible sinks or traps, this reaction is sometimes referred to as trapping. One is interested in the time dependence of concentration of A* following excitation by a delta function light pulse and in the steady-state rate constant that describes the behavior of the system when excited states are being produced by constant illumination. This trapping mechanism for the nonradiative decay of an excited state is valid only when the concentration of A* is sufficiently low so that a B molecule can dispose of the energy it received from an A* before it has an opportunity to quench again. To describe the fluorescence intensity that is monitored experimentally, one must of course allow for the possibility that an A* can also decay to A radiatively. If the two competing pathways are independent, as is usually assumed, this can easily be incorporated into the formalism. The zeroth-order description on the above process is based on the phenomenological rate equations of chemical kinetics. These predict that the fluorescence intensity decays as a single exponential and that the ratio of the steady-state intensity in the absence and presence of quencher varies linearly with quencher concentration. The ultimate goal of theory is to provide a first principles description of the kinetics, to establish the conditions under which the rate equations are valid, and to express the phenomenological rate constants in terms of microscopic parameters. A more modest goal is to carry out this program within the framework of a microscopic model of the dynamics. The simplest model of this type assumes that the dynamics of all the molecules in the system is diffusive in nature and that quenching can occur with a certain rate whenever an A* and a B are in close proximity. As is well-known, the diffusive model breaks down on sufficiently short time and length scales. Nevertheless, one is still faced with a many-particle problem that can be exactly solved only in special cases. Within the framework of the diffusive model of dynamics, there are numerous ways of attacking the many-particle problem that arises in the treatment of fluorescence quenching. The most familiar, but not necessarily the best understood, is along lines that can be traced back to the seminal contribution of Smoluchowski.14 Other computationally viable approaches include the

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formalism of Wilemski and Fixmanlo based on a closure approximation, the mean field theory of Cukier,I**l2 and the statistical nonequilibrium thermodynamic theory of Kei~er.’*’~In addition, the problem of trapping by stationary, random hard-sphere sinks has been studied by using a variety of sophisticated tools.1b22 The purpose of this paper is to provide a unified treatment of some of these approaches, to compare their predictions, to establish interrelationships among them, to discuss their limitations and range of validity, and to clear up some widely held misconceptions. Although experimental data are not analyzed here, an effort has been made to provide explicit results that should prove useful in this regard. The outline of this paper is as follows. Section I1 deals with the Smoluchowski approach. We begin by summarizing the conventional formulation based on the solution of the Smoluchowski equation for the time-dependent reactive pair distribution function. In section 11.2 we derive this formalism starting from a many-particle description. This analysis highlights both the limitations (i.e., the various approximations that are made) and the strengths of this approach. For example, it constitutes an exact treatment of the many-particle diffusive dynamics when the B’s do not interact with each other and the ratio of the A* to B

(1) Smoluchowski, M. Z . Phys. Chem. 1917, 92, 129. (2) Collins, F. C.; Kimball, G. E. J . Colloid Sci. 1949, 4 , 425. (3) Noyes, R. M. J. Am. Chem. SOC.1957, 79, 551. (4) Weller, A. Z . Phys. Chem. 1957, 13, 335. (5) Yguerabide, J.; Dillon, M. A.; Burton, M. J . Chem. Phys. 1964, 40, 3040. Yguerabide, J. J . Chem. Phys. 1967, 47, 3049. (6) Nemzek, T. L.; Ware, W. R. J. Chem. Phys. 1975,62, 477. (7) Noyes, R. M. Prog. React. Kinet. 1961, 1 , 129. (8) Rice, S. A. Diffusion-Limited Reactions. In Bamford, C. H., Tipper, C. F. H.; Compton, R. G., Eds. Comprehensiue Chemical Kinetics; Elsevier: Amsterdam, 1985; Vol. 25. (9) Gosele, U. M. Prog. React. Kiner. 1984, 13, 63. (10) Wilemski, G.; Fixman, M. J . Chem. Phys. 1973, 58, 4009. (11) Cukier, R. I. J . Am. Chem. SOC.1985, 107, 4115. (12) Yang, D. Y.; Cukier, R. I. J. Chem. Phys. 1987, 86, 2833. (13) Keizer, J. J . Phys. Chem. 1981, 85, 940; 1982, 86, 5052. (14) Keizer, J. J . Am. Chem. SOC.1983, 105, 1494; 1985, 107, 5319. (15) Keizer, J. Chem. Rev. 1987, 87, 167. (16) Calef, D. F.; Deutch, J. M. Annu. Reo. Phys. Chem. 1983, 34,493. (17) Felderhof, B. U.; Deutch, J. M. J. Chem. Phys. 1976, 64, 4551. (18) Bixon, M.; Zwanzig, R. J . Chem. Phys. 1981, 75, 2354. (19) Muthukumar, M.; Cukier, R. 1. J . Stat. Phys. 1981, 26, 453. (20) Cukier, R. I.; Freed, K. F. J . Chem. Phys. 1983, 78, 2573. (21) Cukier, R. I. J. Chem. Phys. 1985, 82, 5457. (22) Mattern, K.; Felderhof, B. U. Physicu 1986, A135, 505.

This article not subject to U S . Copyright. Published 1989 by the American Chemical Society

6930 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 diffusion coefficients approaches zero. This result has recently been obtained by Szabo, Zwanzig, and A g m ~ from n ~ ~a rigorous density expansion for the time dependence of the concentration of A*. Section 11.3 discusses the calculation of the time-dependent rate constant in d dimensions. Sections 11.4 and 11.5 consider the calculation of the steady-state rate constant (k,) for the trapping reaction and the steady-state fluorescence intensity. The conventional approach to the calculation of k, is based on the solution of the steady-state Smoluchowski equation. This procedure predicts that k , is independent of the concentration of B in three dimensions and fails completely in lower dimensions. However, when k, is calculated starting from a time-dependent point of view, completely analogous to the way the steady-state fluorescence intensity was obtained in the pioneering work of Noyes3 and Weller4, k,, is well-defined and concentration dependent in all dimensions. Section 111 deals with mean field approaches. The mean field theory of CukierI1J2 for the steady-state rate constant of the trapping reaction is reformulated, and it is shown that in general the self-consistency condition that determines k , can be expressed solely in terms of the Laplace transform of the Smoluchowski time-dependent rate constant. His treatment of the fluorescence quenching is modified so that both the radiative and nonradiative decay rates of A* are treated on an equal footing. Section 111.1 establishes the formal relation between the mean field and Smoluchowski approaches. In section IV we compare the steady-state rate constant for the trapping reaction calculated using the Smoluchowski, mean field, statistical nonequilibrium thermodynamics, and some other approaches with available exact results for one-, two-, and threedimensional systems. In section V we make the corresponding comparison for the steady-state fluorescence intensities in the absence and presence of quencher. In addition, we examine models in which the nonradiative lifetime of an A*-B pair depends on the interparticle separation over a finite range and discuss the implications that such models have on the steady-state and transient behavior of the fluorescence intensity. Finally, in section VI we make some concluding remarks. 11. The Smoluchowski Approach

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Consider the reaction A* B A B where A* (A) is an excited (ground)-state molecule and B is a quencher or trap. Suppose at t = 0 one prepares a homogeneous system in which the concentrations of A* and B are [A*(O)] and c, respectively. For the sake of simplicity, assume that both species are spherically symmetric and interact via a potential of mean force, U(r), that depends only on the A*-B distance. The above reaction occurs with a certain rate whenever A* and B diffuse together and come into contact. What is the time dependence of the relative concentration of A*, [A*(t)]/[A*(O)]? This quantity is equivalent to the survival probability of an excited-state molecule in the presence of quenchers at concentration c and will be denoted by &t). 1 . Conventional Formulation. The survival probability is assumed to satisfy dcr"(t)/dt = -k(t)&(t)

(2.1)

where the time-dependent rate constant, k(t), is calculated from the time-dependent radial distribution function of reactants, denoted by p(r,t), as follows. One of the reactants, say A*, is placed at the origin of a coordinate system and p(r,t) is assumed to satisfy the Smoluchowski equation ap/at = DV-e-@uV@up

(2.2a)

where p = (ken-' and D is the relative diffusion coefficient of an A*-B pair (Le., D = DA* De). In d spatial dimensions, eq 2.2a becomes

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(23) Szabo, A.; Zwanzig, R.;Agmon, N. Phys. Reu. Lert. 1988.61, 2496.

Szabo The above equation is solved subject to the initial condition p(r,O) =

(2.3)

and subject to appropriate boundary conditions (see below) at contact (Le., when r = a where a is the sum of the radii of A* and B). The time-dependent rate constant is obtained by integrating the normal component of the flux over the surface of contact

If reaction always occurs when A* and B come in contact, then eq 2.2 is solved subject to the purely absorbing or Smoluchowski boundary condition p(a,t) = 0

(2.6)

The resulting rate constant is denoted here by ks(t). A finite rate of reaction at contact is described by using the partially reflecting or radiation boundary condition of Collins and Kimbal12

- - -+

=k d a 4

(2.7)

where ko is the intrinsic rate of reaction. (When ko m reaction always occurs at contact whereas when ko 0 no reaction occurs.) ko has units of a bimolecular rate constant. In the context of fluorescence quenching it can be interpreted as follows. Suppose the lifetime of an A*-B pair is 7 when a I r I a E . Then ko is the limit of (V(a+t) - V(a))/7 as 7 0 and 6 0, where V(r) is the volume of a hypersphere of radius r. As emphasized by Wilemski and Fixman,lo an alternate, more physical, way of incorporating reaction in the formalism is to add a sink term, - ~ ( r ) p ,to the right-hand side of the evolution equation (2.2a). The unimolecular rate constant K(r) is the reciprocal of the nonradiative lifetime of A* in the presence of a single quencher at distance r. The Collins-Kimball boundary condition (2.7) corresponds to the special case where K(r) = koS(r- a ) / A ( a )where A ( a ) is the area of a hypersphere of radius a. Let kCK(t) be the time-dependent rate constant calculated from the radiation boundary condition (2.6) aFd let k C ~ ( zbe ) its Laplace transform. (We use the notationfiz) = 1; exp(-zt)flt) dt.) For sphFrically symmetric systems kCK(z)can be expressed in terms of ks(z) as

This relation between the two rate coefficients can be readily verified by first noting that the solutio? of the Laplace transform of eq 2.2b can be written as z-l( 1 - Cfr,z)) exp(-@U(r)) where Xr,z) is the solution of the homogeneous equation which is regular at infinity and C is a constant to be determined by ushg the boun$ary condition a t contact. The calculation of both kc&) and ks(z) is then formally carried out. Comparison of these results immediately leads to eq 2.8. 2. Many-Particle Reformulation. We now derive the above formalism in a way that clearly highlights the nature of the approximations involved. In particular, it will be seen that the Smoluchowski approach is exact in the limit that (1) the diffusion coefficient of A* approaches zero and (2) the quenchers do not interact with each other. Consider a homogeneous system containing M A*'s and N B s in volume V. If the A*'s are sufficiently dilute so that possible A*-A* interactions are negligible, it is sufficient to consider the fate of a single A* surrounded by N B's. For the sake of simplicity, we assume for the moment that the particles do not interact with each other and that the reaction always occurs when A* comes in contact with a B. Assuming that the dynamics is diffusive, the probability density that A* is at f A * and the ith B is at fi,denoted by P'(fA*,f,, ...f N , t ) , satisfies

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6931

Diffusion-Influenced Fluorescence Quenching N

a p t / & = (DA.VA.z

+ DBCV:)P’ i= 1

(2.9)

The initial condition is P’(t=O) = Y(N+I), and the boundary condition is that P’vanishes whenever a B touches the A*. Introducing relative coordinates ri = rli - rlA., eq 2.9 becomesz3 dP(rl,...rN,t) N N = (DXV? DA. Vi-V,)P(r,, ...rN,t) (2.10) at i= 1 i#j=l

+

+

where D = DA. DB and P = VP’. For a system containing a single B, the cross term in eq 2.10 is absent. Thus, as is wellknown, it is possible to describe the dynamics of a single A*-B pair by fixing one of the molecules and letting the other diffuse around it with relative diffusion coefficient D. As suspected by Noyes in his classic review article,’ this is no longer rigorously true when more than one B is present, unless, of course, DA. = 0. We now neglect these cross terms. This crucial approximation is analogous to the Born-Oppenheimer approximation in atomic physics (Le., A*, the B’s, and D correspond to the nucleus, the electrons, and the reciprocal of the reduced mass, respectively). Now suppose that the A* interacts with the B’s and that the B s interact among themselves. The simplest way of generalizing the above formalismz4 is to replace Vi2 by the Smoluchowski operator involving the pairwise potential of mean force between A* and B, i.e. N

a P / a t = ( D E Vi.e-Bu(rJVieBu(rJ)P

- -

(2.11)

i= I

where U(r) 0 as r m . Thus, the explicit interaction among the Bs is ignored and enters the formalism only through the A*-B potential of mean force which can depend on the B concentration. In applications, this potential is often approximated by the bare interaction potential. Equation 2.1 1 also assumes that there is no hydrodynamic interaction between A* and B. This effect can be modeled by using a distance-dependent relative diffusion coefficient. I6vz4 The survival probability of an A* in the presence of an initial equilibrium distributions of N B’s is sN(t) = l d r ,...dr, P(rl, ...rN,t)

(2.12)

where P is the solution of eq 2.1 1 with initial condition

where the integral is over the volume accessible to the B’s. It follows from eq 2.1 1 that P is separable and thus sN(t) = ( J d r

p(r,r))N

(2.14)

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where P(r,t) satisfies eq 2.11 with N = 1 . Let the limit of S d t ) as N m, V - m, and N / V = c be 8(t). We shall now show that 8 ( t ) satisfies eq 2.1 with k(t) calculated according to the Smoluchowski prescription (see eq 2.4). As V m, S d r exp(-pU(r)) diverges as V since V(r) 0 as r m, so that

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P(r,O) = e-flu(r)/V

(2.15)

Comparing eq 2.1 1 with N = 1 with eq 2.2 and eq 2.15 with eq 2.3, it follows that P(r,t) = p(r,t)/V where p(r,t) is the radial distribution function that occurs in the formulation of the previous subsection. Thus, eq 2.14 can be rewritten as S d t ) = (Jdr p(r,t)/V)N

(2.16)

Differentiating this with respect to time, one finds

(24) Northrup, S. H.; Hynes, J. T. J . Chem. Phys. 1979, 71, 871.

As V - m, the integral over r is equal to -k(t), given in eq 2.4, as can be seen by integrating both sides of eq 2.2a over r and using Green’s theorem. Thus

-

(2.18)

which is identical with eq 2.1 when N , V m and N / V = c. In summary, the Smoluchowski approach is exact when the diffusion coefficient of A* approaches zero and the B’s do not interact with each other (although they may interact with A*). Clearly, this approach is expected to be poor for the problem of absorption of a diffusing particle (Le., A*) by static sinks or traps (Le., the B’s). Nevertheless, as will be elaborated on below, the Smoluchowski approach is pretty good even in this demanding context. For example, in three dimensions it correctly predicts the first correction to the dilute limit of the steady-state rate constant. Recently, a density expansion, analogous to the viral expansion, has been derivedz3for the time-dependent survival probability of A*. Its leading term is the Smoluchowski result, and the first correction is expressed in terms of exact survival probability of A* in the presence of two B’s. This formalism has been applied to the trapping problem in one dimension where the exact result is known25for DB = 0. It was found that the Smoluchowski result is exact at short times and remains accurate for longer and longer times as the ratio DAI/D decreases. For example, for equal diffusion coefficients the error is 110% as long as $(t) 2 0.05. While one dimension is of limited interest from the experimental point of view, it is theoretically interesting because the errors inherent in the Smoluchowski approach are magnified. In the theory of lattice random walks, the case DB = 0 is called a trapping problem (Le., the A*’s are point random walkers and the B’s are static traps) whereas the case DA. = 0 corresponds to a target problem (Le,, the B’s are random walkers which annihilate static targets). Blumen et a1.26have shown that the target problem admits an analytic, albeit implicit, solution in all dimensions. Their derivation is in essence the discrete time and space realization of the Smoluchowski approach. They also simulatedz7 the survival probability for various integer ratios of the mobilities of A * and B in d = 1 and 2. They found that the difference between the target and trapping survival probabilities in two dimensions is considerably smaller than it is in one dimension. Moreover, as the mobility of the B’s increases, the target solution is approached more rapidly in two dimensions. For equal mobilities and large c (Le., half the lattice sites are occupied by B’s) in two dimensions, the target solution is virtually exact when the survival probability is greater than lo4. On the basis of these trends, one expects that in three dimensions the target (Smoluchowski) solution is highly accurate, except at very long times where the survival probability is insignificantly small, as long as the B’s are not absolutely static and are noninteracting (Le., points). Thus, in three dimensions, the major deficiency of the Smoluchowski approach is the neglect of correlations in the motion of the B’s due to interparticle interactions. 3. Time-Dependent Rate Constants. When U = 0, the solution of the Laplace transform of eq 2.2b can be expressed in terms of modified Bessel functions of the second kind, K,, and it can be shown that 2d/2Dad-2~Kdlz(~) L,(z) =

zr(d / 2, Kd/2-1 ( x )

(2.19)

where x = (za2/D)Il2. The special cases d = 1, 2, 3 are wellknown, and we present them here for future reference. For d = 1 &(z) = (4D/z)I/’

(2.20)

k,(t) = ( 4 D / a t ) l / 2

(2.21)

(25) Agmon, N. J . Stat. Phys. 1986, 43, 537, and references therein. (26) Blumen, A.; Zumofen, G.; Klafter, J. Phys. Rev. 1984, B30, 5379. (27) Blumen, A.; Zumofen, G.; Klafter, J. J . Phys. 1985, 46, C7-3.

6932 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989

Szabo

For d = 2 one has28 R,(z) =

(2.30)

2 r D ( ~ a ~ / D ) ' / ~ K ~D() (' /z2a) ~ / zKo(( z a 2 / D ) ' / 2 )

(2.22) as found previously be Pedersen and Sibak3O The above result is exact when U = 0 ( a , = a ) . It correctly behaves as ko X exp(-@U(a)) at very short times and is exact at long times. It is least accurate at moderately short times. One can obtain an alternate expression for kCK(t) that is better in this time range 0 and as follows. Combining the leading terms in both the z z m expansions given in eq 2.26 and 2.28, one has

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k S ( z )= 4 r D a e / z

+ 4 r ~ ~ e - @ ~ ( ~ ) ( D / (2.3 z ) ' 1) /~

When this is substituted into eq 2.8 and the transform is inverted, one finds that k&K(f)has the same functional form as eq 2.29 but with y replaced by y' where y' = yae2e@u(a)/a2

where y = 0.5772156... and T = D t / a 2 . The approximate expression (2.23d) reproduces the first two terms in both the shortand the long-time expansions and is accurate to 1.3% for all times.29 For d = 3 , one has

+ (za2/D)II2)

(2.24)

+ (a2/rDt)1/2)

(2.25)

ks(z) = ( 4 r D a / z ) ( l

ks(t) = 4rDa(l

For an arbitrary potential, U(r), it is not possible to solve eq 2.2b analytically. However, a number of terms in boih the long-time ( z 0) and short-time ( z m) expansions of ks can be found. Pedersen and have shown that for d = 3

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lim &z) = ( 4 r D a e / z ) ( l 2-0

-

+ ( z a 2 / D ) 1 / 2+ ...)

(2.26)

where the effective radius is given by

Using their work it can also be shown that lim Ls(z) = 1-"

(2.28) where the prime indicates a derivative. Note that when U = 0 both eq 2.26 and 2.28 reduce to the free diffusion result given in eq 2.24. Note also that the forms of eq 2.26 and 2.24 are identical. Thus, the long-time behavior of ks(t) in the presence of a potential is given by the free diffusion result (eq 2.25) with a replaced by a,.

When the radiation boundary condition is used, LCK(z)can be obtained from ks(z) by using eq 2.8. Recall that this is true for arbitrary d and U as long as the problem is (hyper)spherically symmetric. Here we shall examine only some useful approximations for k C K ( t )for d = ?. Using the two terms in the small z (long time) expansion of k s ( z ) given in eq 2.26 and inverting the transform, one has

(2.29) where (28) Carslaw, H. S.; Jeager, J. C. Conduction of Heat in Solids, 2nd ed.; Oxford University Press: New York, 1959; Section 13.5. (29) Szabo,A.; Cope,D. K.; Tallman, D. E.; Kovach, P. M.; Wightman, R. M.J . Electroanal. Chem. 1987, 21 7 , 4 11. (30) Pedersen, J. B.; Sibani, P. J. Chem. Phys. 1981, 75, 5368. (31) Sibani, P.; Pedersen, J. B. Phys. Reu. Lett. 1983, 51, 148.

(2.32)

By use of a different approach, this result has been obtained by Weller4 and subsequently by F l a n n e r ~ . The ~ ~ above argument clearly shows that the price one pays for the improved short-time behavior is that k&K(t), in contrast to kCK(t),does not approach the t = m limit correctly (Le., it is worse at moderately long times). 4 . Steady-State Rate Constant. In the conventional formulation, the steady-state rate constant, denoted by k,, is obtained from the long-time limit of the time-dependent rate constant k , = lim k ( t ) (2.33) f-"

For d = 3 , U = 0, and ko = m, this leads to the classic result k, = 4rDa (see eq 2.25). For d = 1 and d = 2, this procedure leads . is to the conclusion that k,, = 0 (see eq 2.21 and 2 . 2 3 ~ ) This commonly taken as an indication that the formalism is not applicable to these dimensions. However, there is an alternate, more physical, way of calculating k, that not only leads to well-defined results for d = 1 , 2 but also predicts the dependence of k,, on concentration (Le., c) in three dimensions. Suppose that A*'s are added to the system a t rate K per unit volume. Since the A*'s are being removed by the trapping reaction, the A* concentration will eventually reach a steady-state value denoted by [A*],,. The steady-state rate constant is obtained from this concentration by using k , = K/c[A*lss (2.34) We now calculate [ A * ] , within the Smoluchowski framework. The survival probability of an A* surrounded by an equilibrium distribution of B s is the solution of eq 2.1 with the initial conditions S(0) = 1 : S ( t ) = e x p ( - c l f k ( r ) d7)

(2.35)

This quantity can be viewed as a response function. If A*'s are added to the system at rate K ( t ) in such a way that each newly added A* is initially surrounded by an equilibrium distribution of B's, [A*(t)]is given by the convolution of the rate of production and the response function [ A * ( t ) ]= L ' K ( T ) S(t--7) d7

(2.36)

It is interesting to note that this relation cannot be derived by simply adding K ( t ) to the right-hand side of eq 2.1. However, it can be obtained by adding K ( t ) / V " to the many-particle evolution equation (2.11) and carrying out the subsequent analysis. m, and using Setting K ( t ) = K = constant in eq 2.36, letting t the resulting steady-state concentration in eq 2.34, one finally has

-

= (cJmexp(-cJ'k(T)

d7) dt)-'

(32) Flannery, M. R. Phys. Reu. 1982, A25, 3403.

(2.37)

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6933

Diffusion-Influenced Fluorescence Quenching Note that when k ( t ) = k = constant, k, = k . In other cases, k , will be a function of c. Equation 2.37 reduces to eq 2.33 in the limit c 0. Applications of the above expression will be given in section IV. 5. Fluorescence Quenching. To describe a fluorescence quenching experiment, one must include the possibility that A* can decay not only nonradiatively via the trapping process but also radiatively. Since these processes are independent

-

[A*(t)]/[A*(O)] = exp[-(t/ro

+ cJ‘k(r)

dr)]

(2.38)

where we have assumed that A* has a single radiative lifetime, T,,. (Multiple lifetimes can also be easily handled.) Since the intensity of fluorescence is proportional to the concentration of excited states, Z(t) following a delta function excitation at t = 0 is I ( t ) / Z ( o ) = exp[-(t/ro

+ cJ‘k(r)

dr)]

(2.39)

In a steady-state fluorescence quenching experiment, the sample is illuminated by a constant light source. With commonly used sources, only a tiny fraction of the available ground-state molecules (the A’s) are excited. The chance that an A, which has recently been produced by quenching, is reexcited is very small, and thus, to a good approximation, the distribution of B’s around a newly formed A* is the same as the equilibrium distribution of B’s around A. If it is further assumed that the A-B and A*-B potentials of mean force are the same (or if the relaxation is instantaneous), the steady-state concentration of A* is [A*], = K I m e x p [ - ( t / r o

+ c J f k ( s ) d r ) ] dt

(2.40)

where K is the rate at which excited states are produced. The ratio of the steady-state intensity in the absence (lo)and the presence (Z) of quencher is then given by

zo/z

=

T O [Jmexp[-(t/rp

+ cJ‘k(r)

= 1 + cL(ro-I)

dr)] d t ] ’

+ ...

where f&) is the Laplace transform of the time-dependent rate constant calculated by using the perfectly absorbing or Smoluchowski boundary condition at contact. Using the above expressions, one can immediately obtain the fin$ result of a mean field calculation for any problem for which ks(z) is known. In treating steady-state fluorescence quenching, Cukier11J2*21 focuses on the dependence of the fluorophore lifetime on quencher concentration. Since the decay of the fluorescence intensity in the presence of quencher is in general not exponential, the emphasis should be on the steady-state intensity. His formalism appears to be equivalent to the nonlinear Stern-Volmer equation

zo/z

(2.4 1b)

and the plot is linear in c. In other cases, the plot is nonlinear. The above analysis goes back to the early work of Noyes3 and Weller! Wilemski and Fixmanlo have discussed the complications that arise when reexcitation of recently quenched excited states is significant. 111. Mean Field Approaches

The mean field approach of Cukier1I*l2to the calculation of the concentration dependence of the steady-state rate constant assumes that the deviation from the bulk value of the pair distribution function of B’s around a fixed A* satisfies DV.e-8UVe@USp= k,,cSp (3.1) This equation is solved subject to either the Smoluchowski or Collins-Kimball boundary conditions, and the steady-rate constant is equated to the flux at contact. Since Sp is a function of k,,, this procedure leads to an implicit equation for k , that is characteristic of mean field theories. Note that eq 3.1 is formally identical with the Laplace transform of the Smoluchowski equation when z = k,,c. If one compares the mean field calculation and the calculation of Laplace transform of the time-dependent rate constant, it immediately follows that the mean field self-consistency condition can be expressed as k, = [ z m z = k 4 (3.2a) or equivalently (3.2b)

=1

+

k,,ToC

(3.4)

where k, is the concentration-dependent steady-state rate constant discussed above. However, this relation is not correct, even within the mean field framework, because it does not consider the influence of the intrinsic lifetime, r0, of A* on Sp. In other words, since A* can decay by two distinct pathways, eq 3.1 should contain an additional term, r0-’Sp. When this is done, eq 3.4 changes to where k‘, =

[~~(~)l,=v,c+ro-~

-

(3.6)

Thus, k’, becomes a function not only of c but also of r0. In the limit c 0, eq 3.5 and 3.6 reduce to

zo/z

(2.41a)

The plot of Zo/Z vs c is called the Stern-Volmer plot. When k ( t ) = k = constant Zo/Z = 1 + kT0C (2.42)

cR(k,c) = 1

where L(z) is the Laplace transform of k ( t ) calculated by using the formalism discussed in the previous section. The above relations are valid for arbitrary d and U(r) and for both the perfectly absorbing and radiation boundary conditions. For example, in the case of the latter, using eq 2.8, eq 3.2a becomes

= 1

+ C L ( T 0 - I ) + ...

(3.7)

which is identical with the order shown with eq 2.41b obtained by using the Smoluchowski approach. The above mean field theory for k, is closely related to the effective mean lifetime approach of Gosele! Specifically, in three dimensions, his results are equivalent to those obtained from the first iteration of the mean field self-consistency equations (3.2a) and (3.6) (i.e., by setting k, = k’, = k ( t = m ) on the right-hand side). The mean field approach is also related to the statistical nonequilibrium thermodynamic theory of K e i ~ e r . ’ ~In - ~fact, ~ he discusses1Sa simple example involving eq 3.1 in order to motivate his more elaborate approach. The basic idea of Keizer’s theory is to first calculate the steady-state radial distribution function using his formulation of statistical nonequilibrium thermodynamics. The resulting distribution contains the yet unknown steady-state rate constant. Finally, k, is expressed in terms of this distribution, leading to an implicit equation for k,. A particularly attractive feature of his formalism is that it can naturally handle reactions with complex stoichiometry. Results obtained by using this theory for the simple reaction schemes considered in this paper will be discussed in the following sections. 1. Relation to the Smoluchowski Approach. We now consider the relationship between the Smoluchowski (based on eq 2.37) and the mean field (based on eq 3.2a) approaches to the calculation of k,. We will show that, in a sense discussed below, the latter can be regarded as an approximation to the former. We begin by considering the integral

which equals (kJI (see eq 2.37) in the Smoluchowski approach. Let us set k ( r ) = k* ( k ( 7 ) - k*) = k* + A k ( r ) where k* is an, as yet unspecified, constant chosen so as to make Ak(7) “smaller” than k ( r ) . For example, when d = 3 k* could be the t = limit of k ( t ) . We then manipulate J as follows:

+

Q)

6934

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 (3.9a) = c&me-ck“(l - c&‘Ak(T)dT

= (k*)-I(l - cA,$(k*c)

+ ...)

+ ...)

Szabo TABLE I: Comparison of k,/2zD in Two Dimensions, Calculated Using a Variety of Approaches, as a Function of the “Area Fraction”,

m

(3.9b)

(2d)’I2

Smoluchowski

mean field

asymptotic

MFPT

(3.9c)

10-4 10-3

0.095 0.12 0.18 0.33 0.44 0.78 1.51 3.67

0.095 0.12 0.18 0.33 0.46 0.85 1.77 4.91

0.107 0.14 0.21 0.41 0.60 1.24 8.63

0.114 0.15 0.24 0.52 0.80 2.13 14.67

= (k*)-I( 1 + cAi(k*c))-’

(3.9d)

= (k*ci(k*c))-’

(3.9e)

On going from eq 3 . 9 ~to eq 3.9d, we assumed that the two terms in eq 3.9clfrom a-geometric series. In the final step, we used_the identity Ak(z) = k(z) - k*/z. How should k* be chosen? If Ak(z) were zero, J would equal (k*)-’. Therefore, a reasonable choice is the value of k* for which eq 3.9e equals (k*)-l, i.e. k* = k*c,$(k*c) (3.10) Since J = (kJ1 by definition and is approximately equal to (k*)-l by the above argument, it follows that k, E k* where k* satisfies eq 3.10. But eq 3.10 is identical with the self-consistency condition of the mean field theory (see eq 3.2). Therefore, to the extent that the above manipulations are valid, the two steady-state rate constants are the same. The approximations made in the evaluation of the integral in eq 3.8 are expected to be valid when (1) c is small and (2) k(t) approaches a constant as t m (so that it makes sense to add and subtract k*). The latter condition is clearly violated in one dimension (see eq 2.21) and, indeed, k,, values calculated by the two different approaches are not the same 0 limit (see below). even in the c The above analysis shows that one can “derive” the principal result of the mean field theory by approximately evaluating the integral that determines k, in the Smoluchowski approach. Should one regard the mean field k g F as an approximation to the Smoluchowski approach, k:A? There are a number of arguments that can be made to support this view. First, when DA. = 0 and the B’s are noninteracting, kA : is exact. Second, since only the relative diffusion coefficient and A*-B potential of mean force enter the mean field theory, this theory does not contain any of the ingredients that would be needed to improve the Smoluchowski approach. Third, in one dimension, where the approximations used to get mean field theory from the Smoluchowski approach are the most dubious, kzA is more accurate that kgF (see below). Finally, we note if the analysis following eq 3.8 is repeated for the integral

-

-

that determines the steady-state fluorescence intensity (see eq 2,41a), one obtains eq 3.5 and 3.6 of the mean field approach. IV. Steady-State Rate Constants In this section we compare selected results for k, obtained using the Smoluchowski, mean field, and some other approaches for one-, two-, and three-dimensional systems. 1 . One Dimension. The A*’s and B’s are assumed to be noninteracting point particles. The B’s may pass through each other. Reaction occurs whenever an A* and a B come in contact. (a) Smoluchowski Approach. Using eq 2.21 in eq 2.37 and evaluating the integrals, one has

10-2 0.1 0.2 0.5 1.o 2.0

(c) Exact Results. For static quenchers (Le., DB = 0), the problem is exactly s0lvable2~and the exact time-dependent survival probability is IDX eXp(-(T2DAd2t/X2))

sinh (x)

dx

(4.3)

Using this result in eq 2.37 and evaluating the integral, one has k,, = 2 D ~ d

(4.4)

When both DBand DA. are nonzero, a highly accurate expression for k, can be obtained by using the leading correction to the Smoluchowski survival probability determined by Szabo, Zwanzig, and A g m ~ n .In~ this ~ way, one finds ~ ( D A . DB)C k, (4.5) ~ ( +1 0 . 2 3 2 ~+~0.034~‘)

+

where I* = DA*/(DA*+ DB)

(4.6)

The accuracy of the above expression improves as p decreases. For p = 1, it overestimates the exact result in eq 4.4 by 0.5%. For DA. = DB, it predicts that k,, = 2.40(DAt + DB)c (4.7) The error in the numerical coefficient in eq 4.7 is expected to be less than fO.O1. From the comparison of the above results one may conclude that (1) for static quenchers the Smoluchowski result is considerably better that the mean field result (25% vs 100%); (2) when both species are mobile, the accuracy of the Smoluchowski result improves dramatically, while the mean field result remains poor (for DA. = DB, the errors are 6% and 67%, respectively); and (3) when DA. = 0, the Smoluchowski result is exact and the mean field prediction errs by 57%. 2. Two Dimensions. We assume that U(r) = 0 and that the reaction occurs with unit probability at contact. The generalization to incorporate partial absorption at contact using the radiation boundary condition is straightforward. (a) Smoluchowski Approach. The steady-state rate constant as a function of quencher concentration c can be found by using eq 2.37 and 2.23. The results shown in the first column of Table I were obtained from the approximate expression for ks(t) given in eq 2.23d. James Kiefer has shown that numerical integration using the exact expression given in eq 2.23a gives essentially identical results. The “area fraction” is defined as

k,, = ( ~ J ~ e x p ( - ( 1 6 D c ~ ? / a ) ~dt)-’ /~)

4 = Ta2c

= ( ~ / T ) D c 2.55(DA* + DB)C

where a is the sum of the radii of A* and B. The Smoluchowski k,, is exact when DA. = 0 and when the radius of B is zero. In this case, 4 is obviously not the physical area fraction of quenchers. Indeed, the k,, found by using the Smoluchowski approach is well-defined for 4 > 1. 4 becomes the area fraction in the opposite limit where the radius of A* is zero (a is now the radius of B). In this case, one of the fundamental assumptions of the theory, namely, that the B’s are not interacting, is violated and the Smoluchowski k,, is accurate only for small 4. ( b ) Mean Field Approaches. If one uses eq 2.22 in eq 3.2a, the mean field self-consistency condition becomes

(4.1)

This result is exact when DA* = 0. ( b ) Mean Field Approaches. Using eq 2.20 in eq 3.2b and solving for k,,, one has k,, = ~ ( D A-k* DB)c (4.2) In this case, the statistical nonequilibrium thermodynamic theory of Keizer yields the identical e x p r e ~ s i o n . ~ ~ (33) Peacock-Lopez, E.; Keizer, J. J . Chem. Phys. 1988, 88, 1997.

(4.8)

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6935

Diffusion-Influenced Fluorescence Quenching

where erfc is the complementary error function and the “volume fraction” 4 is

4 = 4ra3c/3

The values of k,, found by iteratively solving this equation are given in the second column of Table I. It can be seen that the mean field and Smoluchowski results are equivalent over a fairly large concentration range. (For 4 = 0.02 and 0.5 the differences are 4% and 17%, respectively.) In this range, eq 4.9 in essence (almost) analytically evaluates the very complicated multiple integral that arises in the Smoluchowski formulation. This is in contrast to the situation in one dimension and shows that the approximations, used in section 111.1 in “deriving” the mean field self-consistency relation from the Smoluchowski approach, are reasonable in two dimensions. When 4 0, since K , ( x ) x-I and Ko(x) -(y + In (x/2)) as x 0, eq 4.9 becomes

(4.15)

where a is the sum of the radii of A* and B (a = aA* + aB). As in the two-dimensional case discussed above, eq 4.14 IS exact when DA. = 0 and aB = 0. Richards,37 based on a rather involved analysis, recently obtained this result for the case of static overlappingtraps (quenchers) having spheres of influence of radius a. Since this is equivalent to the case where DB = 0, a~ = 0, and aA* = a, eq 4.14 is not exact for this problem; however, it is a good approximation. The most demanding context for the Smoluchowski approach is the one where DB = 0 and a = aB (i.e., static quenchers with excluded-volume interactions) since both of the fundamental assumptions used in its derivation are invalid. This problem has been tackled by many approaches of varying rigor and sophis(4.10) k,/2rD r (!I2 In 2 - y - !I2In 4)-’ tication.1622 All of these predict that, to the lowest order in the ( l (34)II2 + ...). Thus, even in concentration, k,, = 4 i f D ~ a ~ + as found previously by Keizer.” The results obtained by using this case, the Smoluchowski approach correctly gives the first two this expression are shown in column three of Table I. It can be terms in the concentration expansion of k,,. Recall that in one seen that eq 4.10 is a useful approximation only for very small dimension the Smoluchowski approach predicted an incorrect, quencher concentrations. albeit a reasonable, coefficient of the leading (and only) term in For the reaction scheme considered in this paper, the statistical the concentration expansion of k,. Thus, this approach improves nonequilibrium thermodynamic theory of Keizer predicts that34 as the dimensionality increases. We note in passing that as the dimensionality of the system (4.1 1) increases beyond three, k,, becomes progressively less sensitive to c. For example, using eq 2.19 it can be shown that, in d = 4 Note that, in contrast to one dimension, eq 4.9 and 4.1 1 are not and 5, the correction factors in the 4 0 limit are ( 1 - 24 In identical except at low concentrations where it is permissible to 24 ...) and (1 54 + ...), respectively. replace K l ( x ) in eq 4.9 by its small x limit (Le., x-l). As 4 When the potential of mean force is nonzero, the principal increases, the Keizer k,,)s are larger than the mean field or obstacle in the calculation of k, is the determination of ks(t). For Smoluchowski ones. For static A*’s and noninteracting quenchers, low concentrations, a useful approximation of k, can be obtained this difference is a reflection of the inaccuracy of eq 4.1 1 since, by noting that in this limit the value of the integral in eq 2.37 in this case, the Smoluchowski result is exact and so is the mean depends on the long-time behavior of ks(t). As discussed in section field result for a reasonably wide range of 4. 11.3, this is given by the free diffusion result in eq 2.25 with a ( c ) Mean First Passage Time Approach. Berg and P ~ r c e l l ~ ~ replaced by a, as defined in eq 2.27. Thus, k,, may be approxattempted to get around the problem that the traditional procedure imated by using eq 4.14 and 4.15 with a replaced by a,. In the for obtaining the Smoluchowski k, (Le., by solving the steady-state static quencher limit, Cukier,21using a sophisticated argument, diffusion equation) does not work in two dimensions, by using the showed that the above procedure gives the correct result to the concept of mean first passage times36 (MFPT). The basic idea lowest nontrivial order (Le., (34~,)’/~).As a simple example, is to calculate the mean lifetime of a single particle that is initially suppose that the B’s are hard spheres with radius aBand aA* = uniformly distributed between an absorbing boundary at r = a 0. It can be shown that, to first order in c$~, a, = aB( 1 + ’/24e( 1 and a reflecting one at r = R . The outer radius is chosen so that - 3 / 4 In 3)). Since the potential of mean force between hard on the average only one quencher is found in the region. This spheres is attractive, a, > aB. When (bB = 0.2, the use of the above amounts to choosing R so that a, in eq 4.14 and 4.15 predicts that k,/4aDaB is 17%larger than the result obtained by simply using a = aB. nR2c = 1 (4.12) To generalize these results to partial reactivity at contact within The steady-state rate constant is approximated as the reciprocal the framework of the radiation boundary condition, one uses eq 2.29 in eq 2.37. The integral over T can be done analytically, but of the product of c and the single particle mean lifetime. This procedure gives the integral over t must be done numerically. The resulting k,, is approximate when U # 0. For low quencher concentrations, 4(1 - 4) -k,,- the long-time behavior of kCK(t)dominates, so one should use y (4.13) 27rD (4 - 3)(1 - 4) - 2 In 4 given by (2.30). This procedure yields

-

--

-

+

Numerical values of k, obtained by using this equation are shown in the last column of Table I. It can be seen that this approach gives only qualitative results even when 4 is relatively small (e.g., when (24)’12 = 0.2 or 4 = 0.02, it errs by almost a factor of 2). 3. Three Dimensions. (a) Smoluchowski Approach. We begin by considering the case where U = 0 and ko = m . Using eq 2.25 in eq 2.37 and evaluating the integrals, one has

-

+

k,, = 4aDa,’(l

+ (341)1/2 + ...)

where (4.17) and

4; = 47r(a,’)3c/3 = (1 - (34)’12 exp(34/7r) erfc ( ( 3 4 / ~ ) l / ~ ) ) - l = 1

+ (34)’12 + 3(1 - 2 / ~ ) 4+ ...

(4.14)

(4.18)

Equation 4.16 is exact for arbitrary U to the order given and has been obtained by G o ~ e l eCukier,” ,~ and Felderhof and Deutch17 (U = 0). As 4 increases, the short-time behavior of kCK(t)becomes increasingly important. Thus, one should use y’ given in eq 2.32 for high concentrations. The range of validity of these approx-

(34) Kcizcr, J. Personal communication. (35) Berg, H. C.; Purcell, E. M. Biophys. J . 1977, 20, 193. (36) Szabo, A.; Schulten, K.; Schulten, Z. J. Chem. Phys. 1980, 72, 4350.

(4.16)

~~~~~~

(37) Richards, P. M . J . Chem. Phys. 1986, 85, 3520

Szabo

6936 The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 4nDa(l + (a2/Ds0)1/2)k0r0c I_ O -1+ + ... I ko 4nDa(l (u~/DT~)'/~)

imations remains to be investigated. ( b ) Mean Field Approaches. For U = 0 and ko finite, using eq 2.24 in the self-consistency relation (3.3), one has

+ (k,ca2/D)'I2) = ko + 4nDa(l + (k,ca2/D)'/2) ko4xDa( 1

k,, When ko =

a, this

(4.19)

equation is easily solved to give

kJ4nDa

= 1

+ 3/4 + (34 + y442)1/2

ko4nDa exp(k,ca2/D)'i2

For low concentrations, the exponential in eq 4.21 may be replaced by the first two terms of its Taylor series expansion, and eq 4.21 reduces to eq 4.19. Thus, at low concentrations all three approaches give the same kss. As the concentration increases, Keizer's k, grows faster than either the Smoluchowski or mean field ones. This is the case even when DAW= 0 and the B's are noninteracting point particles (Le., when the Smoluchowski result is exact for all c ) . V. Fluorescence Quenching The calculation of the ratio of the steady-state fluorescence intensity in the absence and presence of quenchers (Io/l) is similar to the calculation of k,, for the reaction A* B A + B with the added feature that A* can decay via an independent process (i.e., A* has a radiative lifetime so). We consider only d = 3, 2 since these results are the most relevant to experiment (e.g., membranes can be regarded as pseudo-two-dimensional systems). 1. Three Dimensions. ( a ) Smoluchowski Approach. When U = 0 and ko = a,I o / I can be obtained by using eq 2.25 in eq 2.41a. Evaluating the integrals, one has 1 4nDas0c 10

+

_ -I

= 1

For future reference, we note that as D reduces to I o / I = 1 k070C

+

-

+

+ 4nDaroc(l + (34 + U ~ / D T ~ )+' /3(1 ~ - 2 / n ) 4 + ...)

where (5.2) and 4 is given by eq 4.15. This result appears to have been first derived by Weller.4 When ko is finite, kCK(t)is given by eq 2.29 with a, = a, but the integral over t in eq 2.41a must be done numerically. However, the initial slope of the Stern-Volmer plot can be found analytically. Using eq 2.24 and 2.8 (with U = 0) in eq 2.41b, one has to linear order in c

m

(5.3)

the above equation

+ ...

(5.4)

0 one finds (5.5)

independent of ko. When U # 0, useful approximations of I o / I can be found by following the procedures suggested in section IV.3a for k, (e.g., for small c and long so replace a, 4 by a,, 4, in eq 5.1; for small c and long T~ [large c and short so] use ~ ( e 2.30) q [~'(eq 2.32)] in eq 2.29). ( b ) Mean-Field Approaches. As discussed in section 111, for arbitrary U and ko I o / I = 1 + k',TOC (5.6) where

In general, f&) can only be determined numerically. However, just as in the case of k, (section IV.3b), reasonable approximaGons can be obtained by using the various limiting expressions for k,(z) (eq 2.26, 2.28, and 2.31) in the appropriate concentration and lifetime regimes. For U = 0, all these expressions are exact, and eq 5.7 becomes 4nDa(l k'ss =

+ (a2(k',c + ro-')/D)'/2)ko

ko + 4nDa(l

+ (az(k:,c + T ~ - ' ) / D ) ' / ~ ) (5.8)

Solving this equation for k:, when ko = result into eq 5.6, one finds Io/Z = 1 + 4nDa(l

a,and

substituting the

+ 3/4 + (34 + y442+ U ~ / D T ~ ) ' / ~ ) T ~ C (5.9)

which should be compared with the corresponding Smoluchowski result in eq 5.1. In both theories, Io/I is a function of just two dimensionless variables: 4 and a2/DT0. As was the case for the steady-state rate constant, there is good agreement between the two results over a wide range of these variables. For example, for 4 = 0.1 and a2/Dro= 0.1, 1, 10 the differences are 2.5%,0.8%, and 0.1%, respectively; for 4 = 0.5 the corresponding differences are 11%, 8%, and 2%, respectively. Finally, we note that, for small c, k', (see eq 5.8) approaches ko and 0 as D approaches and 0, respectively. Thus, both the mean field and Smoluchowski results for Io/I behave identically in these limits (see eq 5.4 and 5.5). For U = 0, ko finite, the statistical nonequilibrium thermodynamic theory of Keizer predicts that I o / I is given by eq 5.6 with k:, determined by k'ss =

I - (3+)1/2 exp(3+/n) erfc ( 3 $ / ~ ) ' / ~

-

-

I o / I = 1 + 4 ~ a ~ ( D s ~ )+' /...~ ---* c 1

(4.20)

(4.21)

ko + 4nDa exp(k,,ca2/D)'/2

+

On the other hand, when D

which should be compared to eq 4.14 obtained by using the Smoluchowski approach. The range of 4 over which these two results agree is somewhat larger than that found in two dimensions (e.g., for 4 = 0.02 and 0.5 the differences are 3% and 12%, respectively). This is to be expected since the approximations used in section 111.1 to obtain the mean field self-consistency condition from eq 2.37 improve as the dimensionality increases. When ko is finite, the solution of eq 4.19 is expected to agree with the Smoluchowski k,, obtained by numerical integration, over an even larger concentration range. To find k, when U # 0, one needs z&z), which in general must be determined numerically. Just as in the Smoluchowski approach discussed above, one can obtain useful approximations by n9ting that when c is small (large) one should use an expression for ks(z) that is accurate when z is small (large). Thus, when c is small, medium, and large, one should use eq 2.26, 2.3 1, and 2.28 for k&) in the self-consistency condition (3.3). It would be interesting to test this procedure against the numerical results for k, obtained by Yang and Cukier12for the Coulomb potential. For U = 0 and finite ko, the statistical nonequilibrium thermodynamic theory of Keizer gives34 ks, =

+

4nDa exp((a2(k',c

+ s;1)/D)'/2)ko (5.10) + T~-')/D)'/~)

ko + 4nDa exp((a2(k',c

This expression is considerably simpler than the one in his paperI4 on the origin of positive curvature in Stern-Volmer plots, because it was implicitly assumed there that a quencher can only quench 0 n d 5 (Le., the reaction scheme A* + B A C was assumed). We note that if one replaces exp(x) by 1 x, eq 5.10 reduces to the mean field self-consistency relation (5.8). This is formally analogous to the correspondence between eq 4.19 and 4.21 for the steady-state rate constant. However, there is a crucial difference. For sufficiently low c, exp(x) in eq 4.21 is well-approximated by 1 + x, and thus all three (Smoluchowski, mean field, statistical nonequilibrium thermodynamic) approaches agree in this limit. Because of the presence of so-' in eq 5.10, this is no longer the case here even as c 0. To shed some light on

-+ +

-

Diffusion-Influenced Fluorescence Quenching

- -+ -

- -

this, consider the D m and 0 limits. When D =, k’, ko, so that Io/Z 1 ko7g, in agreement with eq 5.4. On the other 0 eq 5.10 predicts that k:, again approaches hand, when D ko so that Io/I is the same in both limits. This disagrees with the prediction of both the Smoluchowski (see eq 5.5) and mean field approaches. Instead of trying to resolve this matter on the basis of physical intuition, consider the following back-of-the-envelope calculation. Let 7 be the nonradiative lifetime of A* in the presence of a single quencher in a small volume u around it. As always, T~ is the radiative lifetime. When D m, the fluorescence intensity is a single exponential

-

I ( t ) 0: e-flroe-(n)t/r

(5.11)

where ( n ) is the average number of quenchers in u. Since ( n ) = uc 10 U _ - 1 + -7oc

(5.12) I T As mentioned previously, ko of radiation boundary condition fame, 0, 7 0. Thus, can be interpreted as the limit of u / 7 as u all four approaches agree in this limit. When D 0, the A* population divides into two noninterconverting groups: those that do not have a quencher in u and those that do. At sufficiently low c, we need only consider the possibility that a single quencher is in u. The probability of this is proportional to uc. Thus, as D

- --

The Journal of Physical Chemistry, Vol. 93, No. 19, 1989 6937 A simple example of this formalism is given below. (d)Configuration-Dependent Nonradiative Lifetimes. One of the important problems in bridging the gap between theory and experiment is to develop a model that describes both time (or frequency)-resolved and steady-state fluorescence using the same set of parameters. In an insightful paper, Andre, Niclause, and Ware39 focused on a key ingredient any such model should have. The basic physical idea is that those newly created A*’s that have a quencher in close proximity (but not necessarily at contact) will quench almost instantaneously and thus are not detectable in a time-resolved experiment with finite time resolution. However, the presence of such a subpopulation clearly increases the ratio Io/I measured in a steady-state experiment. Any model in which the lifetime of an A*-B pair depends on the interparticle separation over a finite range (e.g., exponentially) can predict this effect. As discussed above, it is simple to incorporate a position-dependent nonradiative lifetime 7 ( r ) (K(r)-l)into the Smoluchowski approach. As an example suppose that 7 ( r ) = T when a I r I R and zero otherwise. ( a and R are the physical contact and “cutoff“ distances, respectively.) The limiting case of this model in which 7 is zero has been considered by Andre et al.39 Assuming that the boundary at r = a is reflecting and U = 0, it can be shown that the Laplace transform of the time-dependent rate constant is

-0,c-0

where u

since an A* with a quencher nearby can decay both radiatively and nonradiatively. Thus 1 uc ut70 IO - 1 U C T / ( T T ~ )= I + - 7 7 0 + ... (5.14) I

_-

+

+

+

- -

+

which in the limit u 0, T 0 , u / T = ko goes to 1, rather than 1 + k07$, in agreement with the prediction of the Smoluchowski and mean field approaches. We do not understand the underlying reason for this deficiency of the statistical nonequilibrium thermodynamic approach in the small D limit. ( c ) Wilemski-Fixman Approach. The classic paper of Wilemski and Fixmanlo on the general theory of diffusion-controlled reactions has many different facets. It advocates a many-particle viewpoint. It discusses how to model reaction by incorporating a sink term into the evolution equation. It shows how to handle the reexcitation problem in fluorescence. To obtain a computationally viable scheme, they resort to a “closure” approximation. Unfortunately, by use of this approximation some of the attractive many-particle features of the formalism are lost. In the present context, this is manifested by the fact the Stern-Volmer plot is predicted to be linear for all concentration^.^^ We have seen in section 11.2 how the conventional Smoluchowski formalism can be simply derived starting with a many-particle description. Aside from the question of how the reaction is modeled (i.e., boundary condition or sink term), eq 2.9 is the common starting point. It just turns out that the approximations used to obtain the Smoluchowski formalism are less severe than the closure approximation A B reaction. for the A* + B As mentioned previously, the sink term idea is readily incorporated into the Smoluchowski approach by adding -K(r)p to the right-hand side of eq 2.2a, as can be demonstrated by adding -&(r‘i-r’A.)P’ to eq 2.9 and repeating the subsequent analysis. If one assumes that A* and B behave like two hard spheres at contact (Le., use a reflecting boundary condition), then eq 2.4b for k ( t ) becomes

-

+

K ( z ) = 47DR

= 4?r(R3- a3)/3

eX cosh X

- (e - X2aR/e) sinh X

aX cosh X

+ t sinh X

with

X = (t2(1 + T Z ) / D T ) ’ / ~ t=R-a

(5.17)

1

(5.18)

(5.19) (5.20)

and &(z,R) is given by eq 2.24 with a replaced by R (i.e., it is the Laplace transform of the free diffusion time-dependent rate constant with an absorbing boundary at R). In the limit that T 0 eq 5.16 reduces to the result found by Andre et al.39 In the limit that u 0 and 7 0 so that v / r = ko it reduces to t t e result obtained by using the radiation boundary condition (Le., kCK(z)). We now examine the consequences of eq 5.16 for steady-state and time-resolved experiments. The low quencher concentration