J . Phys. Chem. 1986, 90, 4020-4026
4020
sum of the diffusion coefficients of the ions.* On the basis of this equation, we introduced the following model function:
OD(t) = P I
+ P2(P3+ t ) - ] i 2
(4)
where P I corresponds to the yield of free ions; P2 is proportional to the sum of the diffusion coefficients of the cation and anion; P3 is a small correction for the difference between the apparent t = 0 and true t = 0. The result of the curve fitting in a propionitrile solution is shown in Figure 5. The decay curve obtained by the nonlinear least-squares method reproduced satisfactorily the observed result in propionitrile solution. Similar result was obtained also for n-butyronitrile solution. Moreover, the ratio of parameters, P 2 / P 1 ,which is equal to rc/(rD)112was constant, regardless of the excitation intensity. We can evaluate the diffusion coefficient of ions from the ratio of parameters, P 2 / P 1 ,and the values obtained in this way for propionitrile and n-butyronitrile solutions were about [m2/s], which is almost five orders of magnitude smaller than the usual values. It may not be reasonable to assume that the small diffusion coefficient is due to the extraordinarily small mobility of both cation and anion. It appears that the recombination does not occur immediately when cation and anion encounter each other. The partially diffusion-controlled recombination was treated theoretically by Tachiya and Sano, and their results show that the free ion yield becomes high in this case.* Our results show, however, that the free ion yield must be low since no slow rise of the photocurrent which corresponds to the decay of the cation absorption band was observed in propionitrile as well as n-butyronitrile solutions, although the ion pair yield is rather high in these solvents. This discrepancy suggests that not only the recornbination (E) Sano, H.; Tachiya, M. J . Chern. Phys. 1979, 71, 1276.
but also the dissociation is slow. The dissociation of the ion pair may be difficult because the cation seems to be surrounded by very stable anions similar to the acetonitrile polymer anion observed in the system of TMB in acetonitrile.’ In a previous paper? we have demonstrated that the acetonitrile dimer anion, which shows an absorption band in 500-700-nm region, reacts with acetonitrile to form the acetonitrile polymer anion which shows no absorption band in the visible region. A very similar process seems to take place also in the present system. On the other hand, in alcoholic solutions a geminate recombination type of decay was not observed and the ion pairs seem to decay exponentially and give free ions efficiently. This difference may be. due to the quite different structure and the stability of anion part in the ion pair. The ion pairs studied in the present work have very long lifetimes and show a geminate recombination type of decay in spite of the short Onsager length. In order ‘to elucidate the dynamic behaviors of these ion pairs, a new theory which treats in full not only the slow recombination but also the slow dissociation processes seems necessary. Acknowledgment. The present work was partly supported by Grants-in-Aid (No. 5843003, No. 59045097) from the Japanese Ministry of Education, Science and Culture to N.M. and (No. 58740205) to Y.H. and also by a Grant-in-Aid from Yamada Science Foundation to N.M. Registry No. TMB, 366-29-0; acetonitrile, 75-05-8; propionitrile, 107-12-0; n-butyronitriie, 109-74-0;isobutyronitrile,78-82-0; methanol, 67-56-1; ethanol, 64-17-5; 1-propanol, 71-23-8; 2-propanol, 67-63-0; 1-butanol,71-36-3; isooctane, 540-84-1. (9) Hirata, Y.; Mataga, N.; Sakata, Y.; Misumi, S. J . Phys. Chem. 1983, 87, 1493.
Simulation of Fluorescence Energy Transfer Stuart A. Allison Department of Chemistry, Georgia State University, Atlanta, Georgia 30303 (Received: January 21, 1986)
A simulation method is developed for studying diffusion-enhancedintermolecular fluorescence energy transfer between spherical molecules. It is sufficiently general to model effects of hydrodynamic interaction, arbitrary intermolecular potentials of mean force, and different transfer mechanisms. Simulation results for a variety of different model systems are presented. In general, hydrodynamic interaction reduces the efficiency of energy transfer ( E ) . Direct forces also influence E . This is true, under certain conditions at least, for short-range forces in addition to long-range forces. Diffusion serves to enhance the effects of direct forces.
1. Introduction The role of diffusion in fluorescence energy transfer has been recognized for many years. If signifcant diffusion of donor and/or acceptor occurs during the lifetime of the excited-state donor species, the efficiency of energy transfer is Physically, a donor and acceptor pair initially at large separation may subsequently diffuse toward each other, thereby increasing substantially the probability of energy transfer. Yguerabide and c o - w o r k e r ~studied ~ . ~ the effect of diffusion on quenching using (1) Kilin, S. F.; Mikhelashvili, M. S.; Rozman, I. M. Opt. Spectrosc. (Engl. Transl.) 1964, 16, 516-517. (2) Mullin, C . R.; Dillon, M. A,; Burton, M. J . Chem. Phys. 1964, 40, 3053-3058. (3) Elkana, Y.; Feitelson, J.; Katchalski, E. J . Chem. Phys. 1968, 48, 2399-2404. (4) Yguerabide, J.; Dillon, M. A,; Burton, M. J . Chem. Phys. 1964, 40,
3040-3052.
0022-3654/86/2090-4020$01.50/0
the Smoluchowski model for diffusion-controlled reactions. A general theoretical analysis by Steinberg and Katchalski6 included the decay of fluorescent donors by the Forster mechanism’ of nonradiative energy transfer to acceptors. In an experimental study of energy transfer between chelated Tb(II1) (donor) and rhodamine B (acceptor), Thomas et a1.* obtained excellent agreement with the theory of Steinberg and Katchalski over a broad range of solvent viscosities. A limiting case which has received considerable attention is the “rapid diffusion limit”, where the lifetime of the donor is sufficiently long that relative diffusion ( 5 ) Yguerabide, J. J . Chem. Phys. 1967, 47, 3049-3061. (6) Steinberg, I. Z.; Katchalski, E. J. Chem. Phys. 1968, 48, 2404-2410. (7) Fbrster, T. Ann. Phys. (Ldpzig) 1948, 2, 55-75. (8) Thomas, D. D.; Carlsen, W.F.; Stryer, L. Proc. Natl. Acad. Sci. U.S.A. 1978, 75, 5146-5150.
(9) Stryer, L.; Thomas, D. D.; Meares, C. F. Annu. Reu. Biophys. Bioeng. 1982, 11, 203-222.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4021
Simulation of Fluorescence Energy Transfer over distances comparable to or greater than the mean acceptor-donor separation can occur before the donor decays. Since the transfer efficiency is largely determined by the distance of closest approach in this limit, the method has proven to be useful in studies of chromophore accessibility and electrostatic interactions in biopolymers like myoglobinI0 and DNA.” The objectives of this work are (i) to develop a simulation algorithm for studying quenching and energy transfer between spherical molecules and (ii) to study what effect direct forces and hydrodynamic interactions have on fluorescence transfer efficiencies. Except for the rapid diffusion limit, these interactions have been ignored in previous treatments. The theoretical basis for the present algorithm involves a straightforward extension of ref 6. In order to obtain the transfer efficiency, one needs to know the conditional probability that an acceptor, initially at a specified distance, r, from the donor, has “relaxed” the donor at a subsequent time, t. In the past, this conditional probability was solved by analytical or numerical solution of the appropriate time-dependent diffusion e q ~ a t i o n . ~ #In~ this - * work, the conditional probability is obtained by carrying out a simulation consisting of a large number of Brownian dynamics trajectories for the appropriate model system. The outline of this paper is as follows. In the next section, the relevant theory of energy transfer is briefly reviewed followed by descriptions of Brownian dynamics and the decay mechanisms used. The simulation method is described in detail in section 3. In section 4, the simulation method is applied to simple test cases which are compared with analytic results. Effects of direct electrostatic forces, hydrodynamic interaction, and solvent caging are examined in section 5. Section 6 summarizes the main points of this paper and discusses briefly future work. 2. Theory Assume a donor molecule is excited at time t = 0. It can subsequently decay either radiatively with assumed first-order lifetime, r, or nonradiatively by some other process such as quenching or Forster energy transfer to a neighboring acceptor molecule. Let p(t) denote the probability that a donor molecule, excited a t t = 0, remains in the excited state at time t. The experimentally measurable “efficiency of energy transfer”, E, is directly related to p(t) by3
where qA’ and qA represent the quantum yield of donor in the presence and absence of acceptor (quencher), respectively. In the absence of acceptor, p(t) = exp(-t/r) so E = 0 as expected. Following Steinberg and Katchalski,6 it can be shown that where CAis the ambient acceptor concentration. For spherical donors and acceptors which interact via a centrosymmetric potential of mean force, V(r),and which contain chromophores at their respective centers, y(t) is given by y(t) = ;,XR4dg(r)
G(r,t) dr
(3)
the entire system (assumed spherical), g ( r ) = exp[-V(r)/kBT] is the radial distribution function of the acceptor-donor pair a t separation r, kB is Boltzmann’s constant, T i s the absolute temperature, and G(r,t) is the conditional probability that an acceptor, initially at r relative to a particular donor, has “reacted” with that donor by time t. The quantity CAy(t) appearing in eq 2 is identical with eq 10 of ref 6 when g ( r ) is set equal to one. This will only be true if the potential of mean force is negligible, which was the case of primary interest in ref 6. In the past, G(r,t) has been solved by analytical or numerical solution of diffusion equations which contain appropriate reaction terms for quenching4 or Forster energy transfer.6,8 Two integrations as prescribed by eq 1-3 then yield E. In this work, G(r,t) is obtained by carrying out a large number of Brownian dynamics trajectories. The relative motion of a donor-acceptor pair is obtained by using the Brownian dynamics algorithm of Ermak and McCammon.I2 This algorithm has been successfully used to study transport properties of macromolecules,13internal motions in polymer~,’~J~ and steady-state diffusion-controlled reactions.lbls It has recently been generalized to account for rotation and rotation-translation coupling of the spherical subunits which comprise the model system.lg For two spheres, the relative equation of motion is given by12J6 At r = ro + -(Omo - 2D12).F‘ + SI - S2 kBT
(5)
where r is the position vector of sphere 1 in a reference frame with origin at the center of sphere 2, At is the dynamics time step, 0, is the relative diffusion tensor between spheres i andj, Dm0= D l ld D220is the mutual diffusion tensor, F is the direct force acting on sphere 1, and the superscript 0 denotes the value of the quantity just prior to a dynamics step when r = ro. The S, quantities are vectors of Gaussian random numbers with ( S I ) = 0 and variance-covariance
+
(SISI)= 2 DllAt
(6)
They represent the stochastic displacements of the spheres due to solvent collisions, and their construction is described elsehere.'^^'^ The time step, At, must be chosen sufficiently small so that (a) position-dependent quantities do not change significantly, (b) the probability of a reaction is much less than unity, and (c) distances diffused are small compared to distances from absorbing and reflecting boundaries. In some of the simulations, hydrodynamic interaction between the spheres is ignored. In this case, Dl: = ZDl= Z7cBT/6r7alwhere l i s the 3 X 3 identity matrix, 7 is the solvent viscosity, and a, is the radius of sphere i. In simulations where hydrodynamic interaction is included, the Oseen tensorZo with stick boundary conditions is used. The self-diffusion tensors, D,:, are the same as before, but
where rurij denotes the 3 X 3 position dyad corresponding to the displacement vector between subunits i and j . Wolynes and
where 4 S = - r ( R 3 - a3) 3
(4)
S‘ = X R 4 a 3 g ( r ) d r To derive this result, it is necessary to assume a random distribution of donors. In eq 3, a is the distance of closest approach between donor and acceptor, R is the radius of the vessel containing (10) Wensel, T. G.; Meares, C. F. Biochemistry 1983, 22, 6247-6254. (1 1) Wensel, T. G.; Chang, C.-H.; Meares,C. F. Biochemistry 1985, 24, 3060-3069.
(12) Ermak, D. L.; McCammon, J. A. J . Chem. Phys. 1978, 69, 1352-1360. (13) Allison, S.A.; McCammon, J. A. Biopolymers 1984, 23, 167-187. (14) Pear,M.R.; McCammon, J. A. J. Chem. Phys. 1981, 74,6922-6925. (15)-Allison, S. A.; McCammon, J. A. Biopolymers 1984, 23, 363-375. (16) Northrup, S.H.; Allison, S. A.; McCammon, J. A. J . Chem. Phys. 1984.80, 1517-1524. (17) Allison, S. A.; Srinivasan, N.; McCammon, J. A.; Northrup, S. H. J . Phys. Chem. 1984,88, 6152-6157. (18) Allison, S. A.; McCammon, J. A. J . Phys. Chem. 1985, 89, 1072-1074. (19) Dickinson, E.; Allison, S. A.; McCammon, J. A. J . Chem. SOC., Faraday Trans. 2 1985,81, 591-601. (20) Oseen, C. W. In Mathematick and ihre Anwendaugen in Monographien and Lehrbilchern; Hilb, E., Ed.; Akademische Verlagsgesellschaft: Leipzig, 1927; Vol. 1 .
4022 The Journal of Physical Chemistry, Vol. 90, No. 17, 1986
DeutchZ1have pointed out that this tensor overestimates hydrodynamic interaction. Consequently, simulations which include it should be regarded as upper bounds on the effect of hydrodynamic interaction. In this work, three reaction mechanisms are considered. In the case of quenching (section 4), a reaction occurs if and only if the two spheres come to within a distance a of each other. In other words, a spherical, perfectly absorbing boundary is placed at r = a.14*16 Although it is not possible in this case to strictly enforce condition c on the time step as discussed previously, no measurable error is introduced provided At is kept small when r = a.16 For the remaining two reaction mechanisms studied, the probability of a reaction in At can be written p* = k(r)At. For Forster energy transfer, the rate of reaction, k(r), is given
where 7 is the lifetime of the donor in the absence of acceptor and Ro is the distance at which the nonradiative Forster decay rate equals the radiative decay rate, 1/7. Independently, Ro can be determined from the spectral properties of donor and acceptor chromophores as described e l ~ e w h e r e . ~In- ~addition to Forster energy transfer, the mechanism of exchange interaction may be important when the acceptor and donor are within about 10 A of each In this case k(r) = c exp(-2r/d)
(9)
where c is a constant and d corresponds to an effective Bohr radius which is assumed to be 1 A in this work. If both Forster and exchange mechanisms are operative, k(r) is given by the sum of eq 8 and 9. 3. Details of the Simulation In order to determine E and y(t), it is necessary to first know G(r,t). This can be obtained by carrying out a large number of Brownian dynamics trajectories with r initially chosen between a and infinity. Since beyond a certain r there is negligible probability that an acceptor will react with the donor before it decays radiatively, acceptor-donor pairs initially separated beyond some cutoff distance, q, can be ignored. The cutoff distance most frequently used in this work is q = (9D,tm,,
+- 4 R 0 ~ ) ' ~ 2
(10)
+
where Dm = D, D2 is the mutual diffusion constant and t,,, is the maximum time a particular trajectory is allowed to continue. For times greater than several T , the probability that a donor remains in the excited state is small (eq 2). A trajectory can therefore be terminated at sufficiently long time without introducing significant error in E . In this work, t,,, = 5 7 . The initial position of a trajectory, ri, is selected at random from an r-weighted initial distribution ri2 = X ( q 2 - aZ)
+ a*
(1 1)
where Xis a uniformly distributed random number between 0 and 1. For the quenching model discussed in the next section, G(r,t) approaches a/r at long time. Hence, selecting 6 from an rweighted distribution will yield approximately the same number of successful reactions at all ri. Although this is not strictly true for short times, other reaction mechanisms, or models with interparticle forces, this approach was found to yield more accurate and reproducible results than other distributions for a fixed number of trajectories ( 104-105). Since G(r,t) is a conditional probability, however, any initial distribution should be satisfactory provided enough trajectories are carried out. (21) Wolynes, P. G.; Deutch, J. M. J. Chem. Phys. 1976, 65, 450-454, 2030-203 1 . (22) Dexter, D. L. J . Chem. Phys. 1953, 21, 836-850. (23) Meares, C. F.; Yeh, S. M.; Stryer, L.J. Am. Chem. Soc. 1981, 103, 1607-1 609.
Allison As observed p r e v i ~ u s l y , considerable ~~J~ savings in computer time results if a variable time step i s used. On the other hand, At must be selected sufficiently short to satisfy the criteria discussed in the previous section. For the quenching model, the scheme described in ref 16 is used. For simulations in which the donor decayed by Forster energy transfer or exchange interaction, At is assigned as follows: At = q2/240D,; =
(9- a2 + 2)/240D,; = 0.01/20,;
a
r q
>q
> r > a + 10
(12)
+ 10 > r > a
In the event r becomes less than a during a particular dynamics step, that dynamics step is disregarded and the trajectory continued. In other words, acceptor and donor cannot approach closer than a. A particular trajectory is terminated when one of three conditions is met: t exceeds t,,, r becomes sufficiently large so that there is negligible chance of reaction in the time remaining, or a reaction occurs. The second condition is dealt with by terminating the trajectory if r2 > 18Dm(t,,, - t ) + 4R02. In the case of dipolar energy transfer and/or exchange, p* is calculated after each dynamics along with a random number, X, uniformly distributed between 0 and 1. Ifp* exceeds X , a reaction has occurred. In order to construct G(r,t), r and t are divided into equally spaced intervals or "bins" (NR, NT). At the beginning of each trajectory, a single bit is added to array element N(NR) initially set to zero a t the start of the simulation. When the simulation ends, N(NR) represents the number of trajectories initiated in r-bin NR. Another array NG(NR,NT) is incremented, in part, when a reaction occurs. If a reaction occurs in time interval NT1, a single bit is added to all N G array elements provided N T is greater than or equal to NT1. At the end of a simulation, G is approximated G(NR,NT) = NG(NR,NT)/N(NR)
(13)
Because of the manner in which initial positions are chosen, N(NR) could equal zero for certain r-bins. In this case, we set G(NR,NT) = G(NR-1,NT). Because of the large number of trajectories, this procedure was rarely necessary. Numerical integration is then used to obtain y ( t ) (eq 3) and E (eq 1). In order to evaluate the reproducibility of a simulation, each is broken down into four equivalent but independent subsimulations. Overall averages and standard deviations ( u ) are computed from the subsimulation results. If one uses the central limit theorem,24standard deviations for the entire simulation are taken to be u/2. Error bars or uncertainties reported correspond to these standard deviations. 4. Test Cases
The Smoluchowski modelz5for quenching provides a good test for the simulation method since relevant quantities are known analytically. In this model, the two spheres react if and only if they approach to within distance a of each other. Also, there are no interparticle direct forces. The conditional probability is given by4 U
G(r,t) = -[1 - erf ((r - ~ ) / ( 4 D , t ) l / ~ ) ] r
(14)
where erf is the error function.26 Substituting this into eq 3 yields y ( t ) = 4naD,t
+ 8n2n(D,t)1/2
(15)
which is valid provided R >> (4D,t)'12. Results for y ( t ) are (24) Hogg, R. V.; Tanis, E. A. In Probability and Statistical Inference; Macmillan: New York, 1977. (25) Smoluchowski, M. V. Phys. Z . 1916, 17, 557-585. (26) Abramowitz, M.; Stegun, I. A. In Handbook of Mathematical Functions; US.Government Printing Office: Washington, DC, 1964; Natl. Bur. Stand. Appl. Math. Ser. No. 55.
Simulation of Fluorescence Energy Transfer
6
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4023
1
4-
2-
100
200
300
400
500
t (nr) Figure 1. y ( t ) for Smoluchowski model. No direct forces are present. Simulation results are denoted by filled/empty circles without/with hy-
drodynamic interaction. Analytic (X) values were computed from eq 15. TABLE I: Efficiencies of Energy Transfer for Quenching (Smbluchowski) Model CA,M E= E* EC
0.0002 0.0004 0.0006 0.0010 0.0020
0.117 (0.006) 0.210 (0.011) 0.285 (0.015) 0.398 (0.021) 0.568 (0.028)
0.077 (0.005) 0.143 (0.009) 0.200 (0.012) 0.294 (0.016) 0.451 (0.022)
From simulations, no hydrodynamic interaction. tions, with hydrodynamic interaction. CAnalytical.
Figure 2. Stern-Volmer plot for Smoluchowski model without ( 0 )and with (0) hydrodynamic interaction.
0.121 0.217 0.292 0.409 0.584
From simula-
shown in Figure 1 for two identical spheres. Parameters of the model are ai = a2 = 3 A, T = 293 K, 7 = 1.0 cP, a = 6 A, and 0, = 1.428 X lW5 cm2/s. Crosses denote the theoretical estimates of eq 15, and filled circles denote simulation results (lo5 trajectories). Agreement is seen to be very good. The solid line in the figure represents the approximation y ( t ) = 4uaD,t, and this shows that the t1l2term contributes little to y ( t ) . The empty circles are simulation results in which hydrodynamic interaction is included. To the best of our knowledge, y ( t ) has not been solved analytically for this case although Deutch and Felderhofp’ have obtained steady-state diffusionantrolled rate constants for this model. The corresponding results for E are summarized in Table I. The lifetime of the donor, T , was taken to be 100 ns. The third column in this table was calculated by substituting eq 15 into eq 2 followed by numerical integration of eq 1. The simulated results (column 1) agree with the analytic results to within one standard deviation. Data of this form are frequently presented in the form of Stern-Volmer pl0ts.2~ Strictly speaking, a linear plot should only result if y ( t ) = kefft for which E’
1
+ (CAkeffT)-’
(16)
Nonetheless, Figure 2 shows nearly linear Stern-Volmer behavior for the data in Table I. For the case without hydrodynamic interaction, we observed previously that the approximation y ( t ) = 4xD,,,ut works quite well. If we simply assume that y ( t ) = ked = 4uD,Refft and extract Rcff from the Stern-Volmer plots in Figure 2, we obtain Reff = 6.17 and 3.86 A without and with hydrodynamic interaction, respectively. For this particular problem, hydrodynamic interaction reduces Reff(or ken) by about 40%. The cutoff distance used in these simulations was q = (9Dmt,)i/2. When a substantially smaller q was used, simulated y(t)’s fell below the analytic values and the relative difference increased with increasing 1. For q = (3Dmt,)’/2, ysh(f)/yanal(t) = 0.94 and 0.61 at 100 and 500 ns, respectively. This discrepancy (27)Deutch, J. M.;Feldcrhof, B. U. J. Chem. Phys. i973.59.1669-1671. (28)Lakowicz, J. R. In Principles of Fluorescence Spectroscopy; Plenum: New York, 1983. (29)McQuarrie,D.A.In Stutisrical Mechunics;Harper and Row: New York, 1911.
reflects the reaction of donors with acceptors initially separated by distances greater than q. The criteria for defining q (eq 10) was established on the basis of the above results. For Forster energy transfer between spheres, numerical values €or E, but not y ( f ) , are available in the absence of direct forces and/or hydrodynamic intera~tion.’.~In order to compare the Brownian simulation method with the finite-differences a simulation was carried out with a = 5 A, R = 50 A, 0, = lo4 cm2/s, and T = 1000 ns. At CA= 10-4 M, the simulation yielded E = 0.185 f 0.007 in excellent agreement with Thomas et a1.* 5. Results and Discussion Interparticle forces between an ionic donor-acceptor pair are expected to have a substantial effect on E. If one uses the primitive model with Debye-Hiickel screening for finite ions, the potential of mean force is given by
where q, is the charge on ion i in units of the protonic charge, e = 4.8 X esu, e is the solvent dielectric constant, and K is the Debye screening length. For water at 293 K, t = 78 and K = 0.3331 (in units of A-’). 1is the ionic strength in units of molarity (M). In Brownian dynamics, the force, F,appearing in eq 5 is just -V V ( r ) ,where V is the gradient operator. Simulation results of Forster energy transfer with a = 6 A, & = 27 A, T = 100 ns, D, = 1.42 X cm2/s, and I = lo4 M are summarized in Table 11. Parallel simulations of 50000 trajectories with and without hydrodynamic interaction were carried out for each charge model. Quantities in parentheses denote uncertainties of selected data points. Stern-Volmer plots of these results are nearly linear and yield effective reaction radii (Reff)which are listed at the bottom of the table. re^ is defined in the same way as in the preceding section. Increasing the charge on the donor from 0 to +2 when the charge on the acceptor remains -1 leads to a doubling in the effective rate of energy transfer for this system. Hydrodynamic interaction reduces the effective rate as expected, but the effect is somewhat smaller than was observed in the quenching model. Since a reaction occurs at large acceptor-donor separations in the present model, hydrodynamic repulsions are reduced. Unlike the quenching model, energy transfer occurs over a range of distances. Since & (which equals 27 A in this case) represents the distance at which energy transfer equals the radiative decay rate of the donor, mast energy transfer would be expected to m u r at distances equal to or smaller than &. Figure 3 shows how the number of energy transfers varies with distance. Each data point represents the total number of transfers that occurred within a
4024
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986
Allison
TABLE 11: Effects of Charge and Hydrodynamic Interaction for Forster Energy TransfeP E
a
q, = -1, q2 = +2
q, = -1, 92 = +1
q, = -1,q2 = 0
Ca. M
no hi.
hi.
no h i .
hi.
no h.i.
hi.
0.0002 0.0004 0.0006 0.0008 0.0010 0.00 12 0.0014 0.0016 0.0018 0.0020
0.165 (0.005) 0.284 0.374 0.444 0.501 0.547 0.585 0.618 0.646 0.670 (0.006)
0.144 (0.007) 0.254 0.339 0.407 0.463 0.508 0.547 0.580 0.608 0.633 (0.019)
0.210 (0.006) 0.349 0.446 0.519 0.575 0.619 0.656 0.685 0.711 0.732 (0.019)
0.196 (0.008) 0.331 0.428 0.502 0.560 0.606 0.644 0.676 0.703 0.725 (0.019)
0.288 (0.01 1) 0.450 0.554 0.625 0.678 0.7 18 0.749 0.775 0.796 0.813 (0.007)
0.237 (0.007) 0.388 0.491 0.566 0.622 0.665 0.700 0.729 0.752 0.772 (0.009)
Rdf
9.19
7.80
12.40
11.42
18.95
14.49
h i . = hydrodynamic interaction.
TABLE III: Effects of Solvent Model and Transfer Mechanism for Attractive Ions ( I = E
Forster
M) Forster + exchange Drimitive Patev
exchange
10’CA, M
primitive
Patev
primitive
PateY
1.oo 1.11 1.25 1.43 1.67 2.00 2.50 3.33 5 .OO 10.00
0.094 (0.005) 0.103 0.1 15 0.129 0.147 0.172 (0.010) 0.206 0.257 0.340 0.503 (0.036)
0.176 (0.010) 0.192 0.21 1 0.235 0.265 0.303 (0.014) 0.354 0.425 0.530 0.702 (0.015)
0.217 (0.003) 0.235 0.257 0.284 0.317 0.358 (0.005) 0.41 1 0.483 0.585 0.741 (0.015)
0.269 (0.023) 0.291 0.317 0.349 0.387 0.434 (028) 0.494 0.573 0.680 0.827 (0.025)
I .
0.216 (0.013) 0.234 0.257 0.288 0.316 0.358 (0.019) 0.41 2 0.484 0.588 0.747 (0.022)
0.280 (0.004) 0.304 0.331 0.364 0.404 0.454 (0.005) 0.517 0.599 0.708 0.850 (0.01 1)
1-A interval out of a total of 50 000 trajectories. Note that most transfers occur a t distances much less than Ro.Because of the high transfer probabilities at small r, the exchange m e c h a n i ~ m ~ ~ . ~ ~ may be an important factor when significant diffusion can occur 1501 before decay of the donor. We shall return to this point later. Also, the number of transfers increases substantially as the attractive forces between donor and acceptor increase. Attractive forces coupled with diffusion evidently serve to bring the reactive species together, thereby increasing the likelihood of energy transfer. Except for a dielectric constant different from unity, eq 17 ignores the influence of solvent on the potential of mean force. An interesting question is whether or not energy transfer is sensitive to solvation effects. More specifically, under what conditions does solvation affect fluorescence? Patey and co-workersM predicted substantial effects of solvation on N M R relaxation times of attractive ions in water. NMR experiments apparently confirm their prediction^.^^*^' To account for solvation effects on energy transfer, a potential of mean force derived by Patey and *workers was used (Figure 2A of ref 30). Parameters held constant 6 10 15 20 throughout were a, = a2 = 2.8 A, a = 5.6 A, a, = +2, q2 = -1, R (1) 7 = 0.890 cP, T = 298 K, and D, = 1.75 X cm*/s. Figure 4 shows the potential of mean force at zero salt. Figure 3. Number of reactions vs. distance for Forster energy transfer In order to account for ionic screening, we simply assumed ( R , = 27 A, a = 6 A, T = 100 ns). Circles correspond to q , = -1; q2 = 0, and squares to q1 = -1, q2 = +2. Filled/empty systems correspond to simulations without/with hydrodynamic interaction.
where the p subscript refers to the primitive potentials given by eq 17. Provided the ionic strength is low, eq 18 is expected to be reasonably accurate. Simulations of both primitive and “Patey” water models with Ro = 20 A and T = 10, 100, and 1000 ns yielded E’s that were nearly indistinguishable. Only when 7 = lo00 ns was a noticeable difference (about 10%) observed. However, when Rowas reduced (30) Fries, P. H.; Patey, G. N. J. Chem. Phys. 1984, 80, 6253-6266, 6267-6213. (31) Fries, P. H.; Jagannathan, N. R.; Herring, F. G.; Patey, G. N. J . Phys. Chem. 1985, 89, 1413-1416.
.
to 8 A, significant differences became apparent, and the results are summarized in Table 111. In these simulations, T = 10 ns, Z = IO4 M,loo00 trajectories were carried out in each case, and hydrodynamic interaction was ignored. Whefe exchange interaction was included, the constant c appearing in eq 9 was chosen so that the exchange rate equaled the Forster rate (eq 8) at R = 8 A. These results are interpreted as follows. When Ro is small, a significant fraction of successful transfers occurs when the acceptor and donor are very close, and this is where solvation effects are greatest. Indeed, 34.1% and 82.2% of all reactions with Forster energy transfer only (first and second columns) occurred between
Simulation of Fluorescence Energy Transfer
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 4025 TABLE V Effect of V ( r )on Fonter Energy Transfer for Attractive Ions at I = lo4 M E
103cA, M 1 .oo 1.1 1 1.25 1.43 1.67 2.00 2.50 3.33 5.00 10.00
I
I
a
aid
a+2d
l
r
TABLE IV: Effect of Solvent Model on Forster Energy Transfer for Attractive Ions at I = lo-' M E 1 0 3 ~M ~. primitive Patev 1 .oo 0.038 (0.006) 0.066 (0.01 1) 1.11 0.042 0.073 1.25 0.047 0.082 1.43 0.053 0.092 1.67 0.061 0.106 0.073 (0.01 1 ) 0.125 2.00 0.090 2.50 0.151 3.33 0.116 0.192 5.00 0.165 0.263 0.415 (0.049) 10.00 0.286 (0.035)
r = 5.6 and 6 A for the primitive and Patey models, respectively. When Rois increased to 20 A, these fractions drop to about 25% for the primitive model and 45% for the Patey model. When energy transfer is efficient (large Ro and small T ) , it is not necessary for acceptor and donor to be in close proximity for transfer to occur so short-range solvation effects are unimportant. At contact ( r = 5.6 A) the exchange rate is over 14 times greater than the Forster rate for the above models. This helps explain the larger yields obtained in the simulations where exchange interaction was included. The corresponding results at Z = lo-' M are given in Table IV, and it would appear that solvation effects persist at high salt. However, eq 18 assumes that counterions do not perturb the water structure around a central ion, which undoubtedly is not true at high counterion concentrations. A remaining question concerns how sensitive the fluorescence results are on V(r)within the first hydration layer of the ion (out to about r = 7 A). This question has bearing on the accuracy of the potential of mean force of Patey and c o - w ~ r k e r s .Ber~~ kowitz et al.32have used molecular dynamics to estimate V(r) for the sodium ion-chloride ion system in ST2 water. Despite similarities, Berkowitz et al. observe a barrier for transition to the contact region which is about half that of ref 30. Consequently, V(r)was modified as indicated by the dotted line in Figure 4. The effects of this scaling on E are summarized in Table V (entry I). Unscaled results are listed under entry 111. Entry I1 results were obtained by using a V(r) intermediate between the solid and dotted lines with -V(a)/kBT = 5.5. The main conclusion to be drawn (32) Berkowitz,M.; Karim, 0.A.; McCammon, J. A.; Rossky, P.J. Chem. Phys. Len. 1984, 105, 577-580.
I1
I11
0.106 (0.021) 0.1 16 0.128 0.144 0.164 0.190 (0.037) 0.225 0.278 0.361 0.5'15 (0.076)
0.131 (0.012) 0.143 0.158 0.176 0.199 0.229 (0.020) 0.270 0.327 0.416 0.573 (0.031)
0.176 (0,010) 0.192 0.21 1 0.235 0.265 0.303 (0.013) 0.354 0.425 0.530 0.702 (0.015)
TABLE VI: Comparison of Effective Rates for Short-Lived and Long-Lived Donor Models
a+3d
Figure 4. Potential of mean force between -1 and +2 ions in water. The solid line is from ref 30, and the dashed line represents the corresponding primitive potential (eq 17). The dotted line represents a scaled potential discussed in the text. a = 5.8 8, is the distance of closest approach of the ions, and d = 2.8 8, is the diameter of water.
I
I, M
10-l lo-' lo4
solvent model primitive Patey primitive Patey primitive Patey
transfer mechanism'
f f f f ex ex
kerf, M s-' = 10 ns T = lo6 ns 1.05 X 1O'O 0.277 X lo5 2.20 x 10'0 2.62 x 105 0.395 X 1O'O 0.115 X lo5 0.709 X 10" 1.06 X lo5 2.74 X 1 O 1 O 1.76 X lo5 3.85 X loLo 32.14 X lo5 T
Of = Forster energy transfer, ex = exchange interaction.
from this is that E is very sensitive to V(r) at or near contact. It should be kept in mind that the simulations discussed in the two preceding paragraphs dealt with short-lived donor species ( T = 10 ns). What happens when T is large? In the rapid diffusion limits ~ ( t =)
ked
(19)
where
The units of kcff in eq 20 are (molecules s/crn3)-l. To convert this to (M s)-I, eq 20 must be multiplied by NAv/lOOO. The position-dependent rate constant, k(r), is given by eq 8 for Forster energy transfer and by eq 9 for exchange interaction, respectively. For Forster energy transfer, the dependence of keRon solvation (via the integral over g ( r ) ) is independent of Ro.This situation is different from the T = 10 ns cases studied by simulation. However, the rapid diffusion limit requires T 5 1 ms, which yields a much smaller keffif Ro and r are fixed. For this reason, the rapid diffusion limit is sensitive to the distance of closest app r o a ~ h . ~Hence, . ~ we expect the long-lived donor species to also be sensitive to solvation structure. Using the same model as before, but increasing T from 10 to lo6 ns, we determined kcrrby numerical integration of eq 20. Effective rates are tabulated in Table VI for a number of different cases that were also studied by simulation with T = 10 ns. For the T = 10 ns results, rate constants were obtained from Stern-Volmer plots. It is evident that increasing T increases the sensitivity of fluorescence to solvation effects. In nonaqueous solvents, or aqueous solvents in which the acceptor and donor repel each other, fluorescence energy transfer appears to be fairly insensitive to these solvation effects. Entrapment to form a long-lived acceptor-donor complex, which requires strong attractive forces (i.e., a deep well in V(r)at contact) seems to be a prerequisite. 6. Summary
Electrostatic forces and hydrodynamic interaction have been shown to have a significant effect on fluorescence energy transfer. Also, it would appear that the greater the diffusion rate of the species, the greater the effect of direct forces on transfer efficiency ( E ) . When the rate of energy transfer is low, which for Forster energy transfer can occur if Ro is small and/or T is large, short-range forces can also influence E . This will also be true
4026
J . Phys. Chem. 1986, 90,4026-4032
for exchange interaction. Consequently, it should be possible to study solvation structure around ions by using fluorescence. As pointed out by a reviewer, the centrosymmetric models studied in this work could also be studied by the finite-differences However, for complex model systems that are noncentrosymmetric, the trajectory method must be used. A specific example of such a system is intramolecular energy transfer between acceptors and donors located a t various positions along a linear polymer chain.33 Work on this problem is currently under way and involves straightforward modification of the techniques described above. (33) Haas, E.; Katchalski-Katzir, E.; Steinberg, I. Z . Biopolymers 1978, 17, 11-31.
The amount of computer time required to carry out these trajectory simulations depends, to a large extent, on the specific model under study. Most of the simulations carried out in this work required approximately 2-3 h of computing time on a Univac 1100 computer. When a faster system such as a CYBER 205 supercomputer is used, a reduction in computation time by a factor of 100 can be anticipated. Other things being equal, computation time increases as the dimensionless quantity D , T / s ~increases. (s is the mean distance between donors and acceptors.) The first two paragraphs of section 3 help explain why this is the case. Acknowledgment. This work was supported by a grant from the Research Corporation. S.A.A. is a recipient of a Presidential Young Investigator Award and Camille and Henry Dreyfus Grant for Newly Appointed Faculty in Chemistry.
Molecular Beam Infrared Laser Photodlssoclatlon of van der Waals Molecules Contalnhg SF, T. E. Gough, D. G. Knight, P. A. Rowntree, and G. Scoles* Centre for Molecular Beams and Laser Chemistry,t University of Waterloo, Waterloo, Ontario, Canada N2L 3Gl (Received: February 13, 1986)
Molecular beam infrared photodissociation spectra of (SF& and SF6Ar have been obtained by using line tunable C 0 2 and N20lasers and bolometric beam detection. By comparing spectra at different dilution and source pressures, we show that the (SF& spectra are inhomogeneously broadened. However, saturation experiments show a signal vs. laser power behavior of a pure exponential type. Clearly, more work is needed with continuously tunable lasers or with time-resolved techniques before we can claim to know the photodissociation lifetime of this dimer. The angular distribution of the fragments in the photodissociation of (SF& (SF,),, and (C2H4)2 has also been measured with an angular resolution better than 0.8O. Only in the case of (SF,), do we find that the angular distribution has a maximum, this being located at 1.5". The implications of these results for the dynamics of the dissociation process are discussed.
1. Introduction Photodissociation spectroscopy of van der Waals molecules has been, during the past few years, one of the most lively areas of research in chemical physics.' Interest in this field stems not only from general interest in the process of photodissociation, which has been revived by the recent, massive application of lasers to the study of chemical processes, but also from the relationship between this type of predissociation and the relaxation of the internal degrees of freedom of a molecule in a fluid phase.2 When considering infrared photodissociation, van der Waals molecules have, as a model system for the process, the specific advantage that the interactions between the fragments in both the ground and excited state are, in principle, accessible to our knowledge. This represents a definite advantage since the dynamical theory of the process is difficult and often only approximate solutions are offered which cannot tolerate the added uncertainties due to the incomplete knowledge of the interactions. If the available dynamical theories could be tested with a few systems where the interactions are relatively well-known, their application to the photodissociation of chemically bound molecules could be made with greatly increased confidence and would probably lead to a much improved fundamental understanding and practical utilization of this field. The first molecular beam infrared photodissociation measurements on a van der Waals molecule were carried out several years ago in our laboratory3 using an infrared diode laser and bolometric detection of the decrease in the molecular beam flux of (N20)2 via the photodissociation process. Since then, the technique has been adopted in several laboratories4 and has been 'An interdisciplinary center whose members participate in the GuelphWaterloo Program for Graduate Work in Physics (GWP)2 and the GuelphWaterloo Centre for Graduate Work in Chemistry (GWC)*.
0022-3654/86/2090-4026$01.50/0
used to study a relatively large number of van der Waals systems mainly with the help of color center lasers5 Not long after our 1978 paper several other molecular beam groups reported lowpower infrared photodissociation measurements, which were carried out with the help of mass spectrometer detectors, extending rather rapidly the number of systems studied in this way.' Among this second category of experiments we can find the work of the Nijmegen group, where after an initial effort concentrated mainly on the molecule of interest in this study (i.e., SF6)several other molecular combinations have been studiede6 The obvious advantage of mass spectrometers &e., their mass selectivity) is somewhat compensated by the limited sensitivity and stability that they present when operating near a large flux of molecules which can give rise to mass peaks overlapping those (1) For an overview of the field, good starting points are: Faraday Discuss. Chem. Soc. 1982, No. 73; Janda, K. C., to be submitted for publication in Adv. Chem. Phys. (2) Ewing, G. E. Chem. Phys. 1978, 29, 253; J . Chem. Phys. 1980, 72, 2096; Faraday Discuss. Chem. SOC.1982, No. 73, 325. (3) Gough, T. E.; Miller, R. E.; Scoles, G. J. Chem. Phys. 1978,69, 1588. (4) (a) Brechignac, Ph.; DeBenedictis, S.;Halberstadt, N.; Whitaker, B. J.; Avrillier, S.J. Chem. Phys. 1985,83, 2064. (b) Miller, R. E.;Vohralik, P. F.; Watts, R. 0.J. Chem. Phys. 1984,80,5453. (c) Spector, G. B.;Brady, B. B.; Flynn, G. W. J. Phys. Chem. 1985,89, 1875. ( 5 ) (a) Coker, D. F.;Miller, R. E.; Watts, R. 0.J. Chem. Phys. 1985,82, 3554. (b) Fischer, G.; Miller, R. E.; Vohralik, P. F.;Watts, R. 0.J. Chem. Phys. 198583,1471. (c) Fischer, G.; Miller, R. E.; Watts, R. 0.Chem. Phys. 1983, 80, 147. (6) (a) Geraedts, J.; Setiadi, S.;Stolte, S.; Reuss, J. Chem. Phys. Lett. 1981, 78, 277. (b) Geraedts, J.; Stolte, S.;Reuss, J. 2.Phys. A . 1982, 304, 167. (c) Geraedts, J.; Waayer, M.; Stolte, S.; Reuss, J. Faraday Discuss. Chem. SOC.1982, No. 73, 375. (d) Geraedts, J.; Stolte, S.;Reuss, J. Chem. Phys. Lett. 1983, 97, 152. (e) Geraedts, J. Thesis, Katholieke Universiteit, Nijmegen, 1983. (f) Geraedts, J.; Snels, M. N. N.; Stolte, S.;Reuss, J. Chem. Phys. Leu. 1984, 106, 377. (9) Snels, M.; Geraedts, J.; Stolte, S.; Reuss, J. Chem. Phys. 1985, 94, 1.
0 1986 American Chemical Society
