J. Phys. Chem. 1993, 97, 1823-1832
7823
Luminescence Quenching of a Ru( bpy)32+ Moiety, Covalently Attached to a Polyelectrolyte, by Noncovalently Attached Multivalent Anions Richard E. Sassoont and G. L. Hug' Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 Received: March 22, 1993; In Final Form: May 17, 1993
A ruthenium tris(bipyridine) excited chromophore that was covalently bound to a poly(3,34onene) was quenched by the multivalent anions Ni(CN)42-, Fe(CN)a3-, IrCl&, and PtC1d2-. Various kinetics models were used to fit the luminescence decay traces. The classical kinetics models that were employed did not fit well. The most successful nonclassical model was one that allowed for incoherent transfer via the exchange interaction. In this model the quencher anions were taken to be electrostatically bound to the positively-charged sites of the polyelectrolyte. The polymer backbone and the quencher anions were assumed to not move significantly during the excited-state lifetime. However, the distributionof distances of sites from the excited moieties was accounted for through the use of a density of sites, based on a two-dimensional fractal space of a tetrahedrally-restricted Gaussian linked polymer chain. A more general incoherent-transfer model that allowed for the diffusion of the quencher anions did not significantly change the tits and gave a wide scatter of the fitted diffusion constants. This was taken to show that diffusion was not contributing to the decay.
Introduction
Reactions in restricted geometries' have recently been of wide interest, especially since Mandelbrot published his ideas on the fractal geometry of nature.2 Polyelectrolytes form a class of limiting spaces which are fractals in two dimensions. In addition, their charged backbone produces an intense electric field which has the potential for influencing chemical reactions taking place on or near their surface. One application of this has been to improve conversion and storageof solar energy by the addition of certain polyelectrolytes to appropriate photochemical systems. The electrostatic field of polyelectrolytes may be exploited in order to enhance the yield of charge-separated, photosensitized electron-transfer products and to inhibit their back-reaction. Factors of inhibition of the back-reactionof between 1 and 2 orders of magnitude are usually achieved with the addition of polyelectrolytes to such systems.' However, a factor of more than 5 orders of magnitude has been obtained in a system involving photochemical generation of electron-transferproducts and their stabilization on separate likecharged polyelectrolyte^.^ Basic understanding of the properties of polyelectrolytes in solution is necessary in order to build the optimum photochemical systems for solar energy conversion and storage. One way to monitor the motion and/or configuration of polyelectrolytes is to use luminescence probes to monitor the dynamics of quenchers. In this work, a luminescence probe was covalently bound to the polyelectrolyte backbone, and the quenchers that were employed were of opposite charge to the polyion but were not bound covalently to its backbone. Various dynamicalmodelswere used to fit the time-resolved luminescence decays from classical kinetics of micelles (with two fitting parameter@ to models of diffusion (with four fitting parameters) on fractal structure^.^.^ All of the other models have three fitting parameters, one of which was always taken to be the initial luminescence intensity. The terms "static quenching" and "dynamic quenching" are used in the work in the following sense. Static quenching is that quenching which follows from time-independent effects and is associated in homogeneous media by a lack of relative motion of the chromophore/quencher pairs. In this paper, dynamic + Current addrtss: Science Applications International Corp., 555 Quince Orchard Road, Suite 500, Gaithersburg, MD 20878.
0022-3654/93/2097-7823%04.00/0
quenching is that quenching which follows from time-dependent effects and is not limited to chromophore/quencher pairs with relative motion. In this sense, chromophore/quencher pairs that are fixed relative to one another can be dynamically quenched if it takes time for the quenchingprocess to develop. Mechanisms that show this more general characteristic of dynamic quenching are discussed herein.
Experimental Section The luminescent polyionic probe, abbreviated Poly3,3-Ru( b p ~ ) 3 ~used + , was a Ru(bpy)32+ chromophore covalently linked to a poly(3,34onene) polyelectrolyte. (For structure, see Figure 1.) It was prepared as described el~ewhere.~ Poly3,3-Ru(bpy)s2+ was eluted down an ion exchange column containing amberlite IRA 400 ion exchanger in the chloride form in order to remove any negatively charged impurities and to convert all of the polyelectrolyte to the chlorideform. The polyelectrolyte solution was thendialyzed against pure water using Spectra/Por 3 dialysis tubing (molecular weight cutoff 3500) for 48 h to yield stock solutions. When more concentrated stock solutions were necessary, their volumes were reduced down on a rotary evaporator at 60 OC. The concentrations of the solutions were determined by titration of the ionic chloride content with AgNO3 using potassium chromate as an indicator. The molecular weight of Poly3,3-R~(bpy)3~+was determined to be 9700 k 3000 from viscosity measurements using an Ubbelohde viscometer.*.g K ~ N ~ ( C N ) ~ . Hwas Z O prepared according to the literature methodlo while all the other quenchers were obtained commercially. They were of the highest purity available and were used without further purification. All other compounds used in this study were also of the highest purity grades and were used as received. Absorption measurements on the sample solutions were recorded on a Cary 219 absorption spectrophotometer. All solutions were prepared in deionized water (passed through a Millipore Milli-Q water purification system) and deaerated by vigorous bubbling with prepurified argon gas (Linde) for 15 min in rubber-capped 1.0 X 1.0 cm quartz cells. Time-resolved fluorescence measurements were performed on a computer-controlledflash photolysis system." The laser used for excitation was a Molectron nitrogen laser with a pulse of about 5-ns half-width and a pulse fluence of about 3 mJ/pulse. Excitation/analysis geometry was right angle. The decay traces 0 1993 American Chemical Society
Sassoon and Hug
7824 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 0.1
V'
0.0 -0.1
(CI-1"
Figure 1. Structure of Poly3,3-R~(bpy),~+.
were analyzed according to the various models to be discussed below. The parameters were estimated from a nonlinear leastsquares program12which employed the Marquardt13 algorithm. The weights and goodness of fit, xr2, were computed assuming that the noiseof a particular luminescencetracecould beestimated from the base line before the time of the laser pulse. The average of the noise from these 15-20 time channels (100 total) was taken to be the standard deviation of the Gaussian noise. Some of the fits that are reported herein with relatively large X F Shad good fits on visual inspection of the fitted curves. One source of this discrepancy could be that the small sample of points before the flash gave an inconsistent empirical measure of the Gaussian noise in a decay trace. Steady-state emission measurements were made on the same solutions used in the flash photolysis system using an S.L.M. 8000 photon counting spectrofluorimeter. The emission intensity was determined at a 90° angle to that at which the sample was excited, and all emission spectra were corrected.
Results Solutionsof Poly3,3-R~(bpy)3~+ (3.75 X lC5M) wereexcited by pulses from thenitrogen laser (337 nm). Luminescencetraces were recorded and analyzed with the models that are described below. When no quenchers were present, the luminescencedecay of Poly3,3-Ru(bpy)32+ was exponential with a lifetime of 515 f 7 ns. The time-resolved luminescence decay of Poly3,3-Ru( b ~ y ) ~ 2was + also followed in the presence of four different quenchers, Fe(CN)&, IrCl6>, Ni(CN)d2-, and PtC1d2-. These multiply-charged anions were not covalently bound to the polyelectrolyte. A typical nonexponential decay trace is shown in Figure 2. Four different concentrations of each quencher, e.g. (0.5-2.0) X 1 V M for Fe(CN)a', were employed. The concentrationsof quenchersand Poly3,3-R~(bpy)3~+ were chosen such that cross-linking of the polymer did not occur in solution. In addition, the decay traces, for each quencher and each concentration, were recorded on two different time scales to make sure that the models were sampling all the relevant intervals of the time profiles. Finally, experiments were done with each of the four quenchers at the highest concentrations employed as above but where NaCl was added in six different concentrations. This was to test whether there was a salt effect. In the experiments with the four multiply-charged quenchers, the initial intensity of luminescence was observed to decrease as the concentration of quenchers was increased. For the lowest concentration of quencher to the highest, the intensities of luminescence decreased by up to 50%. A more precise estimate of this decrease can be obtained by noting the variation of the fitted initial intensity of luminescence after the flash, with quencher concentration in Table I or 11. In the discussion that follows, this decrease will be interpreted as the extent of static quenching, namely, the quencher anions in contact with the Ru(bpy)32+site. Furthermore, such a chromophore/quencher pair will be assumed to be totally nonemitting so that the fitting procedures will only deal with the dynamic quenching. This assumption is not necessary in all the models, e.g. the incoherenttransfer model with the exchange interaction.
c,
I .I
I
-.z
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (ps)
Figure 2. Quenching of the luminescenceof Poly3,3-Ru(bpy),2+ by 1.2 X l (r M Ni(CN)d2-in aqueous solution. [Polymer] = 3.75 X 10-5 M. Top panel: difference between the fitted curve (model from q 17,
incoherent transfer with no diffusion)and the experimentaldata. Middle panel: semilog display of decay trace with solid fitted curve. Bottom panel: linear display of decay trace and fitted curve. Parameters from fit: xrz= 1.5,j3 = 6.4nm-I, and r1= 6.9 X 106 s-1.
TABLE I: Poly3,3-Ru( bpy)s*+Luminescence Quenching. Fittings from Classical Kinetics Mocllfied Poissonian Distribution Model [QI I kd time quencher (l(rM) @mV I/az (106s-I) x? range/ps Fe(CN)6& 0.5 11.8 0.062 6.4 2.8 1156 0.5 1.0 1 .o 1.5
1.5
IrCla' Ni(CN)$
PtC14'-
2.0 2.0 1.8 1.8 0.4 0.4 0.8 0.8 1.2 1.2 1.6
10.6 10.0 8.9 8.7 7.3 7.3 5.8 12.8 9.9 11.6 10.2 8.1 6.9 6.6 5.5
5.1
1.6
3.1
3.0 3.0
9.8 8.4
0.068 0.14 0.14 0.21 0.20 0.28 0.30 45 16 0.17 0.17 0.32 0.28 0.42 0.38 0.52 0.48 0.46 0.51
6.6 6.9 5.8 7.6 6.6
8.1 6.2 0.37 0.40 6.7 6.1 8.0 8.0 10 8.9 12 9.9
6.3 5.3
4.2 6.4 7.3
7.1 6.7 9.8 8.4 20 8.7 10 6.1 11 8.5 13 10 14 4.4 21 19
3.91 1.56 3.91 1.56 3.91 1.56 3.91 1.56 3.91 1.56 3.91 1.56 3.91 1.56 3.91 1.56 3.91
1.56 3.91
Adding sodium chloride seemed to reverse, at least partially, the static quenching. When varying amounts of sodium chloride were added to solutions, each of which contained one of the four quenchers, the initial luminescenceintensity generally increased. That of Ni(CN)42- doubled, IrCl63- increased by lo%, and PtCL2increased by 50% on adding 1 M NaCl. In these salf-effect experiments, the concentration of each of the quencher anions was held constant at the highest values used in the above studies of the concentration dependence of the anions themselveson decay rates of the luminescence of the polyelectrolyte. While effects were observed on the initial luminescenceon adding NaCl to the
Luminescence Quenching of a Ru(bpy),2+ Moiety
c.
The Journal of Physical Chemistry, Vol. 97. No. 30, 1993 7825
0.6
\
0
-3 .3
0.4
0.2
V."
1.o
0.0
3.0
2.0
4.0
[Q]/ M x Figure 3. Steady-stateluminescencemeasurements for the quenching of the emission of Poly3,3-Ru(bpy)s2+ by various anionic quenchers. Semilogarithmic plots of the ratio of the intensity of emission in the absenceofquenchertothat in theprtsenceofquencherversus thequencher concentration. Excitation wavelength = 480 nm. Emission wavelength = 620 nm.
TABLE Ik Poly3,3-Ru( b~y)3~+ Luminescence Quenching. Fittings from Classical Dispersive Kinetics Model quencher Ni(CN)42-
IrCl&
(1pL)$/mV 0.4 0.4 0.8 0.8 1.2 1.2 1.6 1.6 0.6 0.6 1.2 1.2 1.8 1.8 2.4 2.4
11.0 10.3 8.0 7.4 6.9 6.2 5.8 4.5 13.0 11.7 12.6 10.9 12.8 10.7 11.6 9.7
(k)/ (lo's-') 6.0f 0.05 2.0f 0.03 8.6f 0.06 3.1 f 0.03 1 1 f 0.03 3.6f 0.03 13 f 0.01 4.1 f 0.08 2.1 f 0.1 0.9f 0.02 4.5 0.06 1.6 i 0.03 5.2 f 0.06 1.8 f 0.03 5.7 0.06 2.2f 0.04
*
*
time
u/(106s-I) X? rangelps 2.7 f 0.01 9.9 1.56 2.6f 0.01 8.3 3.91 3.8 f 0.02 1.7 1.56 3.91 3.5 f 0.01 12.4 1.56 1.2f 0.01 7.3 4.6 f 0.04 9.7 3.91 8.1 f 0.1 5.6 1.56 6.3f 0.1 3.7 3.91 1.56 2.2f 0.01 1.6 2.2 f 0.01 3.1 3.91 3.0 f 0.01 1.2 1.56 2.9 f 0.01 1.5 3.91 3.8 f 0.01 1.5 1.56 3.6 f 0.01 0.9 3.91 4.6 f 0.01 1.6 1.56 3.91 4.4 0.02 1.5
*
reaction solutions, no clear trends were found on applying the various fitting models to the remainder of the luminescencedecays. This observed lack of salt effect is discussed later. Steady-stateexpcriments on the reaction solutions also revealed behavior inconsistent with that expected for dynamic quenching in homogeneous solution. Figure 3, for example, shows semilogarithmic plots of the ratio of the intensity of emission in the absence of quencher to that in the presence of quencher versus the quencher concentration. This ratio is seen to increase in an approximately exponential fashion with increasing quencher concentration, in contrast to a linear fashion expected with SternVolmer behavior.
Discussion Theoretical Models: Classical Kinetics. The first attempt at modeling the nonexponential decays was the classical kinetics model of Tachiya.5 The physical picture behind the model is to consider the excited species to be trapped in relatively small physical volumes or cells corresponding to either the inside of a micelle or the "polymer phase" surrounding a polyion. The quencher species are distributed among the small cells according to Poissonian statistics. These quenchers can either be considered as mobile or immobile relative to the movement between a cell and the intercellular volume. The quenchers inside the cells "react- with the excited species via the law of mass action. When transfer of quenchers into and out of the cells has to be considered,
the differential equations that result are coupled but can be solved exactly by a generating function methodn5 In the case where transfer of quenchers through the cell boundary can be ignored during the lifetime of the excited state, the resulting differential equations are not coupled. The individual solution for n quenchers can be multiplied by the Poissonian probability that a cell has n quenchers, and the sum over all possible quencher occupations of a cell can be made exactly. In addition, a similar generating function approach with a modified distribution of the quenchers was also tried." The physical picture is almost the same as that of the Tachiya model except the distribution of anionic quenchers among the various polyions is modified slightly from the Poissonian distribution. The modification is to account for the electrostatic repulsion of the i - 1 quenching anions already present in the polymer phase of a given polyion to the ith anion being added to the polyion. This modification introduces into the quencher distribution a multiplicative factor,14 for each i, of 1 - (i - l ) a z / l , where a ranges from 0 for purely hydrophobic quencher/polymer interactions to 1 for purely electrostatic interactions, 1 is the number of polyionic charges per polyelectrolyte, and z is the absolute value of the quencher ion's charge. In this model, for quenchers that are restricted to moving in the bound phase of the polyelectrolyte, the luminescence decay traces should foll0wl4
P'(t-tJ =
In eq 1, kq is the first-order rate constant for quenching of one excited probe by one quencher (both probe and quencher associated with the same polyelectrolyte); kfis the luminescence decay rate in the absence of quencher ions; and A, is the average number of quencher molecules per polyelectrolyte. The three times, t, to, and r I , referred to in eq 1 are the time of sampling, the time of the laser flash, and the beginning of the time window of the fitting analysis, respectively. The fitting parameters, A,, were
A, = l / a z
(3)
A, = kq
(4)
Experimental values were used for A, and kf. None of these classical approaches gave fits with reasonable x?. The physical parameters obtained from the decay function in eq 1, which is based on a modified Poissonian distribution with no quenchers entering or leaving the bound phase of the polyions, are given in Table I. The search procedure of the fitting did not convergefor threeoftheconcentrationsofboth IrCla"andPtC42-. The goodness of fit xr2is quite large in all cases represented. In addition, the number of monomers in the polymer, given by 1, is shown to be unreasonably small since a is a parameter that should range between 0 and 1.14 In both the micelle and the modified Poissonian models, the space sampled by the excited state and quenchers was a three-dimensional space. One additional classical kinetics model will be discussed. It has been applied specifically to quenching of excited states on polyelectrolytes. The dispersive kinetics model was originally applied to surface phenomena.15 Wolszczak and Thomas applied it successfully to polyelectrolytes.16 If the luminescence decay
Sassoon and Hug
1826 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 u. I
0.0
.
I UT-
-u.I
1:*; .
-2.0
-3.0 0.0
0.5
,
1.0
,
1.5
,
2.0
,
2.5
,
-2,
3.0
3.5
1 4.0
Time (p) Figure 4. Quenching of the luminescenceof Poly3,3-R~(bpy)3~+ by 3.0 X 10-3 M PtC142- in aqueous solution. [Polymer] = 3.75 X 10-5 M. Top panel: difference between the fitted curve (model from eq 6, dispersive classical kinetics) and the experimental data. Bottom panel: semilog display of decay trace with solid fitted curve. Parameters from fit: xrZ = 3.3, (k) = 2.2 X 107 s-l, and u = 3.5 X 106 s-l. traces followed this model, their functional form would be
where the exponential decay function has been averaged over decay rates from a normal distribution with a central value of (k)and a width of u. As in the original work,l5 the integral in eq 5 was transformed to
lattice. The quenchers were assumed to be immobile on the time scale of the excited state and to be bound to the polyelectrolyte's skeleton by electrostaticforces. In the next section,this restriction will be relaxed so that the assumption will be directly tested. Electrostatic forces also exist between the excited state and the quenchers, but the quenching mechanism is assumed to be an incoherent electron transfer or energy transfer via an exchange interaction. Incoherent, in this context, means that the deactivation effect of the individual quencher proceeds independently of the other quenching species;there is no concerted action of the quenchers on the excited moiety. Multipolemultipole interactions were also tried. A mathematical formulation of the incoherent transfer model described above, can be put into the framework of a master equation. Let there be a set Q of labels defined by
52 = {i1,i2,...,in)
(7)
where then members of the set Q are chosen from the set of labels {1,2, which correspond to the Nsites in a polymer. If there are n quenchers occupying sites labeled by members of the set Q and if there is no back-transfer, then the probability, @(K,,,,,J, that the excitation is on the initial site at time t is given by a master equation,
...,w
where w[R,(t)] is the decay rate of the excitation due to the interaction between the excited site and thejth quencher. The label m enumerates the various quencher configurations of the n quenchers, and the label K,,,,,, stands for the mth configuration of the n quenchers. Equation 8 can be solved by an integrating factor technique to give
(9) where E,(t) is the contribution to the decay of the excited species due to its interaction with the quencher on site j
exP(-(k)tF")I df ( 6 ) In eqs 5 and 6,x and are dummy variables of integration. The integrationsin eq 6 were performed numericallywith the extended Simpson's rule as in ref 15; however, 100 points, as opposed to 10, were used to evaluate the integrals. The results are given in Table 11. is again an overall normalization to the experimental decay trace. From this table it can be seen that about half of the fits, as judged by the X? values, are quite good. One of the poorer fits is illustrated in Figure 4. It was characteristic of most of the fits using this model, even for many of those with X? < 2.4, in that the calculated curve deviated in a positive manner at long times. Theoretical Models: Incoherent Transfer. The failure of the classical kinetics models in three dimensions prompted a search for alternate models among quantum and statistical mechanical theories which have been employed to describe quenching dynamics in reduced dimensions. Although there are many attractive approaches'J'J* to the questions of incoherent energy and electron transfer, with or without motion of the quenchers, the general approach of Blumen19 was chosen. This approach has the features of being exact (even after an ensemble average), of being easily programmable in the exact form, of being extended to moving quenchers, and of there being numerous approximate forms of the general theory includingvarious fractal applications," one of which has been used for describing polymer dynamics.21 The version of the Blumen models that proved most widely applicable will be described first. In order to apply the mathematical models to Poly3,3-Ru(bpy)++, the polyelectrolyte was taken to consist of a nonbranched lattice with a single excited state and a set of n quenchers randomly distributed along that
The explicit time dependence of the vector Rj is included to accommodate motion of the quencher during the lifetime of the excited state. For fixed quenchers, the expression in eq 10 can be integrated to give E,(r) of the form exp[-tw(R,)]. The time dependence of @(Kn,,,,) itself is then also exponential. One of the chief advantages of the Blumen-Manz work is that they were able to do an exact ensemble average of @(K,,J. If there are N total sites, then Blumen and Manz22showed that the average function could be written as N
- P + PE~WI (1 1)
~ ; t )(@(K,,,,;t))= exp(-k+)J-J[l /=l
wherep is the probabilitythat a site will be occupied by a quencher. The ensemble average22 in eq 11 is over all n quenchers and all m configurations of those quenchers, where the quenchers are assumed to be distributed by the binomial distribution
P(K,,,) = p"( 1 - P)~-",m = 1, ...,
N! n!(N- n)!
(12)
Blumen and Manz take the logarithm of eq 11 to give XI
The prime means the excited site is to be excluded; this condition is relaxed in the following for the continuum approximation and the exchange interaction. Since both Ej andp < 1, the logarithm
Luminescence Quenching of a Ru(bpy)32+Moiety
The Journal of Physical Chemistry, Vol. 97, NO. 30, 1993 7827
on the right-hand side of eq 13 can be expanded to give N
-
In @.@;t) = -k$ - x ' F ; [ l /=I
Most of the calculations were done with the exchange interaction,25
k
- exp{-fw(Rj))lk
(14)
=1
In order to get analytical results, Blumen and co-workersusually drop all terms except for k = 1, change the sum over sites into an integral over the space in question, and assume that the interaction is isotropic, i.e. w(R) = w(R). In the current work on Poly3,3-Ru(bpy)++, p sometimes was greater than 0.1, so it was appropriate to keep some higher terms in k to make sure the series was being accurately evaluated. Making the continuum approximation and the isotropic approximation, m
In @(p;t)= -k+ -
-k
- exp{-rw(R)]lkp(R)R2 dR dQ
w ( ~=),-leN-R)
(19) where r1 is the transfer rate when a quencher is located at a distanceR = dfrom theexcitedsiteandpis thecoupling constant for the exchange interaction in units of reciprocal distance. The distance d is usually the intersite distance, but because of the importance of through-bond interactions, it is often taken as the distance along the bonds.26 The latter convention was used in these calculations with d = 0.616 nm. Inserting w(R) for the exchange interaction into eq 18 gives an expression that can be integrated by parts after making the substitution y = d R . The result is
In @(p;t) = -k+ -
(15) where p(R) is the density of sites and dS2 is the angular part of the differential volume element. For a Gaussian-linked polymer, the sites form a fractal of dimension 2. A rationalizationfor this statement and an indication of the density of sites of such an entity are given as follows. An ideal polymer with links of length a has on average a distance squared of (R2) = Nu216 from the center-of-mass to a site N links away (see p 430 of ref 23). This formula is calculated on the basis of a random-walk model of N steps, and such an ideal polymer is said to be Gaussian-linked. If N and (R2) = R2 can be thought of as variables, then ANis proportional to 12RARfa2. In a three-dimensional Euclidian space, AV = 4?rR2AR,and so thedensityofsites A N / A V = 3/(lra2R). Onewaytocharacterize2 the fractal dimension of a space is that its density of points goes as R-" with the fractal dimension being D. This is a heuristic demonstration that a Gaussian-linked polymer is a fractal of two dimensions. A rigorous derivation of the pair correlation function for sites in a Gaussian-linked polymer is given by (see expression in figure on p 36 of ref 24 and the derivation of its Fourier transform on P 261) p(R) = 3/(7ra2R) (16) This is identical to the density of sites from the center-of-mass in the preceding heuristic argument, and p(R) will be taken as the density of sites from the chromophore, which can be at an arbitrary site in the polymer. For a tetrahedrally-restricted Gaussian linked polymer an extra factor of 2 appears in the denominator, because such a restriction will stretch out a polymer by a factor of the square root of 2 compared to a freely-linked Gaussian chain. In addition, for P0ly3,3-Ru(bpy)3~+, the positively charged quarternary ammonium sites are four links apart which introduces another factor of 4 in the denominator. The distance between the chromophore and the first site was approximated as four links also so that a simple density function
p(R) = 3/ (87ra;R) (17) could be used, where a0 was taken to be 0.154 nm, the distance of a C-C bond. For the density in eq 17 and after integrating over angles, eq 15 becomes
The earlier assumptions of quenchers immobilized during the excited-state lifetime and the continuum approximation with the density of sites given by eq 17imply the following physical picture. The individual polymer molecules are assumed to be frozen by the former assumptions, but a full range of polymer conformations are sampled by the latter approximation.
where g2(u) = uSyY,=d(lny)2e-uydy
is one of the In~kuti-Hirayama~'integrals (see eq A2 of ref 28). The two physical quantities that are to be sought in the fitting are p, the reciprocal interaction distance, and 7 , the hopping time to the nearest site. The quantity 0 enters in two places in eq 20. First, it comes as k2in multiplying the various powers of p , which is proportional to the actual concentration of quenchers at sites on the polyelectrolyte. This seems to be the dominant functional dependence which strongly determines 0 in the fitting procedure. Second, @ enters eq 20 as a factor in the argument of an exponential coefficient that scales the time. This seems to be a much weaker determinant of 8. Since this exponential coefficient is multiplying some dependence of the two fitting parameters is introduced. The first functional dependence is so strong that this coupling of the two fitting parameters appears to cause only mild problems. This particular coupling of the fitting parameters can be formally eliminated by choosing d to be zero. This was done as a check on the general procedure. The resultingj3's werethesameaswhend = 0.616nm,and theresulting 4 ' s were equal to the original r1 multiplied by exp(0d). The procedure of setting d = 0 was not done in general because it sometimes led to floating-point overflows in the computer runs. When analytical formulas are wanted, the Inokuti-Hirayama integrals are usually replaced by their asymptotic form (see eq A14 of ref 28)
+,
-
g, (In u ) ' + 1.15443 In u + 1.97811 (22) The parameter u is linearly related to the time. Thus for analyzing decay curves,care must be taken when using the simple asymptotic forms. A valid expansion for small values of u can be found from the functions'
where g2(u) = Gz,l(u). It was found that when u < 5, the asymptotic form began to deviate from the expansion for small u. On the other hand, even though the expansion of G,,(u) is convergent, the algorithm used in this work showed erratic behavior for u > 45. Between 5 < u < 45, the two expansions gave the same results to at least three significant figures. Because of these difficulties the decay curves were analyzed with a switchoverfrom theexpansion of G2.1 to the asymptoticexpansion. Both u = 5 and u = 30 were used as switchover points. As long
7828 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993
c
4.0 3.5 0
0.05
0.10
0.15
P Figure 5. Display of concentration dependence of 8: Best fits of 8 for
incoherent transfer (withoutdiffusion)linearly extrapolated (using linear least squares) to zero concentration of quencher (probability of site occupation p 0).
-
-
TABLE III: Quenching Parameters for Poly3,3-Ru(bpy)P1 [Q] 0 Incoherent-Transfer Model with the Exchange Interaction quencher j3lnm-I l-lJs-1 6.5 f 0.2 (5.5 f 0.6) XIOs Fe(CN)$ (2.7 f 0.9) X lo5 IrC16’ 4.0 f 0.3 5.6 f 0.3 (6.5 i 1.8) X lo6 Ni(CN)4*(5.1 f 0.2) x 105 PtCl424.8 f 0.2 as k = 1 in the expansion of eq 20, very small differences in the resulting fitting parameters were noticed. For the bulk of the computations the switchover was taken as u = 5. The results from this model, of immobile quenchers undergoing incoherent transfer via the exchange interaction, are given in Figure 5 and Table 111. Each point in Figure 5 represents a nonlinear least-squares fit of eq 20 to a specific luminescence decay trace. Of the 32 traces represented, all of the fits had X? < 2.5 except for 6 traces. Twenty-two of the fits had xr2< 2.0. A representative fit is displayed in Figure 2. These fits, as judged by the X? values and visual inspection of the fits, are much better than any of the classical kinetics fits. In addition, the physical parameters show some consistent patterns. If the decay kinetics on the polyelectrolyte obeyed the incoherent-transfer model exactly, then for each quencher both j3 and I-I would be constant for all quencher concentrations. The concentration of the quenchers would be completely accounted for in the calculation of the probability, p , of site occupation. The parameter 0 follows this rule for Fe(CN)63-and IrCls3-. From Figure 5 it can be seen that in contrast to the B’s for Fe(CN)6> and IrC163-, the B’s for Ni(CN)42- and PtC142- increase with an increase in the concentration of the quencher. In addition, there are significant deviations in p when the NaCl concentration is increased. Table I11 gives the j3 and 1-1 parameters which have been linearly extrapolated to zero concentration of quencher. The variation in the physical parameters with concentration was fit to a straight line by a linear least-squares procedure. There is no theoretical justification for choosing a linear as opposed to some other functional form for the concentration dependence of the parameters. A linear form was chosen only for convenience of the presentation. The values for j3 in Table I11 can be compared to ,B’s determined from experiments on the intermolecular electron transfer in bifunctional compounds where the donor and acceptor are coupled together by saturated hydrocarbon structures. Closs et al.29
Sassoon and Hug determined a 0 of 10.1 nm-1 with androstane, cyclohexane, and decalin bridges on the basis of the through-space distance between donor and acceptor. As mentioned by Penfield et aL30 this through-space j3 corresponds to a decrease of a factor of 3.0 per bond; so a through-bond j3 would be about 1.5 nm-l. Penfield et al.30 find the square of the electron exchange matrix element to decrease by about 2.3 per bond with norbornyl spacers between donor and acceptor. This translates to a through-bond j3 of about 5.4 nm-1. These p’s computed on the basis of through-bond distances are in the same range as the B’s listed in Table 111. The hopping time, T,for quenching by Ni(CN)42- was almost independent of the quencher concentration. However the I-’ parameter for the other three quenchers showed a systematic increase with quencher concentration that is contrary to the expectation of the model. Such behavior makes cooperative effects a likely candidate for deviation from model behavior. The current model only deals with direct transfer from each individual quencher. Phenomena such as superexchange could be enhancing the elementary rates for transfer from a specific site to the excited probe or vice versa. For example, in the simple superexchange of McConnelP the interaction is amplified by higher order terms. From second-order time-dependent perturbation theory, the rate constant is given byj2
where x 4th means no fourth-order terms,32 qj represents the energy of the j t h state, pr is the density of intermediate states, s, and p is the corresponding quantity for the final states, m. The Vis are the matrix elements of the perturbation between the zero-order states i and j . For the exchange interaction, the first term in eq 24 will be the direct exchange between donor and acceptor, while the second term will be interpreted as the superexchange. Although McConnell couched his argument in terms of stationary states, half of the splitting between the symmetric and antisymmetric states in his model
is analogous to the second term in eq 24. In the McConnell model, there are N - 2 chemical entities forming a bridge linking the donor and acceptor. The 9’s are the donor and acceptor states, and $j and E 6, are the wavefunction and energy, respectively, of the j t h delocalized virtual state of the bridge. E is the center of gravity of these virtual bridge states, Jon is the transfer integral for the donor (acceptor) to the first localized bridge state, and JB is the transfer integral among nearestneighbor localized bridge states. The quencher anions used in this work have relatively lowlying excited states. They would have small energy denominators in eq 25 if they were thought of as bridge states. As the quencher concentration increases, the number of bridges between the donor and acceptor would increase. The number of pathways among bridge states would also increase. Since the distance between donor and acceptor scales as the number of bridge links, the superexchange term in eq 24 is exponential in distance just like that of the direct term. Hence, the superexchange can affect either j3 or r1 compared to the direct exchange, corresponding to the quencher concentration approaching zero. It should be noted that simply treating each additional quencher ion as interacting with the excited site solely via a single superexchange pathway will not explain the increases in j3 and T-I with concentration. If there were only one single superex-
+
Luminescence Quenching of a R ~ ( b p y ) ’ ~Moiety + change pathway, a definition could be generated from eqs 19 and 25,
Since there is only one pathway, the (-1)N-1 factor disappears through the squaring process in eq 24, and r 1and j3 can be written directly in terms of the McConnell parameters as
As the distance between quenchers becomes smaller, it is expected that both JDJ and JB would increase. However, since eq 25 is valid only for JB values rarely were under 3. For the two chlorides, the X? values were routinely over 10. In addition, for all four quenchers, the uncertainties from the fitsI2 for r1were almost always larger than the fitted parameters themselves. Therefore, in spite of the desirable characteristic of the fitted parameters from the dipolquadrupole interaction being only weakly dependent on the concentration of quenchers, the fits were so poor that it is unlikely that the quenching is proceeding via the dipole-quadrupole interaction. A check was also made to see whether loweringp, Le. that not all the quencher anions were bound to the polyelectrolyte, could account for variations in the physical parameters with quencher concentration. Although counterion condensation is expected for the quencher ions on the basis of Manning’s theory,36it is still possible that not all quencher ions are in the polymer phase. In Manning’s analysis if
.pN’
(32)
where N is the counterion valence and
then counterion condensation occurs. [is a dimensionless measure of linear charge density of the polyelectrolyte, q is the nuclear charge, z is the solvent’s dielectric constant, k~ is Boltzmann’s constant, T is the absolute temperature, and b is the average distance between the charge sites of the polyelectrolyte backbone. Since the quencher anions had charges of -2 and -3, they fall within the region of counterion condensation. When there is counterion condensation, the charge fraction, f, the limiting fractional charge on the polyelectrolyte not compensated by counterions, is given by ([N)-l. For Poly3,3-Ru(bpy)32+, f 1, so the charge fractions,f, are approximately 0.33,0.5, and 1 for -3, -2, and -1 charged anions, respectively. For the concentration of Poly3,3-R~(bpy)3~+ of 3.75 X le5M that was used in the experiments and for an average of 80 charged monomer units for each polyion, it can be shown from these considerations that all the multivalent quencher anions should be bound to the polyions for all concentrations of quenchers used in this work. On the other hand, according to Manning’s theory, no single-valent anions are expected to be condensed on the polyion. This explains the lack of any effect of added NaCl on the luminescence decay behavior. For the presence of additional singly charged chloride, anions in solution will remain in the bulk of solution and cannot influence the condensed multicharged quencher anions. However, in order to check whether the absence of some fraction of the multivalent quenchers could account for the concentration dependence, the following tests were run. In the quenching by Ni(CN)42- anions, the occupation probability was reduced by factors of 0.1 and 0.5. The resulting physical parameters from the fitting procedure were different in magnitude than the calculations above, but the qualitative trend, namely, that some of the ps and most of the 4 ’ s increased with concentration, still held. One additional conclusion can be drawn from the recalculation of j3 and r l on the basis of lower input values of p . Another possible physical cause of the lowering of p could be the lower attraction for the next anion after the bonding of a subsequent anion.14 Such a mechanism cannot be the cause of the observed
-
Sassoon and Hug
7830 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993
concentration dependence. When two fits of a specific decay trace are made, starting with two different values ofp, the smaller value of p gives a smaller value of j3 but a larger value of 4. However, the trends in the variations of the parameters with concentration persisted. One additional possibility for an explanation for the concentration dependence of the parameters B and r 1is that the configuration of the polymer is changing with quencher concentration. It is not unreasonable to expect the chain to coil up as the charge on the backbone of the polymer is neutralized by quencher ions. In the current model, there are two configurational/structural parameters, p, the density of sites, and d, the nearest-neighbor distance. If the polymer were coiling at higher quencher concentration, p should increase and/or d should decrease. On increasing p by a factor of 2, equivalent to changing from a tetrahedrally-restricted to a freely-linked Gaussian chain, the resulting @'sfrom the fit increase by about 40% while the ~ l ' s decreased by a factor of 3-5. On decreasing the nearest-neighbor distance by 1076, the @'sincreased by about 12% while the T I ' S remained the same. For both of the configurational/structural changes, there was almost no change in X> compared to the original fit. Neither of these simple configuration/structure factors by themselves would account for the concentration dependence of both j3 and T . However, the arbitrary variation in d is illustrative of the significance of the choice of nearest-neighbor distance. It shows that a choice of something less than the through-bond distance between charged sites on the backbone would increase j3. Such a slight increase in the @'swould put them right on the of 1 A. conventional exchange distance The density of sites in eq 17 can be contrasted to two other plausible models. First, instead of using a continuum model, which requires a density-of-sites expression, the Blumen-Manz model can be summed over a finite number of sites. Such a procedure was used with Gaussian averaging being accomplished by numerical integration. This numerical procedure greatly extended the computer time and did not lead to results that were substantially different from those which followed from eq 17. That end effects from the infinite chain should cause no trouble is plausible because of the exponential distance dependence of the exchange interaction. Second, a linear model of the polymer was tried. Both continuum and finite lattice models were used. When reasonable intersite distances were used, the resulting /3's were almost 10 times smaller than the ones calculated from eq 17. These were deemed to be physically unreasonable, and the fractal model of the polymer was adopted. Third, nonideal polymers, modeled by self-avoiding random walks, can be described by fractal models.*' These polymers will be on average longer than Gaussian-linked polymers because of the excluded volume. Based on the trends observed in this work with Gaussianlinked polymers and the linear-rod models, it is expected for the exchange interaction that the fitted parameters will increase with concentration of quencher and that the Ps will be somewhere between the results in Table I11 and those of the linear-rod model. Theoretical Models: Incoherent Transfer and Diffusion. The incoherent-transfer models, described above, do not account for any motion of the quenchers. Although with exchange interaction, the model gives relatively consistent fits over a range of quencher concentrations for all four quenchers, no account was taken of diffusion in this model. Allinger and Blumen extended the incoherent-transfer model to moving quenchers.6~7 They discuss two extremes. In the first case, diffusion is very rapid compared to the transfer process. Under these extreme conditions, the quenchers sample all the sites of the polymer, and the time integral in eq 10 is replaced by a volume integral multiplied by t. Such a computation leads to a simple exponential decay, which is not observed in Poly3,3-R~(bpy)3~+atallconcentrationsofquenchers.
The other extreme for the motion of quenchers, that was analyzed by Allinger and Blumen,6+'was for the diffusion being slow compared to the transfer. Their derivation of the important equations was quite lengthy, and only a summary and the final results will be repeated here. The slowness of the moving quenchers allowed them to expand w(R,,t) as a Taylor series about the initial positions of the sites. The resulting expression can be expressed as two factors, one depending on the velocities of the quenchers and the other depending on the starting configurations. After the motional averages were taken and the continuum approximation was made, the average over starting configurations disappears. The expression is exactly analogous to eq 18 with exp{-tw(R)) replaced by (exp{-tw(R,t))),, an ensemble average over velocities. The ensemble average was found by assuming the quenchers were Brownian particles which followed the Langevin equations. Following their development an expression analogous to their6 eq 3.13 for a three-dimensional problem can be found. For a twodimensional fractal with density given by eq 17 and with only the k = 1 power of p remaining, the decay function is
&d "
n=l
'
where D is the diffusion constant, f is defined as
(35) and K!,;") relation6
is given, for two dimensions, by the recurrence
which is subject to the initial values, K13
= 1;
K(1:1)
= -1;
K 2.0 (l)
= 2.,
K 1.(1 l )
= 0, 1 # 1 # 2 (37)
The functions Co,/(ft) are the incomplete gamma functions. They can be evaluated numerically with the series in eq 23, and their asymptoticformsareI'(I) = (I- l)!. Gl,/(ff)canalsobeevaluated by eq 23, and their asymptotic forms are given by
-
Gl,/({f)
(I- l)![ln({f) -0.57721
+
ck-'] I- 1
(38)
k- 1
The asymptotic forms were necessary because the numerical evaluation of the series solution became erratic for large ft. The diffusion calculations are of particular interest in resolving twoissues left in the fitting to the nondiffusiveincoherent-transfer models. The first issue is whether diffusive effects make a significant contribution to the decay, and the second issue is whether the general increase in B and/or 7-l with concentration observed in Figure 5 for most of the quenchers could be ascribed to diffusive effects. Fits for all the quenchers showed much of the same behavior using both thediffusive (four parameters) and nondiffusive (three parameters) fitting procedures. In most cases values of j3 and 4 (parameters common to both diffusive and nondiffusive fitting procedures, along with were obtained from the two fitting procedures that agreed within 10% of each other. Values of diffusion constants (the fourth parameter in the diffusive model), obtained from fitting the decay curves to eq 34, exhibited very large uncertainties. However, they were always found to be at
e)
The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 7831
Luminescence Quenching of a Ru(bpy)32+Moiety least 5 orders of magnitude lower than the diffusion constants of the quenchers in water (typically 10-5 cm2 s-1).37 These values are consistent with estimates of expected diffusion constants of the quencher anions within the polymer phase that can be made from polyelectrolyte films. A recent'" data compilation gives a critical review of diffusion coefficients in polyelectrolyte coatings on electrodes. All of the diffusion coefficients for ions similar to the quencher anions used in this work have diffusion coefficients at the highest l/Soth of the diffusion constant in water. In one of the polyelectrolytes that was most analogous to the one used in this work, Le. (-N+(CH3)2C6H4-)n, Fe(CN)6' had a diffusion constant3gthat was about 0.0001 that of its diffusion in water. It may therefore be concluded that on accounting for the diffusion of the quencher ions on applying the incoherent-transfer model, the fits are very similar to those obtained from applying the nondiffusive model. Diffusive effects are found to be very small in the time range of the luminescence decays, and it is therefore reasonable to assume that the quenchers are essentially immobile in this time frame. Static Quenching. In time-resolved methods, static quenching shows up as a decrease in the initial luminescence intensity, as quencher concentrations increase. The steady-state quenching in Figure 3 contains the effects of both static and dynamic quenching (time-independent and time-dependent effects, respectively). A plot very similar to Figure 3 was obtained by integrating sets of the experimentaldecay traces over all time. These latter artificial "steady-state" luminescence plots could be reproduced very well by numerical integration of the incoherenttransfer model of eq 20, where the three parameters 8, and rl)were taken from the individual fits of the decay traces. This indicates that the incoherent-transfer model of eq 20 can be used to rationalize the dynamic quenching as above while, at the same time, it is safe to relegate the static quenching to Attempts to gain additional insight into the, as yet, uninterpretated parameter, were found to be unsuccessful since no analytical expression exists for the time-integrated form of eq 20. It is therefore not possible to isolate physical quantities of the model from plots of the steady-state data. Even though it is possible to separate out the effects of static quenching by using time-resolved methods, there is still one way static quenching might spoil the analysis of this paper. If the nonluminescing complex (which we speculate to be the source of the nonluminescence) is made up of many quenchers clustered about the excited site, then the calculation of p, the probability of occupation of the quaternary ammonium sites, might be drastically altered from that used to obtain the fittings of the experimental decay traces. However, one can easily calculate how many quenchers would be needed to deactivate (via static quenching) the proper number of polymer chains to match the decrease in For 1:l complexes the number is quite small and does not effect the overall calculation of p for the nonexcited sites, This is because per polymer there are few excited sites (0 or 1) compared to the numbers of quencher anions (1-10) or nonexcited sites (80).
e,
e,
e.
e,
e.
Summary In the quenching of the luminescence of Poly3,3-R~(bpy)3~+ by multivalent anions, several theoretical approaches were used in an attempt to fit the time-resolved decays. All of the models, except for the classical micelle model (two fitting parameters) and the incoherent transferwith the exchangeinteractionincluding diffusion (four fitting parameters), had three fitting parameters, one of which was always the initial luminescence intensity, $. None of the classical theories worked well, but the most successful one was the model employing a Gaussian distribution of decay rate constants.
Various incoherent-transfer models of Blumen and co-workers were applied to solve the same problem. The model that worked the best was a model for incoherent transfer via the exchange interaction with the quencher anions bound to the polyelectrolyte backbone at charged sites. The continuum approximation for summing over sites and the isotropicinteraction model were used to simplify the equations. The configuration/structure of the polymer was dealt with through the density of sites within the continuum approximation. The space over which the continuum approximation was made was a two-dimensional fractal, and the polyelectrolytewas taken to be a tetrahedrally-restrictedGaussianlinked polymer. The physical parameters from the fit had interactiondistancesthat varied from 0.16 to 0.25 nm and hopping times to nearest neighbors from 0.2 to 3 ps. This model was only partially successful because, contrary to expectations,the physical parameters varied with concentration of quencher. It was shown that this variation could not be due to diffusion of quenchers along the polyelectrolyte backbone, single-pathway superexchange interactions, or insufficient multivalent anions attached to the backbone. One possible rationalization of the concentration dependence of the parameters is still a multipathway superexchange. Transfer among quenchers is only one of several pathways left out of the truncated master equation (eq 8), which is the basis for the developmentof Blumen and co-workers that is used in this work. This approach also leaves out back-transferprocesses. The concentration dependence of the fitting parameters may ultimately be due to a neglect of any, or all, of these processes. Even with these limitations, the incoherent-transfer model, with the exchange interaction and immobile quenchers, gives the best fits to the widest range of data of all the models examined. The interaction parameters, 8, and hopping times, T , from this model can be taken as a reasonable first approximation to the dynamics of the quenching in these polyelectrolyte systems. The dependence of the parameters on quencher concentration may also be accounted for by configurational changes of the polyion which will affect its site density. Increase in quencher concentrationwill lead to additionalneutralizationof the polymer charge, reduced internal charge repulsion along the chain, and thus more coiling of the polymer. Refining the model by introduction of appropriate polymer shape factors will be investigated in the future.
Acknowledgment. The work described herein was supported by the Office of Basic Energy Sciences of the Department of Energy. This is Document No. NDRL-3550 from the Notre Dame Radiation Laboratory. The authors would like to thank Dr. I. Carmichael for his advice and many discussions on the numerical procedures in this work. One of us (G.L.H.) would like to thank Dr. J. R. Miller and Professors S.E. Webber and M. A. Winnik for critical and instructive discussions on the physics of the systems discussed herein. References and Notes (1) Klafter, J.; Drake, J. M. Molecular Dynamics in Restricted Geometries; Wiley: New York, 1989. (2) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, 1982. (3) Rabani, J.; Sassoon, R. E. J. Photochem. 1985,29,7, and references cited therein. (4) Sassoon, R. E.; Gershuni, S.;Rabani, J. J. Phys. Chem. 1992, 96, 4692.
( 5 ) Tachiya, M. Chem. Phys. Lett. 1975, 33, 289. (6) Allinger, K.; Blumen, A. J . Chem. Phys. 1980, 72, 4608. (7) Allinger, K.;Blumen, A. J. Chem. Phys. 1980, 75, 2762. (8) Rembaum, A.; Noguchi, H. Mucromolecules 1972, 5, 261. (9) Casson, D.;Rembaum, A. Macromolecules 1972, 5, 75. (10) Fernelius, W. C.; Burbage, J. J. Inorg. Synrh. 1946, 2, 227. (11) Encinas, M.V.;Scaiano, J. C. J. Am. Chem. Soc. 1979,101,2146. (12) Bevington,P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969; Chapter 6. (13) Marquardt, D. W. J. SOC.Ind. Appl. Math. 1963, 11, 431. (14) Sassoon, R. E. Chem. Phys. Lett. 1986, 125,74.
7832 The Journal of Physical Chemistry, Vol. 97, No. 30, 1993 (15) Albcrv. W. J.; Bartlett, P. N.; Wilde, C. P.; Danvent, J. R. J. Am. Chem..Soc. 1& !, 107,1854. (16) Wolszczak, M.; Thomas, J. K. Radiat. Phys. Chem. 1991,38, 155. 117) Song. L.: Dorfman. R. C.: Swallen. S. F.; Faver, M. D. J . Phys. Chem.'1991;95,3454. (18) Boulu, L. G.; Kozak, J. J. Mol. Phys. 1988, 65, 193. (19) Blumen, A. Nuovo Cimento 1981, 638, 50. (20) Klafter, J.; Blumen, A. J . Chem. Phys. 1984,80, 875. (21) Roy, A. K.;Blumen, A. J. Chem. Phys. 1989, 91,4353. (22) Blumen, A,; Manz, J. J . Chem. Phys. 1979, 71, 4694. (23) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (24) de Gennes, P.-G. Scaling Concepts in Polymer Physics; Cornel1 University Press: Ithaca, NY, 1979. (25) Dexter, D. L. J. Chem. Phys. 1953, 21, 836. (26) Newton, M. D. Chem. Reo. 1991,91,767. (27) Inokuti, M.; Hirayama, F. J. Chem. Phys. 1965,43, 1978. (28) Blumen, A. J. Chem. Phys. 1980, 72, 2632.
Sassoon and Hug (29) C l m , G. L.; Calcaterra, L. T.; Green, N. J.; Penfield, K.W.;Miller, J. R. J. Phys. Chem. 1986,90, 3673. (30) Penfield, K. W.; Miller, J. R.; Paddon-Row, M. N.; Cotearis, E.; Oliver, A. M.; Hush,N. S . 1.Am. Chem. Soc. 1987, 109,5061. (31) McConnell, H. M. J. Chem. Phys. 1961, 35, 508. (32) Bates. D. R. Ouantum Theow I. Elements: Academic Press: New YO&, i96i; S& eq 183, p 284. (33) Endiwtt, J. F. Coord. Chem. Rev. 1985.64, 293. (34) Ferraudi, G. J. Elements of Inorganic Photochemistry;Wiley: New YorL, 1988; p 81. (35) Abramowitz, M.; Stegun, I. A. HandbookofMathcmaficalFunctcIlon; Dover: New Yorlc, 1965. (36) Manning, G. S . Ace. Chem. Res. 1979, 12, 443. (37) Rice,S.A. Di/fusion-LimitedReactioru;Elsevier: Amsterdam, 1985. (38) Oyama, N.; Ohsaka,T. In Techniques of ChemistrySeries; Murray, R. W., Ed.; Wiley: New York, 1992; Chapter VIII. (39) Ohsaka, T.; Okajima, T.; Oyama, N. J. Electrwnal. Chem. 1986, 215, 191.
