- Email: [email protected]
Electron transfer in a disordered polymeric system: fluorescence spectroscopy study
9 July 1999
Chemical Physics Letters 307 Ž1999. 367–372 www.elsevier.nlrlocatercplett
Electron transfer in a disordered polymeric system: fluorescence spectroscopy study Marcelo K.K. Nakaema, Rosemary Sanches
)
Instituto de Fısica de Sao ´ ˜ Carlos, UniÕersidade de Sao ˜ Paulo, Caixa Postal 369, 13560-970 Sao ˜ Carlos, SP, Brazil Received 18 November 1998; in final form 25 April 1999
Abstract Experimental studies of electron transfer in a polymeric matrix were made using fluorescence spectroscopy. Donor molecules were excited in the presence of acceptor molecules and the decay time of the donor fluorescence was measured. The data were analyzed using the model for electron transfer in a disordered system. The idea was to verify the validity of the model in the very large disorder limit. Using a random sample, the electron transfer rate obtained could be compared to the one obtained considering the random distribution of acceptors. A good agreement was obtained. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer is a fundamental mechanism in many biological and chemical processes w1x. In most systems the electron transfer rate is described in a nonadiabatic formulation, which is valid for a large separation between donor and acceptor sites, and is written as w2x k ET s Ž 2 pr" . < T DA < 2 Ž FC . , Ž 1. where ŽFC. is the Franck–Condon factor associated with the nuclear motion along the reaction coordinate, and TDA reflects the electronic coupling between the donor and acceptor sites. The simple models w2–4x assume that this coupling decays exponentially with the donor–acceptor distance, T DA A ) Corresponding author. Fax: q55 16 271 5381; e-mail: [email protected]
expŽya rr2., or k ET s Z expŽya r .. Systems for which the donor–acceptor distances are accurately known provide useful models for understanding the electron transfer mechanism, and several have been studied w5x. However, many systems do not possess sufficient structural constraints and the distance between a donor and an acceptor can suffer small variations. It is certainly relevant to understand how this disorder affects the electron transfer process. A recent paper by Pande and Onuchic discusses this situation and proposes equations to analyze fluorescence decay measurements w6x. To find out the effect of the disorder on the electron transfer process, a completely random system was studied. The intention was to verify if the disordered model was still valid in this extreme situation of disorder. That was possible because the result could be compared to that obtained considering the real random distribution of acceptors. Oc-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 5 2 7 - 8
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
368
taethylporphyrin ŽH 2-OEP., at a fixed concentration, and duroquinone ŽDQ., in different concentrations, were randomly distributed in polystyrene films. Fluorescence lifetime measurements of H 2-OEP were made and they were analyzed considering that an electron could be transferred from H 2-OEP to the nearest DQ molecule. The nearest-neighbor assumption was used in the analysis of trapped electron scavenging reactions w7,8x and it was shown that this approximation is quite good. The distribution of nearest neighbors was then approximated by a Gaussian and the data were analyzed considering the model for a disordered system. The electron transfer rate obtained is in good agreement with the rate obtained considering the random distribution of acceptors w9x.
One case to consider is if there is a Gaussian distribution of lengths, with a mean distance rc and dispersion s 2 . Then, 1
`
QŽ t . s
H0 exp yk
=exp y
ET
Ž r.t
Ž r y rc .
'2 p s
2
dr ,
2s 2
Ž 3.
and k ET Ž r . s k c expwya Ž r y rc .x, where k c is the electron transfer rate for a donor–acceptor distance equal to rc . Using a new variable x s yŽ ar2.Ž r y rc ., one gets k ET s k c expŽ2 x ., and Q Ž t . s exp yk c t exp Ž 2 x .
H
ž
= y 2. Theoretical
2 x2
s 2a 2
/
2
'2 p sa exp
dx,
Ž 4.
which can be written as 2.1. Electron transfer in a disordered system
Q Ž t . s exp yk c t exp Ž 2 x .
H
In the theoretical development of Pande and Onuchic w6x the disorder is described in terms of Green’s function with the pathway strategy. Here a different approach is used to deduce the equations to analyze fluorescence decay measurements taking into account the disorder of the system. In time-dependent fluorescence experiments, one measures the decay with time of the population of the fluorophore excited state, which is given by: expŽyk 0 t ., where k 0 is the decay rate. In a system where electron transfer is possible, the population of the excited state can decay also via electron transfer with a rate k ET . If the donor–acceptor distance is fixed, a single decay rate exists, and the fluorescence signal is given by expŽyk 0 t . P expŽyk ET t .. However, if the possibility of a ‘disorder in length’ is considered Ža variable donor–acceptor distance r ., then the fluorescence signal depends on the distribution of lengths, g Ž r ., and is written as: `
F Ž t . A exp Ž yk 0 t .
H0 exp yk
s exp Ž yk 0 t . Q Ž t . .
ET
Ž r. t gŽ r. dr Ž 2.
ž
=exp y
x2 2s X2
/
1
'2 p s X
dx
s exp yk c t exp Ž 2 x . P Ž x . d x ,
Ž 5.
H
where s X s asr2 and P Ž x . is the Gaussian distribution for x. This integral was calculated by Pande and Onuchic w6x and, assuming that expwyk c t expŽ2 x .x P Ž x . looks Gaussian in x, it yields Q Ž t . s Ž 1 q 4s X 2 k c t .
ž
y1 r2
=exp yk c t 1 y
2 s X2 kc t 1 q 4s X 2 k c t
/
.
Ž 6.
In the long time limit Ž t ) 1rŽ s X 2 k c .. this approximation breaks down, i.e., expwyk c t expŽ2 x .x deforms P Ž x . away from Gaussian, and the integral yields w6x QŽ t . f
1 2
1 y erf
ž
ln Ž k c t . 2'2 s X
/
.
Ž 7.
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
Using Eq. Ž6. or Eq. Ž7. in Eq. Ž2., the parameters k c and s X can be obtained from the fluorescence data. 2.2. Random distribution of donors and acceptors In a system with a random distribution of donors and acceptors, if N is the number of acceptors per unit volume, the probability that a donor has a nearest acceptor molecule at a distance r is given by w10x: g Ž r . s 4 p Nr 2 exp Ž y4p Nr 3r3 . ,
Ž 8.
and the average distance to the nearest acceptor is `
rs
H0 rg Ž r . d r s
0.17
ž / N
1r3
.
Ž 9.
The distribution can be written in terms of the average distance as g Ž r . s4p
0.17 r
3
4 0.17 r 2 exp y p 3 r 3 . 3 r
ž
/
Ž 10 .
This function can be approximated by a Gaussian with mean distance rc s 0.985r and dispersion s 2 s Ž0.38 r . 2 . If there is a Gaussian distribution for the donor– acceptor distances, the model for disordered systems can be used, assuming the electron is going to transfer to the nearest acceptor. The parameter k c can be written in terms of Z and s X ln Ž k c . s ln Z y 5.18 s X ,
Ž 11 .
and the parameter s X can be related to the number of acceptor molecules per unit volume, N, through
s Xs
as
0.105 a
. Ž 12 . N 1r3 From Eqs. Ž11. and Ž12. the parameters Z and a can be obtained. s
2
3. Experimental The samples used were made with a fixed concentration of H 2-OEP Ž0.39 mM. and different concentrations of DQ Žfrom 0 to 0.69 M., both from SIGMA. They were dissolved in toluene and mixed
369
with polystyrene and the mixture was spread on a glass plate and extended to form a film ; 70 mm thick. The films were left in a glove box with N2 to evaporate the solvent for ; 12 h. The number of DQ ˚ 3 in the films Ž N . was calculated molecules per A considering the density of polystyrene as 1.06 grml w11x. To verify that there was a random distribution of porphyrin and quinone molecules in the films, the absorption and static fluorescence spectra of different pieces of the films were measured. The spectra of the different pieces were indistinguishable, suggesting that the films were homogeneous. The fluorescence data were obtained using a ISS K2 multifrequency cross-correlation phase and modulation fluorometer w12x. Latex beads in water were used as the scattering solution. The excitation was with a Xe lamp at 498 nm and a high pass filter ŽGG435. was used in the excitation beam. Another high pass filter ŽOG590. was interposed between the sample and the detector to block the scattered light. The modulation and phase of the emitted light were measured as a function of the modulation frequency which was varied from 1 to 120 MHz. Data obtained at higher frequencies Žup to 350 MHz. just confirmed the presence of a small amount Ž- 5%. of scattering.
4. Results and discussion For each film 3 or 4 pieces were cut and 3 measurements were made for each piece. The average value of the best 3 or 4 measurements Žsmallest x 2 values in the exponential fitting procedure. were considered in the analysis. The data for the film with H 2-OEP only were fitted with a single decay time of 19 " 0.3 ns. The value obtained in solution was 18 ns w13x. For the films with H 2-OEP and DQ it was observed, as expected, that as the DQ concentration increases, a stronger fluorescence quenching occurred. For these data, good fittings were obtained considering two decay times. Fig. 1 shows the experimental data obtained for two consecutive measurements of a sample with wDQx s 0.69 M. It is also shown the fitting to obtain the decay times and the residuals of the fitting procedure. The resulting fluorescence decay data plotted as a function of time is
370
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
Fig. 1. Fluorescence data in the frequency domain. Two consecutive measurements of the phase Ž=. and modulation ŽI. as a function of the modulation frequency for a sample with wDQx s 0.69 M. The line is the fitting to obtain the decay times. The figure on the top shows the residuals of the fitting procedure.
as shown in Fig. 2. It is also shown the fittings with the Inokuti and Hirayama model w9x Žexact solution. and the nearest-neighbor approximation Žthis is equivalent to consider only the first term in Inokuti and Hirayama model.. The values obtained for the ˚ y1 . and Z changes parameter a are identical Ž0.76 A 10 y1 Ž from 1.5 = 10 s exact solution. to 2.5 = 10 10
Fig. 2. The fluorescence decay as a function of time for a sample with wDQx s 0.69 M. It is also shown the fitting to the data of the Inokuti and Hirayama model, the nearest-neighbor approximation and the nearest-neighbor approximation with the parameters obtained from the Inokuti and Hirayama model.
sy1 Žnearest neighbor.. If the Z value obtained from the exact solution is used in the nearest-neighbor model, the fitting is not good Žsee Fig. 2.. The slower decay obtained is because the additional quenching due to all other neighbors was neglected. However, a small change in the parameter Z was enough to get a good fitting with the nearest-neighbor model. The use of the disordered model with its nearest-neighbor approximation is then justified. The parameters k c and s X of the disordered model were fitted to the fluorescence decay data obtained for samples with different concentrations Ž N . of DQ, having in mind that Eq. Ž11. had to be obeyed. For the samples with small amounts of DQ ŽFig. 3a. good fittings were obtained with Eq. Ž6. and, for the samples with larger amounts of DQ ŽFig. 3b. Eq. Ž7. had to be used. The values of k c and s X obtained for the samples with larger amounts of DQ indicate the long time limit regime. Fig. 4a shows the plot of ln k c vs. s X obtained from the fitting procedure for the data of samples with different DQ concentrations. A straight line was fitted to the data imposing a slope of 5.18. The parameter Z obtained from this plot was ; 2.5 = 10 10 sy1 . This value is very close to the one obtained from the Inokuti and
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
371
paper ŽEqs. Ž6. and Ž7... This model assumes that there is a Gaussian distribution of donor–acceptor distances, and that the disorder is given by the Gaussian dispersion. The disorder parameter is of great interest when one prepares samples, like growing protein crystals, assembling films to be used as display, and so on. One of the parameters obtained from the disordered model, s X , is related to the product of a , the exponential factor of the electron transfer rate, and s , the square root of the Gaussian dispersion. If a series of samples is made with the same starting material, the parameter s X is a good indicator of the disorder of the samples. On another hand, if the disorder parameter of a sample is known, it is possible to obtain the exponential factor of the electron transfer rate. It seems, therefore, that one can take into account the conformational heterogene-
Fig. 3. The fitting of the disordered model to the fluorescence data in the time domain: Ža. data for a sample with wDQx s 0.17 M Ž . and fitting using Eq. Ž6. in Eq. Ž2. Ž – – – .; and Žb. . and fitting data for a sample with wDQx s 0.35 M Ž using Eq. Ž7. in Eq. Ž2. Ž – – – ..
Hirayama model and identical to the value obtained with the nearest-neighbor model. The plot of s X vs. 1rN 1r3 is shown in Fig. 4b, which also shows the fitting of Eq. Ž12. to the data. ˚ y1 . This The value of a obtained was ; 0.74 A value is very close to those obtained with the Inokuti and Hirayama model and with the nearest-neighbor model, and is comparable to those reported for other donorracceptor systems, where a is in the range ˚ y1 w2,14,15x. The excellent agreement ob0.3–1.2 A tained for the two parameters validates all the assumptions made for the disordered model and the use of this model in the very large disorder limit. The idea of considering how the disorder of the system affects the electron transfer process was introduced by Pande and Onuchic w6x. Using a different approach, equivalent equations were obtained in this
Fig. 4. The analysis of the disordered model: Ža. the logarithmic X plot of the parameter k c vs. s obtained from the fittings to the fluorescence data Žthe straight line is the fitting using Eq. Ž11..; X and Žb. the parameter s plotted as a function of 1r N 1r3 , and the linear fitting using Eq. Ž12..
372
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
ity of a protein or the disorder in an assembled film in the electron transfer process. In this study the extent of the disordered model was investigated in the very large disorder limit. The system used allowed the comparison of this model with the well-known random model w9x. Two approximations were made: transfer occurred only from the donor to the nearest acceptor and the distribution of nearest acceptors was approximated by a Gaussian. The nearest-neighbor approximation was shown to be quite good. It can be observed, graphically, that the assumption of a Gaussian distribution of donor– acceptor distances also is quite reasonable. In fact, it can be confirmed from the good agreement obtained for the exact and the disordered models. Finally, the agreement of the two models also indicates that the disordered model can be used even for samples with a lot of disorder, provided that there is a Gaussian distribution of donor–acceptor distances.
Acknowledgements This work was supported by the Brazilian agencies FINEP and CNPq.
References w1x J.R. Bolton, N. Mataga, G. McLendon ŽEds.., Electron Transfer in Inorganic, Organic and Biological Systems, Advances in Chemistry Series 228, Am. Chem. Soc., Washington, DC, 1991. w2x R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 Ž1985. 265. w3x S. Larsson, J. Am. Chem. Soc. 103 Ž1981. 4034. w4x D.N. Beratan, J.J. Hopfield, J. Am. Chem. Soc. 106 Ž1984. 1584. w5x M.R. Wasielewski, Chem. Rev. 92 Ž1992. 435. w6x V.S. Pande, J.N. Onuchic, Phys. Rev. Lett. 78 Ž1997. 146. w7x M. Tachiya, A. Mozumder, Chem. Phys. Lett. 28 Ž1974. 87. w8x R.C. Dorfman, M. Tachiya, M.D. Fayer, Chem. Phys. Lett. 179 Ž1991. 152. w9x M. Inokuti, F. Hirayama, J. Chem. Phys. 43 Ž1965. 1978. w10x S. Chandrasekhar, Rev. Mod. Phys. 15 Ž1943. 1. w11x M.K.K. Nakaema, V.B.P. Leite, R. Sanches, Arch. Biochem. Biophys. 344 Ž1997. 61. w12x E. Gratton, D.M. Jameson, R.D. Hall, Annu. Rev. Biophys. Bioeng. 13 Ž1984. 105. w13x K. Kalyanasundaram, Photochemistry of Polypyridine and Porphyrin Complexes, Academic Press, New York, 1992, p. 407. w14x J.R. Miller, J.V. Beitz, J. Chem. Phys. 74 Ž1981. 6746. w15x H. Kuhn, J. Photochem. 10 Ž1979. 111.
Chemical Physics Letters 307 Ž1999. 367–372 www.elsevier.nlrlocatercplett
Electron transfer in a disordered polymeric system: fluorescence spectroscopy study Marcelo K.K. Nakaema, Rosemary Sanches
)
Instituto de Fısica de Sao ´ ˜ Carlos, UniÕersidade de Sao ˜ Paulo, Caixa Postal 369, 13560-970 Sao ˜ Carlos, SP, Brazil Received 18 November 1998; in final form 25 April 1999
Abstract Experimental studies of electron transfer in a polymeric matrix were made using fluorescence spectroscopy. Donor molecules were excited in the presence of acceptor molecules and the decay time of the donor fluorescence was measured. The data were analyzed using the model for electron transfer in a disordered system. The idea was to verify the validity of the model in the very large disorder limit. Using a random sample, the electron transfer rate obtained could be compared to the one obtained considering the random distribution of acceptors. A good agreement was obtained. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction Electron transfer is a fundamental mechanism in many biological and chemical processes w1x. In most systems the electron transfer rate is described in a nonadiabatic formulation, which is valid for a large separation between donor and acceptor sites, and is written as w2x k ET s Ž 2 pr" . < T DA < 2 Ž FC . , Ž 1. where ŽFC. is the Franck–Condon factor associated with the nuclear motion along the reaction coordinate, and TDA reflects the electronic coupling between the donor and acceptor sites. The simple models w2–4x assume that this coupling decays exponentially with the donor–acceptor distance, T DA A ) Corresponding author. Fax: q55 16 271 5381; e-mail: [email protected]
expŽya rr2., or k ET s Z expŽya r .. Systems for which the donor–acceptor distances are accurately known provide useful models for understanding the electron transfer mechanism, and several have been studied w5x. However, many systems do not possess sufficient structural constraints and the distance between a donor and an acceptor can suffer small variations. It is certainly relevant to understand how this disorder affects the electron transfer process. A recent paper by Pande and Onuchic discusses this situation and proposes equations to analyze fluorescence decay measurements w6x. To find out the effect of the disorder on the electron transfer process, a completely random system was studied. The intention was to verify if the disordered model was still valid in this extreme situation of disorder. That was possible because the result could be compared to that obtained considering the real random distribution of acceptors. Oc-
0009-2614r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 9 9 . 0 0 5 2 7 - 8
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
368
taethylporphyrin ŽH 2-OEP., at a fixed concentration, and duroquinone ŽDQ., in different concentrations, were randomly distributed in polystyrene films. Fluorescence lifetime measurements of H 2-OEP were made and they were analyzed considering that an electron could be transferred from H 2-OEP to the nearest DQ molecule. The nearest-neighbor assumption was used in the analysis of trapped electron scavenging reactions w7,8x and it was shown that this approximation is quite good. The distribution of nearest neighbors was then approximated by a Gaussian and the data were analyzed considering the model for a disordered system. The electron transfer rate obtained is in good agreement with the rate obtained considering the random distribution of acceptors w9x.
One case to consider is if there is a Gaussian distribution of lengths, with a mean distance rc and dispersion s 2 . Then, 1
`
QŽ t . s
H0 exp yk
=exp y
ET
Ž r.t
Ž r y rc .
'2 p s
2
dr ,
2s 2
Ž 3.
and k ET Ž r . s k c expwya Ž r y rc .x, where k c is the electron transfer rate for a donor–acceptor distance equal to rc . Using a new variable x s yŽ ar2.Ž r y rc ., one gets k ET s k c expŽ2 x ., and Q Ž t . s exp yk c t exp Ž 2 x .
H
ž
= y 2. Theoretical
2 x2
s 2a 2
/
2
'2 p sa exp
dx,
Ž 4.
which can be written as 2.1. Electron transfer in a disordered system
Q Ž t . s exp yk c t exp Ž 2 x .
H
In the theoretical development of Pande and Onuchic w6x the disorder is described in terms of Green’s function with the pathway strategy. Here a different approach is used to deduce the equations to analyze fluorescence decay measurements taking into account the disorder of the system. In time-dependent fluorescence experiments, one measures the decay with time of the population of the fluorophore excited state, which is given by: expŽyk 0 t ., where k 0 is the decay rate. In a system where electron transfer is possible, the population of the excited state can decay also via electron transfer with a rate k ET . If the donor–acceptor distance is fixed, a single decay rate exists, and the fluorescence signal is given by expŽyk 0 t . P expŽyk ET t .. However, if the possibility of a ‘disorder in length’ is considered Ža variable donor–acceptor distance r ., then the fluorescence signal depends on the distribution of lengths, g Ž r ., and is written as: `
F Ž t . A exp Ž yk 0 t .
H0 exp yk
s exp Ž yk 0 t . Q Ž t . .
ET
Ž r. t gŽ r. dr Ž 2.
ž
=exp y
x2 2s X2
/
1
'2 p s X
dx
s exp yk c t exp Ž 2 x . P Ž x . d x ,
Ž 5.
H
where s X s asr2 and P Ž x . is the Gaussian distribution for x. This integral was calculated by Pande and Onuchic w6x and, assuming that expwyk c t expŽ2 x .x P Ž x . looks Gaussian in x, it yields Q Ž t . s Ž 1 q 4s X 2 k c t .
ž
y1 r2
=exp yk c t 1 y
2 s X2 kc t 1 q 4s X 2 k c t
/
.
Ž 6.
In the long time limit Ž t ) 1rŽ s X 2 k c .. this approximation breaks down, i.e., expwyk c t expŽ2 x .x deforms P Ž x . away from Gaussian, and the integral yields w6x QŽ t . f
1 2
1 y erf
ž
ln Ž k c t . 2'2 s X
/
.
Ž 7.
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
Using Eq. Ž6. or Eq. Ž7. in Eq. Ž2., the parameters k c and s X can be obtained from the fluorescence data. 2.2. Random distribution of donors and acceptors In a system with a random distribution of donors and acceptors, if N is the number of acceptors per unit volume, the probability that a donor has a nearest acceptor molecule at a distance r is given by w10x: g Ž r . s 4 p Nr 2 exp Ž y4p Nr 3r3 . ,
Ž 8.
and the average distance to the nearest acceptor is `
rs
H0 rg Ž r . d r s
0.17
ž / N
1r3
.
Ž 9.
The distribution can be written in terms of the average distance as g Ž r . s4p
0.17 r
3
4 0.17 r 2 exp y p 3 r 3 . 3 r
ž
/
Ž 10 .
This function can be approximated by a Gaussian with mean distance rc s 0.985r and dispersion s 2 s Ž0.38 r . 2 . If there is a Gaussian distribution for the donor– acceptor distances, the model for disordered systems can be used, assuming the electron is going to transfer to the nearest acceptor. The parameter k c can be written in terms of Z and s X ln Ž k c . s ln Z y 5.18 s X ,
Ž 11 .
and the parameter s X can be related to the number of acceptor molecules per unit volume, N, through
s Xs
as
0.105 a
. Ž 12 . N 1r3 From Eqs. Ž11. and Ž12. the parameters Z and a can be obtained. s
2
3. Experimental The samples used were made with a fixed concentration of H 2-OEP Ž0.39 mM. and different concentrations of DQ Žfrom 0 to 0.69 M., both from SIGMA. They were dissolved in toluene and mixed
369
with polystyrene and the mixture was spread on a glass plate and extended to form a film ; 70 mm thick. The films were left in a glove box with N2 to evaporate the solvent for ; 12 h. The number of DQ ˚ 3 in the films Ž N . was calculated molecules per A considering the density of polystyrene as 1.06 grml w11x. To verify that there was a random distribution of porphyrin and quinone molecules in the films, the absorption and static fluorescence spectra of different pieces of the films were measured. The spectra of the different pieces were indistinguishable, suggesting that the films were homogeneous. The fluorescence data were obtained using a ISS K2 multifrequency cross-correlation phase and modulation fluorometer w12x. Latex beads in water were used as the scattering solution. The excitation was with a Xe lamp at 498 nm and a high pass filter ŽGG435. was used in the excitation beam. Another high pass filter ŽOG590. was interposed between the sample and the detector to block the scattered light. The modulation and phase of the emitted light were measured as a function of the modulation frequency which was varied from 1 to 120 MHz. Data obtained at higher frequencies Žup to 350 MHz. just confirmed the presence of a small amount Ž- 5%. of scattering.
4. Results and discussion For each film 3 or 4 pieces were cut and 3 measurements were made for each piece. The average value of the best 3 or 4 measurements Žsmallest x 2 values in the exponential fitting procedure. were considered in the analysis. The data for the film with H 2-OEP only were fitted with a single decay time of 19 " 0.3 ns. The value obtained in solution was 18 ns w13x. For the films with H 2-OEP and DQ it was observed, as expected, that as the DQ concentration increases, a stronger fluorescence quenching occurred. For these data, good fittings were obtained considering two decay times. Fig. 1 shows the experimental data obtained for two consecutive measurements of a sample with wDQx s 0.69 M. It is also shown the fitting to obtain the decay times and the residuals of the fitting procedure. The resulting fluorescence decay data plotted as a function of time is
370
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
Fig. 1. Fluorescence data in the frequency domain. Two consecutive measurements of the phase Ž=. and modulation ŽI. as a function of the modulation frequency for a sample with wDQx s 0.69 M. The line is the fitting to obtain the decay times. The figure on the top shows the residuals of the fitting procedure.
as shown in Fig. 2. It is also shown the fittings with the Inokuti and Hirayama model w9x Žexact solution. and the nearest-neighbor approximation Žthis is equivalent to consider only the first term in Inokuti and Hirayama model.. The values obtained for the ˚ y1 . and Z changes parameter a are identical Ž0.76 A 10 y1 Ž from 1.5 = 10 s exact solution. to 2.5 = 10 10
Fig. 2. The fluorescence decay as a function of time for a sample with wDQx s 0.69 M. It is also shown the fitting to the data of the Inokuti and Hirayama model, the nearest-neighbor approximation and the nearest-neighbor approximation with the parameters obtained from the Inokuti and Hirayama model.
sy1 Žnearest neighbor.. If the Z value obtained from the exact solution is used in the nearest-neighbor model, the fitting is not good Žsee Fig. 2.. The slower decay obtained is because the additional quenching due to all other neighbors was neglected. However, a small change in the parameter Z was enough to get a good fitting with the nearest-neighbor model. The use of the disordered model with its nearest-neighbor approximation is then justified. The parameters k c and s X of the disordered model were fitted to the fluorescence decay data obtained for samples with different concentrations Ž N . of DQ, having in mind that Eq. Ž11. had to be obeyed. For the samples with small amounts of DQ ŽFig. 3a. good fittings were obtained with Eq. Ž6. and, for the samples with larger amounts of DQ ŽFig. 3b. Eq. Ž7. had to be used. The values of k c and s X obtained for the samples with larger amounts of DQ indicate the long time limit regime. Fig. 4a shows the plot of ln k c vs. s X obtained from the fitting procedure for the data of samples with different DQ concentrations. A straight line was fitted to the data imposing a slope of 5.18. The parameter Z obtained from this plot was ; 2.5 = 10 10 sy1 . This value is very close to the one obtained from the Inokuti and
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
371
paper ŽEqs. Ž6. and Ž7... This model assumes that there is a Gaussian distribution of donor–acceptor distances, and that the disorder is given by the Gaussian dispersion. The disorder parameter is of great interest when one prepares samples, like growing protein crystals, assembling films to be used as display, and so on. One of the parameters obtained from the disordered model, s X , is related to the product of a , the exponential factor of the electron transfer rate, and s , the square root of the Gaussian dispersion. If a series of samples is made with the same starting material, the parameter s X is a good indicator of the disorder of the samples. On another hand, if the disorder parameter of a sample is known, it is possible to obtain the exponential factor of the electron transfer rate. It seems, therefore, that one can take into account the conformational heterogene-
Fig. 3. The fitting of the disordered model to the fluorescence data in the time domain: Ža. data for a sample with wDQx s 0.17 M Ž . and fitting using Eq. Ž6. in Eq. Ž2. Ž – – – .; and Žb. . and fitting data for a sample with wDQx s 0.35 M Ž using Eq. Ž7. in Eq. Ž2. Ž – – – ..
Hirayama model and identical to the value obtained with the nearest-neighbor model. The plot of s X vs. 1rN 1r3 is shown in Fig. 4b, which also shows the fitting of Eq. Ž12. to the data. ˚ y1 . This The value of a obtained was ; 0.74 A value is very close to those obtained with the Inokuti and Hirayama model and with the nearest-neighbor model, and is comparable to those reported for other donorracceptor systems, where a is in the range ˚ y1 w2,14,15x. The excellent agreement ob0.3–1.2 A tained for the two parameters validates all the assumptions made for the disordered model and the use of this model in the very large disorder limit. The idea of considering how the disorder of the system affects the electron transfer process was introduced by Pande and Onuchic w6x. Using a different approach, equivalent equations were obtained in this
Fig. 4. The analysis of the disordered model: Ža. the logarithmic X plot of the parameter k c vs. s obtained from the fittings to the fluorescence data Žthe straight line is the fitting using Eq. Ž11..; X and Žb. the parameter s plotted as a function of 1r N 1r3 , and the linear fitting using Eq. Ž12..
372
M.K.K. Nakaema, R. Sanchesr Chemical Physics Letters 307 (1999) 367–372
ity of a protein or the disorder in an assembled film in the electron transfer process. In this study the extent of the disordered model was investigated in the very large disorder limit. The system used allowed the comparison of this model with the well-known random model w9x. Two approximations were made: transfer occurred only from the donor to the nearest acceptor and the distribution of nearest acceptors was approximated by a Gaussian. The nearest-neighbor approximation was shown to be quite good. It can be observed, graphically, that the assumption of a Gaussian distribution of donor– acceptor distances also is quite reasonable. In fact, it can be confirmed from the good agreement obtained for the exact and the disordered models. Finally, the agreement of the two models also indicates that the disordered model can be used even for samples with a lot of disorder, provided that there is a Gaussian distribution of donor–acceptor distances.
Acknowledgements This work was supported by the Brazilian agencies FINEP and CNPq.
References w1x J.R. Bolton, N. Mataga, G. McLendon ŽEds.., Electron Transfer in Inorganic, Organic and Biological Systems, Advances in Chemistry Series 228, Am. Chem. Soc., Washington, DC, 1991. w2x R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 Ž1985. 265. w3x S. Larsson, J. Am. Chem. Soc. 103 Ž1981. 4034. w4x D.N. Beratan, J.J. Hopfield, J. Am. Chem. Soc. 106 Ž1984. 1584. w5x M.R. Wasielewski, Chem. Rev. 92 Ž1992. 435. w6x V.S. Pande, J.N. Onuchic, Phys. Rev. Lett. 78 Ž1997. 146. w7x M. Tachiya, A. Mozumder, Chem. Phys. Lett. 28 Ž1974. 87. w8x R.C. Dorfman, M. Tachiya, M.D. Fayer, Chem. Phys. Lett. 179 Ž1991. 152. w9x M. Inokuti, F. Hirayama, J. Chem. Phys. 43 Ž1965. 1978. w10x S. Chandrasekhar, Rev. Mod. Phys. 15 Ž1943. 1. w11x M.K.K. Nakaema, V.B.P. Leite, R. Sanches, Arch. Biochem. Biophys. 344 Ž1997. 61. w12x E. Gratton, D.M. Jameson, R.D. Hall, Annu. Rev. Biophys. Bioeng. 13 Ž1984. 105. w13x K. Kalyanasundaram, Photochemistry of Polypyridine and Porphyrin Complexes, Academic Press, New York, 1992, p. 407. w14x J.R. Miller, J.V. Beitz, J. Chem. Phys. 74 Ž1981. 6746. w15x H. Kuhn, J. Photochem. 10 Ž1979. 111.