Distribution of the decay rate constants of individual excited probes surrounded by randomly distributed quenchers Maria Hilczer and M. Tachiya Citation: J. Chem. Phys. 130, 184502 (2009); doi: 10.1063/1.3131169 View online: http://dx.doi.org/10.1063/1.3131169 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v130/i18 Published by the American Institute of Physics.
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THE JOURNAL OF CHEMICAL PHYSICS 130, 184502 共2009兲
Distribution of the decay rate constants of individual excited probes surrounded by randomly distributed quenchers Maria Hilczer1 and M. Tachiya2,a兲 1
Institute of Applied Radiation Chemistry, Technical University of Lodz, Wroblewskiego 15, 93-590 Lodz, Poland 2 National Institute of Advanced Industrial Science and Technology (AIST), AIST Tsukuba Central 5, Tsukuba, Ibaraki 305-8565, Japan
共Received 12 March 2009; accepted 19 April 2009; published online 11 May 2009兲 Recent development in single molecule spectroscopy enabled us to measure the decay kinetics of individual excited probes surrounded by randomly distributed quenchers. Since the distribution of quenchers around individual excited probes change from one excited probe to another, the quenching rate constant also changes from one excited probe to another. We calculated the distribution of quenching rate constants of individual excited probes theoretically and analyzed the observed distributions of quenching rate constants, which were recently measured by Lupton et al. 关J. Phys. Chem. C 111, 11511 共2007兲兴 by using single molecule spectroscopy. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3131169兴 I. INTRODUCTION
The decay kinetics of excited probes due to quenching by quenchers that are randomly distributed around the excited probes in a rigid matrix is important in various fields. A very general theory for this decay kinetics was developed by Tachiya and Mozumder1 some time ago. If the concentration of excited probes is much lower than that of quenchers, the ensemble-averaged decay kinetics of excited probes is given by1
冋
P共t兲 = exp − k0t − c
冕
册
共1 − e−k共r兲t兲dV ,
共1兲
where P共t兲 is the survival probability of excited probes at time t, k0 is the unimolecular decay rate constant of individual excited probes, c is the concentration of quenchers, k共r兲 is the rate constant for quenching of an excited probe by a quencher located at distance r, and dV is the volume element. This is a very general result. The well-known expressions2,3 for the decay kinetics of excited probes due to energy transfer via Förster and Dexter mechanisms are easily obtained by introducing the Förster and Dexter equations for k共r兲 in Eq. 共1兲. Equation 共1兲 was also applied to discuss the effect of an external electric field on electron transfer between donors and acceptors in a rigid matrix by using the Marcus equation for k共r兲.4 Recent development in single molecule spectroscopy enabled us to measure the decay kinetics of individual excited probes. For the time being we exclude the unimolecular decay of excited probes and consider only the decay via quenching. The former can be easily included later. If quenchers around one excited probe are located at r1 , r2 , . . . , rN, the excited probe decays exponentially with the N total quenching rate constant given by kq = 兺i=1 k共ri兲, where a兲
Author to whom correspondence should be addressed. Electronic mail:
[email protected].
0021-9606/2009/130共18兲/184502/6/$25.00
k共ri兲 is the rate constant for quenching of an excited probe by a quencher located at distance ri 共Fig. 1兲. Since the distribution of quenchers around individual excited probes changes from one excited probe to another, the total quenching rate constant kq also changes from one excited probe to another. In other words, if we observe the rate constant kq for individual excited probes, we should have a distribution of kq, which defines the probability 共kq兲dkq that the total quenching rate constant of an individual excited probe will take a value between kq and kq + dkq. Recently Lupton et al.5 measured this distribution experimentally by using single molecule spectroscopy for the system of individual semiconductor nanocrystals as probes and dye molecules as quenchers, both dispersed in a rigid polymer film 共50 nm thick兲 at low temperature. The objective of this paper is to theoretically calculate the distribution of quenching rate constants 共kq兲 of individual probes and analyze experimental data of Lupton et al.5 on the basis of the theoretical results. In Sec. II we present a theory to calculate 共kq兲 and calculate it in the case of energy transfer via the Förster mechanism. In Sec. III we analyze experimental data on the basis of the theoretical results. Concluding remarks are given in Sec. IV.
II. THEORY
The distribution 共kq兲 of total quenching rate constants kq is in general given by
冕 冕 冋兺 N
共kq兲 =
¯
␦
i=1
k共ri兲 − kq
册
⫻u共r1,r2, . . . ,rN兲dV1dV2 ¯ dVN ,
共2兲
where ␦共x兲 is the Dirac delta function, u共r1 , r2 , . . . , rN兲 denotes the distribution function of quenchers, and dVi is the volume element. The general method of calculating the
130, 184502-1
© 2009 American Institute of Physics
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FIG. 1. 共Color online兲 Distribution of quenchers around an individual excited probe Pⴱ and the schematic distribution 共kq兲 of the total quenching rate constant kq = 兺ik共ri兲 of individual excited probes.
distribution 共kq兲 on the basis of Eq. 共2兲 is presented in Appendix A. There is a simpler way of calculating the distribution 共kq兲 of total quenching rate constants, if the ensembleaveraged decay kinetics P共t兲 of excited probes due to quenching is known. It is obvious that the decay kinetics P共t兲 is related to the distribution 共kq兲 through P共t兲 =
冕
e−kqt共kq兲dkq .
共3兲
Therefore, if P共t兲 is known, one can calculate 共kq兲 by inverse Laplace transformation of P共t兲. If quenching occurs by Förster energy transfer to quenchers, the rate constant k共r兲 is given by k共r兲 = k0共R0/r兲6 ,
共4兲
where 1 / k0 is the lifetime of an excited probe in the absence of quenchers and R0 is the Förster radius. For quenchers that are randomly distributed in two dimensions, the ensembleaveraged decay kinetics of excited probes due to quenching is calculated as
再
P共t兲 = exp − c2D
冕
2关1 − e−k共r兲t兴rdr
冎
= exp关− ⌫共2/3兲c2DR20共k0t兲1/3兴,
共5兲
where ⌫共x兲 is gamma function. According to Eq. 共3兲, the distribution of total quenching rate constants is obtained by inverse Laplace transformation6 of Eq. 共5兲,
共kq兲 =
冉冊
1 k0 关⌫共2/3兲c2DR20兴3/2 3 3 kq
再
1/2
⫻K1/3 2关⌫共2/3兲c2DR20兴3/2
冉 冊 k0 27kq
1/2
冎
,
共6兲
where K1/3共x兲 is the modified Bessel function of the order of 1/3.
FIG. 2. 共Color online兲 Distribution 共kq兲 of total quenching rate constants of individual excited probes surrounded by quenchers distributed randomly in two dimensions 共full symbols兲. Calculations are performed for different val0 0 , where c2D = 共R20兲−1 and R0 ues of the quencher concentration, c2D / c2D = 6 nm. Open symbols show the distribution of kq = k共rnq兲 for quenching due to the nearest quencher.
On the other hand, if quenching occurs by Förster energy transfer to quenchers that are randomly distributed in three dimensions, the ensemble-averaged decay kinetics of excited probes is calculated as
再
P共t兲 = exp − c 3D
冋
= exp −
冕
4关1 − e−k共r兲t兴r2dr
册
冎
4 c3DR30共k0t兲1/2 . 3
共7兲
Inverse Laplace transformation6 of Eq. 共7兲 yields the following equation for the distribution of total quenching rate constants:
共kq兲 =
冉冊 再 冉
2 k0 c3DR30 3 3 kq
1/2
exp −
4 c3DR30 3
冊
2
冎
k0 . 4kq 共8兲
Energy transfer from an excited probe occurs to all quenchers surrounding the excited probe. However, since the energy transfer rate constant in general decreases rapidly with increasing distance, one may be able to approximate the total quenching rate constant kq by the quenching rate constant due to the nearest quencher. In Appendix B we calculated the distribution of total quenching rate constants by using the nearest neighbor approximation. The results are given by Eqs. 共B4兲 and 共B6兲 in two and three dimensions, respectively. Figure 2 shows the distribution 共kq兲 of total quenching rate constants kq of individual excited probes in two dimensions for several values of the quencher concentration ex0 0 , where c2D = 共R20兲−1 and R0 = 6 nm. The pressed as c2D / c2D distribution 共kq兲 obtained from the nearest neighbor approximation 关Eq. 共B4兲兴 is also included for comparison. As expected, at low values of kq, the distribution 共kq兲 given by Eq. 共B4兲 is considerably higher than that given by Eq. 共6兲, while at high kq the former distribution is slightly lower than
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J. Chem. Phys. 130, 184502 共2009兲
Distribution of quenching rate constants
FIG. 3. 共Color online兲 Distribution 共kq兲 of total quenching rate constants of individual excited probes surrounded by quenchers distributed randomly in three dimensions 共full symbols兲. Calculations are performed for different 0 0 , where c3D = 共4R30 / 3兲−1 and values of the quencher concentration, c3D / c3D R0 = 6 nm. Open symbols show the distribution of kq = k共rnq兲 for quenching due to the nearest quencher.
the latter one. Figure 3 shows the distribution 共kq兲 of total quenching rate constants kq in three dimensions for several 0 , values of the quencher concentration expressed as c3D / c3D 0 3 where c3D = 共4R0 / 3兲−1. The distribution 共kq兲 obtained from the nearest neighbor approximation 关Eq. 共B6兲兴 is also included for comparison.
FIG. 4. 共Color online兲 Distributions ⌽共kt兲 of total decay rate constants kt of individual excited probes surrounded by quenchers distributed randomly in two dimensions 共full line兲 and in three dimensions 共line with open symbols兲 at different quencher concentrations. R0 = 6 nm. The column plots show the normalized experimental distributions of kt.
⌽共kt兲 = ⌿共1/kt兲/k2t . III. ANALYSIS OF EXPERIMENTAL DATA A. Distributions of total decay rate constants of individual excited probes
The total decay rate constant kt of an individual excited probe is given by kt = k0 + kq, where k0 is the unimolecular decay rate constant of an excited probe. Therefore, the distribution ⌽共kt兲 of total decay rate constants of individual excited probes is given by ⌽共kt兲 = 共kt − k0兲. If the unimolecular decay rate constant k0 is distributed, the distribution of total decay rate constants of individual excited probes is given by the following convolution: ⌽共kt兲 =
冕
f共k0兲共kt − k0兲dk0 ,
共9兲
where f共k0兲 denotes the distribution of unimolecular decay rate constants of individual excited probes. Lupton’s experimental data are presented in terms of the distributions of lifetimes of individual excited probes. However, theoretically it is more convenient to consider the distributions of total decay rate constants of individual excited probes. We can convert the distribution of lifetimes to that of total decay rate constants in the following way. Since the lifetime of an excited probe is related to the total decay rate constant through = 1 / kt, the distribution of total decay rate constants of individual excited probes is related to the distribution ⌿共兲 of lifetimes through
共10兲
The experimental distributions of total decay rate constants are obtained by converting the observed distributions of lifetimes shown in Fig. 2 of Ref. 5 with the aid of Eq. 共10兲. Figure 2共a兲 corresponds to the distribution of lifetimes in the absence of quenchers. Conversion of the distribution of lifetimes in Fig. 2共a兲 gives the distribution f共k0兲 of unimolecular decay rate constants of individual excited probes. As already mentioned, the theoretical distribution of total decay rate constants of individual excited probes is given by Eq. 共9兲. Since 共kq兲 and f共k0兲 are already known, one can calculate the theoretical distributions of total decay rate constants by using Eq. 共9兲. When calculating Eq. 共9兲, we first assume that quenchers are randomly distributed in two dimensions and that 共kq兲 is given by Eq. 共6兲. However, since in the experiment of Lupton et al.5 the polystyrene film in which semiconductor nanoantennae and dye molecules are dispersed has some thickness, rigorously speaking, the spatial distribution of probes and quenchers may not be purely two-dimensional 共2D兲. So, we also consider the case in which quenchers are randomly distributed in three dimensions and 共kq兲 is given by Eq. 共8兲. Figure 4 compares the experimental distributions of total decay rate constants of individual excited probes with the theoretical distributions obtained by assuming 2D and three-dimensional 共3D兲 distributions of quenchers. Lupton et al. obtained the values of the approximate mean spatial separation 具d典 between quencher molecules by examining the 2D projection of the 50 nm thin polymer-film samples. If we assume that quenchers are dis-
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tributed two-dimensionally around a probe, the quencher concentration is estimated by c2D = 具d典−2. On the other hand, if we assume that they are distributed three-dimensionally, the quencher concentration is estimated by c3D = 具d典−2l−1, where l is the thickness of polymer-film samples. As can be seen from Fig. 4, the theoretical results based on 2D distribution of quenchers reproduce the experimental results more satisfactorily, compared with that based on 3D distribution. Agreement between the experimental results and the theoretical ones become less satisfactory when the concentration of quenchers becomes high. B. Distributions of stationary fluorescence intensities of individual excited probes
If the lifetime of an excited probe is given by , the stationary fluorescence intensity I of the excited probe is given by I = ␣ ,
共11兲
where ␣ is proportional to the radiative rate constant of the excited probe. Let ⍀共I兲dI denote the probability that the stationary fluorescence intensity of an individual excited probe will take a value between I and I + dI. The distribution ⍀共I兲 is related to ⌿共兲 through ⍀共I兲 = ⌿共I/␣兲/␣ .
共12兲
Equation 共12兲 shows that there is a simple relation between the distribution ⍀共I兲 of stationary fluorescence intensities of individual excited probes and the distribution ⌿共兲 of lifetimes of individual excited probes. This relation holds irrespective of the quenching model adopted. We checked the experimental data of Lupton et al.5 against this relation. First we determined the value of ␣ such that ⍀共I兲 calculated from ⌿共兲 in the absence of quenchers 关Fig. 2共a兲 of Ref. 5兴 by using Eq. 共12兲 will fit to the corresponding ⍀共I兲 directly measured 关Fig. 1共g兲 of Ref. 5兴. The estimated value of ␣ equals 24.5. Then we calculated ⍀共I兲 in the presence of quenchers from ⌿共兲 关Fig. 2共b兲 of Ref. 5兴 by using Eq. 共12兲 together with the determined value of ␣, and compared it with the corresponding ⍀共I兲 directly measured 关Fig. 1共j兲 of Ref. 5兴. Comparison is made in Fig. 5. As we can see, agreement between the calculated and measured distributions of ⍀共I兲 is not good, especially when the concentration of quenchers is high. IV. CONCLUDING REMARKS
The decay kinetics of excited probes due to quenching by quenchers, which are randomly distributed around the excited probes in a rigid matrix, is important in various fields. Since the distribution of quenchers around individual excited probes changes from one excited probe to another, the quenching rate constant also changes from one excited probe to another. Lupton et al.5 recently measured the distribution of total decay rate constants of individual excited probes and that of their stationary fluorescence intensities by using single molecule spectroscopy. In the present paper we calculated these distributions theoretically by assuming 2D and 3D distributions of quenchers and compared the obtained
FIG. 5. 共Color online兲 Comparison between the distributions ⍀共I兲 of stationary fluorescence intensities of individual excited probes calculated from the distribution ⌿共兲 of their lifetime 共full line兲 and the corresponding distributions ⍀共I兲 共column兲 directly measured for quencher concentrations of 0 equal to 0 and 0.442. c2D / c2D
results with the experimental results.5 For the distribution of total decay rate constants of individual excited probes, the theoretical results based on 2D distribution of quenchers reproduce the experimental results more satisfactorily, compared with that based on 3D distribution. Agreement between the theoretical results and the experimental ones becomes less satisfactory when the concentration of quenchers becomes high. For the distribution ⍀共I兲 of stationary fluorescence intensities of individual excited probes, agreement between the theoretical results and the experimental ones is not good, especially when the concentration of quenchers is high. The reason for this discrepancy may be partly due to blinking effect. It is well known7 that semiconductor nanoparticles exhibit blinking phenomenon. The distributions of lifetimes of individual excited nanoparticles shown in Fig. 2 of Ref. 5 are constructed by measuring the lifetimes of only those nanoparticles which are in bright state. On the other hand, the distributions of stationary fluorescence intensities of individual nanoparticles shown in Fig. 1 of Ref. 5 are constructed by measuring fluorescence intensities of nanoparticles over a certain measurement time. In the presence of blinking, individual nanoparticles may be partly in dark state during this measurement time, which decreases the measured fluorescence intensity. This seems to be one of the reasons the distributions of stationary fluorescence intensities directly measured have a peak at much lower intensities, compared with those calculated from the distributions of lifetimes.
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J. Chem. Phys. 130, 184502 共2009兲
Distribution of quenching rate constants
In our theoretical treatment it is assumed that the distribution of quenchers around each excited probe is circularly or spherically symmetric in the statistical average. However, rigorously speaking, this may not be true. In the experiment of Lupton et al.5 the probe is a spherical CdSe core connected on one end to a rodlike CdS structure. Therefore, the spatial distribution of quenchers around each probe may not be circularly or spherically symmetric even in the statistical average. This may affect our theoretical results, but we hope its effect is not so large. According to Fig. 2共a兲 of Ref. 5, the lifetime of individual excited probes is broadly distributed even in the absence of quenchers. More interestingly, the average lifetime 共⬃2 ns兲 is much shorter than what was previously reported by many research groups 共⬃20 ns兲. If there exist unidentified quenchers more or less randomly distributed in the polystyrene film, even in the absence of identified quenchers excited probes are quenched by these unidentified quenchers. However, the concentration of unidentified quenchers has to be very high in order to explain the observed lifetime and its width of individual excited probes in the absence of identified quenchers. As we already pointed out, if the ensemble-averaged decay kinetics P共t兲 of excited probes due to quenching is known, the distribution 共kq兲 of total quenching rate constants can be calculated by the inverse Laplace transformation of P共t兲 through Eq. 共3兲. When quenching occurs by Förster energy transfer, the analytical expression for P共t兲 is available. However, when quenching occurs by Dexter energy transfer or by Marcus electron transfer, one cannot get the exact analytical expressions for P共t兲. In these cases one can calculate the distribution 共kq兲 of total quenching rate constants by using Eq. 共A4兲 together with Eq. 共A5兲.
ACKNOWLEDGMENTS
共kq兲 = =
=
1 2VN 1 2 1 2
冕 冕
冕
⬁
e−ikqx
−⬁
⬁
冋冕
冋 冕 冋 冕
e−ikqx 1 −
−⬁ ⬁
eik共r兲xdV
1 V
dx
where c = N / V is the concentration of quenchers. It is convenient to rewrite the last equation of Eq. 共A3兲 as
共kq兲 =
1 2
冕
⬁
共A4兲
e−ikqxe−cg共x兲dx,
−⬁
where g共x兲 is given by g共x兲 =
冕
关1 − eik共r兲x兴dV.
共A5兲
The distribution of total quenching rate constants can be generally calculated by using Eq. 共A4兲 together with Eq. 共A5兲.
APPENDIX B: NEAREST NEIGHBOR APPROXIMATION
In this approximation the total quenching rate constant kq is approximated by the quenching rate constant due to the nearest quencher, i.e., kq = k共rnq兲,
共B1兲
where rnq is the position of the nearest quencher. If quenchers are randomly distributed in two dimensions with the concentration c2D, the probability w共rnq兲 that the nearest quencher will be found between rnq and rnq + drnq from the excited probe is given by9 共B2兲
With the aid of Eqs. 共B1兲 and 共B2兲 the distribution of quenching rate constants kq is expressed as
共kq兲 = w共rnq兲兩drnq/dkq兩. APPENDIX A: CALCULATION OF THE DISTRIBUTION „kq…
If quenchers are randomly distributed, u共r1 , r2 , . . . , rN兲 is given by u共r1,r2, . . . ,rN兲 = 1/VN ,
共A1兲
where V is the volume of the system. The delta function is expressed as8
␦
冋
册 冕 再 冋兺
兺 k共ri兲 − kq = i
1 2
⬁
exp ix
−⬁
i
k共ri兲 − kq
册冎
dx. 共A2兲
dx
共1 − eik共r兲x兲dV dx, 共A3兲
2 w共rnq兲 = 2c2Drnq exp共− c2Drnq 兲.
We thank Professor J. M. Lupton and Dr. K. Becker for kindly providing us with supplementary experimental data of Ref. 5.
册 册
N
共1 − eik共r兲x兲dV
e−ikqxexp − c
−⬁
册
N
共B3兲
In the case of energy transfer via the Förster mechanism, we have the following expression for the distribution of kq: −4/3 共kq兲 = 31 c2DR20k1/3 exp关− c2DR20共k0/kq兲1/3兴. 共B4兲 0 kq
If quenchers are randomly distributed in three dimensions with the concentration c3D, the probability w共rnq兲 that the nearest quencher will be found between rnq and rnq + drnq from the excited probe is given by9
冉
2 w共rng兲 = 4c3Drnq exp −
冊
4 3 , c3Drnq 3
共B5兲
so the distribution of quenching rate constants due to energy transfer via the Förster mechanism has the form
共kq兲 =
冋
册
2 4 −3/2 exp − c3DR30k1/2 c3DR30共k0/kq兲1/2 . 0 kq 3 3
Substituting Eqs. 共A1兲 and 共A2兲 in Eq. 共2兲, one obtains
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共B6兲
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M. Tachiya and A. Mozumder, Chem. Phys. Lett. 28, 87 共1974兲. T. Förster, Z. Naturforsch. A 4, 321 共1949兲. 3 M. Inokuti and F. Hirayama, J. Chem. Phys. 43, 1978 共1965兲. 4 M. Hilczer, S. D. Traytak, and M. Tachiya, J. Chem. Phys. 115, 11249 共2001兲. 5 D. Soujon, K. Becker, A. L. Rogach, J. Feldmann, H. Weller, D. V. Talapin, and J. M. Lupton, J. Phys. Chem. C 111, 11511 共2007兲. 1 2
J. Chem. Phys. 130, 184502 共2009兲
M. Hilczer and M. Tachiya
G. E. Roberts and H. Kaufman, Table of Laplace Transforms 共Saunders, Philadelphia, 1966兲. 7 P. Frantsuzov, M. Kuno, B. Janko, and R. A. Marcus, Nat. Phys. 4, 519 共2008兲. 8 L. I. Schiff, Quantum Mechanics, 3rd ed. 共McGraw-Hill, New York, 1968兲, p. 56. 9 S. Chandrasekhar, Rev. Mod. Phys. 15, 1 共1943兲. 6
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