Diffusion-controlled reactions: Analysis of quenched fluorescence decay data

Chemical Physics 136 (1989) 361-378 North-Holland, Amsterdam

DIFFUSION-CONTROLLED ANALYSIS OF QUENCHED Ranjan

REACTIONS: FLUORESCENCE

DECAY DATA

DAS and N. PERIASAMY

Chemical Physics Group, Tata Institute ofFundamental

Research, Colaba. Bombay 400 005, India

Received 8 January 1989

The rate coefficient for a diffusion-controlled reaction, according to some models, is given by the following equation: k(t) =a+ b exp(c*t) erfc(ct ‘I’) where a, b and c are constants. The method of non-linear least-squares has been used for the analysis of time-resolved quenched fluorescence decay data simulated for the above equation. The method is capable of giving unique values for the parameters a, band c under certain conditions; for example, when the exact value of the time-shift parameter, 6 is known. In the absence of an exact value for 6 it becomes necessary to evolve a procedure for the analysis of experimental data, based on the results of analysis of appropriate simulated data. The method is successfully applied for the analysis of experimental data.

1. Introduction

The Smoluchowski-Collins-Kimball model is generally considered more realistic for rapid bimolecular reactions in the absence of an intermolecular attraction/repulsion potential [ 2,3 1. The rate coefficient for this model is given by

Diffusion-limited bimolecular chemical reaction continues to be an interesting subject for theoretical study [ l- 17 1. The primary objective of the theory has been to predict the rate of the reaction which can be experimentally verified. For a few models of diffusion-limited reactions it has been possible to obtain analytic equations for the rate coefficient which is a function of time. Experimental verification of the time-dependent rate coefficient has been, till recently, limited to the simplest of all available equations, namely, the Smoluchowski equation, k(t)=a+/?t-“2.

k(t)=a+bexp(c2t)

(1.1)

03.50 0 Elsevier Science Publishers Physics Publishing Division )

(1.2)

where a, b and c are constants. Eq. ( 1.2) is also the predicted equation for ionic reactants under certain conditions [ 5,131. Eq. ( 1.1) is an approximation of eq. ( 1.2) at large t (see section 2). Eq. ( 1.2) has now been directly verified in frequency domain [ 23-25 ] and time-domain [ 261 quenched fluorescence studies. The time-resolved quenched fluorescence data obtained using time-correlated single-photon counting (TCSPC) technique has an advantage for testing assumed models since the noise associated with the data is known exactly [ 27,28 1. It is also possible to simulate fluorescence data for a particular model and the simulated data can then be tested to ensure that a particular method of data analysis is successful to extract the parameters of the model. Such tests are especially important, perhaps imperative, when new models are being tested using time-domain data. In this paper we describe a method for analysing quenched fluorescence time-domain data to fit eq.

Eq. ( 1.1) is important because several models other than Smoluchowski predict eq. ( 1.1) as a reasonable approximation. Indeed, experimental data for a wide variety of rapid reactions (several examples can be found in refs. [ 1,2,18-221) were found to be in good agreement with eq. ( 1.1). However, experimental studies which probed beyond the mere agreement of eq. ( 1.1) to the experimental data have revealed that the diffusion coefficient calculated using experimentally determined (Yand /3was generally unsatisfactory [ 18,2 1,221. This indicated a difficulty which was not easy to identify. 0301-0104/89/$ ( North-Holland

erfc(ct’/2),

B.V.

R. Das, N. Periasamy

362

/Diffusion-controlled

( 1.2 ). The method is successfully used in the analysis of simulated and experimental data.

2. Time-dependent rate coefficient 2.1. The model and analytical equations

When one considers reactions where A and B are ions, Coulombic interaction between the reactants affects the initial distribution at t=O and the dynamics of the reaction. The rate of the reaction then depends on an additional parameter, the Onsager length r, besides R, D and k,, r, = Z, ZBe2/4nekB T,

The dynamics of a rapid chemical reaction can be replaced by a physical model with the following features explicitly defined: (i) distribution of the reactants at time t = 0, (ii ) a reaction mechanism that defines the number of chemical species, (iii) reaction rate as a function of separation distance, and (iv) dynamics of all the species. An equation for the rate coefficient can then be obtained which will be a function of the parameters which define the above features. A large number of theoretical models have been considered in the literature [ 1- 17 1. For the simplest model the reaction mechanism is given by A+B+products.

reactions

(2.8)

where Z,e and Z,e are the charges on the ions, kB is the Boltzmann constant, E is the dielectric constant of the solvent and T is the temperature. It has not been possible to obtain an unrestricted solution for the rate equation for ionic reactants with CollinsKimball boundary condition. Flannery [ 51 obtains an exact solution for k(t) valid for the condition r> R when R > I r, 1, where r is the separation distance between the reactants. The Flannery equation is identical to eq. (2.2) with the following definitions for a, b and c: (2.9)

(2.1)

Considering that, (i) the spatial distribution of A and B is uniform at t ~0, (ii) the reaction rate k, ( cm3 s- ’ molecule-’ ) is isotropic and orientation independent at a separation distance r= R, and the reaction rate is zero for r> R, and (iii) microscopic molecular dynamics is determined by the isotropic diffusion coefficients (DA and DB), the dynamics of the chemical reaction is governed by three parameters: R, k, and D ( = D,+D,). The time-dependent rate coefficient derived by using the Collins-Kimball boundary condition is k(t)=a+bexp(x’)

erfc(x),

b=_t?i.(YR+a,

(2.10)

’

(2.11) % =k exP( -rc/R),

(2.12)

aT=4xDrC[exp(rC/R)-11-l.

(2.13)

Hong and Noolandi [ 4 ] have obtained tion for k(t) which is valid at “long-time” D, r,2lD),

(2.2) (2.14)

where a=k,(

an equa(t x=-r2/

1 +k,/47cRD)-‘,

b=k,( 1+4nRD/k,)-‘, c= ( 1+ kJ4xRD)D

(2.3) (2.4)

l12/R

(2.5)

and x=ct ‘/2.

(2.6)

When k, tends to infinity, one obtains from eq. (2.2) the well-known Smoluchowski equation, k(t)=4xRD[

where

1 +R/(xDt)“‘].

(2.7)

RnN = (1 +4arCD/k,yexp(rC/R)-

1.

(2.15)

The condition under which a long-time approximation of the Flannery equation leads to the HongNoolandi equation has been examined before [ 22 1. Green [ 13 ] has obtained an equation for k(t) for ionic reactions applicable for solvents in which r, 5 10 A. The Green equation (eq. (24) in his paper) is similar in form to eq. (2.2). The difference between

R. Das, N. Periasamy / Di$iuion-controlled reactions

Flannery equation and Green equation lies in the definition of the argument x in exp (x2) erfc( x) . The Green and Flannery equations become identical for r,/2R+z 1. (Note an error in the equation for x in ref. [ 13 ] : it should be rf in the denominator and not r, as given. ) Eq. (2.2) appears to be the most important equation for diffusion-limited reactions. It is significant to note that the function exp ( x2 ) erfc (x) appears in the analytic equation for k(t) in other models [ 141 also. We now examine some approximations for exp(x2) erfc( x) which will be useful either for its numerical computation or for modifying k(t) to a simpler form. 2.2. Approximation for exp(x’) erfc(x): short-time domain In the time domain in which x < 1, several reasonable approximations are available. Expanding erfc( x) as e&(x)=1--$

g x .=on!(2?2+1)

-exp(x2)

(

1- -$

>

(2.17)

(2.18)

If one expands exp (x2 ) as a Maclaurine series [ 18 1, eqs. ( 2.17 ) and ( 2.18 ) lead to simpler equations, exp(x2) erfc(x) N 1- -$

(2.19)

and k(t)=a+b-

gt112.

1+-

2x2 + (2x2)2

+ ~ 1X3X5 *** >’

1x3

(2.21)

we have for x CK1 exp(x2) e&(x)

-exp(x2)

- -$

(2.22)

and k(t)=a+bexp(c2t)-

$tli2.

(2.23)

So far there have been no experimental verification for the “short-time” approximation for k(t). “Long-time” domain. For XB 1 the asymptotic expansion of erfc(x) is [ 301

e&(x)

N

exp( -x2) XX’/=

1

exp(x2) e&(x)

1x3 (Zx2)2

(2.20)

Considering another series expansion for erfc(x) 1291,

1 = x7c’/z.

(2.25)

It is to be noted that the series in eq. (2.24) diverges for every x, but the accuracy of eq. (2.25) is better than l%forx>7. k(t) isgivenby k(t)=a+

k(t)=a+bexp(c2t)

exp(c2t).

(

Retaining the first term, one gets

and

2b - -ct’12 x1/2

x

-x2)

(2.16)

one may retain terms only up to the first power of x. Then exp(x2) e&(x)

erfc(x) = 1- Ixexp( K,,2

1-s+

2n

363

b cx’/2 t-‘12.

(2.26)

Eq. (2.26) is similar in form to that of Smoluchowski’s “all-time” k(t) for neutral reactants and to the Hong-Noolandi equation for k(t) at “long-time” for ionic reactants. “All-time” domain. The motivation for considering the “short-time” and “long-time” domain approximation for exp(x2) erfc(x) was essentially to obtain simple and reasonably accurate equations for k(t) in the respective time domains. We now consider approximate equations for exp(x2) erfc(x) valid for O
364

R. Das. N. Periasamy /Diffusion-controlled reactions

example, forx=25, exp(x2) -2.71 X 102”, e&(x) N 8.30~ 1O-274 and exp(x*) erfc(x) ~0.0225. If one were to compute erfc( x) and exp ( x2 ) independently by standard subroutines, the product exp( x2) erfc (x) may be difficult to compute because of computational limitation beyond a value for x. There are approximate analytical expressions for erfc (x) using the Chebyshev polynominal [ 32,33 ] which are of the form erfc(x) =exp( -x’)f(x).

(2.27)

These can be used to compute e&(x) with a fractional error as low as lo-l9 for the entire range 0 ix< co. According to Hastings formula [ 32 ] f(x)

=

-&

(a,

v+a2v2+w13

+a4r14+a515)

(2.28)

with q=(l+px)-‘.

e&(x)

1 -x&,(x2),

N

(2.32)

where &,(z) are rational functions. The standard IMSL (International Mathematical and Statistical Library) routine uses Cody’s equation to calculate erfc(x). We have estimated the accuracy in the calculationofexp(x2) erfc(x) byusingeqs. (2.28) and (2.30) in a comparison with the values obtained by using the standard IMSL subroutine which is modified to compute exp(x2) erfc(x) directly. It was found that in the range 0 G x< 100 the value off(x) obtained by eq. (2.30) differed from that of Cody by less than lo-’ whereas the value obtained from eq. (2.28) differed substantially from that of Cody. In the latter case the difference kept on increasing with x and at x= 100, the deviation was about20% of the actual value! We recommend eq. (2.30) as a practically useful approximation for exp(x2) erfc(x).

(2.29)

The numerical values for p, a,, u2, a,, u4 and u5 are given in ref. [ 321. Similarly, the expression for erfc( x) given by Press et al. [ 33 ] gives an exponential of a polynomial as an approximate form forf(x) in eq. (2.27) with a fractional error less than lo-‘,

3. Fluorescence decay equations When a rapid chemical reaction involves a fluorescent excited singlet state as one of the reactants, the fluorescence intensity variation with time after excitation by a &function-like laser pulse is given by I

f(x)rTexp(b,+b,T+b,T2+b3T3+bqT4 +b5T5+b6T6+b7T7+bgT8+b9T9),

Z(t)=lo

T=(1+0.5x)-’

(2.31)

and b. = - 1.2655 1223,

b, = 1.00002368,

6, =0.37409196,

b3 =0.09678418,

b, = -0.18628806,

bS =0.27886807,

b6=-1.13520398,

b,=1.48851587,

bg=-0.82215223,

b9=0.17087277.

Yet another expression for erfc(x) due to Cody [ 341 gives numerical values with a maximal relative error between 6x lo-l9 and 3x lo-*‘. The expressionforerfc(x) isintheformofeq. (2.27)forx>0.5. ForxG0.5

k(t) dt

(

(2.30)

where

-t7{‘-Co

exp

I0

, >

(3.1)

where 7. is the lifetime of the excited singlet state in the absence of the quenching reaction, Co is the bulk concentration of the quencher, which is far greater than the concentration of the fluorophore so that Co is time independent, k(t) is the time-dependent rate coefficient and IO is the intensity at t = 0. Z(t) is experimentally measured in the convoluted form as F(t) (eq. (5.1) below) and hence k(t) is verifiable. Substitutingeq. (2.2) fork(t) in eq. (3.1) one gets Z(t)=Zo exp

-bC,

--t(T,’

j exp(c2t) 0

leading to

+ucQ)

( erfc(ct’12)

dt >

(3.2)

R. Das, N. Periasamy/ Diffiion-controlled reactions

Z(t) =I0

exp

(

--t(r;’

+aco,

ct’/2

-- 2bC, c2

xexp(x’)

I

e&(x)

dx

0

>

.

(3.3)

After integration [ 5 ] one gets Z(t) =I0

exp

[

--t(.ro’ +ac,>--

9

(

X exp(c2t) erfc(ctLj2) - 1+ $

>I.

(3.4)

4. Simulation of fluorescence decay data The details of the simulation of the quenched fluorescence decay data are given below: Excitation function R ( t ) . We have chosen two different excitation functions for the simulation of quenched fluorescence data. In the first case, the function is a Gaussian, R(t)=Wexp[-(t-x)2/2a2].

(4.1)

365

tion data at 5 ps/channel which led to a significant improvement in the results. Decay equation, Z(t). The decay equation (3.4) is used for the simulation of quenched fluorescence data. The following values were used for the calculationofa, bandcineq. (3.4): D=2x10-5cm2s-‘, R=7x 10-s cm, r,= -7.2~ 10m8 cm and k,= 3x lo-“~m~rnolecule-~s ‘.Thevalueofr,was chosen to be 2.00 ns. The chosen values of D, R and Zc.are typical values for most fast chemical reactions in water at room temperature. The chosen value of r, corresponds to a case of Coulombic attractive reactants (Z,J, = - 1) in water at room temperature. Simulation was done for different values of quencher concentration, Co=O.O25,0.05,0.075 and 0.1 M. The value of IO( zA ) in eq. (3.4) will be determined when simulated emission data are normalised to a specific peak height. The value of T, and concentration were chosen so that the simulated data are similar to experimental data for cresyl violet/iodide reaction in water. Emission data, F(t). F(t) is calculated by numericalintegrationofeq. (4.2). F(ti=O)=R(ti=O)=O is the first datum and F( ti=25, 50, ... p), I

W determines the peak height. For 0=0.2 ns and x= 1.2 ns, the fwhm of the excitation function is 0.47 ns with peak at 1.2 ns. Excitation data at 0.25 ps interval were used to calculate emission data at 25 ps interval. Such a small interval for excitation data is necessary to ensure that the simulated data are realistic and comparable to experimental data for which the exciting light pulse is continuously varying in time. For deconvolution analysis the excitation data at 25 ps interval were used. It is the usual practice to keep identical time interval for excitation and emission decay data. In the second case, the excitation data are collected experimentally at 1 ps/channel resolution in the TCSPC experimental setup [ 221 using MCP-PMT (Hamamatsu 1564 U-01 ). The fwhm of this excitation data is 77 ps. The experimental data are smoothened several times in the multichannel analyser (Tractor Northern TN 7200) and the smooth data are used for the calculation of emission data at t = 25 ps. For deconvolution analysis excitation data were selected at 25 ps interval with a fwhm of 3 channels. Deconvolution analysis was also done using excita-

F(t)=

j R(s) Z(t-s)

ds,

(4.2)

0

were calculated numerically (trapezoidal rule) with

the excitation data created in the interval of 0.25 or 1 ps. Calculation of the decay data was continued until F(t) decays to O.lOhof the peak value. The emission data were normalised to a peak value of 2 x 104. The normalising constant is the simulation value for the parameter A ( =ZO) in the decay equation. Addition of noise. Smooth data of R (ti) and F( ti) were made noisy by addition of Poisson noise to resemble experimental TCSPC data. For ZV>20, Poisson distribution with a standard deviation of N ‘I2 is similar to Gaussian distribution of identical standard deviation [ 27 1. Addition of noise to R ( ti) and F( ti) follows the recommended procedure [ 27 ] according to Ni =Ni +pNf”,

(4.3)

where Ni ( > 20) is the data in the ith channel and p is the Gaussian distributed random number with a mean of zero and a standard deviation of one. p is

R. Das, N. Periasamy / Diffusion-controlled reactions

366

calculated by the Box-Muller-Marsaglia algorithm [ 271. The use of different sets of random number sequences ensures that the random noise pattern in no two excitation or emission data is likely to be identical [ 35 1.

were calculated according to the following equations: aF/&

aF -=aA aF

-

’ R(s+S)Z(t--s)

d.Y,

(5.1)

0

using excitation function or instrument response function R(t) and the intensity decay equation Z(t) given in eq. (3.4). In eq. (5.1) 6 is the time-shift parameter. The parameter 6 is important for the analysis of experimental fluorescence data to correct for certain instrumental artifacts [ 28 1. In the case of simulated data the value of S is zero. It was found that 6 as a free parameter helps the recovery of correct values for other parameters in the case of simulated data also. The rationale for the result of nonzero 6 for simulated data is explained later. The other free parameters in eq. (5.1) are to be found in Z(t) (eq. ( 3.4) ): I,, a, b and c. We found it convenient to treat UC, ( =p) and bC, ( =q) as the free parameters instead of a and b. Choosing time in nanosecond the units for p, q and c are ns-‘, ns-’ and ns-‘I*, respectively. The partial derivatives aF/aA, aFlap, aF/aq and

ds,

(5.3)

( $> -

0

x exp(x:) erfc(x,)(

aF - =

ac

I

1+ $$

>

d.s,

(5.4)

3

R(s+d)Z(t--s)

0

X exp(x%)erfc(x,)(l-x:)-l+ (

$$

>

d.s, (5.5)

where x,=c(t--s)“*.

(5.6)

aF/a6 was calculated using aF -=

as

I

R(s+S)Z(t-s)[-(t-s)]

I

a4

The aim of the analysis of simulated fluorescence data is to establish a procedure by which one can recover the values of the parameters used in the simulation. The method used by us for the analysis of data is the iterative reconvolution method by adjustment of free parameters according to the Levenberg-Marquardt procedure, until the sum of squares of the residuals changes minimally or the adjustment in each parameter is insignificantly small. This method has been described in detail [ 361 and discussed [ 27,28,37]. The method requires calculation of fluorescence data, F(t), and its partial derivatives with respect to each of the free parameter in the equation used to calculate F( t ). The fluorescence data, F(t), is calculated according to

(5.2)

0

aF - =

5.1. Method

JR(s+G)Z(t--s)

.

I

=

ap

5. Analysis of data: method and results

F(t) =

F(t) A’

F(t+d)-F(t) A

’

(5.7)

where A is one tenth of time per channel. The convolution integral in the above equations was calculated numerically by the trapezoidal rule which tends to keep the value of S close to zero, as it should be for the simulated data. The value of exp (x2 ) erfc (x) was calculated by using eq. (2.30). The choice of p, q and c as the optimisable parameters permits one to test the consistency of the quenched fluorescence data to the basic form of the predicted equation. It is not necessary to give the value of Co as input. The values of D, R and k, are calculated from p, q and c. On the other hand one may also choose D, R and k, as the optimisable parameters. This approach is less satisfactory because the experimental error in the mandatory input value for Co may disproportionately influence the values of R, D and k.. In the method of non-linear least-squares by lterative reconvolution, the global minimum in the chisquare surface is sought to be attained. In addition to

R. Das, N. Periasamy / Di’iion-controlled

the global minimum it is likely that there are several local minima especially when chi-square is a function of several parameters. In addition to local minima, the likelihood of the global minimum occurring in a shallow valley( chi-square is invariant for one parameter or ratio of two parameters) or a plateau (chisquare is invariant for two or more parameters) is also to be examined in fitting the data to a function such as eq. (3.4). The global minimum in the chisquare surface is recognised by the conventional quantitative statistical tests [ 28,381: calculation of’ reduced chi-square (also referred to simply as chisquare or x2), Durbin-Watson test parameter (DWP) to check serial correlation and standard normal variate of ordinary Runs test (ZRUN). In case of a suspected local minimum (usually indicated by a value of reduced chi-square >> 1) the search for the global minimum is repeated by starting with different initial values for the parameters. As a routine, data analysis is repeated for two or more sets of initial values. Negative values for A, p, q and c are physically meaningless for the model to be tested and such negative values, whenever they occur, are modified to zero for the next iteration. The search is terminated when the sum of squares of residuals does not change by more than 0.1 (the fractional change in reduced chi-square is less than 0.00 1) or if the fractional change in the values of parameters is less than 1O-) in successive iterations. In view of the substantial computation time the maximum number of iterations (NITER) is limited to 20. When NITER exceeds 20 the search is stopped and the final optimised values for the parameters are accepted if the reduced chi-square is acceptable. The search is also stopped when I, the multiplication factor used to modify the diagonal elements of the Hessian matrix exceeds IO. The starting value for 1 is lo-* which is either decreased by 3 if the chi-square decreases or increased by 10 if the chi-square increases in successive iterations. It is important to estimate the uncertainty (standard deviation) associated with the optimised values of the parameters. It is well known [ 39,401 that statistically significant confidence tests are time consuming even for simpler decay equations such as a biexponential function. We adopt, therefore, the simple method of estimating the uncertainties using the

reactions

361

diagonal elements of the error matrix [ 36,37 1. 5.2. Analysis of simulated data

The fluorescence data simulated for the Gaussian excitation function and for the following values of parameters were chosen for extensive tests: 6=0, A=3.56,p=0.635 ns-‘, q= 1.892 ns-’ and c=4.601 ns- I/*. This corresponds to Co = 0.05 M for the values of D, R, r, and k, given earlier (section 4 ) . Table 1 shows the results of analysis for eight different sets of initial values for p, q, c, A and 6. The values were chosen arbitrarily except the eighth set for which the initial values were nearly equal to the values used for simulating the fluorescence data. In all cases, except the sixth, convergence in the search was achieved after a few iterations. An iteration is counted only if the set of corrections for the values of parameters reduces the chi-square. In the case of the sixth set the choice of the initial values is extremely poor and a second iteration was not possible. In the case of the third set the value of chi-square was high and hence this set of optimsed values ought to represent a local minimum. In all the other six cases convergence was attained satisfactorily and the value of chi-square is in the range 0.881-0.888. The values of DWP and ZRUN are also in the acceptable range. Fig. 1 shows the excitation curve, emission curve and the distribution of weighted residuals for all the six cases. The distributions of residuals are acceptably random in all the six cases. It is reasonable, therefore, to expect that the six sets of optimised values for 6, A, p, q and care in the vicinity of the global minimum. Based on the criteria of qualitative and quantitative tests on the randomness of weighted residuals all six sets are acceptable. An examination of the six sets of optimised values for the parameters reveals puzzling features. The values of A and p are not badly off from the values used for the simulation. The relative uncertainties associated with these parameters are also small. In contrast, the optimized values of S, q and c are widely varying with relative uncertainties often exceeding their values. This indicates that the information content in the fluorescence data is not sufficient to give unambiguous values for S, q and c. In addition correlation among the parameters (especially S, q and c) may be responsible for a large, flat valley in the chi-

R. Das. N. Periasamy /Diffusion-controlled reactions

368

Table I Results of analysis of simulated fluorescence data using Gaussian excitation function and eq. (3.4) for I( I): p=O.635 ns-‘, q= 1.892 ns-’ andc=4.601 ns-‘/2. Peakcount=Zx IO4 No. p, q. c, A, 6 a’

Comergence b,

6 (ps)

A ‘)

B(20)

0.07 (1.8) 0.9 (2.4) -12.2 (0) 1.16 (2.8) 0.88 (2.37) 65.7 (0) 0.82 (2.32) 0.43 (1.98)

3.53 (0.11) 3.59 (0.21) 3.14 (0) 3.62 (0.29) 3.59 (0.208) 2.97 (0.056) 3.59 (0.198) 3.56 (0.141)

I

0.5, os,os,

2

1.0, 1.0, 1.0, 5.0, 0.0

A(l0)

3

0.2, 1.0, 5.0, 1.0, 0.0

A(6)

4

0.5, 5, 5, 5, 10.0

A(5)

5

1,2,3,4,5

A(l0)

6

5,4,3,2,

7

5,5,5, 5,O

A(12)

8

0.635, 1.892,4.601, 3.56,0.0

A(3)

5,0.0

1

C(l)

P c’

4 =’

c c’

(ns-‘)

(ns-‘)

( ns-‘I’)

0.640 (0.024) 0.648 (0.021) 0.845 (0.005) 0.650 (0.020) 0.648 (0.021) 0 (0.001) 0.647 (0.021) 0.644 (0.023)

1.58 (1.30) 2.66 (5.35) 2.06 (0.017) 3.46 (11.2) 2.62 (5.10) 0 (0.0006) 2.52 (4.59) 1.92 (2.18)

3.88 (3.85) 6.95 (15.0) 0.0 (0.005) 9.15 (31.0) 6.82 (14.3) 18.1 (0.007) 6.54 (12.9) 4.85 (6.29)

b=

2.0 ns, LO ps, A= 3.56,

Chisquare

DWP

ZRUN

0.888

1.81

- 0.403

0.882

1.82

-0.641

3.46

0.45

- 10.0

0.881

1.82

-0.90

0.883

1.82

-0.64

5.9x lo4

0.007

- 15.5

0.883

1.82

-0.641

0.885

1.81

-0.641

‘) Initial values for the parameters; units as given in the caption. b, Convergence attained: A means that the decrease in the sum of squares of weighted residuals is minimal. B means NITER > 20 and the optimised values at the 20th iteration are acceptable. C means i exceeded 10. The number of iterations (NITER) is given in parentheses. ” The value in parentheses is the estimated uncertainty. Zero for the value of parameter indicates a negative value. Zero for the uncertainty indicates that the diagonal element of the error matrix is negative.

square hypersurface along S, q and c axes. In such a situation it is impossible to get reliable values for the parameters by the non-linear least-squares method. It is interesting to note that in the case of the eighth set (table 1) for which the starting values are nearly equal to the values used for the simulation of data, the optimised values are only marginally different from the expected values. The relative uncertainties in 6, q and c do not differ significantly from the other five sets of starting values. The marginal shifts in the optimised values of parameters from the true ones and the large relative uncertainties in S, q and c are definitely due to the random noise which reduces the information content in the data. The information content in the fluorescence data can be increased by increasing the signal-to-noise ratio. This is achieved by increasing the peak count in the fluorescence data and maintaining Poissonian noise characteristics. A set of excitation and quenched fluorescence data were simulated with a peak count of 1 x lo6 for the same values of D, R, r,, k, and Co. The pattern of noise and the number of data points

in the simulated fluorescence data (including rising edge and decay up to 0.1% of the peak value) was kept identical to that of the low-precision (peak: 2 X 1O4 counts) fluorescence data for which the results are given in table 1. This helps to make an objective comparison with the results given in table 2 for the high-precision data. As seen in table 2 all the six trials led to satisfactory convergence and the sets of optimised values for S, A, p, q and c are consistently similar in all except one, in contrast to the wide variation observed in table 1. The relative uncertainties.in 8, q and c are also less. However, the optimised values of S, q and c are still significantly different from the expected values, and hence a straightforward analysis of high-precision data cannot be relied upon to yield correct results. It is worth noting that in one case (No. 1) the optimised values for 8, A, p, q and c are satisfactorily close to the expected values even though the value of chi-square is relatively large, but the value of ZRUN is excellent. It appears from the result that ZRUN may be a better criterion for selection than chi-square for high-precision data. More

R. Das, N. Periasamy / Di’ion-controlled

Fig. 1.The simulated excitation and emission data usingeqs. (4. I ) and (3.4) are shown in the lower part. See text for the details. Deconvolution analysis using six different initial values (see table 1) for the five free parameters resulted in randomly distributed weighted residuals which are shown in the upper part. Ap proximately 94% of the weighted residuals were in the range from + 2 to - 2, in all cases. The calculated emission curve for one case is also shown (smooth emission curve) in the lower part.

extensive analysis on simulated data is required to establish this point. Analysis of the low-precision simulated data (see table 1) is then carried out by keeping one of the three weak parameters, S, q and c, at a fixed value. By keeping one of these parameters fvted it was found that the optimisation of the remaining four parameters proceeds rapidly (fewer iterations) and the scatter in the optimised values of parameters is less for several trials. A priori it is not possible to assume a value for

369

and c whereas it is possible in principle to estimate a value for S. For example, 6% 0 in experiments using MCP PMT. Hence, we choose 6 as a fixed parameter for the analysis. Table 3 gives the results obtained when the simulated data are analysed for fixed values of 6: 0, 5 and - 2 ps. For each value of S, three trials were made with different sets of initial values for A, p, q and c. For CL 0, the optimised values and statistical parameters are consistently closer irrespective of the choice of the starting values. Moreover, the optimised values were in good agreement with the expected values and with the relative uncertainty in cx 50% and in qz 40%. For 6= -2 ps, the optimised values and relative uncertainties were reproducible for different starting values. In this case the optimised values were far off from the expected values and the statistical test parameters indicate that randomness of residuals is poor in comparison to those obtained for 6~0 choice. For 6= 5 ps, the optimised values of q and c were widely fluctuating with poorer randomness of residuals. This analysis reveals that unless one knows the correct value of 6 with an accuracy of a picosecond or less it is not possible to optimise the values of other parameters using low-precision data. The data analysis with 6 as a fixed parameter was repeated for various values of S in the range from - 10 to + 10 ps for one set of starting values. The optimised values ofp, q, c, chi-square and DWP are plotted against 6 in fig. 2. It is seen that variation of chi-square is parabolic in shape and chi-square is less than 1.2 for a large span of 6 values. In the region in which chisquare is minimum (0.88 < chi-square < 0.95 ) the value of 6 varies from - 3 ps to 5 ps. The DWP shows a similar variation with S, with a flat maximum in the region where chi-square has a flat minimum. In this region of 6, - 3 0, respectively, besides erroneous values for q and c for these values of 6. q

TIME (NS)

reactions

R. Das, N. Periasamy / Difffusion-controlled reactions

370

Table 2 Results of analysis of high-precision simulated fluorescence data: peak count in excitation and emission data= 106. See table 1 for the values of p, q, c, A, 6 and r. for the simulation No.

P> 4,

c,A, 6 ”

Convergence b,

6 (PS)

A ”

0.66 (0.28)

3.57 (0.02) 3.63 (0.037) 3.63 (0.038) 3.63 (0.037) 3.63 (0.036) 3.63 (0.038)

1

0.5,0.5,0.5,5,0.0

R(20)

2

1, 1, 1, 5,0.0

R(20)

3

5, 5, 5, 5, 10

A(6)

4

1,2,3,4,5

R(20)

5

5, 5, 5, 5,O

R(20)

6

0.635, 1.892,4.601, 3.56,O

.4(18)

1.32 (0.38) 1.34 (0.38) 1.32 (0.38) 1.31 (0.37) 1.34 (0.37)

P c,

4 c’

c c’

(ns-‘)

(ns-‘)

(ns-‘12)

0.634 (0.003) 0.640 (0.003) 0.641 (0.003) 0.640 (0.003) 0.640 (0.003) 0.641 (0.003)

1.96 (0.29) 3.27 (1.21) 3.33 (1.27) 3.26 (1.19) 3.23 (1.16) 3.34 (1.28)

4.74 (0.81) 8.30 (3.22) 8.46 (3.38) 8.27

(3.19) 8.20 (3.11) 8.49 (3.42)

Chisquare

DWP

ZRUN

0.892

1.75

-0.23

0.853

1.83

- 1.35

0.852

1.83

- 1.35

0.853

1.83

-1.35

0.853

1.83

- 1.37

0.852

1.83

- 1.35

a) Initial values of parameters. b, Mode of convergence and number of iterations. See table 1 for explanations. ‘) Value in parentheses is the estimated uncertainty. Table 3 Results of analysis of simulated fluorescence data (see table 1 for details) when 6 was used as a fixed parameter. Peak count = 2 X 10“ No.

Convergence b,

A ”

0.5,0.5,0.5,5,0.0-

Nl6)

1.0, 1.0, 1.0,5,0.0 -

A(7)

1,2,3,4,0.0-

A(7)

0.5,0.5,0.5, 5,z

R(20)

1.0, 1.0, 1.0, 5.0, _5

A(l3)

1,2,3,4, I

R(20)

0.5,0.5,0.5, 5,_2.0

A(l6)

1.0, 1.0, 1.0,5.0, -2.0

A(7)

1,2,3,4,

A(5)

3.55 (0.04) 3.55 (0.04) 3.55 (0.04) 3.77 (0.12) 3.82 (0.19) 3.79 (0.14) 3.45 (0.02) 3.45 (0.02) 3.45 (0.02)

P, 4,

c,4 6 ”

-2.0

P =)

4 c’

c c’

(ns-‘)

(ns-‘)

( ns-‘lZ)

0.647 (0.014) 0.647 (0.014) 0.647 (0.014) 0.630 (0.010) 0.632 (0.007) 0.631 (0.009) 0.624 (0.023) 0.624 (0.023) 0.625 (0.023)

1.92 (0.83) 1.82 (0.82) 1.92 (0.83) 10.6 (28.5) 1.0x lo5 (3.5X 108) 18.2 (92.4) 0.976 (0.166) 0.976 (0.166) 0.976 (0.167)

4.96 (2.58) 4.95 (2.57) 4.95 (2.57) 25.3 (69.4) 2.5 x 10’ (8.4x 10’) 43.7 (224) 2.09 (0.67) 2.09 (0.67) 2.09 (0.67)

Chisquare

DWP

ZRUN

0.886

1.81

-0.65

0.886

1.81

-0.65

0.886

1.81

-0.65

0.954

1.69

-2.13

0.941

1.71

-1.60

0.947

1.70

-2.33

0.919

1.75

- 1.38

0.919

1.75

- 1.38

0.919

1.75

- 1.38

a) Initial values of parameters. 6 is held fixed at the value underlined. b, Mode of convergence and number of iterations. See table 1 for explanations. ‘) The value in parentheses is the estimated uncertainty.

Further tests were carried out with three other sets of simulated fluorescence data for which the simulation values were the same but the random noise pattern was different. The results of analysis for these

data followed a similar trend. The optimised values are c=4.96f2.58, 4.24k1.89, 4.08k1.85 ns-‘/2, qz1.98kO.83, 1.74+0.61,1.61f0.57ns-‘and&O, -0.5 and - 1.0 ps, respectively. The relative uncer-

R. Dar, N. Periasamy/ Dijiuion-controlledreactions

371

6.0 6.0

l

c

0

q (K’)

(nZ %

4.0

I.6 I.4

t P

I.2

1 -10

L

-6

I

-6

1

4

1

1

1

-2

0

2.

1

1

1

4

6

6

I

Fig. 2. Results of the fit of eq. (3.4) to the simulated data. The variation of the optimised values (p, g and c) and statistical test parameters (chi-square ( o ) and DWP ( l ) ) with timeshift parameter, 6 are shown in different panels. The horizontal arrows indicate the values of p, q and c used in the simulation of data. The error bars are the calculated uncertainties, which exceed the values for q and c for 635 ps. The vertical arrows in the top panel indicate that the values for q and c (given within bracket) are outside the boundary.

tainties are similar as before. The optimised values of A and p are of course much closer to the expected values with far less uncertainties. Based upon the above analysis of simulated data we conclude that for the level of precision in the data (peak 2 x 104) and At = 25 ps, the best result obtainable will necessarily have fixed relative uncertainties for q and c: = 40% for q and x 50% for c. This approximate quantification of the relative uncertainty

in q and c may serve as the basis for selecting the appropriate set of optimised values of p, q, c, A and S, in addition to the established criteria of randomness of weighted residuals. We now examine the validity of this proposition with quenched fluorescence data simulated for other concentrations of quencher. Quenched fluorescence data were simulated for different values of Co (0.025, 0.075 and 0.1 M). All other parameters are the same as in the case of

372

R. Das, N. Periasamy / Dijiision-controlled reactions

the analysis of quenched fluorescence data simulated using the Gaussian excitation function for which fwhm is 470 ps. Such pulse widths are usually obtained in TCSPC experiments using high-gain photomultipliers such as XP202OQ [ 281. The results presented above show that deconvolution analysis using such a broad instrument response function can be successful in the verification of eq. (3.4) for experimental data. In section 5.3 we demonstrate this with the experimental fluorescence data. We now turn to the results obtained in the analysis of fluorescence data using the experimentally derived excitation function with a fwhm of 77 ps (see section 3 ). The simulation values for the parameters are identical and the time per channel is 25 ps for both excitation and emission data. The results of the deconvolution analysis followed a pattern similar to the one obtained for the “Gaussian-derived” data (tables l-4 and fig. 2). In the upper part of table 5 we give the final results for four concentrations; that is, the optimised values of S, A, p, q and c which are close to the expected values (simulation values)after analysis of data by varying 6 as a fixed parameter with an interval of 0.5 ps. The results can be directly compared with those given in table 4. The values for A and p are very close to the expected values and the relative uncertainties are small. The values of q and c are reasonably close to the expected values with relative uncertainties of 20-40°h and 25-50%, respectively. The relative uncertainties in q and care slightly less in this case. Another common feature between tables 4 and 5 is the deviation of the value for S,,,

Co = 0.05 M. The range of data for analysis (including the rising edge and decay up to 20 counts) varied with concentration: 360 for Co=O.O25 M, 2 10 for Co=O.O75 M and 170 for Co=O.l M as against 260 for Co = 0.05 M. The results of the analysis for these sets of quenched fluorescence data proceeded in a fashion similar to that for Co = 0.05 M, discussed extensively above. From an analysis in which 6 was varied as a fixed parameter at an interval of 0.5 ps, it was possible to pick out the set of optimised values for A, p, q and c which were closer to the expected values. Table 4 shows the collected results and the expected values for 8, A, p, q and c for each concentration. It is observed that the optimised values of A and p are in excellent agreement with the expected values. The values of q and c are somewhat scattered, but reasonably close to the expected values. (It is possible to obtain better values for q and c if analysis had been carried out for still smaller interval in 6 rather than at 0.5 ps.) It is to be noted that the relative uncertainties in q and c are large at all concentrations. The variation in the relative uncertainty of c is 4060% and that in q is 30-50% for all concentrations. It is significant to note also a trend in the deviation of S,,, from the expected value of zero with concentration. This shift may be understood as a compensation for the error in the numerical calculation of the convolution integral when Z(t) has a fast component [ 35 1. This shift is minimised if the excitation data interval is shorter than the emission data interval of 25 ps/channel (see below). The results given in tables l-4 were obtained by

Table 4 Values of 6, A, P, q, c used for the simulation of data using Gaussian excitation function and results obtained in the analysis of data for different values of Co. The values in parentheses are the uncertainties. Peak count = 2 X 1O4 cQ

W)

Simulation values a)

Results a)

6

A

P

4

c

6opt

A

P

4

C

0.025

0.0

3.01

0.3175

0.946

4.601

-1.0

0.05

0.0

3.56

0.635

1.892

4.601

0.0

0.075

0.0

4.12

0.9525

2.838

4.601

0.5

0.1

0.0

4.706

1.27

3.784

4.601

2.0

3.02 (0.02) 3.56 (0.04) 4.12 (0.05) 4.74 (0.07)

0.318 (0.008) 0.639 (0.015) 0.940 (0.027) 1.26 (0.04)

0.847 (0.408) 1.81 (0.65) 2.58 (0.67) 3.87 (1.09)

4.02 (2.37) 4.40 (1.98) 3.95 (1.41) 4.55 (1.74)

a) Units: 6 (ps),p (ns-‘), q (ns-I) andc (ns-1/2).

X2

DWP

1.08

2.06

0.93

2.08

0.95

2.01

0.91

2.00

R. Das, N. Periasamy/ Difiion-controlled reactions

373

Table 5 Values of S, A, p, q and c used for the simulation of data using experimental excitation function and the results obtained in the analysis of simulated data. The values in parentheses are the uncertainties. Peak count = 2 x IO“ cQ

(M)

Simulation values a) 4

c

excitation data: 25 ps/channel 0.025 0.0 12.88 0.3175

0.946

4.601

0.0

0.05

0.0

13.81

0.635

1.892

4.601

0.5

0.075

0.0

14.64

0.9525

2.838

4.601

1.0

0.1

0.0

15.47

1.27

3.784

4.601

2.0

6

A

Results ‘)

P

6OP(

excitation data: 5 ps/channel 0.025 0.0 12.88 0.3175

0.946

4.60 1

-0.25

0.05

0.0

13.81

0.635

1.892

4.601

-0.25

0.075

0.0

14.64

0.9525

2.838

4.601

0.25

0.1

0.0

15.47

1.27

3.784

4.601

0.25

a) Units: 6 (ps),p (ns-I),

A

P

4

12.9 (0.08) 13.7 (0.1) 14.7 (0.12) 15.7 (0.13)

0.32 (0.01) 0.64 (0.01) 0.94 (0.02) 1.21 (0.03)

0.85 (0.31) 1.67 (0.4) 2.66 (0.5) 4.02 (0.63)

4.1 (1.3) 4.1 (1.1)

12.8 (0.1) 13.7 (0.1) 14.6 (0.1) 15.5 (0.1)

0.32 (0.07) 0.64 (0.01) 0.95 (0.02) 1.22 (0.04)

0.96 (0.45) 1.85 (0.55) 2.77 (0.61) 3.72 (0.63)

4.8 (2.7) 4.7 (1.8) 4.5 (1.3) 4.1 (1.0)

X2

DWP

1.16

1.98

1.36

1.69

1.11

1.60

1.45

1.53

1.03

2.16

1.13

1.93

0.86

1.93

0.99

1.87

C

(Y)

q (ns-I) and c (ns-“2).

from the expected value of zero. The cause of this deviation is now examined. It is well known that an experimental fluorescence decay is derived from the light pulse (or a hypothetical instrument response function) which varies continuously with time. Hence, the calculation of F(t) by eq. ( 5.1) will be more accurate if R ( t) is available as an analytic equation. It was shown [ 351 that deconvolution analysis may lead to serious discrepancies when rapid decay components are present in the decay equation if appropriate corrections are not applied. The method of linear approximation of R(t) data suggested in ref. [ 35 ] is not suitable if eq. (5.1) is not integrable for the chosen I(t), such as eq. (3.4). We now adopt an alternative method by using R(t) data at a smaller time interval compared to the time interval for the emission data. Such a method can be readily implemented in the deconvolution analysis. The results obtained by using excitation data at 5 ps/channel for the deconvolution of simulated fluorescence data at 25 ps/channel are given in the lower part of table 5. Significant improvements in the results (chi-square, DWP and SO,,) are observed when

compared with the results given in the upper part of table 5. The relative uncertainties in q and c remain unchanged. It is gratifying to note that S,,,, is very close to the expected value of zero for all concentrations. This result indicates that if one knows the exact value of &opt,then the values of all other parameters can be estimated from an experimentally determined quenched fluorescence decay for a reaction for which k(t) is given by eq. ( 1.2). Acquisition of excitation and emission data at different time resolutions is easily implemented in the TCSPC experiment, but a value for SO,,has to be judiciously chosen. 5.3. Analysis of experimental fluorescence data The method of data analysis tested for the simulated quenched fluorescence decay data can be directly applied to experimentally measured fluorescence data for rapid bimolecular fluorescence quenching reactions. If the chosen reaction system follows the theoretical model which predicts eq. (3.4) as the decay equation, then an analysis of the experimental decay data of precision and range similar to

R. Das, N. Periasamy / DiJiision-controlled reactions

374

0-

C (ns-1/2) 0 q (ns-‘) l

6-

,,

0 & u

4-

P

Z-

Fig. 3. The results of the tit of eq. (3.4) to the experimental quenched fluorescence data (CV/KI(49.3 mM)/HzO). The variation of the optimised values (p, g and c) and statistical test parameters (chi-square (o ) and DWP (a ) ) are shown in different panels. The variations are similar to those observed in the case of simulated data. See the caption to fig. 2 for an explanation of error bars, vertical arrows and the values in brackets.

the simulated data ought to give results similar to those described in section 5.2. In ref. [26], the method has been applied to the analysis of data in the investigation of two diffusion-limited fluorescence quenching reactions involving ionic reactants in water: cresyl violet cation(CV 590 and its analogue CV 585) and potassium iodide. It was demonstrated that the values of p, q and c obtained from the experimental fluorescence data at various quencher concentrations were in good agreement with the theoret-

ical expectation that (i) p and q are linearly dependent on concentration and (ii) c is concentration independent. In addition, it was possible to obtain the values for D, R and k. by using the value of r, calculated according to eq. (2.8 ) . In ref. [ 26 1, an MCP-PMT was used for which the fwhm of instrument response function was x 110 ps. The results of analysis of simulated data indicated that a wider instrument response function, such as the one obtained using XP 20204 tube, is not a hindrance to test the

R. Das. N. Periasamy / Di$‘iuion-controlled reactions

315

Table 6 Values a) of S, A, p, q and c obtained in the analysis of experimental data. The values in parentheses are the uncertainties. Peak count =2 X IO4 CQ

(mM)

A

P. 4, c, A =)

28.1

0.5, 1, 1, 5

49.3

0.5, 1, 1,s

16.6

1, t,5,5

99.1

1, 3, 5, 5

a) Units: 6 (ps),p (ns-I), q (m-r), b, See table 1 for explanations.

-2.0 2.0 10.0 9.0

5.06 (0.04) 5.83 (0.05) 6.49 (0.07) 6.53 (0.07)

P

4

C

(ns-‘)

(ns-‘)

(ns-‘I*)

0.334 (0.07) 0.562 (0.01) 0.802 (0.02) 1.08 (0.025)

0.755 (0.367) 1.51 (0.56) 2.72 (0.73) 3.02 (0.75)

3.779 (2.22) 4.68 (2.10) 4.88 (1.62) 4.47 (1.47)

Chisquare

DWP

1.26

1.73

1.46

1.62

1.64

1.60

1.72

1.48

c (ns-1/2).

validity of eq. (3.4). We give below the results of an analysis of quenched fluorescence decay data of cresyl violet (CV 590)-potassium iodide-water system measured with a XP202OQ tube. The TCSPC experimental set up using XP202OQ has been described earlier [ 221. With XP202OQ the instrument response function has a fwhm of x 280 ps for the excitation wavelength in the region of 580620 nm. The excitation and emission data were collected at 21 ps/channel resolution. The quenched fluorescence data for cresyl violet (Lambda Physik) in water were obtained for various concentrations of potassium iodide: 28.1,49.3,76.6 and 99.1 mM. The fluorescence decay data for Co = 49.3 mM were analysed by keeping 6 as a fixed parameter and the results are plotted versus S in fig. 3. The variations of chi-square, DWP, p, q or c with 6 are similar to those observed for the simulated data (fig. 2). It is observed that for 6= 2.0 ps, the relative uncertainty in c is z 50% and hence S,,,, is taken to be 2.0 ps. The results of analysis of the quenched fluorescence data for other concentrations are similar to those observed for Co = 49.3 mM. The optimum value for 6 was chosen on the basis of the criterion established using the simulated data; that is, the relative uncertainty in c is in the range of 40-6OW, and the value of c itself is concentration independent. (It is to be noted that in ref. [ 26 1, the criterion adopted for selecting optimum S is based on an average value x 50% for the relative uncertainty in c for all concentrations. This criterion leads to slightly larger (and smaller) S,,, values at higher (and lower) concentration and consequently the optimised value of c tends to in-

crease with Co. It was shown, however, that for a value of c> 5.2 ns- ‘I* a physically realistic value for R was not to be found. The criterion used in this paper is more appropriate since c is a concentration-independent constant according to the model being tested. ) The results obtained for the experimental data for all concentrations are given in table 6 and fig. 4. The non-random distribution of weighted residuals obtained by fitting the fluorescence decay for Co = 49.3 mM for a one-exponential decay equation is also shown at the top of fig. 4 for a comparison with other residual distributions. The optimised values of p, q and c given in table 6 are plotted against Co in fig.5 The values for a and b were obtained from the slopes of p versus Co and q versus Co plots, respectively. The values of a and b so obtained, and that of c are ~~1.77~ lo-” cm3 s-’ molecule-‘, b=5.Ox lo-” cm3 s-’ molecule-’ and ~~3.8 to 4.9 ns-‘I*. These values compare very well with those obtained using MCP-PMT for the same system (CV 590-iodide in water): a= 1.9x lo-” cm3 s-l molecule-‘, b=4.75 x lo-” cm3 s-’ molecule-’ and ~~3.5-5.0 ns-‘I*. These values of a, b and c have led to reasonable valuesforR,Dandk, [26].

6. Conclusions The method of analysis of quenched fluorescence decay data described in this paper is suitable for verifying the “all-time” equation (eq. ( 1.2) ) predicted to be the rate coefficient for diffusion-controlled re-

376

R. Das, N. Periasamy /Diffusion-controlled reactions

21 PS/CH

TIME (NS) Fig. 4. Quenched fluorescence decay curves of cresyl violet in water (lower panel) for four concentrations of the quencher KI: 28.7 mM (A), 49.3 mM (B), 76.6 mM (C), and 99. I mM (D). Curve E is the excitation function for one emission decay, namely, curve A. The smooth lines passing through the emission decay curves are the calculated curves after fitting the decay data using eq. (3.4) for the values of S.,,,, A, p, q and c given in table 6. The distributions of the weighted residuals for all the emission decays (A, B, C and D) are shown in the middle panel of the figure. The values for chi-square and DWP for the fitting are given in table 6. The range for fitting was chosen to include the significant part of the rising edge (2 10 ps before the peak of emission) and decay up to 20 counts, which is 0.1% of the peak value. The excitation and emission wavelengths are 585 and 620 nm, respectively. The residual distribution shown in the top panel is the result of fitting the emission decay curve B (Co=49.3 mM) to a one-exponential decay function, I(t) =A exp( -t/r), with time-shift optimization. The results for this fit are as follows: lifetime, r=0.864 ns, pre-exponential factor A= 5.30, time-shift = - 7.1 ps, reduced chi-square= 3.86 and DWP=0.614.

actions of neutral [ 2,18 ] or ionic reactions [ 5,131. A suitable approximation for exp (x2) -erfc (x) has been used and the method of non-linear least-squares is successful in verifying eq. ( 1.2) for experimental sys-

terns. The emphasis in our method has been to verify the validity of eq. (3.4) to experimental data and to obtain values for p, q and c in eq. (3.4) without any preconditions. The precision in the data and, possi-

R. Das, N. Periasamy/ Diffwion-controlledreactions

6-

317

eral subroutines of which are used in the program for the deconvolution of non-exponential decays. The authors acknowledge the financial support of Department of Science and Technology, Government of India for setting up the Unit on Chemical Dynamics and Picosecond Spectroscopy.

. c o 9

References

CO(M) Fig. 5. The plots ofp (ns-I), q (ns-‘) and c (ns-1/2) against quencher concentration Cq for cresyl violet-KI system in water.

bly, numerical errors involved in the calculation of the convolution integral using data with a large time per channel does not permit a unique tit of eq. (3.4) to either experimental data or simulated data when five free parameters are to be optimised. The analysis of the data becomes considerably unique if the value for the time-shift parameter 6 is known exactly. In the absence of an exact value for 8, it becomes necessary to develop a working procedure by resorting to analysis of simulated quenched fluorescence data with the level of precision and range of data comparable to the experimental data. The working procedure described in section 5.2 may be useful for ionic reaction systems with ZAAB= - 1 in water, such as, cresyl violet and potassium iodide. Further work is in progress to identify more reactions (neutral and ionic) for which eq. ( 1.2) may be applicable.

Acknowledgement

The authors acknowledge helpful discussions with Professor B. Venkataraman, Dr. S. Doraiswamy, Professors G.R. Fleming and J.W. Longworth. The authors wish to thank Professor De Schryver and Dr. N. Boens for the generous supply of the program for the deconvolution for multi-exponential decay, sev-

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