J . Phys. Chem. 1990, 94, 1661-1666
1661
Comparison of the Maximum Entropy and Exponential Series Methods for the Recovery of Distributions of Lifetimes from Fluorescence Lifetime Datat Aleksander Siemiarczuk, Brian D. Wagner, and William R. Ware* Department of Chemistry, Photochemistry Unit, University of Western Ontario, London, Ontario, Canada N6A 5B7 (Received: May 12, 1989; In Final Form: August 22, 1989)
The maximum entropy method and a new more robust exponential series method are described and tested for the recovery of underlying fluorescence lifetime distributions derived from digital fluorescence data. Both real and simulated data are used in this study. Except for the resolution of single-lifetime systems, it is found that the two methods are similar in their resolving powers. The problem of differentiating between the case of a continuous distribution and three discrete components is discussed. The effects of radio-frequency noise, background subtraction, time base, and data precision on the recovery of distributions are described.
Introduction With the advent of the single-photon time correlation (SPTC) methodl for measuring fluorescence decay, it became routine to fit to one- or two-exponential functions by iterative least-squares techniques or some other numerical method. The natural evolution of the SPTC technique soon gave rise to data that failed to fit two exponentials and three were invoked, and it is now commonplace to see data in the literature analyzed with this six-parameter function. When three components fail to fit, one is tempted to try four, a very powerful fitting function indeed. Data collected at a low level of precision, measured for example by the counts in the peak channel (CPC), may be adequately fit by a two-exponential function, but when collected at a higher level of precision may require three components, and as the level of precision is increased, even three exponentials may fail to give random residuals. The precision of the data and the presence or absence of nonrandom noise play a critical role when one is fitting decay data to preconceived models. The SPTC method is now routinely applied to intrinsically heterogeneous systems such as molecules adsorbed on surfaces or contained in micelles, molecules in monolayers or vesicles, and molecules where intramolecular quenching takes place through the interactions of two moieties attached to either end of a chain. In such systems one might expect2-5distributions of lifetimes, and thus the question quite naturally arises as to the level of precision, time base, and total channel number required to expose the presence of a distribution. At low levels of precision many distributions can be fit6 by a two-component decay with visually random residuals and a good chi-squared (x2). At higher levels of precision, three exponentials will provide a good fit to what in fact are distributions of lifetimes6 The parameters obtained from these two- and three-component fits are in general meaningless if in fact the data arise not from discrete components but from a true distribution. Thus, the differentiation of systems obeying a decay law made up to two or three discrete components from systems where there exists a continuous distribution of lifetimes, or a distribution plus one or more discrete components, has become an important problem in the area of applications of the SPTC method to complex systems. The popularity of fluorescence probe techniques in polymer photophysics and in biologically interesting systems makes this an even more pressing problem. The recovery of shapes of distributions of fluorescence lifetimes from fluorescence decay data presents a nontrivial problem in numerical analysis, a problem closely related to the inversion of Laplace transforms which is notoriously ill-conditioned. Similar problems arise in light scattering,’ image recovery in astrophysics,* etc. Methods involving the minimization of x2 are commonly used, but recently the maximum entropy method (MEM) has gained Inherent in the MEM method is the theoretical Contribution No. 425 from the Photochemistry Unit, Department of Chemistry, University of Western Ontario.
0022-3654/90/2094-1661$02.50/0
lack of bias and the potential for recovering the coefficients of an exponential series with fixed lifetimes which are free of correlation effects and artificial oscillations. Two hundred or more terms can be used in the MEM trial function. In this paper we will compare the MEM with a new version of the exponential series method (ESM). The new ESM is much more robust than the version previously described13 and allows use of the same size probe function as the MEM. Comparative studies will be reported for real and simulated data for one-component decays and for twecomponent decays with several different separations between the two lifetimes. In addition, data derived from both real and simulated distributions will be analyzed with both methods. Distributions based on real data have been derived from (a) molecules in micellar systems, (b) lifetime distributions associated with Forster transfer, and (c) file summations creating distributions by adding decay curves with known decay times and amplitudes. Data simulation was done in the following fashion. The delta pulse decay function was constructed from a sum of exponentials with appropriate amplitudes. Sufficient terms were used to provide quasi-continuous distributions. Discrete lifetimes were of course represented by only one term. This decay law was then convoluted with the instrument response function (IRF) with added Poisson noise to give the synthetic decay curve for MEM or ESM analysis. Real data were collected with a conventional laser based SPTC instrument. A Coherent argon ion mode-locked laser synchronously pumped a cavity-dumped dye laser, the output of which was frequency doubled. The IRF was between 200 and 300 ps.
The MEM and ESM Methods In the problem of recovering the shape of a lifetime distribution in pulse fluorimetry, we face the task of selecting one solution out of a set of all feasible solutions. The feasible set comprises the solutions that are in agreement with experiment; Le., they pass a x 2 test. Such a set can be very large indeed. Out of a feasible (1) OConnor, D. V.;Phillips, D. Time Correlated Single Proton Counting Academic: London, 1984. (2) Siemiarczuk, A.; Ware, W. R. Chem. Phys. Lett. 1987, 140, 227. (21 (3) Wagner, B. D.; James, D. R.; Ware, W . R: (3) R.Chem. Phys. Lett. 1987,
138. 181. (4) Ware, W. R.; Liu, Y. S.; Peterson, N. 0.; Siemiarczuk, A,; Wagner, 8. D. Proc. SPIE-Int. Soc. Opt. Eng. 1987, 743, 117.
(5) James, D. R.; Turnbull, J. R.; Wagner, B. D.; Ware, W. R.; Peterson, N. 0. Biochemistry 1987, 26, 6272. (6) James, D. R.; Ware, W. R. Chem. Phys. Lett. 1985, 120, 455. (7) Livesey, A. K.; Licino, P.; Delaye, M. J. Chem. Phys. 1986.84, 5102. (8) Gull, S. F.; Daniell, G. J. Nature 1978, 272, 686. (9) Skilling, J.; Bryan, R. K.Mon. Not. R. Astron. SOC.1984, 211, 11 1. (10) Smith, C. R.; Grady, W.T., Jr., Ed. Maximum Entropy and Bayesian Methods in Inverse Problems; Reidel: Boston, 1985. (1 1) Livesey, A. K.; Skilling, J. Acta Crystallog. 1985, A41, 113. (12) Livesey, A. K.; Brochon, J. C. Biophys. J . 1987, 52, 693. (13) James, D. R.; Ware, W. R. Chem. Phys. Lett. 1986, 126, 7.
0 1990 American Chemical Society
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The Journal of Physical Chemistry, Vol. 94, No. 4, 1990
Siemiarczuk et al.
set of solutions the MEM selects the one that maximizes the function Q=XS-C
(1)
where S is the entropy-like function (Shannon-Jaynes entropy’O) N
s = -Cat 1
log (aJat0t)
(2)
I=
where the a, are preexponential factors in the model decay function F(t) containing N terms, i.e. N
F(t) = Cui exp(-t/ri)
(3)
I=I
The factor X is a Lagrange multiplier and atot= Xu,. The 7i are logarithmically spaced.I2 The term C guarantees that the solution is bound by the x2 statistic. With experimental data subject to Poisson scatter k
(4) where Yi represents the number of photons counted in the ith channel, n is the number of channels, DF is the convolution matrix
DF = &‘iL(li - t’) exp(-t’/~,) dt’
(5)
and L ( t ) comprises the convolution of excitation pulse and instrument response function. Our numerical procedure follows the general algorithm proposed by Skilling and Bryan9 A local quadratic approximation is applied to both S and C, and amplitude increments x, are calculated from the condition aQ/ax, = o
CaimlOC
(8)
o
log
T
aREC d(log
7)
= LczSIM d7
(10)
and the data recovered by the MEM have to be divided by 7 to facilitate comparison with the input distribution. Similarly, if a simulated distribution has been constructed linearly in k = 1 / 7 space, then
and instead of (7) the global minimization of C is employed, i.e. =
s,
(7)
is ~atisfied.~ This condition reflects the fact that the entropy and x2 gradients should be antiparallel at Q,,,. By dropping the condition that S be maximized, we revert to what has been termed the ESM.6 In this latter approach eq 1 is replaced by
ac/ax,
intensities (Cai7i)are represented correctly in this particular case only in log 7 space. However, if a decay was constructed out of exponents equally spaced in 7 space (e.g., a simulated Gaussian), the recovered MEM or ESM distribution will be skewed due to the variable density of the probe function. In order to compare shapes, the recovered distribution has to be transformed to linear 7 space or the input function to log 7 space. For any 7 interval, the following equation is obeyed
are where aSIMare amplitudes in the simulated decay and aREC those recovered by either MEM or ESM. Hence $lM = aREC/7 (11)
CminloC is the local minimum of C. Aiming at a higher C than CminloC allows the effective maximization of S. In practice, this constraint due to x2 is equivalent to maximizing S for x2 equal to approximately unity. A distance constraint is imposed on the amplitude increments which prevents the total change of Cui from exceeding 20% in a given iteration. After reaching an aimed value of x2 (usually =l), the program keeps maximizing S with C kept constant until the condition OS[(grad C/lgrad q) - (grad S/(grad SI)] Q 0.1
Q = -C
Figure 1. (a) MEM analysis of a simulated single-exponential decay at 383 ns (dotted line), CPC = 3 X los. Probe function contained 80 terms over range of 300-500 ns. Fitting range of 460 channels with 6 ns/ channel. (b) Same as (a) but with ESM. (c) MEM analysis of real data for pyrene decay in cyclohexane. Conditions the same as (a). (d) Same as (c) but with ESM.
(6)
with the Lagrange multiplier X adjusted to satisfy
c=
LIFETIME/nS
(9)
which corresponds to achieving a local value of Cminin each iteration. This approach allows the direct comparison of the MEM with the ESM using essentially the same algorithm. Both approaches have been used to analyze the same data sets. In all cases a flat distribution with logarithmically spaced lifetimesI2 has been used as the initial guess. In the case of MEM this is equivalent to having no a priori knowledge about the shape of the distribution. For both real and simulated data, the MEM and ESM in general yielded observed x2 values within fO.l of unity. Both methods are capable of handling up to 200 exponential terms in the probe function. Physical significance can be attributed only to amplitudes directly recovered, Le., represented in log 7 space. Fluorescence
s,
log
aREC d(log
T
7)
= s,,uSIM dk
(12)
and a valid comparison requires multiplication by of aREC by 7 . Calculations were performed on the University of Western Ontario VAX 8550 and more recently on the VAX 6230 which is approximately 30% faster in executing the MEM and ESM programs. In general, the ESM is about 5 times as fast as the MEM in execution.
Results and Discussion In Figure la,b the results are presented for both ESM and MEM analysis of simulated decays following a single-exponential with a 383-11s lifetime at a CPC of 3 X IO5. In this case an 80-exponential probe function was used in the range of 300-500 ns. In Figure lc,d are illustrated the same analyses with real data from a sample containing pyrene in cyclohexane with the same lifetime and CPC. Differences between simulated and real data are clearly minimal. The MEM in both cases recovers a narrower pattern of lifetimes than the ESM. The relative width d is 0.02 (Figure la,c) and 0.05 (Figure Ib,d) for the MEM and ESM, respectively, where d = A7/7,,, and A7 is the full width at half-maximum. The ability of the two methods to recover a smooth distribution is illustrated in Figure 2a,b. A decay curve was constructed by convolution of the IRF with a multiple-exponential decay with amplitudes such that the distribution of lifetimes was Gaussian ( ( 7 ) = 50 ns, u = 20 ns) at a CPC of 2 X lo5. Analysis was then carried out with a probe function of 80 exponentials covering the range of 0.5-150 ns. Included in Figure 2 are the predicted amplitudes. Both MEM and ESM appear equivalent in their
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1663
Maximum Entropy and Exponential Series Methods
ai So
0
100
'
So
'0
100
ai
I
.
LIFETIME/nS Figure 2. MEM (a) and ESM (b) analysis of simulated decays containing a Gaussian distribution (dotted line) of lifetimes. ( T ) = 50 ns, u = 20 ns, CPC = 2 X los. Probe function of 80 terms over 0.5-150-11s range. Fitting range of 256 channels with 2.5 ns/channel. 0
20
40
60
0
20
40
60
80
LlFETlME/nS Figure 5. Analysis of simulated data with an underlying distribution ) 40 ns, composed of two Gaussians (dotted line). ( T~) = 20 ns, ( T ~ = (a, b) or 60 ns (c, d), u = 5 ns, CPC = 2 X lo5, 256 channels. (a) MEM with 80-term probe function over 3-80 ns; (b) ESM with 80-term probe, same range; (c) MEM with 120-term probe function over range of 1-100 ns; (d) ESM with 120-term probe function over same range. 2 ns/ channel (a, b) and 3 ns/channel (c, d).
ai
i
ai
5
LIFETIME/
ns
Figure 3. MEM and ESM analysis of a distribution composed of real and simulated data made up of the sum of 11 discrete decays of 3methylindole in water at various temperatures. CPC = 8 X lo5, 153 ps/channel, 256 channels, T,,, = 4.6 ns. Probe function contained 70 terms equally spaced in log 7. Solid lines, the MEM or ESM results; dotted lines, the input distribution. (a) Real data, MEM; (b) real data, ESM; (c) simulated data, ESM; (d) simulated data, ESM, time per channel doubled to 306 ps/channel.
ai
-
LIFETIME/nS
Figure 4. MEM and ESM analysis of a distribution composed of real and simulated data made up of the sum of 11 discrete decays of 3methylindole of water at various temperatures. CPC = 8 X los, 153 ps/channel, 256 channels, T~~~ = 9.73 ns. Probe function contained 70 terms equally spaced in log T. Solid lines, the MEM and ESM results; dotted lines, the input distribution. (a) Real data, MEM; (b) real data, ESM; (c) simulated data, ESM; (d) simulated data, ESM, time per channel doubled to 306 ps/channel.
ability to recover the detailed shape of a smooth unimodal distribution. In Figures 3 and 4 the recovery of a smooth distribution from real data is illustrated and compared with simulated data. Here the decay curve was made up of the summation of single-exponential SPTC files where the decay times and amplitudes are known such that the resultant file contains a decay curve representing a known input distribution. As can be seen, the recovery is also satisfactory for real and simulated data with the shape adequately recovered by both methods.
0
50
100
50
100
LIFETIME/tlS Figure 6. Analysis of a simulated three-exponential decay with lifetimes and amplitudes taken from the three-component fit (dotted lines) to the Gaussian in Figure 2. a, = 0.17 ns, T~ = 20.8 ns; a2 = 0.58, T~ = 47.4 ns; u3 = 0.253, T~ = 78.6 ns. CPC = 8 X los. Probe function composed of 80 terms with range of 0.5-150 ns. (a) MEM and (b) ESM.
The ability to recover the shape of a bimodal distribution is of course dependent on the separation of the maxima and the widths. In Figure 5a,b are illustrated the results for a simulated decay based on the summation of two Gaussian distributions. As before, a multiexponential decay was constructed, convoluted with the IRF, and submitted to both ESM and MEM analysis. From Figure 5 it is clear that both methods have about equal success in resolving the bimodal distribution with eTcellent resolution of the leading and falling edges but not complete recovery of the distribution in the valley. If one increases the separation of the maxima, the fit in the valley improves dramatically whereas if the maxima are closer together the resolved curve will eventually appear to be unimodal. This is illustrated in Figure 5c,d. For a distribution recovery protocol to be really useful, it must enable one under suitable circumstances to distinguish between a system that decays with, for example, three distinct exponential functions and one where there is a continuous distribution of lifetimes. If one constructs a decay curve from a broad Gaussian distribution of lifetimes and then attempts to fit with two or three exponentials, one will find that if the CPC is high, two components will fail but three will fit rather well. In general, the three lifetimes are situated with one at the maximum and the other two at approximately f a . Only at very high CPC will one in general find that the residuals are not random for the three-component fit and thus the model fails. If one now takes the three components recovered from the discrete fit to what in fact is a distribution (the distribution shown in Figure 2), constructs a new decay curve based only on these three discrete components, and subjects it to MEM and ESM analysis, one should get back not the Gaussian but three peaks. That this is true is illustrated in Figure 6a,b. The three input lifetimes were 20,47, and 79 ns, and while neither method resolves them completely, there can be no question that if one had data generated by three components one would see
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Siemiarczuk et al.
ai
LIFETIME
I1
I
ns
Figure 7. Gaussian distribution (( 7 ) = 20 ns, = 5 ns) with added single lifetime component (dotted line) at 50 ns (a, b) and 200 ns (c, d) analyzed with MEM (a, c) and ESM (b, d). CPC = 8 X lo5,256 channels. Fitting range 3-60 ns with 80 exponentials, 3 ns/channel (a, b), and 3-240 ns with 120 exponentials, 6 ns/channel (c, d).
evidence for their existence, whereas if one had had a Gaussian distribution, it would have been recovered (see Figure 2). Thus, in this particular case, if one has data generated by a Gaussian distribution of lifetimes, one can distinguish this case from the alternative hypothesis that the system is described by three discrete components. Obviously, extending this type of argument to cases of four components will only prove satisfactory if the separation of lifetimes in the discrete set is sufficient to yield resolution. Four lifetimes close together will merge into a smooth distribution in most cases, and it is not possible by present MEM and ESM techniques to resolve them even at high CPC. Both MEM and ESM have difficulty recovering the combination of a single discrete lifetime and a distribution unless they are well-separated. This problem exists when the single discrete lifetime is either to the short or long side of the distribution. The latter case is illustrated in Figure 7 for simulated data where poor resolution is obtained with a lifetime at 50 ns and a Gaussian centered at 20 ns with c = 5. However, if the discrete lifetime is at 200 ns, then the recovery is excellent. Again, the ESM and MEM methods are virtually identical. In Figure 8 are shown a set of distribution analyses based on real and simulated data arising from micellar kinetics. The system consists of pyrene in 0.1 M sodium dodecyl sulfate (SDS) quenched by cupric ions. For this type of system the decay law is givenl4Js I F ( ? ) = Ai exp{-A2t - A,[ 1 - exp(-A4t)]}
(13)
which is equivalent to m
IF(?) = AlCPO',(n)) ex~{-(ko+ jk,M j=O
(14)
where
ko = A2 jk, = A4
( n ) = A3 P u , ( n > ) = (n)je-(n)/j!
(15)
In this case one has a discrete lifetime at j = 0 well-separated from a closely spaced discrete distribution of lifetimes for j = 1, 2, ..., n. For the system in question, there is negligible amplitude beyond j = 5-1 0. (This depends on the quencher concentration.) The results obtained with MEM analysis of real decay data are shown in Figure 8a,c,e. One can from these data obtain ko (from ( 14) Kalyanosundraram, K. Photochemistry in Microheterogeneous Systems; Academic: London, 1984. (1 5) Dederen, J. C.; Van der Auweraer, M.; De Schryver, F. C. J . Phys. Chem. 1981,85, 1198.
10
100
300
10
100 300
LlFETlME/nS Figure 8. Pyrene decay in 0.1 M SDS solution with cupric ion quencher analyzed with MEM. Quencher concentration: (a) 5 X IO4 M; (c) IOd3 M. (b, d, f) represent results for MEM analysis of M, (e) 3 X simulated Poisson distributions with ( n ) , k,, and ko obtained from (a), (c), and (e), respectively. CPC = 3 X lo5, 516 channels, 4.7 ns/channel.
the long-lived spike) and ( k ) , the average of k over j = 1, ..., n. Taking as a first approximation (n) = [Q]/[M] where [Q] is the quencher concentration and [MI the micelle concentration, k, can be estimated from k, = ( ( k ) - ko)(l - e d n ) ) / ( n )
(16)
Once the approximate value of k, is known, one can fit the data to
Cbi exp{-(ko + ikq))
(17)
where the only parameter is kT This parameter is varied to obtain the best fit to the decay data, and the bi values are then used to construct a discrete distribution consisting of spikes located at ko + ik, with amplitude bi. These distributions can then be compared with those predicted from Poisson statistics (eq 15) and the value of (n) varied to obtain the best fit. Once one has ko, (n), and k, the decay can be simulated and analyzed with MEM. The recovered distributions agree well with those recovered from the real data, as can be seen in Figure 8b,d,f. Note that as the cupric ion concentration increases, the ( k ) shifts but ko also increases. This is predicted from the general mechanism for intramicellar quenching with micelle-micelle collisions inducing transfer of quencher.15 It is thus possible by using either ESM or MEM to recover the discrete distribution and test independently the hypothesis that the quenching is proportional to the term jk,. and that the quenchers are distributed according to Poisson statistics.16 The resolution of the ESM and MEM, as assessed by the ability to resolve two close-lying discrete components, is strongly influenced by the separation and by the background but little affected by radio-frequency (rf) noise. This is illustrated for the MEM in Figure 9 with real data from the acid quenching of indole. The data submitted for MEM analysis consisted of the sum of two files, each consisting of a single exponential. Background was determined either from the leading edge of the decay or from the tail after the fluorescence had completely decayed. For r = 7 1 / 7 2 = 1.41, only fair resolution is obtained at CPC = 5 X lo5. At CPC = 5 X lo4 the resolution deteriorates significantly. If the background of 20 counts per channel is not subtracted in the case of CPC = 5 X lo5, all resolution is lost. This underscores the importance of careful determination of background, especially when a large CPC is involved. In Figure 9d the effect of the presence of R F noise is shown to be insignificant. In Figure 9e,f are illustrated the effects of increasing slightly the separation of (16) Siemiarczuk, A.; Ware, W. R. Chem. Phys. Lett. 1989, 160, 285.
The Journal of Physical Chemistry, Vol. 94, No. 4, 1990 1665
Maximum Entropy and Exponential Series Methods
c
1
1
ai
Ill,
.&l -
the two components. Already at r = 1.53 the resolution is quite good and becomes excellent at r = 1.89. Another interesting case where distribution analysis can play a significant role is in long-range electronic energy transfer. The Forster decay law in rigid media
IF(?) = Io exp{-(at
+ bt1l2))
(18)
The parameters a and b are functions of the critical transfer distance Ro,the acceptor concentration, and the unquenched lifetime. Since the delta pulse response of the fluorescence system is related to the distribution of lifetimes in 1/7 = k space by an inverse Laplace transform, it follows that the Forster distribution of lifetimes given by F ( k ) is as follows.
+ bt1/2))=
F(k) expi-kt) dk Lm
(19)
exp{-b2/4(k- a ) )
(20)
and
F(k) =
et
it,
.09
L I FETIMEAlS
Figure 9. Resolution of two discrete decays (dotted lines) by MEM as a function of CPC, background, rf noise, and separation of the two decay times. Constructed from real data in 256 channels for various r = (a) r = 1.41, CPC = 5 X IO5; (b) r = 1.41, CPC = 2 X 10“; (c) r = 1.41, CPC = 5 X IO5, background not subtracted; (d) r = 1.41, CPC = 5 X IO5, rf noise in data; (e) r = 1.53, CPC = 5 X lo5; (f) r = 1.89, CPC = 5 x 105.
expi-(at
’
b
2[7r(k - a)3]’/2
Also, Ro can be determined from k,,, since
k,,, = a + (1/6)b2
(21)
Figure 10 illustrates the success of ESM and MEM in recovering this type of distribution for simulated data. Various spacings of the probe function are shown as well as data for 256 and 512 channels, the latter corresponding to a doubled time range for the data acquisition. It appears that the best recovery in general is obtained with ESM with a probe that is evenly spaced in log 7 space with 5 12 channels of data. One hundred terms were used in the probe function. The difference between 512 and 256 channels is evident in the goodness of fit to the position of the maximum which is located at the long lifetime side of the distribution. A detailed study with real data is currently being prepared for publication which examines the applicability of distribution analysis to Forster transfer both in rigid media and (17) Fiirster, Th. Z.Nururforsch. 1949, 49, 321. (18) Birks, J. B. J . Phys. B, Ser. 2 1948, I , 946.
Figure 10. MEM and ESM recovery of rate constant distributions from a simulated Forster energy-transfer system ( a = 0.0185 ns-* and b = 0.2 ns-’; eq 18), 1100 ps/channel. (a) 5 X lo6 CPC, 512 channels, log 7 probe, ESM; (b) 5 X lo6 CPC, 512 channels, log 7 probe, MEM; (c) 5 X lo6 CPC, 256 channels, log 7 probe, ESM; (d) 5 X lo6 CPC, 512 channels, linear 7 probe, ESM; (e) 5 X lo5 CPC, 512 channels, log 7 probe, ESM; (f) 5 X lo4 CPC, 512 channels, log 7 probe, ESM. Probe function composed of 100 exponential terms.
in the case where diffusion changes the decay law.19 Conclusions The above examples illustrate what is in fact a general conclusion from extensive studies in our laboratory; Le., the ESM and MEM are similar in their ability to recover underlying distributions from fluorescence decay data. Only in the case of a single exponential is there a significant difference. One can conclude from this that the x2 condition dominates over the entropy factor in this type of analysis. In fact, a considerable gain in entropy can be achieved by minor adjustments in the amplitudes which in turn have no visual effect on the recovered distribution. Also, both methods start from a flat distribution model and in small steps modify this flat distribution, moving from “high” to “low” entropy until the x2 condition is satisfied. Imposing a maximum entropy condition does not appear as significant as the simultaneously applied condition on the x2. Since ESM is much faster and thus much less costly to run, it appears to be the method of choice. It must be emphasized that the successful use of either method depends on the interplay of the analysis of real and simulated data. Each case is different, and when one thinks that a distribution has been recovered from real data, a simulated distribution must be run to demonstrate that one can recover the shape in question and differentiate between the recovered distribution and a twoor three-component model. In this context it is frequently desirable to be able to fix the lifetimes of discrete components known a priori to be present and analyze only for the distribution component of the decay. Rejecting the hypothesis that a system is described by say three discrete components requires one to show that when the three components are present, MEM or ESM will fully or at least partially resolve them. Then when the real data yield a distribution, one can assert that had the real data really been composed of the three components in question, one would not have obtained a distribution but rather the three components. Our experience with simulations is that when Poisson noise is added, the results can be relied upon to give guidance in the analysis of real data. However, in the case of real data, scattered light, high rf interference, or incorrect reckoning of background can render this interplay between simulation and the analysis of real data misleading. Also, great care must be exercised in the selection of the CPC and time base, and again simulations can (19) Wagner, B. D.; Ware, W. R. To be submitted for publication.
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be of great value in demonstrating that these conditions are satisfactory for the problem a t hand. When engaged in the analysis of real data pertaining to potentially heterogeneous systems, one may be tempted on the basis of low-precision data to espouse a simple model that predicts the two or three components observed in the analysis of the decay data. It is important to collect data to high precision in order to examine critically the proposed model. Even then, it may not be possible to rule out the presence of an underlying distribution of lifetimes since the success of the MEM and ESM in differentiating the discrete case from the continuous distribution is critically dependent on the separation of lifetimes. The great danger lies in attaching physical significance to the discrete components recovered in the study of complex heterogeneous systems. If the system in fact is described by a continuous distribution of lifetimes, then a two- or three-component fit merely provides an empirical
representation based on what is in fact an inadequate fitting function and the parameters are in general physically meaningless. It perhaps should be mentioned that we find the four-component model to be in general indistinguishable from a continuous distribution unless the components are widely separated in lifetime space. High-precision data will in general not reveal the fact that the four-component exponential function is not a good model, nor will the MEM or ESM resolve four closely spaced discrete components. Simulations are required to establish whether the separation between lifetimes required for resolution is present.
Acknowledgment. The authors acknowledge the financial support of the Natural Science and Engineering Research Council of Canada. Registry No. SDS,151-21-3;pyrene, 129-00-0;3-methylindole, 8334-1; CU", 15158-11-9.
Solubltity of CoCI, in Molten NaCI-AICI, Chien Wai: Ira Bloom, Deanna Caveny,l and Milton Blander* Argonne National Laboratory, Chemical Technology Division/Materials Science Program, 9700 South Cass Avenue, Argonne, Illinois 60439-4837 (Received: May 15. 1989)
Measurements of the solubility of CoCl, (in mole fraction units) in molten NaCI-AlC13 solutions ranged from 2.2 X IO-" to 6.1 5 X lo-, in solvent compositions ranging from a mole fraction of A1Cl3 (XAlcl,)of 0.502-0.669. The steep increase in the solubility of CoCl2 as a function of XNcb is well-represented by the coordination cluster theory. A solubility product (in ion fraction units) for CoC1, of 4.7 X IS deduced from the measurements. The insolubility of CoClz near the 50-50 solvent composition is a consequenceof the fact that the complexing of solid CoCl, by AIC13 to form Co(AlCl,), in solution is considerably weaker than the complexing of 2NaCl(I) to form 2NaAIC14(1).
Introduction Molten chloroaluminates are ordered liquids and have striking physicochemical properties.'" For example, at the 50-50 mol % NaCI-AlCI, composition, one can describe the ordering phenomenon in the melt in terms of the formation of a very stable tetrahedral A1Cl4- anion to form molten NaA1CI4. Charge ordering of this 1:l salt leads to the high probability that the Na+ cations are nearest neighbors of the AICl, anion and, hence, the next-nearest neighbor of an AI3+cation. Thus, charge ordering leads to the type of physical ordering of the cations described in one dimension by ABABAB .... This phenomenon can be correlated with both chemical and physical One of the most striking properties observed is the very sharp minimum in solubility and maximum in activity coefficients of solutes at the 50-50 mol % composition. Activity coefficients are very large and can change by more than 1 order of magnitude going from 50 to 60 mol % A l a 3 . In fact, solutes, which interact with NaCl and AIC13 more weakly than these two solvent components interact with each other, are often insoluble at the 50-50 mol % composition because of a large maximum in the activity coefficients of the solute. This property of the activity coefficients is general for ordered liquids, and the magnitude of the effect is larger the stronger the interaction between the monovalent chloride and AIC1, (Le., the more negative is the excess free energy of mixing of the solvent components). In this paper we describe the influence of ordering on the solubility of CoCI, in NaCI-AICI, melts as a function of composition. Since the solvent can be 'Present address: Department of Chemistry, University of Idaho, Moscow, ID 83843. 'Summer 1985 Student Research Participant from University of Southern Mississippi, Hattiesburg, MS 39401.
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described by both physical and chemical models, we will examine both types of models. It is well-known that the tetrachloroaluminate ion, A1C14-, disproportionates very little at the 50-50 mol % composition of the NaCI-AlCl, system.'-" Values of this disproportionation constant for the reaction
+
2AlC14- s A12C17- CI-
(1)
range from 1.06 X at 175 OC to 5.83 X 10" at 355 0C.4 In physical terms, the disproportionation constant is a measure of the disorder in this ordered system; the larger this constant, the greater the disorder. This disproportionation reaction can be redefined in physical terms. Because of charge ordering, the nearest neighbors of anions are predominantly cations. Thus, the species AIC14-, Al,Clf, and CI- all have sodium cations as nearest neighbors. If Na+ is A and A13+ is B, then AICl, designates AB next-nearest-neighbor pairs, AI2Cl7- designates BB next-nearest-neighbor pairs, and C1- designates AA next-nearest-neighbor pairs where the two A's are Na+ cationic nearest neighbors of C1-. Consequently, eq 1' is a chemical way of expressing this physical 2AB
$
BB + AA
(1')
( I ) Tremillon, B.; Letisse, G. J. Electrocmnal. Chem. 1968, 17, 371. (2) Torsi, G . ;Mamantov, G . Inorg. Chem. 1971, I O , 1900. (3) Fannin, Jr., A. A.; King, L. A.; Seegmiller, D. W. J . Electrochem. Soc. 1972, 119, 801. (4) Boxall, L. G.; Jones, H.L.; Osteryoung, R. A. J . Electrochem. Sot.
1913, 120, 223. ( 5 ) Cyvin, S . J.; Klaeboe, P.; Rytter, E.; 0ye, H. A. J . Chem. Phys. 1970, 52, 2116. ( 6 ) Saboungi, M.-L.; Rahman, A.; Blander, M . J . Chem. Phys. 1984,80, 2 141-50.
0 1990 American Chemical Society
