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Langmuir 1996, 12, 4358-4371
Time-Resolved Fluorescence in Micellar Systems: A Critical Application to the Partitioning of Naphthalene in Aqueous Sodium Dodecyl Sulfate† A. A. Kowalczyk,‡,§ J. Vecˇerˇ,‡,| B. W. Hodgson, J. P. Keene, and R. E. Dale*,‡ Cancer Research Campaign Department of Physics & Instrumentation, Paterson Institute for Cancer Research, Christie Hospital NHS Trust, Manchester M20 4BX, U.K. Received March 21, 1996X Static and dynamic aspects of the partitioning of naphthalene into the hydrocarbon phase in aqueous sodium dodecyl sulfate micelles have been critically examined by nanosecond time-resolved pulse fluorometry. It is shown that, in such a system, a properly detailed analysis of the excited-state decay behavior as a function of progressive partitioning of the solute into increasing concentrations of micelles is capable of revealing and directly quantitating the underlying kinetics of the reversible transfer of naphthalene in its excited state between aqueous and micellar phases. The time-resolved data indicate that, in the aqueous sodium dodecyl sulfate system, the partition coefficient for excited-state naphthalene can differ only marginally from that for the ground state. From the estimate obtained for the rate coefficient for entry of excited-state naphthalene into micelles, it would appear that any barrier to the crossing of excited naphthalene into the micelle, given that a collision has taken place, is likely to be rather small. The contrasting time-resolved behavior of a system in which the basis lifetime of the fluorescence probe in the micellar environment is shorter than that in the bulk environment rather than longer, as in the case examined experimentally, is also modeled. The effects of quenching of the fluorescence at the interface between the micellar and aqueous phases are also examined and discussed in the context of the aqueous cetyltrimethylammonium bromide system.
Introduction Steady-state and time-resolved fluorescence spectroscopies have found extensive application in a variety of studies of aqueous micellar systems.1,2 Among these, some considerable interest has focused on the partitioning of neutral arenes into detergent micelles, particularly of naphthalene into aqueous sodium dodecyl sulfate (SDS) and cetyltrimethylammonium chloride (CTAC) or bromide (CTAB).3-7 In the fluorescence studies, since only rather small changes are noted in the emission (as also in absorption) spectral distributions of naphthalene between aqueous and hydrocarbon solvents, use has been made of the much larger changes observed in lifetime3-5 and quantum yield.6,7 The time-resolved fluorescence studies3-5 revealed biexponential decays in the CTAB system in which considerable quenching of the emission, greater in the micellar environment than in the aqueous phase, appears * To whom correspondence should be addressed. Present address: The Randall Institute, King’s College London, 26/29 Drury Lane, London WC2B 5RL, U.K. † A preliminary report of part of the work described here was presented previously at the Advanced Study Institute on Structure, Dynamics and Equilibrium Properties of Colloidal Systems, University College of Wales, Aberystwyth, September 1989, Poster Contribution 25. ‡ Biophysics Research Section, Paterson Institute for Cancer Research. § Permanent address: Institute of Physics, Nicolas Copernicus University, ul. Grudzia¸ dzka 5/7, 87-100 Torun´, Poland. | Permanent/address: Institute of Physics, The Charles University, Ke Karlovu 5, 121 16 Prague 2, Czech Republic. X Abstract published in Advance ACS Abstracts, August 15, 1996. (1) Grieser, F.; Drummond, C. J. J. Phys. Chem. 1988, 92, 5580. (2) Malliaris, A. Int. Rev. Phys. Chem. 1988, 7, 95. (3) Hautala, R. R.; Turro, N. J. Mol. Photochem. 1972, 4, 545. (4) Hautala, R. R.; Shore, N. E.; Turro, N. J. J. Am. Chem. Soc. 1973, 95, 5508. (5) van Bockstaele, M.; Gelan, J.; Martens, H.; Put, J.; Dederen, J. C.; Boens, N.; de Schrijver, F. C. Chem. Phys. Lett. 1978, 58, 211. (6) Almgren, M.; Grieser, F.; Thomas, J. K. J. Am. Chem. Soc. 1979, 101, 279. (7) Abuin, E.; Lissi, E. J. Phys. Chem. 1980, 84, 2605.
S0743-7463(96)00272-7 CCC: $12.00
to occur. The measured lifetimes assigned to the aqueous and micellar phases were thus, respectively, somewhat shorter (up to ∼80%) than those in a pure aqueous environment and considerably shorter (up to 15-fold5) than those in a pure deoxygenated hydrocarbon environment. In the CTAC and SDS systems, however, only a monoexponential decay, intermediate between that observed in pure aqueous and pure deoxygenated hydrocarbon solutions and essentially identical for the two systems examined, was resolved.8 It was suggested that the component assigned to the aqueous environment is not resolvable in presence of the intense emission assigned to the micellar environment unless the latter is heavily quenched, as in CTAB or in CTAC or SDS to which additional bromide or cupric ions, respectively, have been added.5 On general grounds, however, it is desirable to obtain estimates of such parameters in as unperturbed a system as possible, particularly since the quenching of fluorophore emission in micellar systems can lead to rather complex, nonexponential decays under many circumstances (e.g. ref 9 and references therein). Close inspection of published time-resolved data for SDS5 indicates that the fit to a single exponential is not an adequate description of the decay, even over the rather truncated range (∼1.5 orders of magnitude of decay) considered. This is not surprising, since at the concentration of 0.01 M SDS concerned, with a (pseudo)equilibrium constant on the order of 3 × 104 M-1 for the partitioning of naphthalene into micelles,6 quite similar amounts of naphthalene should be present in the aqueous and the micellar environments (quod vide). Even under these conditions, however, a precise quantitative resolution of two components whose lifetimes evidently differ only by a factor of two or so (∼35 ns and (8) The reported discrepancy between CTAC data (quenched3,4 vs unquenched5) appears reasonably explained in favor of the latter. (9) Van der Auweraer, M.; Dederen, C.; Palmans-Windels, C.; De Schryver, F. C. J. Am. Chem. Soc. 1982, 104, 1800.
© 1996 American Chemical Society
Time-Resolved Fluorescence of C12H10 in Aqueous SDS Scheme 1
Langmuir, Vol. 12, No. 18, 1996 4359
identical micelles (e.g. refs 14 and 15), the ratios of these rate coefficients define the (pseudo)equilibrium constants for the exchange process:
kin k*in ) K (M-1); ) K* (M-1) kout k*out
(1)
The in-rate coefficients will be governed by diffusion and possible (e.g. geometrical or electrostatic) barriers, so that
kin ) κkdiff; k*in ) κ*k*diff
(2)
∼60-70 ns) is likely to be difficult for individual data sets. On the other hand, linked analyses of the decay behavior of a series of samples containing a range of detergent concentrations over which there is an appreciable change in the ratio of micellized to aqueous fluorophore, applying the principles of global and/or target analysis (e.g. ref 10), should render such quantitation possible. As will become evident, this approach also enables provision of a critical test of the tacit assumption made in obtaining partition coefficients for the solute from changes observed in steady-state fluorescence intensities as a function of detergent concentration,6,7 namely, that the system is effectively stationary on the fluorescence time scale, emission from each compartment being independent of the presence of the other compartment, although it appears for the systems examined here that intercompartmental exchange of the excited fluorophore only minimally affects those determinations.
where κ and κ* represent the efficiencies, for ground and excited states, respectively, with which the interface is crossed given that a diffusional collision, whose frequency is governed by the mutual diffusion coefficient for solute and micelle (kdiff, k*diff), has occurred. It is emphasized that the efficiency factors and rate coefficients, and therefore also the equilibrium constants, are not necessarily identical for the ground and excited states of the distributing solute. Under the assumption of weak excitation ([N*] , [N]) and in the absence of any quenching at the interface, the differential equations describing the kinetics, i.e. changes in concentration, of excited states following absorption of a monochromatic light pulse having a time profile I(t) are given by
Theory of Fluorescence for Partitioned Systems
d[N*w](t)/dt ) -(kw + k*in[M])[N*w](t) + k*out[N*m](t) + ln(10)w[Nw]lI(t) (4)
Two regimes of partitioning behavior are considered: a dynamic regime in which the rates of exchange of the (excited-state) fluorophore are comparable with its intrinsic (basis) deactivation rates in either or both phases and a static one in which the exchange rates are very much slower than the latter. The simplest complete description of the fluorescence kinetics of a fluorophoric solute having different emission properties in two phases between which it partitions is presented in Scheme 1, referring specifically here to naphthalene distributed between aqueous (Nw) and micellar (Nm) compartments, m denoting micelles and excited states produced by absorption of a photon (hν) of appropriate wavelength being distinguished from the corresponding ground state by asterisks (*). This scheme corresponds with the general formulation for reversible two-state excited-state kinetics developed by Birks11 for excimer kinetics and is generally applicable to other kinds of two-state reversible excited-state processes such as energy transfer (e.g ref 12) and proton transfer (e.g ref 13). Here, km and kw are the overall excited-state (basis) deactivation rate coefficients, i.e. reciprocals of the excitedstate lifetimes τm and τw respectively, specific to each of the phases in the absence of exchange between them. Under the assumptions that kin, k*in (M-1 s-1) and kout, k*out (s-1), the “in-” and “out-”rate coefficients, respectively, for ground and excited states, are independent of the occupation number of the micelle and that there is a Poisson distribution over the population of otherwise (10) Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Instrum. 1985, 14, 379. (11) Birks, J. B. Photophysics of Aromatic Molecules; Wiley Interscience: New York, 1970. (12) Porter, G. B. Theor. Chim. Acta 1972, 24, 265. (13) Laws, W. R.; Brand, L. J. Phys. Chem. 1979, 83, 795.
d[N*m](t)/dt ) -(km + k*out)[N*m](t) + k*in[M][N*w](t) + ln(10)m[Nm]lI(t) (3) and
where m and w are the molar absorptivities of naphthalene in micellar and aqueous phases, respectively, at the excitation wavelength and l is the path length (ln(10)m[Nm]l, ln(10)w[Nw]l , 1). Steady-state Behavior. For a constant excitation intensity I(t) ) I0, once any transients associated with attainment of a steady excitation level have died away, d[N*m]/dt ) d[N*w]/dt ) 0 and eqs 3 and 4 lead to
[N*m] ) ln(10)w[Nw]I0l{Ak*in[M] + (m/w)K[M]kw}/B (5) and
[N*w] ) ln(10)w[Nw]I0l(Ak*out + km)/B
(6)
where A ) 1 + (m/w)K[M] and B ) kmk*in[M] + kwk*out + kmkw. The overall fluorescence signal F in the micellar solution is given by
F ) kfm[N*m] + kfw[N*w]
(7)
while F0, that originating from a pure aqueous solution containing the fluorophore at concentration [N], is
F0 ) kfw[N*w,0] ) ln(10)(kfw/kw)w[N]I0l
(8)
where kfm and kfw are the intrinsic (“natural”) radiative rates of the fluorophore in the two environments, relating the respective quantum yields of fluorescence φm and φw (14) von Bu¨nau, G.; Wolff, T. Adv. Photochem. 1988, 14, 273. (15) Yekta, A.; Aikawa, M.; Turro, N. J. Chem. Phys. Lett. 1979, 63, 543.
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to the overall (basis) deactivation rates km and kw defined above by
φm ) kfm/km; φw ) kfw/kw
(9)
Experimentally, there are two useful limiting conditions with respect to fluorophore content. Either (a) the solutions are saturated in naphthalene, when its concentration [Nw] in the aqueous phase is constant or (b) a constant (low) overall concentration of naphthalene is maintained, in which case its concentration in the aqueous phase depends on the micelle concentration. At low enough micelle concentrations that the micellar volume (Vm) can be neglected in comparison with that of the whole sample, the fluorescence ratio F/F0 for saturated solutions (case a) is obtained from eqs 5-8 as
F ) {kw[(kfm/kfw)(kw(m/w)K + {1 + F0 (m/w)K[M]}k*in)[M] + km + {1 + (m/w)K[M]}k*out]}/ {kmk*in[M] + kwk*out + kmkw} (10) Since the fraction of light absorbed by the fluorophore in each environment remains proportional to its fractional absorbance, no matter what the overall absorbance may rise to in such a situation, eq 10 will remain valid at all micelle concentrations for which Vm , sample volume, (i) in the absence of other possible changes induced in the system and (ii) provided secondary effects such as reabsorption of emission can be neglected or corrected for. A more complicated expression is obtained when a constant overall concentration of the fluorescence probe, [N0], is maintained: [N0] ) [Nm] + [Nw] (case b). Under this condition, the fluorescence ratio F/F0 is given by
F ) {K[M](F∞/F0){1 + (k*in[M]/kw) + F0 (kfw/kfm)(k*out/kw)} + {1 + (k*out/km) + (kfm/kfw)(k*in[M]/km)}}/{(1 + K[M]){1 + (k*out/km) + (k*in[M]/kw)}} (11)
for the saturated sample and constant dye concentration cases (a and b), respectively. Equation 14 can be rearranged to a well-known form linear in micelle concentration
F - F0 ) K[M] F∞ - F
and equivalent to a reciprocal form used earlier.6 In contrast, eq 11 for the dynamic exchange case cannot be re-expressed in such a linear form. Time-Resolved Behavior. The general solution for the coupled differential equations, eqs 3 and 4, when I(t) is a δ-pulse excitation is well-known (e.g. refs 11-13), taking the form
(16)
[N*w](t) ) Rw1 exp(-λ1t) + Rw2 exp(-λ2t)
(17)
Rm1 k*in[M] Y - λ1 Rm2 k*in[M] Y - λ2 ) ) ; ) ) (18) Rw1 X - λ1 k*out Rw2 X - λ2 k*out where X ) km + k*out and Y ) kw + k*in[M], leading directly to
1 -1 ) λ1,2 ) [(X + Y) - x(X - Y)2 + 4k*outk*in[M]] τ1,2 2 (19) Since Rm1 + Rm2 ) [N*m](0) and Rw1 + Rw2 ) [N*w](0), the pre-exponential factors are also simply obtained and can be expressed in several equivalent ways, for example:
where F∞ denotes the fluorescence intensity of the probe in a pure micellar environment
(λ2 - X)[N*m](0) + k*in[M][N*w](0) ; λ2 - λ1 (X - λ1)[N*m](0) - k*in[M][N*w](0) (20) Rm2 ) λ2 - λ1
(12) and
and F0 denotes that in pure aqueous solution, as defined in eq 8 with [N0] replacing [N], leading to the expected ratio: F∞/F0 ) (m/w)(φm/φw). Equations 10 and 11 represent the steady-state fluorescence behavior in the dynamic regime, i.e., when the exchange rates of the excited fluorophores between aqueous and micellar environments are comparable with those of their deactivation kinetics. In the static regime, when the exchange rate coefficients can be neglected in comparison with the basis fluorescence rate coefficients, i.e. k*out, k* in[M] , km, kw, eqs 10 and 11 simplify to
φm m F )1+ K[M] F0 φ w w
[N*m](t) ) Rm1 exp(-λ1t) + Rm2 exp(-λ2t)
The characteristic shared rate coefficients λ1,2 (reciprocal -1 lifetimes τ1,2 ) and pre-exponential weights Rm1,2 Rw1,2 may be obtained via substitution of eqs 16 and 17 into eqs 3 and 4:
Rm1 )
F∞ ) kfm[N*m,∞] ) ln(10)(kfm/km)m[N0]lI0
(13)
Rw1 )
(λ2 - Y)[N*w](0) + k*out[N*m](0) ; λ2 - λ1 (Y - λ1)[N*w](0) - k*out[N*m](0) (21) Rw2 ) λ2 - λ1
The overall (evolution and) decay of fluorescence of the system is then given by
F(t) ) kfw[N*w](t) + kfm[N*m](t)
F ) F0
R12 )
1 + K[M]
1 + K[M](F∞/F0) )
1 + K[M]
(14)
(22)
while the overall ratio R12 of pre-exponential terms associated with τ1 and τ2 is
and
m φm 1+ K[M] w φw
(15)
R1 kfmRm1 + kfwRw1 ) R2 kfmRm2 + kfwRw2
(23)
which finally, taking the initial conditions in the form [N*m](0)/[N*w](0) ) (m/w)K[M], becomes
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
R12 ) {(kfm/kfw)[M]{(λ2 - X)(m/w)K + k*in} + (λ2 - Y) + k*out(m/w)K[M]}/{(kfm/kfw)[M]{(X - λ1)(m/w)K k*in} + (Y - λ1) - k*out(m/w)K[M]} (24) It is important to note that, in the dynamic exchange regime dealt with above, both the amplitudes and the lifetimes of the two observed kinetic components of emission contain the rate coefficients of all the individual processes occurring in both excited states: the timeresolved fluorescence behavior of one compartment is not independent of those in the other compartment in the dynamic exchange regime. This interdependence is lost in the static regime in which the exchange rate coefficients are negligibly small compared with the intrinsic (basis) excited-state deactivation rates: the two compartments become effectively independent with respect to the excited-state decay kinetics of the partitioned solute. The impulse response is biexponential, each component labeling emission originating only from excitation of fluorophores in its respective phase, the two characteristic lifetimes (the basis lifetimes τ1 ) τm ) km-1 and τ2 ) τw ) kw-1) being constant and independent of the micelle concentration. The ratio of the pre-exponential terms now reflects only the ground-state partition of the fluorophore between the phases and contains all the information on micelle concentration, of which it is a linear function
R12 ) (kfm/kfw)(m/w)K[M]
(25)
Determination of the fluorescence decay parameters of such systems as a function of micelle concentration thus provides, in principle, a sensitive method of detecting and quantitating rapid (sub-microsecond) exchange of excitedstate fluorophores such as naphthalene between the micellar and aqueous phases, thereby enabling estimation of the (pseudo)equilibrium constant for excited-state as well as ground-state exchange, as well as testing the validity of the assumption made in steady-state fluorescence determinations,6,7 namely, that the compartmentalization can be considered essentially static for this purpose. Fluorescence Behavior of Model Partitioned Systems As seen above, both steady-state and time-resolved fluorescence data, in principle, carry information about the possible presence of dynamic exchange processes in micellar systems. As a result of the particular form of complexity obtained in the dynamic exchange regime, it is not immediately evident to what extent “dynamic” behavior can be differentiated from “static” nor if this information can be retrieved accurately enough in practice to enable, first, the identification of such an excited-state exchange process and, second, a quantitative description of the exchange mechanism. The fluorescence behavior of systems where exchange is occurring is dependent on quite a large number of parameters. To examine quantitatively some basic characteristics of this behavior, two model systems with the following parameters will be discussed: (I) km = 0.0167 ns-1 (τm ) 60 ns), kw = 0.0278 ns-1 (τw ) 36 ns), k* in ) 2 × 1010 M-1 s-1, k*out ) 1 × 106 s-1, K ) 2 × 104 M-1, m/w ) 1, kfm/kfw ) 1 and (II) km ) 0.05 ns-1 (τm ) 20 ns), other parameters as in (I). As seen from the parameters chosen, the first model system should approximately simulate the behavior of naphthalene in aqueous SDS or CTAC micellar solutions,8 while the second should approximately simu-
Langmuir, Vol. 12, No. 18, 1996 4361
late that in CTAB.5 The in-rate coefficient is taken to be on the order expected for the mutual diffusion coefficient of naphthalene and SDS micelles in water (quod vide), and the out-rate is set to obtain the same equilibrium constant in the excited state as in the ground state, K* ) K. The immediate difference between the two model systems chosen is that the basis micellar lifetime values fall well above and well below the common basis aqueous lifetime: τm(I) > τw > τm(II). For the given common excitedstate coefficient k*out, this results in the critical difference expressed by (km(I) + k*out) < kw < (km(II) + k*out), a condition which leads to both qualitative and quantitative differences in the observed fluorescence kinetic behavior. In these model calculations, it is assumed that the rates km and kw are intrinsic to the hydrocarbon and aqueous environments, respectively, and that no quenching occurs at the aqueous-micellar interface (consideration of the possible relevant effects of additional quenching at the interface in this case is deferred to the Discussion). Steady-State Behavior. (a) Samples Saturated by the Fluorescent Probe (Constant Aqueous-Phase Probe Concentration). On comparing the predictions of eq 10 for the dynamic case with those of eq 13 for the static one, it is found, remarkably, that the presence of exchange in either of the cases examined produces no appreciable change from the dependence in the static case, even up to very high micelle concentrations (e.g. 50 mM). It is evident that there is no practical possibility of detecting the presence of exchange of the realistic magnitude modeled here in saturated samples. Indeed, the slopes are not appreciably affected by increasing k*in by two orders of magnitude and only slowly respond to increases in k*out. In the latter case, the nonlinearity of the relationship cannot be distinguished. Additionally, in case I for instance, a three-orders-of-magnitude increase in the excited-state out-rate coefficient, i.e., a difference of 103 in the excited-state equilibrium (in favor of the aqueous phase) compared with that of the ground state, results in only about a 25% decrease in slope. Patently, this kind of experimental steady-state data in such dynamically exchanging systems cannot be expected to lend itself to a quantitative evaluation of the exchange parameters. The very presence of exchange will be difficult to establish unequivocally by this means unless the excited-state exchange equilibrium is very different from that of the ground state. Even then, the practical situation complicates matters still further, the most important disadvantage experimentally being the sharp increase in total concentration, and therefore absorbance, of the probe in the sample as the micelle concentration is increased, necessitating very precise corrections for inner filter effects which are neither simple nor, particularly in the case of secondary absorption of the fluorescence and its reemission, always unequivocal. (b) Constant Overall Concentration of the Fluorescent Probe. When the probe concentration is held constant, the fluorescence ratio F/F0 follows eqs 11 and 14 for the dynamic and static cases, respectively. On examining the predictions of these equations for both systems I and II, however, it is evident that, within any conceivable experimental accuracy, it would be impossible to differentiate dynamic from static behavior in either case. The practical difficulty stems partly from the rather precise estimate of the value of F∞ needed to obtain an appropriate linear plot which in principle shows a rather large deviation from linearity in the dynamic exchange case: according to eq 14, a reasonably accurate experimental estimate of F∞ could only be guaranteed at very high micelle concentration, where other changes in the system (e.g. micelle size and shape, gelling) can be expected to
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affect the measured value. Secondly, the differences between the static and dynamic intensity ratios F/F0 are less than 1%, well inside any reasonable level of experimental error, which would again render their differentiation in a linear plot equivocal. As in the case of saturated samples, neither the quantitation of excited-state exchange nor even a qualitative indication of its presence in these systems can reasonably be expected to be obtainable from such steady-state emission measurements. Time-Resolved Behavior. A complete description of the fluorescence kinetics of these model micellar systems is encapsulated in
F(t) ) R1 exp(-λ1t) + R2 exp(-λ2t)
(26)
-1 given by eq 19 and the ratio R1/R2 of the with λ1,2 ) τ1,2 pre-exponential terms given by eq 24. The dependence of lifetimes on micelle concentration defined in eq 19 is displayed for the two cases I and II considered above, in Figures 1 and 2, respectively. The initial behavior of the lifetimes as the micelle concentration is increased under the two conditions is conveniently indicated to first order, i.e for low micelle concentrations where x ) 4k*outk*in M]/(X - Y)2 , 1, by Taylor expansion of the square root. Case I. The condition (km + k*out) < kw here leads to Y > X so that, for small micelle concentration,
λ1 = km + k*out -
k*outk*in[M] kw + k*in[M] - km - k*out
(27)
and
λ2 = kw + k*in[M] +
k*outk*in[M] kw + k*in[M] - km - k*out
(28)
with positive denominators. As seen in Figure 1, for the values assigned to the parameters of this model, the mild increase in the value of the longer lifetime τ1 and quite sharp fall in that of the shorter one with increasing micelle concentration predicted from these approximations are confirmed. The longer lifetime τ1 starts with the value (km + k*out)-1 at nominally zero micelle concentration and approaches the value km-1 as the micelle concentration becomes infinite. The shorter lifetime τ2 starts at the value kw-1 and decreases to zero at inifinite micelle concentration. The ratio R12 ) R1/R2, of amplitudes associated with the long and short lifetimes, respectively, increases very much faster than in the static case. Case II. Here the condition kw < (km + k*out) applies, leading to X > Y so that, in the case for very low micelle concentrations,
(30)
Figure 1. Lifetimes and pre-exponential terms as a function of micelle concentration for a probe partitioning in a micellar system, according to eqs 19 and 24 for model parameters of a system with (km + k*out) < kw (see text). In this dynamic case (d), the long lifetime (τ1) starts at the value of (km + k*out)-1 and, rising slowly, approaches the micelle-associated value τm for high micelle concentrations, while the short lifetime (τ2) initially decreases quite sharply from the aqueous-solution value τw and then more slowly approaches a zero limiting value as the micelle concentration becomes infinite. A sharp nonlinear increase in the ratio R12 of the pre-exponential terms corresponding to the longer (τ1) and shorter (τ2) lifetimes is also seen. The constant lifetimes and linear R12 dependence also depicted (s) correspond to the behavior of the static system in the absence of dynamic exchange on a time scale comparable with those of emission.
again with positive denominators. In contrast with case I, however, the initial relationship between X and Y, here X > Y, does not remain valid over the whole micelle concentration region. The two lifetimes again start out as kw-1 (here the longer one) and (km + k*out)-1, but now both decrease with increasing micelle concentration, the former initially much more rapidly than the latter. Their
courses as a function of micelle concentration, calculated from the full expressions and presented in Figure 2, however, show that they do not cross. The longer lifetime (τ1) starts out as kw-1 in the absence of micelles but ends up with a limiting value of km-1 as the micelle concentration becomes infinite. Conversely, the shorter lifetime (τ2) starts off with the value (km + k*out)-1 but as the micelle concentration becomes infinite, reaches zero. Corre-
λ1 = kw + k*in[M] -
k*outk*in[M] km + k*out - kw - k*in[M]
(29)
and
λ2 = km + k*out +
k*outk*in[M] km + k*out - kw - k*in[M]
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
Langmuir, Vol. 12, No. 18, 1996 4363
with the shorter (τ2) and longer (τ1) lifetimes, R21 ) R2/R1, rises from zero, reaches a maximum which is far below the “static value” expected in the absence of exchange, and dies back rapidly toward zero, the limiting value at infinite micelle concentration. Pathological Condition. It is also noted as a third case that, even if the basis lifetimes in the two phases are identical (τm ) τw ) k-1), lifetime and pre-exponential factor dependences on micelle concentration may be observed under certain conditions. In this case eq 19 yields λ1 ) k and λ2 ) k + k*out + k*in[M], while the ratio of the preexponential terms is obtained from eq 24 as
R12 )
{1 + (m/w)[M]K}{(kfm/kfw)[M]k*in + k*out} [M]{(kfm/kfw) - 1}{(m/w)Kk*out - k*in}
(31)
The second component will evidently be present only if there is a change in natural lifetime between the phases and the special condition K* ) (m/w)K is not fulfilled. Under those conditions, the longer lifetime will remain constant, while in absolute terms the ratio of its amplitude to that of the shorter one (which may be negative) will decrease rapidly from infinity, pass through a minimum, and finally increase linearly with micelle concentration. Materials and Methods
Figure 2. Lifetimes and pre-exponential terms as a function of micelle concentration for a probe partitioning in a micellar system, according to eqs 19 and 24 for model parameters of a system with (km + k*out) > kw (see text). In this dynamic case (d), the longer lifetime τ1 starts at the aqueous-solution value τw but rapidly approaches the micelle-associated value τm corresponding to infinite micelle concentration. The shorter -1 lifetime τ2 takes the initial value (km + k* at vanishingly out) small micelle concentration and decreases, slowly at first but then more rapidly, finally approaching zero as the micelle concentration becomes infinite. As more trivially evident in the previous case (Figure 1) for (km + k*out)-1 and τw, the lifetime -1 region from (km + k* to τm is forbidden. The ratio of preout) exponential terms, here presented as that for shorter- to longerlifetimes, R21, rises initially at the same rate as in the static case, and then more and more slowly, reaching a maximum value and then decreasing to zero. The constant lifetimes and linear R21 dependence also depicted (s) correspond to the behavior of the static system in the absence of dynamic exchange on a time scale comparable with those of emission.
sponding to the range of lifetimes between kw-1 and (km + k*out)-1 in case I, the lifetime region from km-1 to (km + k*out)-1 is forbidden in this case. The here more conveniently considered ratio of pre-exponentials associated
Sample Preparation. Microanalytical grade (Hopkin and Williams, U.K.) or specially zone-refined (Aldrich Chemical Company) naphthalene and SDS specially purified for biochemical use (British Drug Houses, U.K.) were used as obtained. Aqueous solutions were made subsequently from water doublydistilled from alkaline permanganate in an all-glass still. Two types of sample were utilized: aqueous SDS solutions saturated with naphthalene and solutions with a constant low concentration of naphthalene. In the latter the naphthalene concentration of solutions of different SDS concentrations was checked by absorption measurements before and after each fluorescence decay measurement. Spectral Measurements. Absorption spectra were determined on a Shimadzu UV2100 UV-visible double-beam recording spectrophotometer against the appropriate blank “solvent” (water or aqueous SDS) in standard Teflon-stoppered 1 cm path length Suprasil absorption cuvettes (Hellma, U.K.). Emission spectra of samples in standard fluorescence cuvettes (as above) were recorded on a Shimadzu RF 5000 fluorimeter, using a Polaroid HNP′B ultraviolet-transmitting dichroic polarizing film sandwiched between quartz disks to select vertically polarized excitation and, at right angles to the excitation beam direction in the horizontal plane, emission polarized at the “magic” angle of ∼54.7° to the vertical in order to record a signal proportional to the total emission in all directions unbiased by its polarization, which would be expected to change with micelle concentration. All measurements were made on air-saturated solutions at room temperature (∼25 °C). Nanosecond Time-Resolved Fluorescence Measurements. The fluorescence decay behavior of the set of samples described above was determined in air-saturated solutions in 1 cm2 cross-section tightly Teflon-stoppered standard Suprasil fluorescence cuvettes (Helma, U.K.) held at (25 ( 1) °C by single-photon-counting delayed-coincidence nanosecond-pulse fluorometry. The pulse fluorometer was constructed and run essentially in accord with established principles,16 including rapid alternation of collection of the excitation pulse (“lamp”) profile using a dilute scattering solution of Ludox WP (kind gift of Du Pont de Nemours & Co.), the decay profile, and a blank (i.e., water or appropriate aqueous SDS “solvent”). Details of the design and operation of the system are contained in the Supporting Information to this article. Only two spectral lines, at ∼297 and ∼316 nm, are available for excitation of naphthalene by the nitrogen-filled flash lamp used here, both only weakly absorbed (vide infra, Figure 3). The 316 nm line was employed to maximize the fluorescence intensity (lower absorbance, but (16) Badea, M. G.; Brand, L. Meth. Enzymol. 1979, 61, 378.
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Figure 3. Comparison of absorption spectra of naphthalene in water and 0.5 M aqueous SDS (red shifted spectrum) at the same naphthalene concentration (∼60 µM) at 25 °C determined at a spectral resolution of ∼0.5 nm. much higher lamp intensity). Even so, however, the intensity of fluorescence was quite low, resulting in the necessity of long periods of data collection: up to almost 3 days for samples with the low naphthalene concentration eventually employed. This line, using a nominal bandwidth of ∼20 nm (bandwidth of line ∼7 nm), gave an excitation profile of ∼1.8 to 2.2 ns fwhm. Observation of the emission was at 300 nm with a bandwidth of ∼16 nm. Data were typically collected at less than 0.5% of the excitation rate of 20 kHz, reducing pile-up error17 to negligible levels, over a period of 10-60 h, depending on sample fluorescence intensity, usually to between about 2 × 104 and 6 × 104 counts at the peaks of both the excitation (lamp) and emission profiles. To avoid polarization effects, which potentially constitute a more serious source of error in these measurements than for steadystate fluorescence data, vertically polarized excitation was selected by a long-form Glan-Thompson UV prism polarizer (Karl Lambrecht, Chicago, IL) and “magic-angle” polarized emission observed via a Polaroid HNP′B UV polarizing film, as for the steady-state spectral measurements. Data Analysis. The parameters corresponding to several models for analysis of the predicted biexponential impulse responses of naphthalene fluorescence in aqueous micellar solutions of various concentrations of SDS were recovered from the experimental excitation (lamp) profile and the blanksubtracted convoluted response data, collected at a resolution of 0.5 ns/channel, using appropriate versions of a Marquardt-type nonlinear least squares reconvolution and fitting program.18 These were adaptations of an original routine designed for straightforward multiexponential impulse responses kindly supplied by Professor Ludwig Brand19 and set up for interactive use on the Institute’s MicroVAX 3600 minicomputer system. Further details and the requirements for adequate fitting of this kind of data are presented in the Supporting Information. The statistical accuracy of the result at each iteration was judged from the value of the reduced χ2 parameter for goodness-of-fit: χ2ν ) χ2/ν, where χ2 is the sum of the weighted squares of residuals between the data and the fitted value in each channel and ν is the number of degrees of freedom in the fit, corresponding to the number of data points considered less the number of freely-fitted parameters. Confirmatory visual evidence of the quality of fit is obtained by inspection of the randomness of weighted residuals (square roots of the weighted squares of residuals with algebraic sign conserved) and their autocorrelation function.20 The uncertainties of the values of the parameters recovered were estimated from the inverse of the curvature of the χ2 hypersurface.18 To check as directly as possible the appropriateness of the exchange model described above, global10 analysis routines by (17) Coates, P. B. J. Phys. E: Sci. Instrum. 1968, 1, 878. (18) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969. (19) Easter, J. H.; DeToma, R. P.; Brand, L. Biophys. J. 1976, 16, 571. (20) Grinvald, A.; Steinberg, I. Z. Anal. Biochem. 1974, 59, 583.
Kowalczyk et al. which data at all micelle concentrations investigated could by analyzed simultaneously to target the parameters relevant to both static and dynamic models directly were developed. In the former, the two lifetimes of an assumed biexponential decay were freely optimized but held in common throughout, and the preexponential ratio was determined by the product of a freely optimized ground-state equilibrium constant and the micelle concentration according to eq 25. In the latter, the predictions of the biexponential decay of eq 26 for the “dynamic exchange” model of the system expressed in eq 19 for the lifetimes and eq 24 for the pre-exponential ratio were invoked, the parameters k*out and the ratio of radiative rates for emission in the micellar and aqueous environments not being directly included in the search. While the radiative-rate ratio can be expected to be close to unity and to have only a small influence on the analysis (c.f. Results section), k*out is a major determinant of both the lifetimes and the pre-exponential ratios in this model. Its evaluation was accomplished by a step search: the other parameters (K, k*in, τm, and τw) were optimized for a range of fixed values of k*out. The χ2 profile obtained around the minimum χ2 yielded the required estimates of the optimum value of k*out and of its standard error, taken from the values of k*out corresponding to χ2 values higher by χ2ν than the minimum.18
Results Absorption Measurements. Previous absorption measurements for naphthalene in aqueous SDS solutions showed only a small difference from the absorption of naphthalene in aqueous solution.6 Although the higher resolution spectra presented in Figure 3 show a clear 1 nm red shift of the absorption spectrum of naphthalene in aqueous 0.5 M SDS in comparison with that in pure water, the difference is not large enough to provide a satisfactory method for quantitating the partition of naphthalene at a constant low (i.e. water-saturating) concentration between the two phases. However, when the solubilities of the probe are different in the two environments, absorption measurements can convey useful information even when the absorption spectra are identical. Absorption spectra of a set of aqueous SDS solutions at concentrations ranging from 0 to 0.03 M, all saturated with naphthalene at (25 ( 1) °C, were measured to obtain estimates of the critical micelle concentration (cmc) and (pseudo)equilibrium constant K for the partitioning of ground-state naphthalene. Approximating the SDS micelle by a sphere of radius ∼1.8 nm containing about 68 SDS molecules,21 the volume of the micellar phase at a concentration of 30 mM SDS in water is less than 0.5% of the total volume of the sample. At least up to this concentration, the volume of the micellar phase can be neglected without appreciable error in comparison with that of the bulk environment. In this approximation, the optical density of naphthalene in the bulk solution (i.e. the aqueous part) at all SDS concentrations is the same as that in pure water and the ratio of the absorbances of naphthalene in the aqueous SDS sample and the aqueous “reference” solution, A/A0, takes the form
A/A0 ) 1 + (m/w)K[M]
(32)
where the micelle concentration [M] is given by
[M] ) ([SDS] - cmc)/n
(33)
where n is the aggregation number. On inserting eq 33 into eq 32, a linear dependence of the absorbance ratio on SDS concentration is obtained and the critical micelle concentration can be determined by linear regression (Figure 4). The value of the cmc obtained in this way was (21) Almgren, M.; Swarup, S. J. Phys. Chem. 1982, 86, 4212.
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
Figure 4. Concentration dependence of the relative absorbance A/A0 at 300 nm obtained from absorption spectral measurements of aqueous SDS solutions saturated by naphthalene for SDS concentrations of 10, 15, 20, and 30 mM. The dashed line represents the best fit calculated by linear regression according to eq 32 and gives a critical micelle concentration of (7.7 ( 0.2) mM.
Figure 5. Uncorrected fluorescence spectra of ∼110 µM naphthalene in water and 0.5 M aqueous SDS solutions (the upper curve of higher overall intensity) at 25 °C. Excitation wavelength 300 nm; bandwidths 3 nm in excitation and 1.5 nm in emission.
(7.7 ( 0.2) mM, in good agreement with the previously determined values of ∼8 mM.22 Taking m/w ) 0.88 at 300 nm (Figure 3) and an aggregation number of 68, the (pseudo)equilibrium constant K was calculated to be (25 ( 1) × 103 M-1, corresponding well with earlier values.6,7 Steady-State Fluorescence Measurements. Compared with absorption, the fluorescence of naphthalene is more sensitive to the change from aqueous to micellar phase. As seen in Figure 5, the quantum yield of fluorescence in micelles is appreciably higher than that in aqueous solution, in agreement with earlier data.6 However, again the spectrally well-resolved data of Figure 5 show a red shift of the “micellar” emission to about the same extent as in absorption and in addition some sensitivity of the fluorescence peak at ∼336 nm to the micellar environment, although neither effect is really large enough to be useful for quantitative determination of the partition coefficient. Time-Resolved Fluorescence Measurements. As indicated by the results of the simulations presented (22) Mukerjee, P.; Mysels, K. J. Critical Micelle Concentrations of Aqueous Surfactant Systems; National Bureau of Standards: Washington, DC, 1971.
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earlier in the text, steady-state fluorescence measurements at low constant fluorophore concentrations can be ued, in principle, to detect the presence of dynamic exchange in micellar systems, but in practice their facility to do so is very limited. In particular, using reasonable parameters for the case under investigation, the simulations indicate that extremely precise measurements would be needed to identify even the presence of exchange in such systems. Time-resolved fluorescence decay data, on the other hand, offer a very much greater capacity not only to indicate the presence of dynamic exchange but also to describe it quantitatively. For the naphthalene/aqueous SDS system of concern here, the simulations carried out show that, if such an exchange takes place, a marked decrease in the shorter and a small increase in the longer lifetimes should be observable with increasing micelle concentration, in contrast with the static case, for which the two lifetimes are constant. Concurrently, the ratio R12 of the pre-exponential factors defined previously should increase nonlinearly and much faster than the linear increase expected in the static case (Figure 1). Initial time-resolved fluorescence measurements were made using samples saturated with naphthalene. Over a range of concentrations of SDS, the behavior predicted for a dynamically exchanging system was qualitatively observed but quantitative analysis proved unsatisfactory.23 Among the potential reasons for this, the influence of reabsorption on at least the long lifetime measured was established. For a fluorophore-saturated 0.1 M SDS solution, stepwise physical dilution with 0.1 M SDS to a 30-fold lower concentration led to a reliably detectable decrease of ∼1 ns in the longer lifetime.23 Measurements made on “optically diluted” samples, using 3 mm (instead of the normal 1 cm) cross-section cuvettes, confirmed that the longer apparent lifetimes observed in the more concentrated solutions resulted, principally at least, from secondary (tertiary, etc.) emission of reabsorbed primary fluorescence: an approximately threefold decrease in the apparent lifetime-lengthening effect expected as a result of the optical dilution was observed.23 In order to avoid these problems, a set of eight aqueous solutions containing a constant concentration of about 110 µM naphthalene with SDS concentrations of 0, 6, 10, 15, 20, 30, 50, and 100 mM was measured. The curves obtained were analyzed individually for the parameters of a monoexponential and a biexponential impulse response, i.e. respectively for one or two pre-exponential (R) and lifetime (τ) values, and also globally for both dynamic and static behavior, as detailed in the Materials and Methods. Initially, each of the data sets was analyzed individually for the parameters of a monoexponential impulse response (plus scatter). This proved adequate for the pure aqueous sample, for the sample containing the submicellar concentration of 6 mM SDS, and for the highest concentration of SDS employed (0.1 M). No statistically significant improvement in fit was obtained for these data sets on analyzing for a double-exponential model plus scatter. For all the intermediate micellar samples, however, highly significant improvements were observed for the latter model compared with the former. These were readily attestable from the improvement in the reduced χ2 value, which decreased from values far from unity to values close to unity, and confirmed by visual inspection of the randomness of the weighted residuals and of their autocorrelation functions. An example of an individual (23) Vecˇerˇ, J.; Kowalczyk, A. A.; Dale, R. E. NATO Advanced Study Institute, Aberystwyth, 1989, poster section no. 25.
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Kowalczyk et al. Table 1. Summary of the Results of the Global Target Analysis of the Experimental Data According to the Dynamic Exchange Modela [SDS] (mM) [M] (mM) χ2ν (free) 6 10 15 20 30 50 100
0.0 0.034 0.107 0.181 0.328 0.622 1.357
1.107 1.007 0.957 0.967 1.025 1.010 1.093
χ2ν 1.111 1.007 0.966 0.986 1.058 1.085 1.128
τ1 (ns) τ2 (ns) 56.17 56.38 56.78 57.10 57.59 58.24 59.06
36.51 35.72 34.18 32.77 30.33 26.48 20.18
R1/R2 0.0 0.63 2.40 4.85 12.03 38.24 216.52
a The micelle concentration was calculated using the critical micelle concentration of 7.7 mM determined (Figure 4) and an aggregation number of 68.21 The goodness of fit parameters χ2ν for the individual curves according to the global analysis are compared with those obtained from free analysis of the individual curves for an arbitrary biexponential model, χ2ν (free). Best-fit global parameters: χ2ν ) 1.043, m/w ) 0.9, kfm/kfw ) 1.0, τm ) 60.57 ns, τw ) 36.51 ns, k*out ) 1.33 × 106 s-1, k*in ) 16.4 × 109 M-1 s-1, K ) 14.5 × 103 M-1.
Figure 6. Fluorescence decay of ∼110 µM naphthalene in 0.03 M aqueous SDS fitted according to (a) a monoexponential decay model, χ2ν ) 1.445, R ) 0.236, and τ ) 56.20 ns, and (b) a biexponential decay model, χ2ν ) 1.095, R1 ) 0.021 ( 0.003, τ1 ) (28 ( 3) ns, R2 ) 0.221 ( 0.003, and τ2 ) (57.6 ( 0.2) ns. The standard errors quoted here are estimates made within the least-squares minimization procedure18 and are meaningful only when the fit is statistically adequate. The appropriately weighted residuals are also displayed along with their autocorrelation function in both cases (upper right inset), confirming the inadequacy of the single exponential model.
analysis of such data is given in Figure 6, for 30 mM SDS. The inadequacy of a monoexponential model and the consistency of a biexponential one with this system is clearly demonstrated. The free double-exponential analysis provided some indication that the lifetimes might not be constant and the ratio of pre-exponentials not linear in micelle concentration, i.e that the simpler model for partitioning, not involving rapid exchange of the solute between micellar and aqueous phases, may be inappropriate. This was confirmed by a constrained analysis in which the two lifetimes for the intermediate range of micelle concentration were fixed at the values obtained in the limiting monoexponential cases, only their pre-exponential factors being free in the fitting procedure. While excellent fits were thus obtainable for all these data sets, the ratios of pre-exponentials obtained were far from linearly dependent on the micelle concentration. In the global analyses, all of the curves were analyzed simultaneously (a) with the R and τ values targeted for the dynamic case, eqs 19 (in the second-order Taylor approximation) and 24, and (b) with the R values targeted for the static case, eq 25, the τ values being held common over the whole concentration range. The results of the global analysis for the dynamic case are presented in Table 1. The parameters obtained were τw ) (36.51 ( 0.03) ns, τm ) (60.6 ( 0.2) ns, K ) (15 ( 2) × 103 M-1, k*out ) (1.33 9 -1 -1 s , yielding ( 0.09) × 106 s-1, and k* in ) (16 ( 2) × 10 M an excited-state equilibrium constant K* ) (12 ( 2) × 103 M-1, the goodness-of-fit parameter being χ2ν ) 1.043. The
Figure 7. Weighted residuals and autocorrelation functions from the global target analyses of fluorescence decay data for ∼110 µM naphthalene in aqueous SDS solutions according to (a) the dynamic exchange model described by eqs 19 and 24, χ2ν ) 1.043, and (b) the static model described by eq 25 with two constant lifetimes, χ2ν ) 1.165. SDS concentrations (top to bottom): 6, 10, 15, 20, 30, 50, and 100 mM.
main contribution to the ground-state equilibrium constant error estimate is given by uncertainty in m/w, which was not determined to better than about (0.9 ( 0.1), for the relevant band of excitation wavelengths around 316 nm (see Materials and Methods and Figure 3), and this arises because only the product (m/w)K can be determined from R12, eq 24, in the decay curve analysis. The radiative rate ratio kfm/kfw appearing in R12 was not included directly in the nonlinear least-squares search. Introduction of values around unity showed that χ2ν is rather insensitive to kfm/kfw, the analysis giving a shallow minimum for kfm/ kfw ∼ 1.0. The global analysis for the static model of this system gives K ) (13.1 ( 0.1) × 103 M-1, τw ) (36.61 ( 0.02) ns, τm ) (59.47 ( 0.02) ns, and R12 ranging from 0 to 17.8, with a goodness-of-fit parameter χ2ν ) 1.165. Time-Resolved Fluorescence Simulations. A “quality comparison” of dynamic and static analyses is presented in Figure 7, where weighted residuals and their auto-
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
Figure 8. Standard error estimates ((σ) of the lifetimes recovered by individual free biexponential analyses of data sets simulated from the parameters recovered in the global target analysis of the real data set according to the dynamic exchange model featured in Figure 7a. The (σ values are indicated by “+” markers above and below the true values of the parameters (continuous curves). If real fluorescence decay data for such a system are analyzed in this way, the parameters recovered can be expected to exhibit values scattered mainly within about (2σ.
correlation functions are compared over the whole range of the data. To evaluate the significance of the difference between the two global fits, a set of fluorescence decay data similar to the experimental data was strictly analytically simulated by a procedure described elsewhere,24 using the parameter values from the dynamic global analysis presented in Table 1 and curve intensities similar to those obtained experimentally. Since these simulated decay curves refer, by definition, to the dynamic situation expressed in eqs 19 and 24, their analysis will yield criteria for the limitations of quantitative evaluation of data obtained experimentally for such systems. These simulated data were first utilized to determine the ability of free biexponential analyses to recover the correct decay parameters. The free biexponential analysis of, for instance, the simulated decay corresponding to the 0.03 M SDS experimental curve (Figure 6) gave the expected excellent fit (χ2ν ) 0.980) with very good recovery of the longer lifetime (τ2 ) 57.8 ns compared with the “true” value of 57.51 ns), while the recovery of the shorter lifetime was much less accurate (τ1 ) 34 ns vs the “true” value of 30.33 ns). In spite of this, however, the results are equivalent to those obtained for the real data (Figure 6), because the standard error (σ) estimates are about 0.3 and 4 ns for the longer and shorter lifetimes, respectively. This example simply illustrates the difficulties connected with free analysis of biexponential decays characterized by lifetimes which are not very different.25 The expected uncertainties in the lifetimes obtained by free analysis of the simulated data are demonstrated in Figure 8, where the true values and their (σ estimates are displayed. There is no σ estimate on the shorter lifetime for the highest SDS concentration because this component is so weakly (24) Vecˇerˇ, J., Kowalczyk, A. A.; Dale, R. E. Rev. Sci. Instrum. 1993, 64, 3403. (25) Knutson, J. R.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1983, 102, 501.
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Figure 9. Weighted residuals and autocorrelation functions for the global target analyses of decay data simulated with parameters recovered for real data analyzed assuming the dynamic exchange model. Analyses according to (a) the dynamic exchange model, χ2ν ) 1.007, and (b) the static model, χ2ν ) 1.168. The presentation corresponds to that for the real data displayed in Figure 7. The systematic misfits to the individual data sets of the best overall fit for the static model, seen in the residuals and, even more clearly, in their autocorrelation functions, behave similarly to those observed for real data, and the increase in χ2ν over that obtained for analysis by the appropriate dynamic model is also very similar.
represented that a monoexponential analysis gives a statistically adequate fit to the data. Taking into account the fact that parameters fitted by nonlinear least-squares routines usually fluctuate more than one σ estimate around the “true” value, it is evident that such “free” analyses can give only a qualitative insight into the concentration behavior of fluorescence and more progress can only be made by global target analysis. A comparison of the results of dynamic and static model global analyses for the simulated data (weighted residuals and their autocorrelation functions) is displayed in Figure 9. The global analysis according to the dynamic model returns τw ) (36.49 ( 0.03) ns, τm ) (60.6 ( 0.2) ns, K ) (14.7 ( 0.1) × 103 M-1, k*out ) (1.3 ( 0.1) × 106 s-1, and k*in ) (14 ( 2) × 109 M-1 s-1 (leading to K* ) (11 ( 3) × 103 M-1) with χ2ν ) 1.007, compared with values entered of 36.5 ns, 60.6 ns, 14.5 × 103 M-1, 1.30 × 106 s-1, and 15.7 × 109 M-1 s-1 (leading to 12.1 × 103 M-1), respectively. The apparent rather high precision of K arises as a result of not taking into account the experimental uncertainty in m/w. The somewhat low value of k* in recovered again reflects the inherent uncertainty of its recovery, indicated also by the large value of its σ estimate: a value of 15.9 × 109 M-1 s-1 was recovered, with the same order of uncertainty, from analysis of the same simulated data set without added noise. This and comparison of the results with real experimental data confirm what was already known from the σ estimates on the parameters recovered there, i.e that the parameter k*in is the most uncertain. Most importantly though, the values of the parameters recovered from the simulated data are identical within the error estimates to the values entered in the simulation. Similarly, practically identical results to those obtained for real data are also returned by a global analysis of the simulated data for the static model: K ) (13.3 ( 0.1) × 103 M-1, τw ) (36.58 ( 0.01) ns, τm ) (59.36 ( 0.01) ns with
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Figure 10. Influence of changes of in- and out-rate coefficients and ground-state equilibrium constant on the concentration dependences of lifetimes and the ratio of pre-exponential factors R12 ) R1/R2 associated with the long (micelle-associated) and short (water-associated) lifetimes, calculated from the parameters obtained from the global target analysis according to the dynamic exchange model and given in Table 1: (a) change of out-rate coefficient from 1.3 µs-1 to (1) 1.0 and (2) 1.5 µs-1; (b) change of in-rate coefficient from 15.7 M-1 ns-1 to (1) 10 and (2) 20 M-1 ns-1; (c) change of the ground-state equilibrium constant from 14.5 mM-1 to (1) 10 and (2) 20 mM-1. The abscissa marker at about 1.35 mM indicates approximately the point at which, for a ground-state equilibrium constant of 20 mM-1, the pre-exponential term R2 becomes zero and beyond which it decreases in the negative direction, eventually bringing the now negative R12 back to zero at infinite micelle concentration.
χ2ν ) 1.168. On comparing the values of χ2ν obtained for real data in the two global analyses with those obtained for the simulated data set, it is evident that for such a system a difference in the global χ2ν greater than about 0.16 between an appropriate analysis assuming the dynamic exchange model and an inappropriate one for the static model cannot be expected. Overall, it is clear that the system under study shows very similar behavior to that predicted by the dynamic exchange model developed here and that it is not consistent with a static model for partitioning of the excited state. Discussion From the results presented above, it is evident that determination of a set of fluorescence decay data over an appropriate range of detergent concentrations can yield valuable information about the partitioning and dynamic exchange of fluorescent probes in aqueous micellar systems; i.e., the ground-state equilibrium constant and the in- and out-rates for the excited state can be accurately recovered from the concentration dependences of lifetimes and pre-exponential factor ratios by appropriate global
target analysis. The basic behavior of the experimental naphthalene/aqueous SDS system reported here, with regard to the effect of changes in the excited-state exchange rates and the ground-state equilibrium constant on the lifetimes and ratio of pre-exponentials, is visualized in Figure 10. The high precision of the out-rate recovered results from the fact that this parameter has a strong influence both on the lifetime of the longer-lifetime component (which is always strongly represented, and therefore well-determined, in the biexponential decays observable at all but negligible micelle concentrations: R12 ∼ 1 at [M] = 50 µM) and on the pre-exponential ratio (Figure 10a). On the other hand, even though changes in the in-rate strongly influence the shorter lifetime (Figure 10b), this component is so weak, and its uncertainty therefore so high, that only very large changes in k*in will be reliably reflected in this lifetime. This, the fact that its influence on the longer lifetime is small in comparison with that of the out-rate, and, importantly, its lack of appreciable influence on the pre-exponential ratio over the range of SDS concentrations up to about 0.05 M provide the main reasons for the rather
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
larger uncertainty in the determination of the in-rate. Its absolute precision is further compromised by the fact that it affects these parameters only by way of its product with the micelle concentration, whose estimation in the important range just above the critical micelle concentration and around the R ratio inflection region is subject to relatively large error due to uncertainty in the critical micelle concentration and aggregation number. The ground-state equilibrium constant K does not affect the lifetimes but has a strong influence on the concentration dependence of the ratio R12 of the longer- to shorterlifetime pre-exponential factors (Figure 10c). As indicated in the figure, a value as high as 2 × 104 M-1 would result in a negative value for R2 at concentrations of micelles in excess of about 1.35 mM. The dependence of R12 on the product (m/w)K offers an interesting possibility to compensate very high values of K, by selecting an excitation wavelength for which the ratio value is as small as possible. The short lifetime component is thereby effectively amplified and it (and also, therefore, the excited state in-rate) can be determined more precisely. From the absorption spectrum of naphthalene (Figure 3), it can be seen that the most suitable wavelength for excitation of the system in this respect is around 309 nm. At this wavelength m/w reaches its lowest value of about 0.6, for which, without of course changing the lifetimes, the concentration dependence of R12 is close to that simulated here (for m/w ) 0.9) for K ) 104 M-1, i.e with a much stronger short-lifetime component. Conversely, for excitation at a wavelength for which m/w > 1 (near the absorption edge at about 290 nm, where the difference in is high due to the red shift) a well-determined negative R2 would be observed. Excited-State Exchange Rates. The values obtained for the out-rate of excited-state naphthalene from SDS 6 -1 micelles, k* out = 1.3 × 10 s , can be compared with that estimated previously for 1-bromonaphthalene in a timeresolved phosphorescence quenching study.6 The rate there was determined to be higher than 5 × 104 s-1, which was the upper limit measurable. The in-rate value, k*in = 16 × 109 M-1 s-1, determined here might appear untenably high at first sight. However, the mutual diffusion coefficient for the small fluorescent probe and the large micelle, kdiff, depends not only on the diffusion coefficients of each of the partners (say, A and B) but also on their dimensions. According to the Smoluchowski relationship, kdiff ) 4πσAB(DA + DB)N′, where DA and DB are the diffusion coefficients of the two moieties and σAB ) rA + rB is an encounter radius equal to the sum of their radii, N′ being Avogadro’s number per millimole. On factoring out the part which depends exclusively on the dimensions of the two partners: kdiff ) (8RT/3000η)f(rA,rB) with f(rA,rB) ) (rA/rB + rB/rA + 2)/4, the effect of different sizes of the partners can be visualized (Figure 11). The function exhibits a minimum value of unity along the line rA ) rB, and around this minimum the surface rises rather slowly. The shaded area shows a region where the function exceeds its minimal value by less than 10%. Close to the borders where the discrepancy in radii begins to become large, the function rises rapidly. Taking the effective radius of an SDS micelle to be about 2 nm and approximating that for the naphthalene molecule as the geometric mean (abc)1/3 of the Van der Waals dimensions, yielding a value 0.3 nm, kdiff can be calculated to have a value of about 14 × 109 M-1 s-1. Since naphthalene is rather flat, a much small cross-section of the molecule for movements in planes parallel to the molecular plane can be expected, which would lead to even higher values of the mutual diffusion coefficient. It follows
Langmuir, Vol. 12, No. 18, 1996 4369
Figure 11. Plot of the dimensionless function f(rA,rB) ) [rA/rB + rB/rA + 2]/4 describing the effect of different relative sizes of spherical collision partners of radii rA and rB on their mutual diffusion coefficient kdiff. The shaded area delineates the region in which f(rA,rB) is less than 10% in excess of the minimum value of unity.
that the estimate of the in-rate recovered from the data presented here is not unrealistic. Partitioning in the Ground and Excited States. The ground-state equilibrium constant for the partitioning of naphthalene between SDS micelles and water determined from time-resolved fluorescence data, taking cognizance of the dynamic exchange between two phases in the excited state, K ) (15 ( 2) × 103 M-1, is considerably smaller than that obtained from the absorption measurements also reported here, K ) (25 ( 1) × 103 M-1. However, there is what would appear to be a critical difference between the systems used for these two determinations. The time-resolved measurements were carried out on samples containing a constant very low concentration of naphthalene and therefore a low average occupation number per micelle at all micelle concentrations employed, decreasing as the micelle concentration increased. The absorption measurements on the other hand utilized samples saturated with naphthalene, the overall naphthalene concentration therefore increasing linearly with micelle concentration with a constant high average occupation number per micelle. There would seem to be no doubt that an equilibrium constant as high as 25 × 103 M-1 is at variance with the time-resolved data reported: as demonstrated above (see Figure 10c), the ratio of pre-exponential terms would increase very much faster initially than is observed, to the extent that the short-lifetime component would already become undetectable at very low micelle concentrations. One possibility that might explain the discrepancy is that the large mole fraction of naphthalene present in the saturated micelle interior (about five per micelle) affects its structure in such a way as to lead to an increased free energy of solubilization, a kind of cooperative solubility, of naphthalene in this “mixed” phase compared with that effective in an otherwise “pure” hydrocarbon interior. An alternative measure of the equilibrium is expressed by the dimensionless partition coefficient Kp between the micellar and aqueous phases:
Kp ) cNm/cNw
(34)
4370 Langmuir, Vol. 12, No. 18, 1996
Kowalczyk et al.
where cNm and cNw are the concentrations of (ground-state) naphthalene relative to the volumes of, respectively, the hydrophobic and aqueous phases themselves, as opposed to the concentrations of fluorophore located in the micellar and aqueous phases, respectively, and of the micellar units defined on the total volume of the sample. The equilibrium constant and the partition coefficient are thus related by
Kp ) K/Vm
(35)
where Vm is the relevant molar volume of the hydrophobic micellar phase. Assuming a molar volume of about 20 L mol-1 to enable direct comparison with previous data, the partition coefficient would appear to have a value of about (730 ( 90) at 25 °C, in reasonable agreement with the previous value of (550 ( 60), measured at 20 °C using a pulse radiolytic method.26 The equilibrium constant for partitioning of excited naphthalene between micellar and aqueous phases, K* ) (12 ( 2) × 103 M-1, is indistinguishable from that for the ground state within the experimental error attained. A potential decrease in the partitioning efficiency of excitedstate naphthalene into the hydrocarbon phase compared with that of the ground state, since in the excited state naphthalene is likely to be somewhat more polar than in the ground state, was therefore not confirmed. Since K and K* are indistinguishable and k*in was found to correspond quite closely with the expectations of diffusion theory with no, or only a rather small, barrier to crossing the interface when a collision occurs (see also further discussion below on quenching), there is no reason to suppose that the exchange rates for ground-state naphthalene are not also similar to those for the excited state. Finally, for the present at least, there seems to be no evidence to justify proposing a more complex partitioning model such as, for instance, invoking a third component corresponding to the polar region at the micelle interface with the aqueous phase, leading to a third lifetime component. It is not possible, of course, to rule out such a model with any great certainty, particularly if the third basis lifetime involved happens to be quite close to either the aqueous or the micellar-interior basis lifetime. However, it is obvious that the extra parameters would prove extremely difficult, if not impossible, to extract with any reasonable degree of certainty even for considerably more accurate data than those presented here. Apart from the third basis lifetime, four exchange rates for the excited state and two equilibrium constants would be featured in a formal kinetic model, giving rise to three lifetimes and two pre-exponential ratios, some or all of which would vary systematically with micelle concentration. Quenching at the Micelle SurfacesNaphthalene in Aqueous CTAB. Quenching at the micellar surface, if it occurs, would result from interaction of the excited fluorophore with a “heavy-atom” or charged group located at the interface, i.e. the sulfate moiety in the case of SDS and the tetraalkylammonium moiety or chloride or bromide counterions in the case of CTAC and CTAB, respectively. Such quenching would lead to a shortening of the lifetimes of the fluorophore irrespective of whether the excited-state solute was exchanging dynamically or not, since additional kinetic terms, kqw[M] in Y and kqm in X, depleting the excited-state populations in the two phases, have to be included. The term kqm does not depend on micelle concentration if the size and shape of micelles (26) Evers, E. L.; Jayson, G. G.; Robb, I. D.; Swallow, A. J. J. Chem. Soc., Faraday Trans. 1 1980, 76, 528.
remain unchanged.27 The rate coefficient kqw, like the in-rate (cf. eq 2), depends on the mutual diffusion coefficient of the fluorophore and micelle:
kqw ) κqk*diff
(36)
where κq is the efficiency of quenching given that a collision has occurred. It is evident that the efficiencies of quenching at the interface and crossing the interface are complementary: κ* + κq e 1. Thus, for a 100% quenching efficiency at the interface, κ* ) 0 leading to k*in ) 0 and k*out ) 0, so that no excited-state exchange behavior will be observed. Under these conditions, the pre-exponential ratio will be linear with micelle concentration, the now specifically and exclusively micelle-associated lifetime will be constant, and the now specifically and exclusively water-associated one will decrease in a Stern-Volmer manner. It was already clear that any such effect must be small for the SDS3-5 and CTAC5 systems, since the micelleassociated lifetimes reported remain very similar to that observed in a hydrocarbon environment, and the work reported here confirms this more quantitatively for the SDS case. In fact for the present data, an additional global target analysis including the additional quenching rate kqw[M] specifically (kqm, whatever its value, manifests itself only as part of the basis lifetime which becomes effectively km + kqm) yielded no significant improvement of the fit. In addition, the inclusion of substantial values for kqw, held fixed in the analysis, resulted in poorer fits to the data. These analyses would appear entirely to rule out the possibility of significant quenching of excited-state naphthalene at the interface between the aqueous and micellar phases in the aqueous SDS system. However, previous lifetime data for naphthalene in CTAB,3-5 or CTAC to which a small fraction of CTAB has been added,5 have shown that both lifetimes observed are distinctly shorter than that in a pure aqueous solution or those in the SDS or pure CTAC systems. From the theoretical consideration of the effects of such quenching presented here, and in view of the comparable drastic reduction of lifetimes observed in CTAB3-5 and in CTAC with which only small amounts of CTAB are admixed,5 it would seem quite possible that a full nanosecond timeresolved examination of the aqueous naphthalene/CTAB system as a function of micelle concentration might indicate by constancy of one lifetime and linear micelle concentration dependence of the inverse of the other that in this heavily quenched system no exchange of excitedstate naphthalene takes place, the equilibrium being confined to the ground-state fluorophore only. If this is indeed the case, it is obvious too, without going into specific details, that steady-state fluorescence measurements must yield an incorrect estimate for the ground-state equilibrium constant in such a 100% quenching efficiency system. From the above results in the SDS system, it is to be expected that naphthalene in aqueous CTAB will exhibit the same phenomenon if the surface quenching efficiency κ* is not too near unity. However, given this added complication of at least considerable quenching at the interface, it may be very difficult to establish the exchange experimentally. To illustrate this, the results of model calculations comparing “pure” dynamic exchange fluorescence behavior with that with 100% quenching on the micelle border are presented in Figure 12. In this figure the curves labeled a are the same as those also displayed (27) In the cases considered, it would seem reasonable to invoke only a single quenching rate kqm appropriate for all micelles rather than a distribution of rates as appropriate for Poisson occupancy of micelles by added partitioning quenchers.9
Time-Resolved Fluorescence of C12H10 in Aqueous SDS
Langmuir, Vol. 12, No. 18, 1996 4371 9 -1 -1 aqueous-micellar interface (k* s , kqw in ) 20 × 10 M ) 0). Curves b, on the other hand, were calculated for the case with the same, uniquely micelle-associated, lifetime τm ) 20 ns (including now all types of quenching of probe fluorescence inside the micelle) but with the assumption of 100% quenching efficiency at the interface (k* in ) 0, kqw ) 20 × 109 M-1 s-1). For the latter case, it is evident that the micelle- and aqueous-phase lifetimes will cross, since quenching in the aqueous phase will decrease the initially longer lifetime while the micellar lifetime remains constant. This is immediately evident in the more normal, Stern-Volmer quenching plot of such data, and the dependence of the R ratio for these lifetimes is also linear with micelle concentration. The curves shown in Figure 12 for this case express the same fact but are presented in the manner of those for the excited-state exchange case (i.e. separating the longer and shorter decay times observed) for more convenient comparison. The sharp inflection point, at a micelle concentration of about 1.1 mM in this case, corresponds to the concentration at which the two lifetimes become identical and switch their assignment, as indicated also by the pre-exponential ratio which inverts at this point. For intermediate cases, as the quenching efficiency at the interface increases, k*in decreases and kqw increases. It is evident from the figure that the changes in lifetimes as a function of micelle concentration in systems having these characteristics will not by themselves provide a good qualitative indication of the presence of excited-state exchange, whether or not quenching is occurring at the interface. On the other hand, the ratio of pre-exponential factors is evidently very much more sensitive to the balance between quenching and exchange, particularly at high quenching efficiencies. As well as giving a qualitative indication of the balance between exchange and quenching, it is this which should ensure that a global target analysis for the appropriate ratio of pre-exponential terms as well as the lifetimes would enable the contribution of both processes to be accurately assessed.
Figure 12. Comparison of time-resolved fluorescence behavior of the case II model system, for which (km + k* out) > kw, in the “pure” dynamic exchange limit (a), with that for the “no exchange” case when 100% quenching takes place at the aqueous-micellar interface (b). An intermediate case with both 9 -1 s-1, k 9 exchange and quenching (k* qw ) 18 × 10 in ) 2 × 10 M 6 s-1) is also illustrated. The “pure” M-1 s-1, k* ) 0.1 × 10 out dynamic curves are the same as in Figure 5. The rest of the parameter values and other details are given in the text. This presentation of R21, the ratio of the pre-exponential terms associated with the short (τ2) and long (τ1) lifetimes, leads to the inversion seen for the 100% quenching case which occurs when the two observable lifetimes become identical at a micelle concentration of about 1.1 mM: below this point, R21 ) Rm/Rw, the ratio of the pre-exponential terms associated with micellar and aqueous environments, respectively, while above it the ratio becomes Rw/Rm. In the more usual (Stern-Volmer) plot for such a case, the ratio Rw/Rm increases linearly with micelle concentration throughout.
in Figure 2 and relate to “pure” dynamic behavior in model system II (kw < (km + k* out)) without any quenching at the
Acknowledgment. This work was supported by the Cancer Research Campaign and The British Council. J.V. wishes to thank The British Council also for personal support. The authors would also like to record their gratitude to Dr. A. J. Swallow, whose request that we try to confirm his partition coefficient for naphthalene in aqueous SDS micelles26 led to this work, to R. N. Zobel for helpful advice on setting up the data transfer system for the nanosecond pulse fluorometer, and to Dr. M. Ameloot for his critical appraisal of an original manuscript which led us better to clarify several points in the text.
Supporting Information Available: More complete detail on the design and operation of the pulse fluorometer used in these studies, as well as on aspects of the least-squares data analysis applied (5 pages). Ordering information is given on any current masthead page. LA9602729
