Fluorescence Emission of Pyrene in Surfactant Solutions

This is the accepted manuscript of the following article: Piñeiro, L., Novo, M., & Al-Soufi, W. (2015). Fluorescence emission of pyrene in surfactant solutions. Advances In Colloid And Interface Science, 215, 1-12. doi: 10.1016/j.cis.2014.10.010. © 2015 Elsevier B.V. This manuscript version is made available under the CC-BY-NC-ND 4.0 license (http://creativecommons.org/licenses/by-nc-nd/4.0/) 1 Fluorescence Emission of Pyrene in Surfactant Solutions Lucas Piñeiro, Mercedes Novo, and Wajih Al-Soufi* Department of Physical Chemistry, Faculty of Science, University of Santiago de Compostela E-27002 Lugo (Spain), Fax: (+34) 982824001, E-mail: wajih.al-[email protected] Abstract The systematic description of the complex photophysical behavior of pyrene in surfactant solutions in combination with a quantitative model for the surfactant concentrations reproduces with high accuracy the steady-state and the time resolved fluorescence intensity of pyrene in surfactant solutions near the cmc, both in the monomer and in the excimer emission bands. We present concise model equations that can be used for the analysis of the pyrene fluorescence intensity in order to estimate fundamental parameters of the pyrene-surfactant system, such as the binding equilibrium constant K of pyrene to a given surfactant micelle, the rate constant of excimer formation in micelles, and the equilibrium constant of pyrene-surfactant quenching. The values of the binding equilibrium constant KTX100 = 3300 103 M-1 and KSDS = 190 103 M-1 for Triton X-100 (TX100) and SDS micelles, respectively, show that the partition of pyrene between bulk water and micelles cannot be ignored, even at relatively high surfactant concentrations above the cmc. We apply the model to the determination of the cmc from the pyrene fluorescence intensity, especially from the intensity ratio at two vibronic bands in the monomer emission or from the ratio of excimer to monomer emission intensity. We relate the finite width of the transition region below and above the cmc with the observed changes in the pyrene fluorescence in this region. Keywords: Surfactants · Pyrene · Critical Micelle Concentration · Fluorescence · Kinetic Description Supporting information for this article is available on the WWW or from the author. Contents 1 Introduction ............................................................................................................................................... 2 2 Theory ....................................................................................................................................................... 3 2.1 Surfactant Concentration Model ...................................................................................................... 3 2.2 Binding Equilibrium ......................................................................................................................... 4 2.3 Dye Distribution ............................................................................................................................... 4 2.4 Excimer Formation ........................................................................................................................... 5 2.5 Pyrene-Surfactant Quenching........................................................................................................... 5 2.6 Steady-State Fluorescence Intensity ................................................................................................. 6 2.7 Time-Resolved Fluorescence Intensity ............................................................................................ 7 3 Results and Discussion.............................................................................................................................. 8 3.1 Steady-State Fluorescence Intensity ................................................................................................. 8 3.2 Time-Resolved Fluorescence Intensity .......................................................................................... 11 3.3 Pyrene-Surfactant Quenching......................................................................................................... 12 4 Conclusion .............................................................................................................................................. 12 5 Experimental Section .............................................................................................................................. 13 5.1 Materials ......................................................................................................................................... 13 5.2 Absorbance and Fluorescence Measurements ................................................................................ 13 5.3 Data Analysis ................................................................................................................................. 13 Acknowledgements.......................................................................................................................................... 14 References ....................................................................................................................................................... 14 Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 2 1 Introduction Pyrene has been used since more than 50 years as fluorescent probe par excellence for microheterogeneous systems such as micelles [1-13], polymers [14-16], proteins [17-19], peptides [20] and biological membranes [18,21- 23]. The sensitivity of the pyrene fluorescence intensity to the solvent polarity is widely used for the determination of the cmc of micellar systems [3,12,13,24-30]. Therefore, it is surprising that there is still no established procedure to extract a cmc value from pyrene fluorescence data that can be related to basic properties of the micellar system and thus be compared to results obtained from other techniques. Currently, ad hoc graphical procedures are used, such as the intersection of straight lines drawn through the experimental data at low and high surfactant concentrations or the fit of arbitrary functions that have no foundation on the well-known fluorescence properties of pyrene in micellar systems. In order to obtain a cmc value consistent with other techniques it is necessary to use a common definition of the cmc that is based on the surfactant concentration and then to deduce the response of the pyrene fluorescence to the changes in the micellar solution around the cmc. The difficulty to do this lies mainly in the first part, the definition of the cmc and the necessary model for the concentrations of monomeric and micellized surfactant around the cmc. Many excellent theoretical and numerical descriptions of surfactant self-aggregation [31- 37] allow now to understand and to predict the properties of micellar systems, but they are still too complex to be used as models for the analysis of experimental data. As a practical solution we proposed recently a manageable model for the surfactant concentration derived from a few empirically established properties of micellar solutions around the cmc [38]. This model reproduces with high precision the concentrations of monomeric and micellized surfactant around the cmc and allows one to deduce unified descriptions for several properties that depend on these concentrations. We obtained consistent cmc values from different surfactant properties such as electrical conductivity, surface tension, NMR chemical shifts, absorption, and self-diffusion coefficients, and also from the fluorescence intensity and the mean translational diffusion coefficient of several fluorescent dyes in surfactant solutions [38,39]. In this contribution we combine this concentration model with the well-known photophysics of pyrene in order to obtain a manageable procedure for the analysis of the fluorescence intensity of pyrene in micellar solutions. The photophysical behaviour of pyrene in microheterogeneous systems is complex and has been object of many detailed studies and reviews. The fluorescence spectrum of pyrene shows characteristic vibronic bands around 370 - 400 nm, whose absolute and relative intensities, width and positions depend sensitively on the polarity of the microenvironment [3,26,40-45]. The pyrene fluorescence emission can be quenched due to the diffusion controlled formation of excimers with a characteristic emission band at longer wavelengths around 500 nm [24,40,46-49]. The time dependence of the pyrene fluorescence intensity after pulsed excitation follows Stern- Volmer quenching kinetics in pure water but has a much more complex behaviour in microheterogeneous environments [26,49-53]. Although pyrene is highly hydrophobic and only sparingly soluble in pure water, it still follows a partition equilibrium between the aqueous phase and the micellar pseudophase with a partition equilibrium constant that is high but not infinite [27,39]. The ratio of the fluorescence intensities of the first and third vibronic bands of pyrene (II/IIII-ratio) increases characteristically with increasing polarity of the probe environment and defines the so called “py-scale” [42,44]. The passage of the hydrophobic pyrene from the aqueous phase to the apolar micellar pseudophase with increasing surfactant concentrations results in a sigmoidal decrease of the II/IIII -ratio around the cmc. Although visually very suggestive, there is not one special point of the sigmoid which can be directly assigned to the cmc of the solution, especially not the centre of the sigmoid. Zana et al. [27] gave a special solution for the limiting case of zero cmc taking into account the partition equilibrium of pyrene between aqueous and micellar phase. Aguiar et al. proposed to use the relative width of the sigmoidal as a criterion for the selection of one of two possible cmc values: the centre of the sigmoid in the case of nonionic surfactants and the intersection of two straight lines drawn through the rapidly changing part and the horizontal part at high concentrations for ionic surfactants with higher cmc values [30]. This leads to cmc values that are in agreement with those obtained with other techniques. However, this phenomenological approach does not take into account the partition equilibrium of pyrene and does not relate the sigmoidal model and its parameters with more fundamental physical properties of the system under study. It is not clear why any of the two selected concentrations should represent the same cmc as determined with other techniques. Pyrene excimer formation is another property that is highly sensitive to the formation of the first micelles around the cmc [8,52,54,55]. Due to the low solubility of pyrene in pure water, the efficiency of excimer formation is low at surfactant concentrations below the cmc. The same is true at high surfactant concentrations where there is a very low probability to find more than one pyrene molecule in a micelle. However, at the cmc the pyrene molecules are crowded into the first few micelles so that excimer formation is favoured and leads to a relatively strong peak in the excimer fluorescence intensity. Again, this peak does not coincide with the cmc itself and the quantitative analysis of the excimer fluorescence for the determination of the cmc has not been fully achieved yet. Misra et al. interpret the fact that the excimer peak appears at concentrations below the cmc as evidence for the formation of pre-micellar aggregates and the gradual decrease of the I1/I3-ratio as a progressive increase of the micellar size [56]. As we will show here, both observations can be readily explained taking into account the partition equilibrium of pyrene and the finite size of the transition Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 3 region around the cmc without the need for pre-micelles or a step-wise growth of micellar size. A full description of the dependence of the pyrene fluorescence intensity and decay on the surfactant concentration near the cmc has to take into account several processes (Scheme 1): (1) the concentration of monomeric surfactant and micelles for a given total surfactant concentration (the surfactant-concentration model), (2) the partition equilibrium of pyrene between bulk water and micelles, (3) the distribution of bound pyrene among the micelles, (4) the quenching of pyrene due to excimer formation and (5) the interaction of pyrene with monomeric surfactant. We will combine these processes step by step and then apply the resulting equations to experimental data. Scheme 1: Processes controlling the dependence of the pyrene fluorescence intensity and decay on the surfactant concentration near the cmc. Free dye (Df) partitions between bulk water and micelles with binding equilibrium constant K. The distribution of the bound dye (Db) among the micelles leads to different local dye concentrations in micelles Mi with occupancy i. Not shown here is the quenching of free dye by surfactant molecules and the formation of dye excimers in the micelles. 2 Theory We distinguish three dominant fluorescent pyrene species in a micellar solution: free pyrene in the aqueous phase emitting as monomer (Df), and pyrene bound to micelles emitting as monomer (Dbm) or emitting as excimer (Dbe) (see Fig. SI2 in the SI). We ignore the very weak contribution of the excimer formed by free pyrene in water. At a given emission wavelength λ the total observed fluorescence intensity I(λ) is the sum of the contributions of each of the species: () () () () f bm be [D ] [D ] [D ] f bm be IF F F λλ λ λ =++ (1) This expression for the fluorescence intensity applies both in the so called “monomer” and in the “excimer” bands of the pyrene emission spectrum, with varying contributions of the species depending on the wavelength. In the “monomer” band the emission is dominated by free and bound monomeric (Ff, Fbm) pyrene with a residual emission of the pyrene excimer. In the “excimer” band the situation is just reversed, with a strong contribution of the excimer (Fbe). The task is now to obtain reasonable model equations for the concentrations of the species (Df, Dbm, and Dbe) as a function of the total surfactant concentration [S]0. A rigorous kinetic description of the pyrene fluorescence in detergent micelles was given by Infelta and Grätzel [5]. We combine their results with our surfactant concentration model, introduce the interaction between pyrene and surfactant monomers observed at low surfactant concentrations and derive and discuss expressions for cases relevant for the determination of the cmc. We also derive equations for the time dependency of pyrene fluorescence. In the following we present the steps in a systematic way. 2.1 Surfactant Concentration Model The established model to calculate the concentration of micelles in solution 0 [M] ([S] ) /cmc n= − from the total surfactant [S]0, the cmc, and the aggregation number n, works well above the cmc region, but is not defined in the important region below and near the cmc. Recently we introduced an improved surfactant-concentration model that has proven to provide an excellent basis for a variety of derived direct properties of a surfactant solution around the cmc and also for the fluorescence intensity and mean translational diffusion of fluorescent probes in micellar solutions [38,39]. The model is still compact enough to be easily implemented in standard fitting software for routine data analysis. At low [S]0 monomeric surfactants dominate whereas at high [S]0 micelles determine the properties of the solution. The transition between these regions around the cmc is not abrupt but gradual within a transition concentration interval with width σ , given by the width of a Gauss function that describes the second derivative (curvature) of [S1] with respect to [S]0 (see SI). The smaller σ is the sharper is the transition between the two linear regions below and above the cmc. In order to facilitate the comparison of widths σ between surfactants with different cmc values, we define the relative transition width r as: r cmc σ = (2) Our model (eq. (3)) for the concentration of surfactant monomers [S1] in a solution with total surfactant concentration [S]0 is based on the “Phillips-condition” [57] and has the form of adequately normalized sigmoidal function centred around the cmc and with relative transition width r and the relative concentration 00 [S]s cmc= [38,39]: 2 120 () 2 [S ] 1 exp 1 11 ( erf 1 2 ) 22 00 ss s A cmc r rr p  −−  +−       =−−−           (3) where A is the normalization constant (for small r <0.5: A≈1): 2 21 ( 0.5) 1 2 1 exp 21 erf 2 Ar rrr p = ≈<  + −+      (4) The concentration of aggregated (micellized) surfactants, [Sm], is the difference between the total concentration and that of monomeric surfactants: m 01 [S ] [S] [S ]= − (5) Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 4 The concentration of micelles, [M], depends on the mean aggregation number n, which is taken to be constant around the cmc: m [M] [S ]/ n= (6) Fig. 1-a shows simulated data of [S1] and [Sm] and the Gaussian function with width r calculated with the concentration model (eqs. (3) to (5) ) for three typical values of the relative transition width r. Details of the surfactant concentration model (eqs. (3) to (6)) have been discussed before and are given in the SI [38]. The concentration of micelles formed already below the cmc increases with the relative transition width r (see Fig. 1-a). For typical small values of r (r<0.5) the fraction of micellized surfactants m0 [S ]/[S] at the cmc is m [S ]( ) / 0./2 4cmcrcm rc p ≈≈ and thus the micelle concentration at the cmc is given by: 1 [M]( ) 0.4 22 cmc r cmc nn n σσ pp ≈=≈ (7) The polydispersity of real surfactant samples is not explicitly taken into account but is captured by the relative transition width r. For example, the conductivity of the polydisperse surfactant mixture LAS is well described by the model with an atypically high value of r of 0.48 [38]. Fig.1: Relative concentrations [S1]/cmc, [Sm]/cmc, molar fractions Xb, Xbe, Xb·Xbe, mean occupancy i and occupancy probability P(i>1) versus relative surfactant concentration s0 = [S0]/cmc for different values of r and K as indicated. The thin lines in panel a are Gaussians with width r and arbitrarily normalized amplitude. (Values used for the simulations: cmc=0.275, n=143, K=3300 and 6000, [D]0=1.6 10-3, Kq=2.2, K’be=0.6). The value of K is normalized to K·cmc/n. 2.2 Binding Equilibrium Near the cmc pyrene coexist in the aqueous phase and in the micellar pseudophase with concentrations of free and bound pyrene given by a 1:1 binding equilibrium as shown in eq.(8), with an apparent (macroscopic) binding equilibrium constant K [4,27,39,58-60]. Some authors prefer the term “partition equilibrium”. However, the binding equilibrium constant K should not be confounded with a partition constant of a phase equilibrium [61]. b f f [D ] D +M D [D ][M] b KK= ���� � � �� (8) The molar fractions of free (Xf) and bound dye (Xb) with respect to the total dye concentration 0f b [D] [D ] [D ]= + are given by eq. (9). The fraction of bound dye Xb near the cmc is shown in Fig. 1-b for different values of r and K. Both K and r determine Xb. For a given value of K=6 (thick lines) an increasing transition width r moves the onset of the transition region towards lower concentrations (to about cmc-2σ or 1-2r in this scale) and thus also, the onset of the increase in Xb. With increasing r the concentration of dye bound to micelles already below the cmc increases. Comparing two values of K (K=6 and K=12) at the same r =0.15, the onset of the change in Xb coincides, but the change is faster for the higher K. In any case, both parameters affect Xb in a rather similar way and are therefore correlated in nonlinear fits. b f 00 [D ] [D ] 1 ·[M] , [D] 1 ·[M] [D] 1 ·[M] fb K XX KK = = = = ++ (9) 2.3 Dye Distribution The mean number of pyrene molecules per micelle (mean occupancy i ) is given by the ratio of the concentrations of bound pyrene and micelles: i b0 f0 i 0 ·[M ] [D ] [D ] [D] · · [M] [M ] ∞ = ∞ = = = = = ∑ ∑ if i i i K KX (10) At very low surfactant concentration (and consequently low [M] and [Db]) the mean occupancy i starts at a limiting value of 0 ·[D]≤iK , and then decreases proportionally to the fraction of free dye Xf as the micelle concentration increases (see Fig. 1-c). Again K and r have subtly different influence on the mean occupancy i . With a broader transition width r the mean occupancy i decreases already at lower surfactant concentrations [S]. Higher values of K move the whole curve up to higher values of i . The distribution of bound pyrene among the micelles is sufficiently well described by a Poisson distribution [5,26,62-65]. The probability Pi that a micelle is occupied by i pyrene molecules (in a micelle Mi) depends on the mean occupancy of the micelles i as in eq. (11). i [M ] [M] ! ii i ie Pi − = = (11) Fig. 1-c shows the probability Pi>1 that micelles are occupied by two or more dye molecules. Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 5 2.4 Excimer Formation k m = k mr + k mnr k-1 k e = k er + k enr D* e Dm+ D* m Dm+ Dm k1 Scheme 2: Mechanism of the formation of excimer e D∗ with rate constants k1 and k-1 after excitation of the monomeric dye Dm. The deactivation rate constants of monomeric dye (km) and excimer (ke) are sums of radiative (kmr, ker) and non-radiative rate constants (kmnr, kenr). The differential equations describing the dynamics of the excimer formation of pyrene in micelles have been presented by Infelta et al. and other authors [5,47,52]. The equations for the case of a homogeneous pyrene solution without surfactants is presented in the SI. Here we show briefly the results in a micellar solution with bound pyrene. The detailed derivation is given in the SI. An excited monomeric dye m D∗ can either deactivate directly to the ground state or it can form the excimer e D ∗ interacting with an unexcited monomeric dye Dm, as shown in Scheme 2. The lifetimes of the excited monomer is 1 1m ( [D ]) mm kk τ − += and that of the excimer 1 )( ee k τ − = . In a micellar solution both free and bound pyrene contribute to the observed fluorescence intensity (eq. (1)). Free pyrene in water forms only a very small fraction of excimer Dfe, which we neglect in our model ([Dfe] ≈ 0, [Dfm] ≈ [Df]). This free excimer Dfe could, however, be easily included in the model. On the contrary, the contribution of the excimer emission from pyrene bound to micelles is important and has to be modelled carefully taking into account the local pyrene concentration given by the occupancy i of the micelles. Without back reaction in the excited state 1 () e kk − � , the fraction of bound excited dye in micelles Mi that decays as monomers () () i bm X or excimers () () i be X is given by: () () () 11 1 ( 1) i ii bm be bm be X XX Ki = = − ′ +− (12) The fraction of bound excimer (1) be X is zero for a micelle M1 occupied by only one dye molecule and increases with higher occupation numbers. The constant Kbe of the formation of excimer in micelles is given by the rate constants of excimer formation k1 and of the monomer deactivation km and is taken to be independent of the occupancy of the micelle: 1be be be mm K k KK kV ′ = = (13) The volume Vm contains the dye bound to a micelle and defines the local concentration of not excited dye in a micelle Mi: () bm [D ] ( 1) / im iV= − . This volume is included in the constants / be be m K KV ′= and 11 / m k kV ′= . Neither the constant Kbe nor the volume Vm depend on [S]0 as long as the structure and the size of the micelles do not change significantly. The total steady state fluorescence intensities Ibm and Ibe of bound monomer and excimer are then given by the sum of the intensities from micelles with different occupation numbers (see SI): 00 [D] , [D] bm bm bm b be be be b I FX X I FXX= = (14) with the fraction b X of bound dye (eq. (9)), and the monomer and excimer fractions of bound dye bm X and be X : 0 1,1 1! i i bm be bm ibe i Xe X X iK i ∞ − = = = − ′ + ∑ (15) As can be seen in Fig. 1-d the excimer fraction of bound dye be X decreases with increasing surfactant concentration following the decreasing mean occupation number. However, more relevant, the overall fraction of bound dye emitting as excimer b be XX shows a sharp maximum near the cmc. This excimer fraction increases at the beginning of the transition region as the first bound dye appears (increasing Xb, high i ), passes through a maximum near the cmc and then decays as i gets smaller. The peak in b be XX results from the competition of two effects. On the one hand, the fraction of bound dye Xb increases as the first micelles appear; on the other hand, the probability that this bound dye emits as excimer (Xbe) decreases with the surfactant concentration since it is distributed among more micelles with decreasing occupancy i . The form of b be XX curve and the position of its maximum depend sensitively and in a complex manner on both on r and K. It is worth noting, that the peak in this curve does not simply results “from the passage of i through a maximum after which all the pyrene in the system is solubilized in micelles”, as previously reported [66]. The occupancy i has no maximum and, more important, the fraction of bound (solubilized) pyrene Xb is still far from reaching saturation (Xb =1) at the maximum of the b be XX curve (see Fig. 1-d). Full solubilisation of pyrene is only reached asymptotically at high surfactant concentrations that depend, of course, on the value of K (see also Figs. 3, 5, 6) 2.5 Pyrene-Surfactant Quenching The emission of dyes may be quenched by surfactant molecules and their counterions [60]. Pyrene is a neutral probe and does not show the complex behaviour observed for charged dyes and ionic surfactants, as for example for the cationic dye Rhodamine 123 [60]. In this section we describe the observed decrease of the pyrene fluorescence below the cmc as a static and/or dynamic quenching process. We discuss the quenching process in section 3.3 Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 6 on the basis of steady-state and time-resolved fluorescence data. The contribution to the fluorescence intensity () f I λ of the unquenched free dye can be expressed by the fraction of not quenched dye nq X and the total concentration of free monomeric dye f0 [D ] [D] f X= : () () 0 [D] λλ = f f nq f I F XX (16) If both static and dynamic quenching occur the fraction ( )( ) ( ) 1 11 1 [S ] 1 [S ] − =++ nq qs qd XK K is second order in [S1] with the quenching constants Kqs and Kqd of static and dynamic quenching, respectively. In the case that one of the two quenching processes dominates a first order fraction with quenching constant Kq is sufficient: ( ) 1 1 1 [S ] − = + nq q XK (17) The nature of the dominant quenching process has to be determined from additional information such as timeresolved fluorescence data. Above the transition region the fraction becomes constant 1 (1 · ) nq q X K cmc − = + . The same quenching formalism applies to the pyrene species bound to micelles, Dbm and Dbe. However, the quenching of these species depends on the local concentration of surfactant and on the microviscosity within the micelles. The size and structure of micelles do not significantly change within the concentration range studied here, and, therefore, within a micelle, both the quenching constant Kq and the local surfactant concentration [ ] m S/ m local nV≈ should be independent from the total surfactant concentration. Then, the fraction of quenched bound dye is constant and can be included in the constants Fbm and Fbe. 2.6 Steady-State Fluorescence Intensity Finally we can express the overall observed fluorescence intensity () 0 ([S] )I λ as a function of the total surfactant concentration substituting in eq. (1) the individual contributions deduced above: ( ) () () () () () () () , 0 ,, [D] f nq bm be f f nq bm f b bm be f b be I I I I F XX q XX q XX λλ λλ λ λ λ = ++= + + (18) Here we define brightness ratios q of the fluorescence intensities of bound monomer and excimer with respect to free dye at a given wavelength λ (see Fig.2-a): () () () () ,, () () , bm be bm f be f ff FF qq FF λλ λλ λλ = = (19) Together with the concentration model of eqs (3)-(6) eq. (18) describes the fluorescence intensity of pyrene in micellar solutions as a function of total surfactant concentration with model parameters cmc, r, n, K, () f F λ , () ,bm f q λ , () ,be f q λ , Kq, K’be, and [D]0. The total dye concentration [D]0 and the aggregation number n are usually known. The fluorescence intensity of free dye in pure water () () () 00 [D] ([S] 0) ff FI I λλ λ = = ≡ , and the ratio () ,bm f q λ can be determined from the intensities at zero and at high micelle concentrations, respectively. The other parameters (cmc, r, K, Kq, K’be, and () ,be f q λ ) have to be determined by nonlinear fit. Traditionally not the absolute intensity of the pyrene emission is analysed but the spectral ratio SR at two wavelengths, λ1 and λ2, for example the monomer/monomer (II/IIII) ratio (“py-scale” [42,44]) or excimer/monomer ratios. We distinguish here between the “spectral ratio” (2,1 )))(2 (1 IISR = of the fluorescence intensities at two wavelengths of the same species and the “brightness ratio” q of the intensities of two different species (eg. free and bound) at the same wavelength. With eq. (18) we obtain for the spectral ratio at λ1 and λ2: (2) (2) (2) (2) (2,1) (1) (1) (1) (1) f f nq bm b bm be b be f f nq bm b bm be b be FXX FXX FXX I SR I FXX FXX FXX ++ = = ++ (20) Neglecting the residual excimer emission in the monomer band and vice versa, we get the following two expressions for the spectral monomer-monomer and excimer-monomer ratios (see SI): (2,1) (2,1) (1) (2) , (2,1) (1) (1), (, ) (, ) ( ) (, ) () ,, (, ) () () , f f nq bm bm f b bm m f nq bm f b bm em em m em ef f nq bm bm f b bm be f b be em emm f nq bm f b bm SR X X SR q X X I SR I XX q XX SR X X SR q X X q X X I SR I XX q XX + = ≈ + ++ = ≈ + (21) The limiting values of the monomer-monomer ratio (1,2) m SR are the spectral ratios of free pyrene, (2,1) f SR at 0 [S] 0= , and of bound monomeric pyrene, (2,1) bm SR at 0 [S] cmc� (see Fig.2): (2) (2) (2,1) (2,1) (1) (1) () () (, ) (, ) () () () (, ) ,() fbm f bm f bm ee f em em bm f bm mm f bm e em be be f m f FF SR SR FF FF SR SR FF F qF = = = = = (22) At very low pyrene concentrations the excimer formation can be neglected in eq. (21) (Xbm≈1, Xbe≈0). Below the transition region (2,1 (21),) m f S SRR ≈ , independently of the quenching fraction Xnq. Above the transition region Xnq is constant. With the constant (1),(1 · ) bm f q c q K K cmc= + and not too high values of r we get a further simplified equation for (2,1) m SR (see SI): Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 7 (2,1) (2,1) (1) (2,1) (2,1) , (1), (2,1) [M] [M] [M] 1 [M] f nq bm bm f f bm nq bm m f SR X SR q K SR SR c X qK SR c ++ ++ ≈≈ (23) Around the cmc both the absolute intensities I(λ) and the spectral ratios (2,1 2) 1)( () m SIIR= and (, ) () ( )em e m e SR I I= depend sensitively on the interplay between the binding equilibrium constant K and the occupancy i (see Fig.2-b and c). The intensities and the spectral ratios follow the fractions Xf, Xb·Xbm, and Xb·Xbe discussed above (Fig. 1). Note that the curves of the intensities and those of the spectral ratios are significantly different, with different positions of the inflection points of the monomer curves and of the peaks of the excimer signals. The cmc does not correspond to any of these special points, but its relative position depends again on the values of K and r. It is also important to note, that neither of these curves has the exact form of a sigmoidal Boltzmann function. Fig.2: a) Detail of the pyrene spectrum with the spectral parameters used in the text. b) Simulated fluorescence intensity () 0 ([S] )I λ of eq. (18) at three wavelengths 372 nm (1), 384 nm (2), and 500 nm (e) as indicated in a). c) simulated spectral ratios (2,1) f SR and (2,1) bm SR of eq. (21) at the three wavelength 1,2, and e as indicated in a). 2.7 Time-Resolved Fluorescence Intensity The time dependence of the pyrene fluorescence had been derived by Tachiya, by Selinger and Watkins, and by Infelta et al. [5,47,50,52,67]. We summarize the main results in our nomenclature and combine them with our surfactant concentration model. The detailed derivation can again be found in the SI. The observed time dependent fluorescence intensity is the sum of the contributions () () ( ( ), ( )) ii bm be itit from micelles Mi with different occupation numbers i: ( ) ( ) ( ) ( ) ( ) ( ) () 10 () 10 () () ( ) e exp ! () e e [D] (1 [D] exp xp 1! (1 bm b m b i ii bm bm be ii i ii be be be beb e m e e im ibe t t F kt i F i I i t Ki i K i Ikt k ti t Ki iikkK ∞∞ − = = ∞∞ − = = ′ = = ′′ = = −+ −− − +′+ − ∑∑ ∑∑ (24) The fluorescence intensity in the monomer band () bm It can be solved analytically ( / be be m K KV ′= , 11 / m k kV ′= ): ( ) ( ) ( ) ( ) ( ) ( ) '' 1bb ( ) [D ] 1exp ex [D ] 1p exp exp bm bm be m m bm m It F i FKk k it t kt kt= = −−−−− − (25) The analytical expression for the excimer intensity () be It is less useful for numerical applications (see SI). In fitting functions it is more reliable to calculate the sums in eq.(24) up to some reasonable occupation number i. Note that () bm It is not monoexponentially with a lifetime 1 1 ( [D ]) mm m kk τ − = + as one would expect in a homogeneous solution. The excimer formation with () 1[D ] i bm k depends on the local dye concentration () [D ] i bm , which varies according to the distribution of the dye among in the micelles. In each micelle Mi the excited dye decays with a different lifetime () 1 1 () ( [D ]) i m bm i bm kk τ − = + . In higher occupied micelles the monomer emission is rapidly quenched due to the effective excimer formation leading to fast decays. At t=0 the decay starts with a rate constant 1 ( ) (1 ) bemm kk kiV Ki= + ′ + . At longer times (1) mbe ktK′� finally only the emission from micelles M1 with a single excited dye is observed that decays monoexponentially with (1) 1 () bm m k τ − = . The time dependence in eq. (25) can be easily confounded with a double exponential decay with (1) 1 () bm m k τ − = and a short lifetime, which has no explanation in a homogeneous system (see Fig. SI1 in SI). The monomer decay () bm It in eq. (25) is furthermore strongly dependent on the micellar concentration. First, trivially, because the concentration of bound dye [Db] depends on [M] (eq. (9)). Second, and more important, because the mean occupancy i in the exponent of () bm It varies with both [Db] and [M] (eq.(10)). At high micellar concentrations 0 [S] )( cmc� the mean occupancy tends to zero ( [ ]/[ ] 0) b iDM= → and () bm It decays monoexponentially with km. At very low micellar concentrations the mean occupancy is limited by 0 [D] ·iK≤ (eq. (10)), although the amplitude of the decay vanishes. The expression of eq. (25) can be used directly as fitting function for () bm It . The excimer fluorescence intensity () be It is zero at time zero, ( 0) 0 be It= = and increases with a complex time dependent rise time given by that of the monomer decay as discussed before. At long times the decay is Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 8 monoexponential with e k . However, at intermediate times the intensity curve follows a more complex function than the difference between excimer and monomer decays. So finally the time resolved fluorescence emission I(t) is given by the sum of the contributions of free dye () f τ , bound monomer Ibm(t) (eq. (25)), and bound excimer Ibe(t) (eq.(24)). Additionally a small contribution of surfactant fluorescence (for example in TX100) and/ro fluorescent impurities with typically short lifetime () s τ is added. The amplitudes Ax contain all constant pre-exponential factors. / / () e e () () f s t t s f bm bm be be It A A A I t AI t τ τ − − =++ + (26) 3 Results and Discussion 3.1 Steady-State Fluorescence Intensity Fig.3 shows spectra and profiles of the fluorescence intensity for pyrene in aqueous solutions of increasing concentrations of TX100. The spectrum of pyrene at the lowest TX100 concentration (black curve in panel A, [TX100] = 0.03mM) presents the known vibronic structure in the monomer band between 370 and 450 nm and no measurable excimer emission above 480 nm. At TX100 concentrations up to about 0.2 mM the fluorescence intensity decreases without significant change in the shape of the spectra. Above 0.2 mM a red-shift of the monomer band of nearly 2 nm and a strong increase of its intensity is observed. Around 0.3 mM a small and broad excimer emission appears with a maximum at 500 nm. The dependence of the fluorescence intensity in the first (II at 372 nm) and in the third vibronic band (IIII at 384 nm) and in the excimer band (Ie at 500 nm) on [TX100] is shown in panel B of Fig. 3. Below the cmc the fluorescence intensity in the monomer band (II and IIII) decreases and reaches a minimum around 0.22 mM. Near the cmc this intensity increases strongly until it reaches about twice the initial value at concentrations above 1 mM. In the excimer band (Ie) the fluorescence emission is nearly zero below the cmc. Around the cmc the excimer intensity increases sharply and then decreases more slowly at higher concentrations. The minimum in the monomer emission does not coincide with the maximum in the excimer emission, and neither of them matches the value of the cmc of TX100 of about 0.26-0.27 mM [26,39]. As can be seen in the inset of Fig.3-a and in the normalized spectra (Fig.4) the spectral shift and the change in the shape of the spectra occurs in a narrow TX100 concentration interval around the cmc. Below about 0.2mM the intensity decreases, but the spectra do not change their shape (Fig.4, dashed orange curve)). This indicates that in this concentration interval the observed emission comes from the same species, namely free pyrene monomer, and that the observed quenching leads to a nonfluorescent species. Around the cmc, between 0.2 mM and 0.4 mM (Fig.4, dashed dotted green curve), follows a fast increase in intensity accompanied by a strong red shift in peak positions and a significant change in the shape of the spectrum. These spectral changes can only be explained by the contribution of at least two fluorescent species that are assigned to free and bound monomer. Above 0.4 mM the shape and position of the spectra stay again mostly constant. Fig.3: Fluorescence emission of pyrene in aqueous solutions of TX100, [TX100] = 0.03 mM – 2.3 mM. Panel A: Fluorescence spectra (thin grey curves) and pure spectra determined from PCGA (thick red dashed curves) of free pyrene (dashed), bound monomeric pyrene (dash dotted) and bound pyrene excimer (dotted). Inset: same as main panel with reduced wavelength scale. Panel B: fluorescence intensity in the monomer band, at peak I (371.9 nm, squares) and peak III (384.2 nm, circles) and in the excimer band at 500.0 nm, (triangles) as function of the TX100 concentration. Panel C: spectral ratios ),( m SR II III I III I = (open circles) and (, e )e e SR II I I = (open triangles). Thick curves in panels B and C: Global fit of equations (18), and (21), with the parameters given in the text. The dashed grey vertical line indicates the cmc. ([Pyrene]=5.2 10-7 M, λexc=319 nm). Note the logarithmic concentration scale and the breaks in the intensity scales. Author post-print Advances in Colloid and Interface Science, 2015, 1-12, 10.1016/j.cis.2014.10.010 9 Fig.4: Normalized spectra of pyrene in TX100 solutions as shown in Fig. 3. The TX100 concentration increases from 0.04 mM (red) to 2.3 mM (blue), with 0.22 mM (dash, orange) and 0.42 mM (dashed dotted, green). Roman numbers and grey rectangles refer to the vibronic peaks in the pyrene spectrum. Principal Component Analysis (PCA) [68,69] of the series of spectra confirms the presence of three fluorescent species in the concentration range studied here (see Figs. SI5 – SI7 in the SI). We assign them to free monomeric pyrene, Df, bound monomeric pyrene, Dbm, and bound pyrene emitting as excimer, Dbe. The observed spectra (or fluorescence intensities at given wavelength) are the sum of the contributions of these three species as given by eq (18). At low [TX100] only free dye Df is present. The addition of TX100 quenches the fluorescence of free pyrene possibly due to formation of non-fluorescent surfactant-pyrene complexes (see section 3.3). Therefore the fraction of not quenched dye, Xnq, decreases with increasing [S1] (eq. (17)) until the cmc is reached (see Fig.6-b). Above the cmc the value of Xnq stays constant since the concentration of monomeric surfactant does not vary ([S1] = cmc). At the onset of the formation of micelles, just before the cmc, a further small decrease in monomer emission is observed, which is not explained by the quenching process. In parallel a sharp increase in the excimer emission, Ie, is found. Both changes are due to the formation of pyrene excimer in higher occupied micelles near the cmc that do not emit in the monomer region. As more and more pyrene molecules bind to micelles the emission in the monomer band is dominated by the higher emissive bound pyrene monomers, whereas the emission in the excimer band vanishes slowly as bound pyrene is distributed among the increasing concentration of micelles. Both the binding equilibrium and the distribution take place within a rather wide concentration interval from just below the cmc up to a multiple of it (note the logarithmic concentration scale in Fig. 3). Table 1. Results of global fits to the experimental data as indicated in the text. TX100 SDS cmc / 10-3mol ℓ -1 0.275±.005 8.1±0.1 r 0.16±0.01 0.07±0.03 K / 103 mol ℓ-1 3300±200 190±40 Kq / 103 mol ℓ -1 2.2±0.2 0.23±0.03 ' 1 3 -1 / 10 mol be be m K KV − = 0.6±0.3 3±1 n 143 64 9 /10 s f τ − 124 127 km /106s-1 9 ( /10 s) m τ − 4.3 (232) 6.3 (160) ke /106s-1 9 ( /10 s) e τ − 9.8 (102) 34 (30) ' 1 6 11 11 / 10 mol s m k kV − −− = 2.4 20 9 /10 s s τ − 5 4 (2,1) f SR / (2,1) bm SR / (1),bm f q / () , e be f q [a] 0.45/0.98/1.5/28 0.44/0.91/1.4/46 ( ,1)e f SR / ( ,1)e bm SR / ( ,1) , e be f q [a] 0.016/0.004/0.37 0.020/.004/0.85 [M](cmc) [b]/ 10-6mol ℓ -1 0.12 3.5 [a] λ1=372nm (peak I), λ2=384nm (peak III), λe=500nm (excimer) [b] Micelle concentration at the cmc estimated from eq.(7). The surfactant concentration dependence of the pyrene fluorescence emission both in the monomer and in the excimer bands are very well described by the model of eq. (18) together with the concentration model of eq. (3). Global fits at three selected wavelength, 371.9 nm, 384.2 nm and 500.0 nm, are shown in Fig.3-b. Global fits of the whole series of spectra by PCGA are given in the SI. The best fit parameter values are given in Table 1. The corresponding fluorescence intensity ratios at 371.9 nm, 384.2 nm and 500.0 nm are () ,bm f q λ =1.49, 3.26 and 0.42, and () ,be f q λ =0, 0, 27, respectively. In the monomer bands the fit reproduces excellently the quenching at low TX100 concentration, the small “dip” just below the cmc due to the formation of excimer and then the sharp increase of the intensity after the cmc. Also the excimer band is very well described, both the sharp increase around the cmc and the approximately exponential decrease at higher concentrations. The fit determines a value of the cmc that is in very good agreement with that determined from the direct TX100 absorption (cmc = 0.270 mM) [39] and with other literature values [26]. The width of the transition region (r) is higher than the value r=0.11 obtained from the TX100 absorption but comparable to that determined for the exchange of other dyes with TX100- micelles [39]. The value of the binding equilibrium constant K = 3.3×106 M-1 is very high and confirms the value published before [39].