Article pubs.acs.org/JPCB
Reversible Excited-State Proton Geminate Recombination: Revisited Ron Simkovitch,† Dina Pines,‡ Noam Agmon,*,§ Ehud Pines,*,‡ and Dan Huppert*,† †
Raymond and Beverly Sackler Faculty of Exact Sciences, School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel Department of Chemistry, Ben-Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel § Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 91904, Israel ‡
ABSTRACT: About three decades ago, Pines and Huppert found that the excited-state proton transfer to water from a photoacid (8-hydroxy-1,3,6-pyrene trisulfonate (HPTS)) is followed by an efficient diffusion-assisted reversible geminaterecombination of the proton. To model the reaction, Pines, Huppert, and Agmon used the Debye−Smoluchowski equation with boundary conditions appropriate for reversible contact reaction kinetics. This reaction model has been used successfully to quantitatively fit the experimental data of the time-resolved fluorescence of HPTS and several commonly used photoacids. A consequence of the reversibility of this reaction is an apparent long-time tail of the photoacid fluorescence signal, obeying (after lifetime correction) a t−3/2 power law asymptotics. Recently, Lawler and Fayer reported that in bulk water the observed power-law decay of the long-time fluorescence tail of HPTS is −1.1 rather than −1.5, as expected from the spherically symmetric diffusion model. In the current study, we reaffirm our previous reports of the power-law behavior of HPTS fluorescence. We also demonstrate that molecularlevel complications such as the deviation from spherical symmetry, rotational dynamics, competitive proton binding to the sulfonate moieties of HPTS, distance-dependent diffusion coefficient, and the initial starting point of the proton can affect the observed kinetics only at intermediate times, but not at asymptotically long times. Theoretically, we analyze the rebinding kinetics in terms of the number of extrema of the logarithmic derivative, showing subtle effects on the direction of approach to the asymptotic line (whether from above or below), which also appears to be corroborated experimentally.
1. INTRODUCTION Photoacids are weak ground-state acids with pKa values in the range of 5−10, which become much stronger acids in their electronically excited singlet state.1,2 ROH photoacids, based on an aryl group (denoted R) to which an OH radical is bound, exhibit (in their excited S1 state) pKa* values in the range of −8 to about 4. Starting from 1983, Pines and Huppert have investigated proton geminate recombination (GR) of photoacids in their excited S1 state.3 It has become apparent by direct time-resolved fluorescence measurements that the excited-state reaction is fully reversible and diffusion-assisted. A back-of-theenvelope solution of the reversible-reaction-coupled diffusion problem was suggested by these authors who argued that at long times the probability of the back-recombination is timedependent, the t−3/2 dependence emerging from the diffusion, which acts to continuously enlarge the effective volume of the reaction space of the geminate pair.4−7 The above kinetic model was fully accounted for by solving the reversible-reaction-coupled diffusion problem using Agmon’s numerical procedure8,9 for the “back-reaction” boundary condition.10 According to Pines, Huppert, and Agmon, the full simulation of the photoacid dissociation included the following kinetic ingredients: Photoprotic excited-state proton transfer (ESPT) to water followed by a R*O− proton GR process, occurring at a finite rate from the same reaction distance, a, to © XXXX American Chemical Society
which the proton dissociation occurs. It follows that the proton may either recombine or diffuse away from a. While recombining, the proton reforms the excited-state protonated photoacid, R*OH, implying that proton recombination occurs at the original dissociation site. The reformed R*OH can then undergo additional cycles of dissociation−GR until either the proton escapes recombination by diffusing to large distances or the photoacid or photobase decay back (radiatively and nonradiatively) to the ground electronic state. The molecular system used as a benchmark platform for demonstrating reversible GR has been 8-hydroxy-1,3,6-pyrene trisulfonate (HPTS);4−9 see Scheme 1. The three negatively charged sulfonate groups of HPTS enhance the Coulomb attraction with the dissociated proton, accentuating the probability for GR. The fluorescence signal , is proportional to the from the R*OH photoacid, IROH F probability of finding the excited pair in its bound state. This has been described as diffusion with back-reaction.10 We denote the probability that the R*O−···H+ pair is bound by P(t) and the probability density that the pair is separated to a distance of r ≥ a by p(r, t), where a is the distance of minimal approach Received: September 7, 2016 Revised: October 30, 2016 Published: November 7, 2016 A
DOI: 10.1021/acs.jpcb.6b09035 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
A complete analytical solution can be obtained when V(r) = 0, in spite of the complicated appearance of the coupled equations (eqs 1 and 2).11,14−16 For the HPTS reaction, this is insufficient because of the strong Coulomb interaction between the solvated proton and the anionic base. The deprotonated form of the HPTS anion has four negative charges. In this case, the Coulomb attraction, V(r) = −RD/r, is large and so is the R*O− proton binding probability, P(t). The Coulomb attraction gauge is the Debye radius
Scheme 1. Molecular Structure of HPTS
RD =
(1)
Here, D is the mutual diffusion coefficient, namely, the sum of the diffusion coefficients of R*O− and H+ (the latter is dominant); V(r) is the interaction potential (in units of the thermal energy, kBT); kd and ka ≡ 4πa2κa are the dissociation and association rate coefficients, respectively; and δ(r − a) is the Dirac delta function that vanishes unless r = a, restricting the reaction to the surface of the sphere. With this “sink term” accounting for the reaction, the boundary condition at contact is reflective, ∂ exp[V(r)]p(r, t)/∂r|r=a = 0. (Alternately, the reaction can be incorporated into the boundary condition.)10 The above partial differential equation is coupled to an ordinary differential equation, namely, a chemical kinetic equation for the bound state dP(t ) = kap(a , t ) − kdP(t ) dt
P(t ) ∼
(4)
ka e−V (a) kd(4πDt )3/2
=
Keq (4πDt )3/2
(5)
Equation 5, which is the exact long-time asymptotic solution of the spherically symmetric DSE for a reversible reaction, eqs 1 and 2, has a remarkably simple form. Like a bimolecular chemical reaction at finite concentrations, whose values at equilibrium depend only on the equilibrium coefficient, Keq = ka e−V(a)/kd, but not on the two rate coefficients separately, so does the long-time behavior of P(t). However, in a GR reaction, P(t) → 0 as t → ∞ because an isolated pair eventually separates to such large distances that the probability of further recombination becomes practically zero. The t−3/2 behavior reflects the probability of a random walker (in the relative separation coordinate, r) to return to the origin of its random walk (the contact distance) where the pair may recombine and regenerate the bound state. Thus, in the ES reaction, the reversibility of the PT reaction regenerates the excited acid (R*OH) whenever the proton recombines (adiabatically on S1) with its conjugate base (R*O −). The repeated backrecombination processes make the undissociated acid live longer in the excited state than it would have under irreversible conditions; thus, at long observation times (typically accessible with our experimental setup), exp(−kdt) ≪ Keq/(4πDt)3/2. If a less-drastic approximation is made on the Laplace transform of eqs 1 and 2, one can obtain approximate solutions that are closer to the exact numerical solution at intermediate times as well.11−13,18
(2)
Although these equations describe a reaction with infinite reactants’ lifetimes appropriate for the ground electronic state of the photoacid, it suffices for HPTS because the fluorescence lifetimes of the excited acid and base are essentially identical, τF = 5.3 ns. If IROH (t) is the (properly normalized) fluorescence F signal from the pure R*OH form, then one can simply correct for the excited-state lifetime and identify ⎛t ⎞ P(t ) = IFROH(t ) exp⎜ ⎟ ⎝ τF ⎠
2
Thus, when r = RD, the Coulomb attraction is equal to the thermal energy. In eq 4, e is the electronic charge, z1 and z2 are the charge numbers of the two reactants, ε0 is the vacuum permittivity, ε is the dielectric constant of the solution (for room temperature water, εH2O = 78), T is the absolute temperature, and kB is the Boltzmann factor. For HPTS in water at room temperature, RD ≈ 28 Å. In this case, the full analytic solution (for all times) of eq 1 cannot be obtained, and a numerical solution is required. It can be obtained, for example, using the Windows application for solving the spherical symmetric diffusion problem (SSDP, ver. 2.6).17 Nevertheless, even in this case, an analytical solution can be obtained for asymptotically long times,9,11 which is in harmony with the heuristic back-of-the-envelope approach.3−7 This can be achieved using Laplace transforms, ∫ 0∞e−stP(t) dt and −st ∫∞ 0 e p(r, t) dt. The partial differential equation (eq 1) becomes an ordinary differential equation in r, with s as a parameter. It can be solved as a power series in √s, its lowestorder term giving, upon Laplace inversion, the longest-order term in t, which is
(the “contact distance”). Assuming that the mechanism for separation is diffusion (in an infinite homogeneous medium), under the influence of the pair interaction potential, and that the reaction between them occurs uniformly on the surface of a sphere for which r = a, p(r, t) is obtained by solving the spherically symmetric Debye−Smoluchowski equation (DSE) in three dimensions8,11,12 ∂p(r , t ) ∂ ∂ = r −2 r 2D e−V (r) eV (r)p(r , t ) ∂t ∂r ∂r δ(r − a) + [kdP(t ) − kap(r , t )] 4πa 2
1 |z1z 2|e 4πε0ε kBT
(3)
To compare with experimental results, P(t) in the above equation should be convoluted with the experimental instrument response function (IRF). If the two excited-state lifetimes are not equal, one can extend the above equations to account for this.12,13 The initial conditions for the problem of ESPT to solvent involve an initially bound state, so that P(0) = 1 and p(r, 0) = 0. B
DOI: 10.1021/acs.jpcb.6b09035 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B Since these theoretical developments took place, the field of ESPT was reviewed several times,19−22 acknowledging that the above theoretical model is in good agreement with ESPT from various ROH photoacids to water23−29 and other solvents.12,30−33 These include photoacids that are much more acidic than HPTS,34 as well as those exhibiting different excited-state lifetimes for acid and base and/or an additional quenching process by the photoemitted proton.12,27−29,33 Subsequently, two additional groups (those of Fayer and Douhal) have confirmed the applicability of the reversible GR diffusion model for HPTS kinetics,35,36 as well as numerous additional measurements of other photoacids from both the Huppert and Pines laboratories.37−42 In a recent article, Lawler and Fayer studied ESPT to solvent in bulk water and in a confined volume of neutral reversible micelles.43 In bulk water, they have found that the HPTS fluorescence decays at long times as t−α with α = 1.1, stating that “the observation in bulk water of a power law t−1.1 for diffusion-controlled recombination is in contrast to the theoretical prediction of t−1.5 and previously reported observations”,43 one of them originating from the same research group (ref 35). This was explained by the deviation of the HPTS molecule from spherical symmetry and because “the attraction to the HPTS sulfonate negative charges may slow the loss of protons from the neighborhood of the HPTS, giving rise to a reduction in the value of the exponent”.43 The motivation of this article is to check once again the time dependence of the HPTS fluorescence in both bulk H2O and D2O and to show that under well-controlled experimental conditions, for two experimental setups at two different laboratories, theory and experiment are in full agreement. We shall also discuss the underlying assumptions of the model, and show that orientational dynamics, competing reversible binding sites, or other deviations from spherical symmetry cannot change the long-time behavior of the observed kinetics. The remainder of the article is structured as follows: Experiments that were independently carried out in the two laboratories are described separately in Sections 2 and 3. Section 4 is devoted to various theoretical examples, analytical and computational, showing that (spatially restricted) complications to the spherically symmetric reversible GR problem can affect the short to intermediate time regimes but not the longtime asymptotic behavior of the binding probability. A novel aspect in the discussion is the classification of the complexity of the kinetic models according to the number (1, 2, or 3) of kinetic “phases” (extrema of the logarithmic derivative (LD)44), which determine whether the approach to the asymptotic behavior is from below or from above.
The home-assembled time-correlated single-photon counting (TCSPC) detection system consisted of a Hamamatsu 3809U multichannel plate (MCP). Sample fluorescence was focused onto the entrance slit of the MCP after passing through a double monochromator (CVI model CM1l2, 1/8 m). The time-to-amplitude converter (TAC; Tennelec 862) was operated in the reverse mode. The absorbance signal was used as the start signal. The pulse-picker signal was detected by a fast (300 ps) photodiode (Newport 818-BB-20) and was used as the stop pulse. The stop and start pulses were discriminated using a modified Tennelec CFD-454 unit. The IRF of the TCSPC apparatus was typically 25 ps full width at halfmaximum (FWHM) in the 200 ns full-scale range of the TAC used for measuring the long-time kinetics of the photoacid. The time resolution of the single-photon-counting apparatus, after data processing, was about 5 ps when the 12.5 ps/channel operation mode of the TAC was used with the full time scale consisting of 16 k channels. The 25 ps IRF appeared as almost a δ-function on the 12.5/ps time scale, spanning only three data channels out of typically 2500 data channels that contained useable kinetic data. For this reason, the exact shape of the IRF had practically no effect on the measured long-time kinetics. Typical counting rates were below 2 kHz. The number of counts at the peak channel was about 600 000 after several hours of data collection by the Tennelec PCA3 card. The long acquisition time of the data required exceptional stability of the laser, which was checked by demanding less than a 10% count drift along the full collection time. As an additional measure, the shape of the instrument function before and after data collection was checked for consistency. Further signal processing and data analysis were done by personal computers. The fluorescence data measurements were performed in a regulated temperature housing, controlled to 20.0 ± 0.5 °C by a Neslab 1040 Chiller. A magnetic stirrer placed within the 1 cm cuvette gently mixed the solution and insured constancy of the temperature and concentration at the laser-irradiated volume. The irradiated volume was determined by a 1 m long focusing lens that focused the 3 mm laser beam with a very small converging angle to about 1 mm width and 0.5 mm height at the center of the cuvette. The collecting optics was placed perpendicularly to the excitation beam and with short focusing arrangement collected the light out of about 0.5 mm of the middle portion of the irradiated path length of the laser beam, making the probed volume of the sample about 0.25 mm3 out of the about 2.5 cm3 total volume of the solution, which was about 1/10 000 of the total solution volume. The concentration of the HPTS (Fisher Scientific, 98%) solution was 2 × 10−5 M measured at pH = 6.1, in which about 97% of the photoacid was in its acidic R*OH form. The relatively high pH was chosen to minimize the probability of homogeneous proton recombination coming from the bulk solution at very long measurement times.9 With the very low laser intensity (about 10 μW for the 0.8 MHz pulse-picked pulse train), the reaction conditions ensured that the probed sample consisted of isolated reaction centers with vanishing probabilities for a cross-talk between the photogenerated ion-pair centers. 2.2. Huppert’s Lab Setup. Fresh solutions of HPTS were used in all measurements. High-pressure liquid chromatography-grade or analytical-grade solvents were used in this study. HPTS and all solvents were purchased from Sigma-Aldrich and used as received. The aqueous HPTS sample (10−4 M and pH ∼ 6.0) was placed in a rotating cell of 6 mm optical path length and 40 mm diameter. The cell was operated at 10 Hz, which
2. MATERIALS AND METHODS 2.1. Pines’ Lab Setup. The setup consisted of a Ti-sapphire laser (Spectra-Physics; Tsunami laser pumped by 10 W Beamlok Ar-ion laser), which was operated in its picosecond lasing mode (1 ps pulses at 82 MHz). The fundamental train of pulses was pulse-selected (model 3980; Spectra-Physics) to reduce its repetition rate down to typically 0.8 MHz and then passed through a doubling LBO crystal. The laser was tuned to 750 nm, and the doubled frequency was used for excitation at 375 nm. This frequency is a minimum point of RO − absorption, and its absorption coefficient is about 1/6 of that of ROH,7 which reduces the directly excited population of RO− to only about 0.5% of the excited ROH. C
DOI: 10.1021/acs.jpcb.6b09035 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
pH is about 6. The measurements reported here utilize emission and absorption wavelengths close to these values. 3.2. Reanalyzing the Data Measured by Pines and Pines. We have reanalyzed the kinetic data of HPTS fluorescence decay reported by Pines and Pines.45 We judge this data set to be of the highest quality. Unbuffered 10−5 M solutions of HPTS were measured at pH = 6.1. The full data setup to 32 ns is shown on a log-linear scale in Figure 2a. The measured amplitude of the fluorescence decay was over almost 5 orders of magnitude. The experimental results were fitted to the solution of the DSE for diffusion in a Coulomb potential with back-reaction boundary conditions,8 using the Windows application developed by Krissinel’ and Agmon17 for solving SSDPs, based on either Euler or Chebyshev propagators for the diffusion operator.46 Of the six parameters listed in Table 1, D was taken from conductivity measurements,47 τF from the R*O− fluorescence decay signal (at 520 nm), and RD from the dielectric constant of water. The contact radius, a, was assigned a fixed value similar to that suggested by Weller.1 Only the two rate coefficients, κa and kd, were then adjusted to fit the data. To check the experimental data against the prediction of the numeric simulation with the SSDP program, we have analyzed the slope of the log−log plot of the data for various time ranges up to 12 ns, which was the limit for an accurate data fit due to the deteriorating signal-to-noise ratio (S/N) (Figure 2b). In this case, the noise is by and large Poissonian, and the S/N deteriorates with decreasing number of counts, n, as n1/2. Before analyzing the kinetic data, we have performed the following procedures: 1. Subtracting “white” noise due to the MCP, firing a signal without receiving a photon. Under our experimental conditions, we have found the ‘white’ noise to be 1 false count to about 125 000 true ones at the peak channel of the HPTS fluorescence signal. 2. Subtracting the overlap of the ROH and RO − fluorescence signals. Figure 3 shows the time-resolved fluorescence of HPTS in the aqueous solution of pH ∼ 6 at five emission wavelengths: 435, 450, 455, 460, and 465 nm. The signals are displayed on linear and semilog scales. As seen in Figure 3b, the ROH signal at 435 nm is (almost) not affected by the RO− emission (seen as an exponential decay − with τRO = 5.4 ns), whereas at 450 nm and longer wavelengths, F the contribution of the RO− emission is noticeable. The longer the monitored wavelength, the larger the RO− emission contribution to the signal. We have analyzed the signals shown in Figure 3 using the SSDP program with the parameters given in Table 2. To fit the R*OH signals at λ ≥ 445 nm, we have added an exponential with a decay time as that of the R*O− form, τF = 5.4 ns, so that
helps in preventing optical degradation of the HPTS sample. All measurements were performed under room temperature conditions (T = 298 K). A cavity-dumped Ti-sapphire femtosecond laser (Mira; Coherent) has been used for sample excitation, which is suited to the high repetition rate needed for the TCSPC technique but still provides rather long waiting times between successive excitations that enable the excited HPTS to relax to the ground state. The cavity dumper operated at a rate of ∼800 kHz. The laser excitation beam at 405 nm is polarized by construction. Its second harmonic output consists of 120 fs pulses over the spectral range of 760−860 nm. The laser power was reduced by density filters to about 10 pJ, to obtain about 500 counts/s in the TCSPC instrument. A 50 mm × 50 mm square plastic polarizer (Kodak) was placed in front of the entrance slits of a double monochromator. The angle between the fluorescence axis and the excitation axis was about 10°. The TCSPC detection system was based on a Hamamatsu 3809U MCP photomultiplier and an Edinburgh Instruments TCC 900 integrated TCSPC system. The time response of the instrument was approximately 40 ps FWHM. The IRF has a broad secondary peak with an intensity of 1/100 of the main response peak and a maximum at about 500 ps. Scheme 2 shows the time-resolved fluorescence setup Huppert’s lab used in the current experiments. Scheme 2. Time-Resolved Fluorescence Front-Face Setup for the TCSPC Experiments
3. RESULTS 3.1. Steady-State Fluorescence Measurements. Figure 1 shows the normalized excitation and emission spectra of HPTS in slightly acidic (pH ≈ 6.5) and basic (pH ≈ 10.7) aqueous solutions. The important point in measuring the ROH time-resolved fluorescence spectrum is to minimize the contribution of the RO− fluorescence to the signal. The optimum emission wavelength at which the RO− contribution is small (about ∼0.1% of the total signal) and the ROH signal count rate is large is about 435 nm (panel a). The optimum excitation wavelength is about 375 nm, for which the excitation is predominantly that of the ROH form (panel b). The optimum
IFROH(t ) = [P(t ) + B] exp( −t /t F)
Table 2 provides the relative amplitude, B, of the base (R*O−) signal at various wavelengths. For λem = 450 nm, the R*O− amplitude is 0.001, about 7 times larger than that at 435 nm. If not taken care of, this small fraction of R*O− signal may easily change the apparent log−log slope at intermediate times (t < 10 ns) and reduce it from −3/2 to a smaller absolute value. In general, the fluorescence overlap is wavelength- (as seen in Figure 1) and also time-dependent. The time dependence is D
DOI: 10.1021/acs.jpcb.6b09035 J. Phys. Chem. B XXXX, XXX, XXX−XXX
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The Journal of Physical Chemistry B
Figure 1. Steady-state fluorescence of HPTS in slightly acidic (pH ≈ 6.5; black line) and basic (pH ≈ 10.7; red line) aqueous solutions. (a) Normalized emission spectra excited at 375 nm at pH ≈ 6.5 and 420 nm at pH ≈ 10.7. (b) Normalized excitation spectra of the acidic and basic samples measured at 530 nm, near the peak of the RO− emission (λmax = 512 nm) in (a).
Figure 2. Time-resolved fluorescence of HPTS in water measured at 420 nm for the ROH form at 20 °C. (a) Semilogarithmic plot of the normalized kinetic data, I(t), after lifetime correction (black dots) as compared with the numeric solution of the SSDP program17 after it was convoluted with the IRF (red line), with parameters summarized in Table 1. (b) Log−log plot of the same data.
contribution to the peak of the R*OH intensity to be 0.05− 0.09% at 435 ± 2 nm, rising to about 0.1% at 420 nm. This was found by two independent methods. The first and the more crude one was from the intensities of the two fluorescence bands measured under steady-state conditions after excitation at the isosbestic point of the ROH and RO− absorption bands. This method was used for estimating the magnitude of the overlap as a function of the fluorescence wavelength. The second and more accurate method was to excite the RO− band directly from the ground state at pH = 11 at the isosbestic point when effectively only RO− was present in the solution and to measure the RO− form contribution at 420 nm (taking the measurement of its fluorescence signal over the same period of time as when exciting the R*OH band). Special care was taken to eliminate the effect of the difference in extinction coefficients of the ROH and RO− bands at the excitation wavelength. In our measurement, the total counts at 420 nm after 3 and 5 ns were 5000 and 1800, respectively, out of which about 350 and 240 counts, respectively, originated from RO− fluorescence. We estimate the total error in evaluating the overlap between the two fluorescence bands to be
