Modeling Studies of Inhomogeneity Effects during Laser Flash

Article pubs.acs.org/JPCA

Modeling Studies of Inhomogeneity Effects during Laser Flash Photolysis Experiments: A Reaction−Diffusion Approach Éva Dóka† and Gábor Lente*,‡ †

Department of Molecular Immunology and Toxicology, National Institute of Oncology, Budapest 1122, Hungary Department of Inorganic and Analytical Chemistry, University of Debrecen, Debrecen 4032, Hungary

‡

S Supporting Information *

ABSTRACT: This work presents a rigorous mathematical study of the effect of unavoidable inhomogeneities in laser flash photolysis experiments. There are two different kinds of inhomegenities: the first arises from diffusion, whereas the second one has geometric origins (the shapes of the excitation and detection light beams). Both of these are taken into account in our reported model, which gives rise to a set of reaction−diffusion type partial differential equations. These equations are solved by a specially developed finite volume method. As an example, the aqueous reaction between the sulfate ion radical and iodide ion is used, for which sufficiently detailed experimental data are available from an earlier publication. The results showed that diffusion itself is in general too slow to influence the kinetic curves on the usual time scales of laser flash photolysis experiments. However, the use of the absorbances measured (e.g., to calculate the molar absorption coefficients of transient species) requires very detailed mathematical consideration and full knowledge of the geometrical shapes of the excitation laser beam and the separate detection light beam. It is also noted that the usual pseudo-first-order approach to evaluating the kinetic traces can be used successfully even if the usual large excess condition is not rigorously met in the reaction cell locally.

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INTRODUCTION Laser flash photolysis (LFP) is a method for generating highly reactive intermediates, transient species, or free radicals by excitation of the samples (either in solution or gas phase) by a very short, high-energy laser pulse. The concentrations of the transients reach a sufficient level to apply spectrophotometric detection to follow their absorbance over time and thus gain kinetic information about their reactions.1−8 The inhomogeneous distribution of the generated intermediates in the sample is a typical and unavoidable characteristic of LFP measurements. This fact should be taken into account during the acquisition and processing of transient absorption data. Sample inhomogeneities can lead to distorted conclusions, especially when the kinetic trace detected is not an exponential curve. Whenever possible, setting pseudo-first-order conditions is desirable in order to avoid the use of actual transient concentrations. Theoretical considerations also show that a spatially inhomogeneous distribution of intermediates does not affect the determination of observed rate constants in pseudo-first-order experiments.9 However, experimental limitations sometimes make it difficult or impossible to set pseudo-first-order conditions. Only a handful of papers deal with the experimental and/or mathematical treatment of potential inhomogeneity issues and the resulting errors in kinetic conclusions. In an early work, J. W. Boag analyzed the effect of a nonuniform distribution of initial transient concentrations along and across the analyzing light beam.10 He considered first- and second-order reactions and made an effort to give analytical expressions for the difference © XXXX American Chemical Society

between the actual and the experimentally determined rate constants. It is worth noting that he used the linearized versions of the integrated rate laws (which are a source of statistical distortions themselves) in his deductions and ignored diffusive motion of the particles within the experimental time. Bazin and Ebbesen studied the error originating from poor overlap between the laser beam and the analyzing (or probe) beam in given experimental arrangements of LFP. They introduced two kinds of correction factors for such bad overlaps in the laser and probe directions and provided technical advice on how to detect them.11 Instead of considering actual kinetic measurements, they focused on the deviation of the measured absorbance (ODexp) from the real value (ODtrue). An important outcome of their calculations is that the ratio ODtrue/ODexp increases with ODexp, and they defined a distortion factor (DF) as the ratio of the relative errors at high and low experimental values. They suggested that the DF tends to cause more trouble in data processing, which confirms that it is preferable to work with low absorbance values as long as the signal-to-noise ratio is acceptable. Cassidy and Long created a mathematical model to test the validity of experimental rate constants determined under putative pseudo-first-order conditions.12 The underlying chemical systems were LFP studies of CO photodissociation reactions Received: January 15, 2017 Revised: March 23, 2017 Published: March 23, 2017 A

DOI: 10.1021/acs.jpca.7b00443 J. Phys. Chem. A XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry A

the entire cell) but locally as well (for concentrations at any given space in the cell).9 In LFP experiments, the transients are typically generated in a reaction chamber that is small compared to the entire cell; therefore, considering local effects and diffusion is highly necessary. The second identified problem is that the measured absorbance signal comes from an analyzing light beam whose size is non-negligible compared to the size of the cell, which results in some spatial averaging in Beer’s law. In this work, we report theoretical calculations aimed at resolving these two problems, both of which are primarily caused by spatial inhomogeneities. A reaction−diffusion partial differential equation is set up and solved in order to model the processes in a LFP experiment. Inspired by the seminal work of Alan Turing on morphogenesis (shape formation in biological systems),21 reaction−diffusion systems have already proved very useful in interpreting chemical and biological pattern formation22−24 as well as other nonlinear dynamic phenomena such as oscillation, chaos, or self-organizing systems up to the most recent literature studies.25−30

from transition metal carbonyl complexes and the subsequent coupling of the M(CO)6 and M(CO)5 forms. Their model implied that M(CO)5 was generated by a laser beam of circular cross section and monitored by a second beam in collinear arrangement, in a cylindrical volume. They also considered the outward diffusion of M(CO)5 molecules from the monitored volume. Mathematically, this system could be described by twodimensional diffusion equations coupled to a second-order reaction. All diffusion coefficients were supposed to be equal. Their results showed that diffusion from the cylinder started to interfere significantly when the relative concentration of M(CO)5 was high (i.e., pseudo-first-order conditions were not valid). The inhomogeneous distribution along the light path (due to Beer’s law) did not cause any distortion as long as the high excess of M(CO)6 was maintained. Finally, the (linearized) pseudo-first-order plot for the determination of the observed rate constant was always perfectly linear, even if a high discrepancy was found between the rate constant used in the simulation and the rate constant obtained from the simulated curves. This was an ominous but disappointingly little-heeded warning against the use of a linearized plot and the correlation coefficient as an indicator for testing pseudo-first-order behavior. The results of the above-described model calculations were not compared to experimental data because LFP measurements were carried out with perpendicular arrangement. Sample inhomogeneity issues can be assessed by experimental approaches as well. Bonneau and co-workers published a detailed manual for the collection and analysis of transient absorption data and emphasized the importance of correct instrument geometry and inhomogeneous transient distributions.13 They suggested that a mirror should be placed at the side of the sample opposite the laser excitation spot in order to reflect the pumping light back and thus increase the absorbed fraction of light and offset inhomogeneities at the same time. Goez et al. used a solid corner-tube retroreflector instead of a mirror as these are less sensitive to alignment imperfections.14 This had a doubly beneficial effect: the total absorbed intensity increased and the inhomogeneity caused by light absorption (sometimes called Beer inhomogeneity) decreased. The authors pointed out that beam shapers or easier experimental settings could correct for the nonuniformity of the beam. Solution inhomogeneities tend to cause distortions in other kinetic measurements as well, although the source of the nonuniform distribution of concentrations is conceptually different than the ones connected to LFP experiments. In the case of the stopped-flow technique, the inhomogeneity arises from the fact that the mixture of the reacting components needs a certain time to fill the observation cell. For very rapid reactions, whose half-life is comparable to the filling time of the cell, a significant difference can be formed between the front and the far face of the observation cell. These effects were carefully studied.15−18 These contributions unveiled the fundamental experimental limitations of the stopped-flow technique and also proposed a number of methods to correct for artifacts. During the evaluation of our recent LFP experiments designed to study the reactivity of the sulfate ion radical,19,20 we identified two major issues that warrant serious theoretical analysis. The first is that it is difficult to verify that pseudo-first-order conditions actually prevail in experiments. The use of the pseudo-first-order method clearly requires that all reactants except the limiting reagent should be used in large excess.1,2,9 It is also understood that in inhomogeneous cases this condition must be met not only globally (for the amounts of substance in

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RESULTS AND DISCUSSION Sulfate Ion Radical−Iodide Ion System. The sulfate ion radical (SO4−•) is a very reactive species that plays a central role as an intermediate in a number of reaction systems, most notably the autoxidation of sulfur(IV)31,32 or one-electron oxidations using the peroxomonosulfate ion.33,34 The iodide ion is also known to catalyze the aqueous autoxidation of sulfur(IV);35 therefore, directly measuring the kinetics of the reaction between SO4−• and I− was important. It was found that the sulfate ion radical can be most favorably generated through the photolysis of a peroxodisulfate ion (S2O82−) initiated by a laser pulse at 266 nm36 hν

S2 O82 − → 2SO4 −•

(1)

−

In the presence I , the following reaction takes place k2

SO4 −• + I− → SO4 2 − + I•

(2)

As a subsequent step, iodine atoms (I•) give iodine molecule ions (I2−•) with excess I− in a reversible process37 k3

I• + I− HooI I 2−•

(3)

k −3

In the last step, I2 recombines to give triiodide ions (I3 ) and I− −•

k4

2I 2−• → I3− + I−

−

(4)

Reactions 3 and 4 are also highly important in the aqueous photochemistry of I−.19,38,39 The progress of reaction 1 is primarily determined by the laser power. The length of a typical pulse is 5−20 ns, which is considerably shorter than the typical lifetimes of reactions 2 and 3, which are around 0.5 μs. Finally, reaction 4 is an entirely separate step on a longer (50 μs) time scale. The present contribution does not give any novel experimental data; it is based on the observations of the instrumental setup of an earlier work in our group, where the LFP experiments in this system were reported in the frame of interpreting the photochemical changes in hydroiodic acid (HI) solutions.19 All of the experimental conditions are given in that earlier paper.19 Here, the main theoretical results will be given using the reactants of reaction 2 so that this process can illustrate the practical use of the theory developed. However, the equations will always be B


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The Journal of Physical Chemistry A formulated in a parametric manner. This has the advantage that the approach presented here will be easily adoptable to any other mixed second-order process; it is only the values of certain parameters that need to be changed. Throughout our earlier experimental investigation, the widely accepted pseudo-first-order approach was applied in the experimental setup and the data analysis.1,2,9 For this, one of the two reactants should be present in at least 10-fold excess over the other so that its concentration stays practically constant in a single kinetic run. When the high excess requirement is not fulfilled, the handling of the time course becomes trickier. Processing of mixed second-order kinetics is already challenging and requires knowledge of the initial and actual concentrations of the reactants at every time points.9 In our experimental work focusing on the oxidation of I− by SO4−•, acceptable pseudo-firstorder behavior was detected in the 0.2−0.8 mM range of I− concentration in a sense that individual curves gave very reasonable fits and the estimated observed rate constants showed the expected concentration dependence.19 Yet, a more careful later analysis of the same results revealed that the basic pseudofirst-order condition (high excess of I− in this case) might have only been met globally for the entire volume but not locally. Estimating the local concentrations of the transient species SO4−• is in fact quite difficult and necessarily involves several assumptions (albeit very plausible ones). The following factors need to be considered: • SO4−• is generated from the photolysis of S2O82− (eq 1) locally only along the path of the laser beam. • The diameter of the laser beam is around 5 mm. The instrument specifications give 6 mm, but our measurements showed that 5.0 mm is much closer to reality. On the other hand, the width of the cuvette (its path length in the direction of detection) is 1.000 cm, and this fact alone introduces high inhomogeneity between those regions within and outside of the beam. The energy distribution of the laser beam in its cross section was also tested somewhat by burning a suitably chosen sheet of black paper. No obvious signs of inhomogeneity were seen; therefore, the energy distribution of the laser beam was assumed to be homogeneous in the detailed calculations. This assumption is also supported by information from the manufacturer of the LFP instrument. • The generation of SO4−• shows Beer inhomogeneity: as the laser light is attenuated by absorption in the direction it travels, its intensity decreases and so does the concentration of the generated transients. • SO4−• is a highly reactive, short-lived intermediate species, and its concentration drops very rapidly because of a second-order self-reaction as well as by the reaction with the given reactant; there is no time to reach homogeneous spatial distribution within an experiment. • Although Beer’s law is valid for the transient as well as for stable species, the molar absorption coefficients are very difficult to determine precisely, which leads to high levels of uncertainty.40 In fact, literature ε values reported thus far for SO4−• span a huge range between 460 and 1600 M−1 cm−1.41−45 • The size of the detection light beam is by no means negligibly small. In our experimental setup, this analysis beam had a circular cross section with a diameter of 3 mm at the surface where it enters the measurement cell. This means that the measured absorbance is a result of spatial

averaging, which must be considered during the evaluation. It should also be noted that it takes about 50 ps for light (both the excitation and detection beam) to cross the full path length of the cuvette. Fortunately, this is not a source of further complications in our experimental setup as the time resolution of the 4 GHz oscilloscope is 250 ps. Yet, this effect will probably have to be handled somehow if the performance of LFP instruments is improved. On the basis of the above considerations and experimental absorbance data measured in our LFP experiments, a rough estimation of the initial concentration of SO4−• at the entry point of the excitation laser beam was carried out at the time instant when the excitation pulse ends, which exceeded 2 × 10−4 M even when ε = 1600 M−1 cm−1 was used, which was published by McElroy45 and was at the higher end of the literature range of values. As the lowest I− concentration was 0.2 mM in the experiments, it is clear that the condition requiring a large excess of reagent necessary for pseudo-first-order evaluation was not met locally. This fact raised some questions: why were the detected curves still exponential and why did the observed rate constants show the expected behavior? The answer to these questions will be sought by developing a detailed spatiotemporal model of the LFP experiments. Reaction−Diffusion Equation Using Cylindrical Coordinates. A general reaction−diffusion equation has the following closed form for two species with concentrations cS and cI ∂cS = DSΔcS + R(cS , c I) ∂t ∂c I = DIΔc I + R(cS , c I) ∂t

(5)

Here, DS is the diffusion coefficient of the first component, DI is the diffusion coefficient of the second component (I), Δ is the Laplace operator, and R is an operator embedding all of the ongoing reactions in the system (in our case, it will be a single reaction). This equation is a second-order, semilinear, parabolic partial differential equation system. The Laplacian (Δ) is a differential operator with a scalar value that gives the divergence of the gradient (∇) of a real valued function. For an f(x,y,z) function in the three-dimensional Euclidean space Δf = ∇2 f =

∂2 ∂2 ∂2 + 2 + 2 2 ∂x ∂y ∂z

(6)

For our purposes, it is favorable to transform the problem into a cylindrical coordinate system because the reaction space is a circular cylinder (see later). Figure S1 in the Supporting Information shows the transformation from Cartesian to cylindrical coordinates, and the form of the Laplace operator in the new coordinate system is as follows Δf = ∇2 f =

∂2 1 ∂ ∂2 1 ∂2 + + 2 + 2 2 2 r ∂r ∂x ∂r r ∂φ

(7)

The angular derivatives (fourth term in this equation) are zero in our model because of the cylindrical symmetry. Thus, applying the transformation in eq 5 yields C


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The Journal of Physical Chemistry A ⎛ ∂ 2c ∂cS ∂ 2cS ⎞ 1 ∂cS ⎟ + R ( c S , c I) = DS⎜ 2S + + r ∂r ∂t ∂r 2 ⎠ ⎝ ∂x ⎛ ∂ 2c ∂c I ∂ 2c ⎞ 1 ∂c I = DI⎜ 2I + + 2I ⎟ + R(cS , c I) r ∂r ∂t ∂r ⎠ ⎝ ∂x

The parameters in these equations are defined as follows: E is the energy of a single laser pulse, λ is the laser wavelength (266 nm), h is Planck’s constant, NA is Avogadro’s constant, c is the speed of light, εS2O82− is the molar absorption coefficient of S2O82− at the excitation wavelength (10.47 M−1 cm−1 = 1.047 m2 mol−1),19 Φ is the quantum efficiency of SO4−• from S2O82−, and [S2O82−] is the concentration of S2O82− in the cell. The composite parameter cS0 also has some physical meaning; it is the maximum concentration of the transient (i.e., at x = 0 and t = 0). If the nominal energy of the laser pulse (40 mJ) and the maximum possible quantum efficiency (Φ = 2) are used for a typical experiment ([S2O82−] = 0.15 M), cS0 is calculated as 3.3 mM, which is an impossibly high value. There are several reasons why eq 9 overestimates cS0. First, the laser energy is in fact smaller than the nominal value claimed by the manufacturer. This was proved by experimentally measuring the pulse energies, which were close to the nominal at the point where the beam left the fourth harmonic generator, but additional optical elements all reduced it somewhat. It was estimated that the actual energy reaching the cell is never higher than 20 mJ. It should be noted that this measured energy also involves the higher-wavelength harmonics, but their contribution is probably small. Second, the assumed value of Φ used in the estimation was the highest possible. The actual quantum yield may be lower. However, we did not find a really reliable experimental determination in the literature; Φ = 2 is generally assumed. Third, it should be taken into account that the laser pulse also takes some finite time, during which some of the transients generated are lost. The nominal pulse width of the laser is 6 ns, but in practice, it was considerably longer (10−12 ns). After considering all of these effects, it should be noted that they only affect the value of cS0 but not the rest of the quantities in eq 10. Therefore, it seems to be a valid approach to try to find a reasonable value of cS0 based on the experimental findings and use it in the theoretical calculations. Solving the Reaction−Diffusion Equation. Partial differential equations such as the one shown in eq 5 can be solved numerically by several standard algorithms. However, as the computation efficiency and high spatial resolution were very important for our calculations, we have decided to implement a solution method for the version based on cylindrical coordinates (eq 8) specifically developed for this problem in Matlab. This was a finite volume method: the cell was divided into small elementary volume units whose shape is shown in Figure 2.

(8)

In the present case, two concentration functions are to be calculated: cS(x,r,t) for SO4−• and cI(x,r,t) for I−. The term R(cS,cI) is simply given by a second-order rate equation; therefore, its contribution is −k2cS(x,r,t)cI(x,r,t) for both variables. Model of the Reaction Space. Our description of the reaction space is a suitably modified version of a model provided by Cassidy and Long (see above).12 Figure 1 shows the

Figure 1. Geometry for modeling the role of diffusion in LFP experiments: perspective view and cross section.

geometric model applied to represent the physical reaction space. The arrow labeled “hν” represents the laser beam, and the blue arrow is the analyzing beam that is used to follow the concentration of the transient species. Adapting the usual notations of Beer’s law, I0 is the intensity of the entering detection light beam, and I is the intensity of the leaving beam. Variables x and r are the spatial coordinates, and L is the path length of the cuvette in the direction of the laser beam. The cross section of the laser beam is considered to be circular with r0 radius, and the origin is fixed to the center of this circle at the entering point of the laser. The shading refers to the decreasing concentration of the transient species along the laser beam. The transient species is generated by the laser pulse traveling along the x axis and fills the cylinder-shaped reaction chamber with radius r0 (2.5 mm in our case). An external radius is also defined around this object (rex, 5.0 mm in our case) to represent the physical limitation of the cuvette wall. Initial Concentrations. The first step of solving the reaction−diffusion equation is calculating the initial concentrations (i.e., those at the end of the laser pulse). I− is present in homogeneous distribution in the entire cell; therefore, cI(x,r,0) = [I−]0 for any pairs of x,r values. SO4−• is generated by the laser beam from S2O82−, and its initial concentration can be calculated as follows cS0 =

ΦEλ ln 10 εS O 2−[S2 O82 −] NAhcr0 2π 2 8

2− ⎧ 2 −[S2 O8 ]x −ε ⎪ c 10 S2O8 0 ≤ r ≤ r0 S0 cS(x , r , 0) = ⎨ ⎪ 0 r0 < r < rex ⎩

Figure 2. Shape of the volume elements used in the solving the reaction−diffusion equation.

The spatial resolution was chosen to be 0.05 mm in both the x and r directions (in Figure 2, this means dx = dr = 0.05 mm). With this spatial resolution, a 200 × 100 matrix could be used to store all of the values cS(x,r,t) at a fixed time t, and the same could be done for cI(x,r,t). The components of the Laplace operator could be calculated by straightforward matrix operations. Finally, the time dependence of these matrixes was calculated by a fourth-

(9)

(10) D


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The Journal of Physical Chemistry A order Runge−Kutta method1,9 using adaptable time steps, which ensured that the highest concentration would not change more than 2% in an integration step. The developed algorithm is described in much more detail in the Supporting Information, and the Matlab code is also given there. Absorbance Calculation. Examination of the detecting beam of the LFP instrument showed that the beam has a circular shape with a 3 mm diameter (w) and the edge of the detection circle is 1 mm (s) away from the front face of the cuvette (sketch given in Figure 3). This is more than half of the diameter of the

Figure 4. Simulated absorbance−time trace for the diffusion-only case in the sulfate ion radical−iodide ion system. cS0 = 0.15 mM, DS = 1.0 × 10−9 m2/s, and ε = 1600 M−1 cm−1 (450 nm).

homogeneous, that is, the intensity per unit area does not depend on the spatial coordinates within the cross section of the beam. It should also be pointed out that the spatial averaging should not be done simply for absorbance contributions; it is the light intensity that has to be dealt with primarily in the calculations, and then, the absorbance must be calculated based on its definition. A long, but routine mathematical derivation (included in the Supporting Information) showed that for this particular geometry the detected absorbance can be calculated by the following integral:

Figure 3. Schematic sketch showing the relative positions and sizes of the excitation and detecting light beams in the LFP instrument.

laser beam (5 mm). Consequently, the effective optical path, which is the length that the detection beam travels through the sample, is shorter than the diameter of the reaction space, and the absorbance values had to be evaluated numerically at each time point. It was assumed that the detection light beam is

Adet (t ) = −log10

8 w 2π

∫0

w /2

⎡ ⎢ ⎢⎣

(s + w)/(2 + w 2 /(4 − r 2) )

∫(s+w)/(2−

w 2 /(4 − r 2) )

The value of this integral was calculated numerically at each time point of the simulations. In the experiments, detection was carried out at 450 nm, where SO4−• has an absorption maximum but I− has no absorption at all.19 Diffusion-Only Test. In the first group of simulations, a diffusion-only process was modeled by setting the rate constant k2 to 0. The diffusion coefficient is known for I− in aqueous solution (DI = 2.0 × 10−9 m2/s).46 No similar literature value was found for SO4−•. However, the diffusion coefficient DS = 1.0 × 10−9 m2/s is very well known for the sulfate ion (SO42−),46 which contains only one more electron compared to SO4−•. Therefore, this value was used in the simulations. For the molar absorption coefficient of SO4−•, a value published by McElroy was chosen, ε = 1600 M−1 cm−1.45 The simulations revealed a very slow change of the detected absorbance even in the absence of a chemical reaction (Figure 4). It should be noted that there is no way to compare this simulation directly with experiments as in the absence of other reagents SO4−• undergoes relatively slow (time scale of ∼1 ms) secondorder recombination, which was not included in the present model. Still, the initial absorbance readings in these simulations were suitable for selecting a cS0 value that was in agreement with our experimental data; cS0 = 0.15 mM was chosen and used in the rest of the simulations. It should be noted that this value critically depends on the assumed value of the molar absorption coefficient. The very long time needed for the absorbance change obvious from Figure 4 implies that the effect of diffusion cannot be very important on the time scales where LFP is usually

e−εSO4

r 2− r 2 −•2 ∫ 0 cS(x , 0

r 2 + ρ 2 , t ) dρ

⎤ dx ⎥ dr ⎥⎦

(11)

used. In the Supporting Information, Figures S3 and S4 show the concentration distributions in a typical diffusion-only process along the x and r axes after 5000 and 15000 s. It is seen that solution homogeneity is not reached even after 4 h. Therefore, it can be safely concluded that diffusion into or out of the reaction channel cannot have a major influence on the kinetics observed during LFP experiments. Full Experiment Simulations. In the most detailed simulations in this study, the value k2 = 5.74 × 109 M−1 s−1 was used as the rate constant for the second-order process between SO4−• and I−, which is exactly the reported experimental value.19 A number of simulations were run with varying initial I− concentrations, which were in the range of the LFP studies. The simulated absorbance−time curves were fitted to exponential functions. Somewhat surprisingly, the exponential fits were quite reasonable even when the initial concentration of I− was not close to the 10 times excess required by the pseudo-first-order approach. As an example, Figure 5 shows an absorbance−time trace simulated for [I−] = 0.3 mM, which is larger than cS0 by only a factor of 2. The simulations reported in the diffusion-only part of this article clearly showed that diffusion is not fast enough to replenish the consumed I− in the reaction channel, and further confirmation of this point is given by Figures S5 and S6 in the Supporting Information, which show the substantial inhomegenities in the concentration profile of I− at the end (3.5 μs) of the simulated curve displayed in Figure 5. As the lowest I− concentration used in the experiments was 0.2 mM, simulations were run at this initial concentration as well, E


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The fact that k2 is not within the uncertainty range of ksim2 illustrates some systematic error introduced by the pseudo-firstorder approach, yet the very small difference between the two values is seen as negligible compared to any reasonable experimental error. The excellent fit of the simulated traces to exponential traces deserves some more attention. Most of these simulations were done under nonflooding conditions, that is, the concentration of the excess species (iodide ion) is less than 10 times that of the other (sulfate ion radical). As already analyzed in the literature,9 the kinetic trace of a perfectly homogeneous mixed second-order process under such nonflooding conditions is still reasonably close to an exponential curve and the pseudo-first-order fit is suitable for extracting meaningful kinetic information. However, the special reaction−diffusion case analyzed here actually gives far better agreement with the exponential fit than the homogeneous analogue. To illustrate this fact, Figure S8 compares the goodness of the exponential fits of the two cases for identical initial concentrations (the same as that shown in Figure 5). From Figure S8, it is clear that the reason behind this somewhat surprisingly excellent agreement cannot be simply the behavior of the homogeneous version.

Figure 5. Simulated absorbance−time trace in the sulfate ion radical− iodide ion system. The markers represent the simulated point (only about 15% of them is shown for clarity), and the solid line is the exponential fit. cS0 = 0.15 mM, [I−] = 0.3 mM, DI = 2.0 × 10−9 m2/s, DS = 1.0 × 10−9 m2/s, and ε = 1600 M−1 cm−1 (450 nm).

which corresponds to a 4:3 [I−]/cS0 ratio. The simulated curve and the exponential fit are shown in Figure S7 in the Supporting Information. The fit of the theoretical curve here would certainly be unacceptable from a strictly kinetic point of view.9 Yet, the actual experiments still showed good agreement with pseudofirst-order kinetics. At this point, it should be recalled that the simulations were only done considering reaction 2. In real experiments, a subsequent reaction (eq 3) also consumes I−, which means that stoichiometric problems also arise here. Therefore, we feel safe to conclude that the actual cS0 value in the experiments should be lower than that estimated here (0.15 mM), which also means that the actual molar absorption coefficient of SO4−• should be substantially higher than even the highest estimate reported in the literature.41−45 In order to follow the usual pseudo-first-order evaluation method all the way through, the observed pseudo-first-order rate constants obtained by fitting the simulated curves were plotted as a function of I− concentration. The resulting plot is shown in Figure 6. The filled circles represent the fitted values, and the larger open diamonds show the theoretical expectation (i.e., k2[I−]). As expected, a straight line without any intercept gave a very good interpretation of the concentration dependence of the kobs values; the slope was ksim2 = (5.67 ± 0.01) × 109 M−1 s−1, which is very close to the input value of k2 (5.74 × 109 M−1 s−1).

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CONCLUSIONS The reaction−diffusion approach used in this work proves that the geometric inhomogeneities necessarily produced during LFP experiments are not substantially changed by diffusion on the typical time scale of a single experiment. Therefore, local concentration effects, which are understandably very large for the transients generated, must be considered in the theoretical interpretation. The simulations in this study show that the pseudo-first-order approach gives reasonable results for the rate constants even if the main condition of its use, that is, one of the two reagents should be in at least 10 times excess, is not met locally in the entire reaction channel. A more important issue that was identified in our simulations is the handling of the measured absorbance. As a consequence of the geometric inhomogeneties, the measured absorbance signal is a rather complicated spatial average, which does not reflect the transient concentration (or even the path length) at any particular point in the reaction cell. Therefore, extreme care must be taken when molar absorption coefficients are estimated based on LFP data. Indeed, failure to recognize the importance of this point may be the reason why molar absorption coefficients reported in the literature span an impossibly large range for a single species. To suppress inhomogeneities, LFP experiments are sometimes done in a way that the overall absorbance of the photolyzed species is low. This approach does indeed decrease geometric inhomogeneites along the excitation path arising from Beer’s law but can do nothing to change the shape of the reaction channel. Furthermore, this method could be useful for photophysical (fluorescence or phosphorescence) measurements but typically not for photochemistry as a low absorbance for the exciting light beam would also mean a very low concentration of the transient.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpca.7b00443. Cartesian to cylindrical coordinate transformation, surface area details, scheme of the exchange of matter in a volume unit, Matlab code for calculations, concentration distribu-

Figure 6. Pseudo-first-order rate constants obtained by fitting the simulated data as a function of initial iodide ion concentration in the sulfate ion radical−iodide ion system. Filled markers mean the fitted points, and the open diamonds are the expected theoretical values (i.e., k2[I−]). F


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tions, the simulated absorbance−time trace, and derivations (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Gábor Lente: 0000-0003-2022-2156 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The research was supported by the EU and co-financed by the European Regional Development Fund under the project GINOP-2.3.2-15-2016-00008.

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REFERENCES

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Modeling Studies of Inhomogeneity Effects during Laser Flash

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