Integral Encounter Theory Research Articles

Kinetics of polaron capture by traps in a lithium niobate crystal

The problem of reversible transformation and trapping of small-radius polarons in a lithium niobate crystal is considered within the framework of the integral encounter theory which is binary in the concentration of reatants. Analytical solutions are obtained for the relaxation kinetics of polarons, their lifetimes, and the rate constants of the corresponding channels of a multistage reaction controlled by polaron mobility. The temperature and concentration dependences of the observed quantities are analyzed. It is shown that at low temperatures polarons accumulate in a bound state characterized by an anomalously low relaxation rate.

Алгоритмы численного моделирования сверхбыстрого фотоиндуцированного межмолекулярного переноса заряда в жидкостях

Предложены подходы к численному решению систем уравнений, описывающих кинетику двухстадийной фотохимической реакции в вязком полярном растворителе. Математическая модель построена на основе расширенной интегральной теории встреч и учитывает диффузионную подвижность молекул-реагентов в жидкости, неравновесность среды и внутримолекулярных степеней свободы, удаленный перенос электрона в донорно-акцепторных парах, разделенных растворителем. В рамках метода броуновского моделирования разработаны алгоритмы расчета безреакционных стохастических траекторий частиц на поверхностях свободной энергии, соответствующих различным состояниям реагентов и продуктов, схемы детектирования реакционных событий и генерации электронных прыжков, а также алгоритмы расчета нестационарных потоков частиц между электронными состояниями и вычисления интегральных ядер кинетических уравнений. Представлены результаты тестовых расчетов, демонстрирующие корректность численного решения и воспроизводящие известные особенности реакций электронного переноса в полярных жидкостях. Efficient approaches to the numerical solution of equations describing the kinetics of two-stage photochemical reactions in a viscous polar solvent are proposed. The mathematical model is based on the extended integral encounter theory and takes into account diffusive mobility of reactants in solution, nonequilibrium of solvent and intramolecular degrees of freedom, and remote electron transfer in solvent-separated donor-acceptor pairs. In the framework of the Brownian simulation technique, a number of numerical algorithms for calculating unreactive stochastic trajectories of particles on free energy surfaces corresponding to different electronic states of reactants and products are suggested, some computational schemes for the detection of reaction events and the generation of electronic hops are developed, and algorithms for calculating the time-dependent reaction fluxes between the electronic states and integral kernels of the kinetic equations are implemented. The results of test simulations demonstrating the validity of the numerical solutions and reproducing well-known features of electron transfer reactions in polar solvents are discussed.

Delayed fluorescence after reversible triplet ionization

A complete set of integral equations is used to describe the kinetics of reversible photo-ionization after instantaneous excitation, including “geminate” and bimolecular charge recombination to either the ground or excited states of neutral reactants. The excitations restored by bimolecular recombination of ions produce the delayed fluorescence which goes to zero as a second power of time.The fluorescence kinetics previously obtained with the spin-less Integral Encounter Theory (IET) and distant dependent rates of electron transfer is essentially corrected using Modified Encounter Theory (MET). The delayed fluorescence dependence on conversion rate is explained analytically on the simpler example of contact ionization and recombination.

Рекроссинг-алгоритмы численного моделирования бимолекулярных фото-химических реакций в рамках обобщенной интегральной теории встреч

Рекроссинг-алгоритмы численного моделирования бимолекулярных фото-химических реакций в рамках обобщенной интегральной теории встреч

Kinetics of exciplex formation/dissipation in reaction following Weller Scheme II.

Creation of exciplexes from the charged products of photoionization is considered by means of Integral Encounter Theory. The general kinetic equations of such a reaction following the Weller scheme II are developed. The special attention is given to the particular case of irreversible remote ionization of primary excited electron donor. Kinetics of exciplex formation is considered at fast biexponential geminate transformation of exciplexes in cage that gives way to subsequent bulk reaction of equilibrated reaction products controlled by power law recombination of ions. It is shown that the initial geminate stage of exciplex kinetics is observed only in diffusion controlled regime of the reaction and disappears with increasing mobility of ions in passing to kinetic regime. The quantum yield of exciplexes is studied along with their kinetics.

Magnetic Field Effects in Fluorescence of Exciplex and Fluorophore for the Weller Schemes I and II: Similarities and Differences

In exciplex-forming donor–acceptor systems the photoexcited acceptor state can be quenched either by exciplex formation at the contact of the acceptor and the donor, followed by reversible dissociation to the radical-ion pair (Scheme I), or by distant electron transfer yielding a radical-ion pair that can reversibly recombine to produce an exciplex (Scheme II). The fluorescence of the photoexcited electron acceptor (fluorophore) and the exciplex is known to be sensitive to the magnetic field, assisting spin conversion in the resulting geminate radical-ion pair (RIP). An increase of the fluorescence yield in high magnetic fields compared to low fields is known as the magnetic field effect (MFE). In this paper theories of the MFE on both the fluorophore and the exciplex within the Schemes I and II are developed. Kinetics of the reversible reactions are described in terms of the integral encounter theory that unlike the differential encounter theory is capable of dealing with such reactions. Regions of the parameters where MFE is observable are discussed. Comparison of functional dependencies of MFE on the model parameters for the Schemes I and II is carried out. Magnitudes and dependencies of MFE on the model parameters are found to be very similar for the Schemes I and II, except the dependencies of the fluorophore and exciplex MFEs on the rate constant of exciplex production from the radical-ion pair, which appear to be opposite for the Schemes I and II.

Hyperfine interaction mechanism of magnetic field effects in sequential fluorophore and exciplex fluorescence

The magnetic field effect on the fluorescence of the photoexcited electron acceptor, (1)A∗, and the exciplex, (1)[D(+δ)A(-δ)] formed at contact of (1)A∗ with an electron donor (1)D, is theoretically explored in the framework of Integral Encounter Theory. It is assumed that the excited fluorophore is equilibrated with the exciplex that reversibly dissociates into the radical-ion pair. The magnetic field sensitive stage is the spin conversion in the resulting geminate radical-ion pair, (1, 3)[D(+)...A(-)] that proceeds due to hyperfine interaction. We confirm our earlier conclusion (obtained with a rate description of spin conversion) that in the model with a single nucleus spin 1/2 the magnitude of the Magnetic Field Effect (MFE) also vanishes in the opposite limits of low and high dielectric permittivity of the solvent. Moreover, it is shown that MFE being positive at small hyperfine interaction A, first increases with A but approaching the maximum starts to decrease and even changes the sign.

Magnetic field effect in fluorescence of excited fluorophore equilibrated with exciplex that reversibly dissociates into radical-ion pair undergoing the spin conversion

The fluorescence of the photoexcited electron acceptor, (1)A∗, and the exciplex, (1)[D(+δ)A(-δ)] formed at contact of (1)A∗ with an electron donor (1)D, is known to be very sensitive to a magnetic field, assisting the spin conversion in the resulting geminate radical ion pair (RIP), (1, 3)[D(+)...A(-)]. The relative increase of the fluorescence in the highest magnetic field compared to the lowest one, known as the magnetic field effect, crucially depends on the dielectric constant of the solvent, ɛ. This phenomenon first studied experimentally is at first reproduced here theoretically by means of the so called integral encounter theory. It was shown to be very sensitive to the position of the exciplex energy level relative to the levels of exciplex precursors and the charged products of its dissociation. The results obtained strongly depend on the dielectric properties of the solvents as well as on the exciplex and RIP formation rates.

Screening of Exciplex Formation by Distant Electron Transfer

The excitation quenching by reversible exciplex formation, combined with irreversible but distant electron transfer, is considered by means of the integral encounter theory (IET). Assuming that the quenchers are in great excess, the set of IET equations for the excitations, free ions, and exciplexes is derived. Solving these equations gives the Laplace images of all these populations, and these are used to specify the quantum yields of the corresponding reaction products. It appears that diffusion facilitates the exciplex production and the electron transfer. On the other hand the stronger the electron transfer is, the weaker is the exciplex production. At slow diffusion the distant quenching of excitations by ionization prevents their reaching the contact where they can turn into exciplexes. This is a screening effect that is most pronounced when the ionization rate is large.

Manifestation of macroscopic correlations in elementary reaction kinetics. II. Irreversible reaction A+B→C

The applicability of the Encounter Theory (ET) (the prototype of the Collision Theory) concepts for widely occurring diffusion assisted irreversible bulk reaction A+B→C (for example, radical reaction) in dilute solutions with arbitrary ratio of initial concentrations of reactants has been treated theoretically with modern many-particle method for the derivation of non-Markovian binary kinetic equations. The method shows that, just as in the reaction A+A→C considered earlier, the agreement with the Encounter Theory is observed when the familiar Integral Encounter Theory is used which is just a step in the derivation of kinetic equations in the framework of the method employed. It allows for two-particle correlations only, and fails to consider the correlation of reactant simultaneously with a partner and with a reactant in the bulk. However, the next step leading to the Modified Encounter Theory under reduction of equations to a regular form both extends the time applicability interval of ET homogeneous rate equation (as for reactions proceeding in excess of one of the reactants), and yields the inhomogeneous equation of the Generalized Encounter Theory (GET) that reveals macroscopic correlations induced by the encounters in a reservoir of free walks in full agreement with physical considerations. This means that the encounters of reactants in solution are correlated at rather large time interval of the reaction course. However, unlike the reaction A+A→C of identical reactants, the reaction A+B→C accumulation of the above macroscopic correlations (even with the initial concentrations of reactants being equal) proceeds much slower. Another distinction is that for the reaction A+A→C the long-term behavior of ET and GET kinetics is the same, while in the reaction A+B→C these kinetics behave differently. It is of interest that just taking account of the above macroscopic correlations in the reaction A+B→C (in GET) results in the universal character of the long-term behavior of the kinetics for the case of equal initial concentrations of reactants and that where one of the reactants is in excess. This is more natural from the point of view of the reaction course on the encounters of reactants in solutions.

Manifestation of macroscopic correlations in elementary reaction kinetics. I. Irreversible reaction A+A→product

Using an modern many-particle method for the derivation of non-Markovian binary kinetic equations, we have treated theoretically the applicability of the encounter theory (ET) (the prototype of the collision theory) concepts to the widely known diffusion assisted irreversible bulk reaction A+A-->product (for example, radical reaction) in dilute solutions. The method shows that the agreement with the ET is observed when the familiar integral ET is employed which in this method is just a step in the derivation of kinetic equations. It allows for two-particle correlations only, but fails to take account of correlation of reactant simultaneously with the partner of the encounter and the reactant in the bulk. However, the next step leading to the modified ET under transformation of equations to the regular form both extends the time range of the applicability of ET rate equation (as it was for reactions proceeding with one of the reactants in excess), and gives the equation of the generalized ET. In full agreement with physical considerations, this theory reveals macroscopic correlations induced by the encounters in the reservoir of free walks. This means that the encounters of reactants in solution are correlated on a rather large time interval of the reaction. Though any nonstationary (non-Markovian) effects manifest themselves rather weakly in the kinetics of the bimolecular reaction in question, just the existence of the revealed macroscopic correlations in the binary theory is of primary importance. In particular, it means that the well-known phenomena which are generally considered to be associated solely with correlation of particles on the encounter (for example, chemically induced dynamic nuclear polarization) may be induced by correlation in the reservoir of free random walks of radicals in solution.

Exciplex Formation Accompanied with Excitation Quenching

The competence of the reversible exciplex formation and parallel quenching of excitation (by electron or energy transfer) was considered using a non-Markovian pi-forms approach, identical to integral encounter theory (IET). General equations accounting for the reversible quenching and exciplex formation are derived in the contact approximation. Their general solution was obtained and adopted to the most common case when the ground state particles are in great excess. Particular cases of only photoionization or just exciplex formation separately studied earlier by means of IET are reproduced. In the case of the irreversible excitation quenching, the theory allows specifying the yields of the fluorescence and exciplex luminescence, as well as the long time kinetics of excitation and exciplex decays, in the absence of quenching. The theory distinguishes between the alternative regimes of (a) fast equilibration between excitations and exciplexes followed by their decay with a common average rate and (b) the fastest and deep excitation decay followed by the weaker and slower delayed fluorescence, backed by exciplex dissociation.

Kinetic Equations for Local Concentrations of Reactants in Spatially Inhomogeneous Solutions

Spatially inhomogeneous dilute solutions of reactants placed in a continual solvent (Waite approach) are examined. Based on the familiar non-Markovian kinetic equations of the Integral Encounter Theory, the Markovian kinetic equations were first obtained that allowed for the force action of reactants on each other due to binary encounters in solution during reactant macroscopic displacement. It has been shown that the value of “collisional integral” of reactants determined by their stochastic motion is small as compared to the “collisional integral” with the solvent molecules in the frame of the binary approach to the description of the solvent evolution. Nevertheless, taking account of the encounters of reactants with each other may turn out to be of primary importance, since it results in new physical effects.

Reversible Charge Separation Followed by Exciplex Formation

The quenching of fluorescence by reversible bimolecular ionization, followed by reversible exciplex formation from an ion pair (Scheme II) subjected to spin-conversion and subsequent radical-ion recombination/separation, has been studied by means of integral encounter theory (IET) and fitted to the available experimental data. Using the incoherent (rate) model of spin-conversion, the ion recombination to the excited triplet products is also accounted for. All of the results are obtained and shown to be different for the pulse excitation of fluorescence and its stationary detection. The free-energy dependence of all of the calculated properties of the forward and backward electron transfer are specified and compared with the conventional free-energy gap (FEG) law.

Luminescence Quenching by Reversible Ionization or Exciplex Formation/Dissociation

The kinetics of fluorescence quenching by both charge transfer and exciplex formation is investigated, with an emphasis on the reversibility and nonstationarity of the reactions. The Weller elementary kinetic scheme of bimolecular geminate ionization and the Markovian rate theory are shown to lead to identical results, provided the rates of the forward and backward reactions account for the numerous recontacts during the reaction encounter. For excitation quenching by the reversible exciplex formation, the Stern-Volmer constant is specified in the framework of the integral encounter theory. The bulk recombination affecting the Stern-Volmer quenching constant makes it different for pulse excited and stationary luminescence. The theory approves that the free energy gap laws for ionization and exciplex formation are different and only the latter fits properly the available data (for lumiflavin quenching by aliphatic amines and aromatic donors) in the endergonic region.

Deviation from the kinetic law of mass action for reactions induced by binary encounters inliquid solutions

In considering the irreversible chemical reaction in liquid solutions two many-particle approaches to the derivation of binarynon-Markovian kinetic equations are compared: simple superposition decoupling and amethod of extracting ‘pair’ channels from three-particle correlation evolution. It isshown that both methods provide an almost identical description of this reaction.However, in studies of reversible reactions in liquid solutions only the channel extractionmethod gives a correct physically clear description of the reaction though it consists of asequence of steps: the development of integral encounter theory (IET), effectivepairs approximation (EPA), modified encounter theory (MET), and the finalregular form (RF) of kinetic equations. It is shown that the rate equations oftenencountered in the literature correspond to the independence of transient channels of‘scattering’ in the bimolecular reversible reaction (), while the independent transient channel of ‘decay’ in the reversible reaction is defined solely by time integral convolution. In the general case transient channels innon-Markovian theory are not independent, and their interference manifests itself as anon-Markovian inhomogeneous source in binary non-Markovian kinetic equations in regularform.Based on the derived equations new universal kinetics (independent of models) of chemicalequilibrium attainment have been obtained. It is shown that these kinetics can differessentially from the kinetics corresponding to the kinetic law of mass action of formalchemical kinetics.

Theoretical study of the CIDNP kinetics of the reaction of a photo-excited dye with a reactive spot on the surface of a sphere: application to CIDNP of proteins

Time-resolved chemically induced dynamic nuclear polarization (CIDNP) in sterically specific photoreactions of proteins with excited dye molecules is studied theoretically by means of the integral encounter theory. The protein is modeled as a chemically inert sphere with one small reactive spot, with steric factor f modeling a single accessible CIDNP-active amino acid residue on its surface. Kinetic equations describing the dye-quenching kinetics and the CIDNP time evolution are obtained, and the non-trivial f dependence of the rate constant of dye quenching, the geminate CIDNP intensity, the radical recombination rate constant and the nuclear T 1 relaxation time are studied. Protein CIDNP kinetics are compared with those of free amino acids.

Theoretical treatment of reversible energy transfer reactions of metastable reactants: Modification of the integral encounter theory

Corrections to the results of the integral encounter theory (IET) of reversible energy transfer reaction provided by two competing approaches of the modified encounter theory (MET) and the many-particle kernel-theory (MPK-theory) are considered in detail and compared with each other. The present treatment establishes that the MET is more preferable for improving the IET at higher reactant densities. Predictions of the MET and MPK-theory for the quantum yields of luminescence and quenching constants are compared with each other, reversible transfer kinetics from metastable reactant is studied in the framework of the MET.

Inhomogeneous electrochemiluminescence. I: Integral encounter theory of kinetics and yield

The generation of excited triplet luminophor by recombination of electrochemically injected ions is calculated. The reversibility of triplet production is shown to affect insignificantly the triplet accumulation and their quantum yield. The free energy dependence of the latter is specified and fitted to the available experimental data. The maximal quantum yield of triplet luminescence found experimentally is sometimes less than 1, even when the excitation is the sole result of charge recombination. This phenomenon may be a result of the biexcitonic annihilation of triplets produced in a narrow strip at the border between the space separated anions and cations. The kinetics of this process is calculated and the maximal yield of fluorescence is shown to decrease with excitation life time and increase with triplet diffusion out of the strip. Alternatively, the maximal quantum yield may be reduced due to the trivial triplet quenching by neutral precursors of ions.

Integral encounter theory of strong electron transfer

The integral encounter theory (IET) has been extended to the reactions limited by diffusion along the reaction coordinate to the level crossing points where either thermal or hot electron transfer occurs. IET describes the bimolecular ionization of the instantaneously excited electron donor D* followed by the hot geminate backward transfer which precedes the ion pair equilibration and its subsequent thermal recombination. We demonstrate that the fraction of ion pairs which avoids the hot recombination is much smaller than their initial number when the electron tunneling is strong.