Onsager Length Research Articles

Injection currents in thin disordered organic films

Analytic model of barrier-limited injection of charge carriers from metal electrodes into organic film, which was introduced by Arkhipov and co-workers, is modified, considering effects of multiple image charges and injection from both electrodes. Limits of applicability of Arkhipov's model are discussed. Variations from Arkhipov's model are important, if film thickness is comparable with Onsager length.

Competition effects in diffusion-controlled bulk reactions between ions

A numerical solution for competitive diffusion into two static charged sinks is presented and compared against an approximate analytical solution. It is shown that for an attractive Coulomb potential the correction to the total diffusive flux of ionic reactants due to diffusion interaction between sinks may be ignored for large enough values of the Onsager length and thus the reaction rate can be described by the classical Debye formula. On the other hand, for repulsive potentials the competition becomes more profound than in the case of uncharged sinks. However, this effect is of little interest because of a small value of the diffusive flux on the reaction surface.

Competition effect in diffusion-controlled reactions between ions

The first correction to the total flux of charged particles into a sink including the mutualinfluence of the other sink for the system of two charged static sinks is obtained. It turnsout that the correction is negative for a high intrinsic rate constant and positiveotherwise. It is shown that for attractive reactants the diffusive interaction effectmay be neglected even in the case of fully diffusion-controlled reactions for largeenough values of the Onsager length. Moreover the monopole approximationformula is suggested to describe the diffusive interaction between closely locatedsinks.

Concentration dependence of fluorescence quenching by ionic reactants

Fluorescence quenching for ionic reactants when the quenchers are nondilute is studiedtheoretically. The fluorophore and quenchers are assumed to interact by Coulomb potential.It has been shown that for oppositely charged ions with large Onsager length the positivedeviation from the linear Stern–Volmer law is negligibly small. Our results are in goodagreement with available experimental data on the luminescence quenching of the ionsW6I142− by bipyridinium ions.

Photoluminescence quenching of porous silicon in organic solvents: evidence for dielectric effects

Photoluminescence (PL) intensity of porous silicon has been recorded in a selected set of organic solvents. PL stability has been systematically examined on the time scale of the measurements in order to carefully exclude any interfering chemical effects. A three-order-of-magnitude drop is found by increasing the value of the dielectric constant of the solvent from 2 to 20. This spectacular result can be quantitatively accounted for in terms of a geminate recombination mechanism and of the variation of the Onsager length with the dielectric constant of the embedding medium. For values of the dielectric constant larger than 20, the Onsager length becomes of the order of the nanostructure size, and no further quenching is observed. These results show that materials of low-dielectric constant should be chosen for encapsulating porous silicon without affecting its PL.

‘‘Order–disorder’’ phenomena in a diffusion-reaction model of interacting dipoles on a surface

We study the reaction efficiency of a surficial process in which a diffusing, tumbling dipole A reacts (eventually and irreversibly) with a stationary target dipole B. In contrast to earlier studies of such irreversible diffusion-reaction events (A+B→C), we consider the situation where at each and every site of the space accessible to the diffusing coreactant A, there is also embedded a fixed dipole. To quantify the influence on the reaction efficiency of (angle-averaged, dipole–dipole) potential interactions between the tumbling dipole A and the ensemble of stationary dipoles, we design a lattice-statistical model to describe this problem and use both analytical methods and numerical techniques rooted in the theory of finite Markov processes to work out its consequences. Specifically, we define the reaction space to be an n×n=N square-planar lattice with the target dipole occupying the centrosymmetric site in that space and determine the mean number of steps required before the irreversible event, A+B→C, occurs. Our results reveal two qualitatively-distinct regimes of behavior for this diffusion-reaction process, a low temperature (or strong coupling) regime dominated by nearest-neighbor excursions only, and a high-temperature (or weak-coupling) regime dominated by non-nearest neighbor excursions of the tumbling dipole A, with the transition between these two regimes occurring over a relatively narrow range of interparticle couplings. This behavior has the character of an ‘‘order–disorder’’ transition and is interpreted here in terms of an ‘‘order parameter’’ W related to a generalized Onsager length. The behavior uncovered is studied as a function of system size and of the boundary conditions imposed.

Diffusion-controlled reaction rate to asymmetric reactants under Coulomb interaction

The rate constant for diffusion-controlled reactions between asymmetric reactants described by the simple model of Solc and Stockmayer under the influence of Coulomb-type interaction is considered. Using the method of dual series relations, we calculate the rate constant with a high accuracy and obtain some approximate analytical formulas. We compare our results with an approximate analytical formula derived before by the constant-flux approximation and with numerical calculations based on the Brownian dynamics simulation. It is shown that in the case of strong attractive potential the rate constant only slightly depends on the size of the active site and tends to the classical Debye result for isotropically reactive particles with a further increase in the Onsager length. Moreover, it is shown that for small-sized active sites the effect of the interaction potential is to scale the rate constant for neutral reactants by a Boltzmann factor, which was first conjectured by Zhou [Biophys. J. 64, 1711 (1993)].

On the solution of the Debye-Smoluchowski equation with a Coulomb potential. II. An approximation of the time-dependent rate constant

The time-dependent Debye-Smoluchowski equation with a Coulomb potential is considered for the condition of a perfectly absorbing sink and random initial distribution. The compound approximation of the reaction rate constant, which works well for whole range of the Onsager length and all time, is suggested. Numerical solution confirmed this approximation.

Observation of singlet exciton photoionization in anthracene single crystal at 2.95 eV

Photo-carrier generation with 420 nm light ( hv=2.95 eV) in an anthracene single crystal has been studied. The main experimental results are the following: (1) The photocurrent waveforms correspond to the penetration depth of the exciting light. (2) The number of carriers generated is proportional to I ex 2 and becomes proportional to I ex 3 4 under strong excitation. (3) The photocurrent has a temperature dependence different from that of the absorption coefficient of the ground state. These observations indicate that charge carriers are generated through photoionization of a singlet exciton, i.e. by sequential two-step excitation. With an applied electric field of 1.5×10 4 V cm −1, the cross section of the singlet exciton photoionization (with λ=420 nm) is 7×10 −19 cm 2. The Onsager length r 0, determined from the temperature dependence of the photocurrent, is 36 Å.

Recombination of electron–ion pairs for arbitrary mean free path

A theory is developed of the dependence of the steady-state electron–ion pair recombination rate constant on the electron mean free path. The problem, in classical mechanics, is reduced to the stochastic dynamics of the electron in the one-dimensional effective potential Ṽ(R)=−kTRc/R −2kT ln R. Rc is the Onsager length Ze2/4πεkT. For a large mean free path λ, the recombination rate is determined by energy relaxation of electrons which cross the transition state of Ṽ(R) at RT=Rc/2, whereas for small λ the Debye result for spatial diffusion-controlled recombination is obtained. The theory gives the dependence of the rate in the crossover regime where λ is comparable to Rc. The results are in good agreement with experiment and Monte Carlo simulations.

Breakdown of the Onsager theory of geminate ion recombination

We investigate how the dynamics of geminate recombination between a positive ion and an electron depends upon the mean free path of the electron for scattering by the solvent. We calculate the escape probability of geminate recombination as a function of the electron mean free path, by use of the Monte Carlo method we previously developed. When the mean free path is negligibly short compared with the Onsager length, the calculated escape probability agrees with the Onsager result. However, it increases with increasing mean free path and becomes much higher than the Onsager result, for long mean free paths. We also study how the effect of an applied electric field on the escape probability is influenced by the length of the electron mean free path. Our theory quantitatively explains recent experimental data on free ion yields in liquids such as methane, argon, and tetramethylsilane, which sustain high electron mobilities or equivalently long electron mean free paths.

Diffusion controlled reactions: Experimental verification of the time-dependent rate equation

The transient effect in a diffusion limited reaction is described by a time-dependent rate coefficient: k(t)=a+b exp(c2t)erfc(ct1/2), where a, b, and c are expressed in terms of the diffusion coefficient (D), the encounter distance (R), and the absolute rate coefficient (ka); and for ionic reactions, the Onsager length (rc). Time resolved fluorescence quenching studies on cresyl violet–potassium iodide system in water confirm the validity of the above equation. The values of D and R obtained from the fluorescence quenching studies are in good agreement with the values inferred from other sources, and the value of ka is reasonable.

Dependence of the recombination kinetics of geminate electron-ion pairs on the mean free path of electrons

We have investigated how the kinetics of geminate recombination of electron-ion pairs is influenced by the mean free path of electrons. By use of the Monte Carlo method previously developed the decay of the pair survival probability was numerically calculated for a variety of values of the mean free path of electrons. When the mean free path is negligibly small compared with the Onsager length, our result is in good agreement with the Hong-Noolandi result based on the Smoluchowski equation. However, as the mean free path increases, it deviates from their result increasingly. This indicates that the treatment based on the Smoluchowski equation is inadequate when the mean free path of electrons is not negligible compared with the Onsager length. We have found that for mean free paths comparable with or larger than the Onsager length the decay kinetics of the pair survival probability approximately follows the first-order kinetics except at very short and very long times. The dependence of the escape probability on the mean free path of electrons is also briefly discussed.

Breakdown of the Debye theory of bulk ion recombination

We have investigated how the rate of diffusion-controlled bulk recombination between electrons and positive ions is influenced by the mean free path of electrons. By use of the Monte Carlo method previously developed we have calculated the rate constant as a function of the mean free path. For mean free paths negligibly short compared with the Onsager length, the rate constant increases linearly with increasing mean free path, in agreement with the Debye theory of bulk ion recombination. However, as the mean free path further increases, the rate constant goes through a maximum and then decreases. The present result quantitatively explains recent experimental data on electron–ion recombination rate constants in liquid methane in which the mean free path of electrons is comparable with the Onsager length.

Geminate recombination of charges in irradiated liquid argon

Contrary to a recent suggestion [Phys. Rev. Lett. 40, 407 (1978)], the Onsager model of charge neutralization is not applicable to liquid argon irradiated by $\ensuremath{\alpha}$ particles. That model treats isolated ion-electron pairs. In $\ensuremath{\alpha}$-irradiated liquid argon the average distance between nearest-neighbor positive ions is about two orders of magnitude smaller than the Onsager length, so a columnar model (Jaffe or Kramers) must be used. The thermalization length of the hot electrons in liquid argon is about 200 nm.