Time operator for diffusion

Volume: 11 Issue: 4 Pages: 465-477 Published: MAR 2000 Times Cited: 9 References: 15 Citation MapCitation Map beta Abstract: We extend the concept of time operator for general semigroups and construct a non-self-adjoint time operator for the diffusion equation which is intertwined with the unilateral shift. We obtain the spectral resolution, the age eigenstates and a new shift representation of the solution of the diffusion equation. Based on previous work we obtain similarly a self-adjoint time operator for Relativistic Diffusion. (C) 2000 Elsevier Science Ltd. All rights reserved.

Extraction of electrode kinetic parameters from microdisc voltammetric data measured under transport conditions intermediate between steady-state convergent and transient linear diffusion as typically applies to room temperature ionic liquids

The extraction of electrode kinetic parameters for electrochemical couples in room-temperature ionic liquids (RTILs) is currently an area of considerable interest. Electrochemists typically measure electrode kinetics in the limits of either transient planar or steady-state convergent diffusion for which the voltammetic response is well understood. In this paper we develop a general method allowing the extraction of this kinetic data in the region where the diffusion is intermediate between the planar and convergent limits, such as is often encountered in RTILs using microelectrode voltammetry. A general working surface is derived, allowing the inference of Butler-Volmer standard electrochemical rate constants for the peak-to-peak potential separation in a cyclic voltammogram as a function of voltage scan rate. The method is applied to the case of the ferrocene/ferrocenium couple in [C(2)mim][N(Tf)(2)] and [C(4)mim][N(Tf)(2)].

A comparative electrochemical study of diffusion in room temperature ionic liquid solvents versus acetonitrile

Measurements on the diffusion coefficient of the neutral molecule N,N,N',N'-tetramethyl-para-phenylenediamine and the radical cation and dication generated by its one- and two-electron oxidation, respectively, are reported over the range 298-348 K in both acetonitrile and four room temperature ionic liquids (RTILs). Data were collected using single and double potential step chronamperometry at a gold disk electrode of micrometer dimension, and analysed via fitting to the appropriate analytical expression or, where necessary, to simulation. The variation of diffusion coefficient with temperature was found to occur in an Arrhenius-type manner for all combinations of solute and solvent. For a given ionic liquid, the diffusional activation energies of each species were not only closely equivalent to each other, but also to the RTIL's activation energy of viscous flow. In acetonitrile supported with 0.1 M tetrabutylammonium perchlorate, the ratio in diffusion coefficients of the radial cation and dication tot he neutral molecule were calculated as 0.89 +/- 0.05 and 0.51 +/- 0.03, respectively. In contrast, amongst the ionic liquids the same ratios were determined to be on average 0.53 +/- 0.04 and 0.33 +/- 0.03. The consequences of this dissimilarity are considered in terms of the modelling of voltammetric data gathered within ionic liquid solvents.

Double potential step chronoamperometry at microdisk electrodes: simulating the case of unequal diffusion coefficients

An efficient approach to the simulation of the double potential step chronoamperometry at a microdisk electrode based on an exponentially expanding time grid and conformal mapping of the space is presented. The dimensionless second potential step flux data are included as a function of the first potential step duration and the ratio of the diffusion coefficients of the reacting species allowing instant analysis of the experimental double potential step chronoamperograms without a need for simulation. The values of the diffusion coefficients are determined for several test systems and found to be in good agreement with existing literature data. (C) 2004 Elsevier B.V. All rights reserved.

Voltammetry of oxygen in the room-temperature ionic liquids 1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)imide and hexyltriethylammonium bis((trifluoromethyl)sulfonyl)imide: One-electron reduction to form superoxide. Steady-state and transient behavior in the same cyclic voltammogram resulting from widely different diffusion coefficients of oxygen and superoxide

The electrochemical reduction of oxygen in two different room-temperature ionic liquids, 1-ethyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)imide ([EMIM][N(Tf)(2)]) and hexyltriethylammonium bis((trifluoromethyl)sulfonyl)imide ([N-6222][N(Tf)(2)]) was investigated by cyclic voltammetry at a gold microdisk electrode. Chronoamperometric measurements were made to determine the diffusion coefficient, D, and concentration, c, of the electroactive oxygen dissolved in the ionic liquid by fitting experimental transients to the Aoki model. [Aoki, K.; et al. J. Electroanal. Chem. 1981, 122, 19]. A theory and simulation designed for cyclic voltammetry at microdisk electrodes was then employed to determine the diffusion coefficient of the electrogenerated superoxide species, O-2(.-), as well as compute theoretical voltammograms to confirm the values of D and c for neutral oxygen obtained from the transients. As expected, the diffusion coefficient of the superoxide species was found to be smaller than that of the oxygen in both ionic liquids. The diffusion coefficients of O-2 and O-2(.-) in [N-6222][N(Tf)(2)], however, differ by more than a factor of 30 (D-O2 = 1.48 x 10(-10) m(2) s(-1), DO2.- = 4.66 x 10(-12) m(2) s(-1)), whereas they fall within the same order of magnitude in [EMIM][N(Tf)(2)] (D-O2 = 7.3 x 10(-10) m(2) s(-1), DO2.- = 2.7 x 10(-10) m(2) s(-1)). This difference in [N-6222][N(Tf)(2)] causes pronounced asymmetry in the concentration distributions of oxygen and superoxide, resulting in significant differences in the heights of the forward and back peaks in the cyclic voltammograms for the reduction of oxygen. This observation is most likely a result of the higher viscosity of [N-6222][N(Tf)(2)] in comparison to [EMIM][N(Tf)(2)], due to the structural differences in cationic component.

Turbulent diffusion and mixing in interstellar dark clouds with adsorption onto grains

This paper reports the results of models of dark cloud chemistry incorporating a depth dependent density distribution with diffusive mixing and adsorption onto grains. The model is based on the approach taken by Xie et al. (1995), with the addition of grain accretion effects. Without diffusion, the central regions of the cloud freeze out in less than 10(7) years. Freeze-out time is dependent on density, so the diffuse outer region of the cloud remains abundant in gas for about an order of magnitude longer. We find that fairly small amounts of diffusive mixing can delay freeze-out at the centre of the model cloud for a time up to an order of magnitude greater than without diffusion, due to material diffusing inward from the edges of the cloud. The gas-phase lifetime of the cloud core can thus be increased by up to an order of magnitude or more by this process. We have run three different grain models with various diffusion coefficients to investigate the effects of changing the sticking parameters.

Dissolved Argon Changes the Rate of Diffusion in Room Temperature Ionic Liquids: Effect of the Presence and Absence of Argon and Nitrogen on the Voltammetry of Ferrocene

This work explores the effects of argon and nitrogen, two electrochemically and chemically inert gases frequently used in sample preparation of room temperature ionic liquid (RTIL) solutions, on the eelectrochemical characterization of ferrocene (Fc) dissolved in the RTIL 1-ethyl-3-methylimidazolium bis(trifluoromethanesulfonyl)imide ([C(2)mim][NTf2]). Remarkably, chronoamperometrically determined diffusion coefficients of Fc in [C(2)mim][NTf2] are found to increase from 4.8 (+/- 0.2) x 10(-11) m(2) s(-1) under vacuum conditions to 6.6 (+/- 0.5) x 10(-11) m(2) s(-1) in an atmosphere of 1 atm Ar. In contrast, exposing a vacuum-purified sample to an atmosphere of 1 atm N-2 resulted in no significant change in the measured diffusion coefficient of Fc. The effect of dissolved argon on diffusion transport is unexpected and has implications in electrochemistry and elsewhere. Fc was found to volatilize under vacuum conditions. We propose, however, that evacuation of the cell by vacuum prior to electrochemical measurements being carried out is the only way to ensure that no contamination of the sample occurs, and use of an in situ method of determining the diffusion coefficient and concentration of Fc dispells,any ambiguity associated with Fc depletion by vacuum.

SO2 Saturation of the Room Temperature Ionic Liquid [C(2)mim][NTf2] Much Reduces the Activation Energy for Diffusion

The physical effect of high concentrations of reversibly dissolved SO2 on [C(2)mim][NTf2] was examined using cyclic voltammetry, chronoamperometry, and ESR spectroscopy. Cyclic voltammetry of the oxidation of solutions of ferrocene, N,N,N',N'-tetramethyl-p-phenylenediamine (TMPD), and chloride in the room temperature ionic liquid (RTIL) 1-ethyl-3-methylimidazolium bis(trifluoromethanesufonyl)imide ([C(2)mim][NTf2]) reveals an increase in limiting current of each species corresponding to the addition of increasing concentrations of sulfur dioxide. Quantitative chronoamperometry reveals an increase in each species' diffusion coefficient with SO2 concentration. When chronoamperometric data were obtained for ferrocene in [C(2)mim][NTf2] at a range of temperatures, the translational diffusion activation energy (29.0 +/- 0.5 kJ mol(-1)) was found to be in good agreement with previous studies. Adding SO2 results in apparent near-activationless translational diffusion. A significant decrease in the activation energy of rotational diffusion with the SO2 saturation of a 2,2,6,6-tetramethyl-1-piperidinyloxyl (TEMPO) solution in [C(2)mim][NTf2] (29.9 +/- 2.0 to 7.7 +/- 5.3 kJ mol(-1)) was observed using electron spin resonance (ESR) spectroscopy. The reversible physical absorption Of SO2 by [C(2)mim][NTf2] should have no adverse effect on the ability of that ionic liquid to be employed as a solvent in an electrochemical gas sensor, and it is possible that the SO2-mediated reduction of RTIL viscosity could have intrinsic utility.

Silica grain labelling and EDX spectroscopy; evidence for inter-grain diffusion of Pt(NH3)(4)(2+) species during Pt/SiO2 catalyst preparation by ionic exchange

Labelling of silica grains and energy dispersive X-ray spectroscopy (EDX) in a TEM-FEG (field emission gun) were used to demonstrate the migration of Pt(NH3)(4)(2+) species from one grain to another during Pt/SiO2 catalyst preparation by the ion-exchange procedure.