Alternative solution for diffusion to two spheres with first-order surface reaction

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.

Section: A Neutral Reactantsmentioning

confidence: 99%

Competition effects in diffusion-controlled bulk reactions between ions

Traytak¹,

Barzykin²,

Tachiya³

2007

“…12 In subsequent papers by others, the effects of a finite, first-order surface reaction rate were examined, both by asymptotic methods, 9 and by exact solution in terms of nested, continued fractions. 10 Two infinitely reactive spheres, each of a different size, were also studied. 2 The case of a finite, first-order surface reaction rate at two identical spheres was extended to treat a first-order reaction rate within the spheres 13 using the method of twin spherical expansions.…”

Section: Introductionmentioning

confidence: 99%

Diffusion and reaction for a spherical source and sink

McDonald¹,

Strieder²

2003

Self Cite

Enhancement of the recollision rate in diffusion-influenced reactions in an inhomogeneous mediumTwo chemically active spheres in an infinite medium, one a zeroth-order reactant source and the other a first-order sink, are considered for various sphere size ratios, center-to-center distances, and sink strengths from chemical to diffusion controlled conditions. This source-sink model simulates some aspects of biological mutualism interactions between different cells. Infinite series expansions in a single index n are obtained for the sink reaction rate and reactant concentration profiles using the bispherical expansion. Each of the coefficients, generated exactly by a matrix elimination method, is expressed in terms of nested, continued fractions easily evaluated for the given n. At intermediate and larger sink-source separation distances the sink reaction rate decays harmonically. For smaller sink-source separations with a highly reactive small sink, a local maximum in the sink reaction rate is found.

“…This behavior is similar to that of two unbounded identical spheres. [7][8][9] It can also be seen from the figure that the results of the reaction rate for the contact case appear finite.…”

Section: Resultsmentioning

confidence: 79%

“…Later, an expression for the reaction rate with recurrence relations was reported in a supplementary note. 8 One major finding is a decrease in the reaction rate with the increasing closeness between the two spheres. Very recently, Tsao 9 applied the method of twin spherical expansion to investigate the behavior for two identical reactant-penetrable spheres, within which the reactant undergoes a first-order reaction as well.…”

Section: Introductionmentioning

confidence: 98%

Diffusion into a nanoparticle with first-order surface reaction confined within a sphere

Chen

Tsao

2002

The exact series solution for the reaction rate of a spherical sink confined within a sphere is presented from reaction- to diffusion-limited condition based on the bispherical coordinate method. The reaction rate varies with the particle location and the size ratio of particle to enveloping sphere. The maximum and minimum rates take place when the particle and the confining sphere are at contact and concentric, respectively. A thin-gap analysis is employed to derive the rate expression analytically for a near contact case. While the reaction rate at near contact diverges logarithmically for a purely diffusion-limited condition, it remains finite for a fast surface reaction with finite rate constant. The average reaction rate is then calculated based on a prescribed particle distribution function. It is found that for a uniform distribution, the mean rate is at most 50% higher than that for the concentric case. Other than the hard-body interaction, additional attractive and repulsive interactions will enhance and reduce the mean rate, respectively.