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Sergey D. Traytak

Publications and source records attributed to Sergey D. Traytak.

9 recordsLinked to original sources
arXiv2026

Coupling of diffusion and reaction in a thin cylindrical tube: Methodological drawbacks of the Fick--Jacobs approach

We investigate a problem, that describes coupling between diffusion and reaction inside a thin circular cylindrical tube. The asymptotic solution of the posed problem is derived by means of the boundary functions method. We perform comparison of this asymptotic solution against corresponding exact solution, which revealed serious methodological drawbacks of known Fick-Jacobs reduction approach. The results obtained may be used to study a wide range of reaction-diffusion problems, when the Fick-Jacobs method cannot be applied.

physics.chem-ph↗
arXiv2025

Diffusion-controlled reaction rate to an active site in a spherical cavity: Extension of Berg's theory

This study is due to various applications in physics, chemistry and especially in biology, where both bounded configuration domain and chemical anisotropy could play a great part. In fact we generalize the well-known Berg theory, which describes diffusion-controlled reactions occurring within a spherically symmetric absorber-cavity system. The trapping probability and the reaction rate at which a small diffusing particle is captured by an axially symmetric one reactive patch absorber inside a spherical cavity were found semi-analytically and numerically by means of the dual series relations method. This approach leads to such incredibly fast convergence, that it may rightly be referred to as exact one. The results obtained can be used to test numerical programmes that describe diffusion-controlled reactions in real physical systems for reactants with arbitrary anisotropic reactivity, which are located inside of various cavities as well as in the unbounded domains. Moreover, we managed to find a close connection between the dual series relations method and the generalized method of separation of variables.

physics.chem-ph↗
arXiv2024

The generalized method of separation of variables for diffusion-influenced reactions: Irreducible Cartesian tensors technique

Motivated through various applications of the trapping diffusion-influenced reactions theory in physics, chemistry and biology, this paper deals with irreducible Cartesian tensors (ICT) technique within the scope of the generalized method of separation of variables (GMSV). Presenting a survey from the basic concepts of the theory, we spotlight the distinctive features of the above approach against known in literature similar techniques. The classical solution to the stationary diffusion equation under appropriate boundary conditions is represented as a series in the ICT. By means of proved translation addition theorem we straightforwardly reduce the general boundary value diffusion problem for $N$ spherical sinks to the corresponding resolving infinite system of linear algebraic equations with respect to unknown tensor coefficients. These coefficients comprise explicit dependence on the arbitrary three-dimensional configurations of $N$ sinks with different radii and surface reactivities. Specific application of the ICT technique is illustrated by some numerical calculations for the reactions, occurring in the arrays of two and three spherical sinks. For the first time we treat the statement and solution to the above reaction-diffusion problem, applying the smooth manifold concept.

physics.chem-ph↗
arXiv2023

Persistent effects of inertia on diffusion-influenced reactions: Theoretical methods and applications

The Cattaneo-Vernotte model has been widely studied to take momentum relaxation into account in transport equations. Yet, the effect of reactions on the Cattaneo-Vernotte model has not been fully elucidated. At present, it is unclear how the current density associated with reactions can be expressed in the Cattaneo-Vernotte model. Herein, we derive a modified Cattaneo-Vernotte model by applying the projection operator method to the Fokker-Planck-Kramers equation with a reaction sink. The same modified Cattaneo-Vernotte model can be derived by a Grad procedure. We show that the inertial effect influences the reaction rate coefficient differently depending on whether the intrinsic reaction rate constant in the reaction sink term depends on the solute relative velocity or not. The momentum relaxation effect can be expressed by a modified Smoluchowski equation including a memory kernel using the Cattaneo-Vernotte model. When the intrinsic reaction rate constant is independent of the reactant velocity and is localized, the modified Smoluchowski equation should be generalized to include a reaction term without a memory kernel. When the intrinsic reaction rate constant depends on the relative velocity of reactants, an additional reaction term with a memory kernel is required because of competition between the current density associated with the reaction and the diffusive flux during momentum relaxation. The competition effect influences even the long-time reaction rate coefficient.

physics.chem-ph↗
arXiv2018

Diffusion to capture and the concept of diffusive interactions

Diffusion to capture is an ubiquitous phenomenon in many fields in biology and physical chemistry, with implications as diverse as ligand-receptor binding on eukaryotic and bacterial cells, nutrient uptake by colonies of unicellular organisms and the functioning of complex core-shell nanoreactors. Whenever many boundaries compete for the same diffusing molecules, they inevitably shield a variable part of the molecular flux from each other. This gives rise to the so-called diffusive interactions (DI), which can reduce substantially the influx to a collection of reactive boundaries depending chiefly on their geometrical configuration. In this review we provide a pedagogical discussion of the main mathematical aspects underlying a rigorous account of DIs. Starting from a striking and deep result on the mean-field description of ligand binding to a receptor-covered cell, we develop little by little a rigorous mathematical description of DIs in the stationary case through the use of translational addition theorems for spherical harmonics. We provide several enlightening illustrations of this powerful mathematical theory, including diffusion to capture to ensembles of reactive boundaries within a spherical cavity.

cond-mat.soft↗
arXiv2018

Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries

We apply the generalized method of separation of variables (GMSV) to solve boundary value problems for the Laplace operator in three-dimensional domains with disconnected spherical boundaries (i.e., an arbitrary configuration of non-overlapping partially reactive spherical sinks or obstacles). We consider both exterior and interior problems and all most common boundary conditions: Dirichlet, Neumann, Robin, and conjugate one. Using the translational addition theorems for solid harmonics to switch between the local spherical coordinates, we obtain a semi-analytical expression of the Green function as a linear combination of partial solutions whose coefficients are fixed by boundary conditions. Although the numerical computation of the coefficients involves series truncation and matrix inversion, the use of the solid harmonics as basis functions naturally adapted to the intrinsic symmetries of the problem makes the GMSV particularly efficient, especially for exterior problems. The obtained Green function is the key ingredient to solve boundary value problems and to determine various characteristics of stationary diffusion such as reaction rate, escape probability, harmonic measure, residence time, and mean first passage time, to name but a few. The relevant aspects of the numerical implementation and potential applications in chemical physics, heat transfer, electrostatics, and hydrodynamics are discussed.

physics.comp-ph↗
arXiv2014

Diffusion-influenced reactions in a hollow nano-reactor with a circular hole

Hollow nanostructures are paid increasing attention in many nanotechnology-related communities in view of their numerous applications in chemistry and biotechnology, e.g. as smart nanoreactors or drug-delivery systems. In this paper we consider irreversible, diffusion-influenced reactions occurring within a hollow spherical cavity endowed with a circular hole on its surface. Importantly, our model is not limited to small sizes of the aperture. In our scheme, reactants can freely diffuse inside and outside the cavity through the hole, and react at a spherical boundary of given size encapsulated in the chamber and endowed with a given intrinsic rate constant. We work out the solution of the above problem, enabling one to compute the reaction rate constant to any desired accuracy. Remarkably, we show that, in the case of narrow holes, the rate constant is extremely well-approximated by a simple formula that can be derived on the basis of simple physical arguments and that can be readily employed to analyze experimental data.

cond-mat.other↗
arXiv2013

Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximation

The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudial coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudial diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.

physics.chem-ph↗