Using an approach based on the diffusion analog of the Cattaneo–Vernotte differential model, we f... more Using an approach based on the diffusion analog of the Cattaneo–Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski’s boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski’s theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice’s formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper ...
The Brownian coagulation of highly dispersed aerosol particles in a stochastic medium with small ... more The Brownian coagulation of highly dispersed aerosol particles in a stochastic medium with small velocity fluctuations is analyzed. The fluctuations of the velocity field are assumed to be Gaussian with a uniform pair correlation function. Exact equations are found for the mean field of the nonuniform size distribution function of the particles. An effective Brownian-coagulation equation is constructed for a velocity correlation function of a specific type. The relationship between the turbulent diffusion coefficient and the effective coagulation kernel is determined.
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable t... more The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal 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 longitudinal 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.
We have presented an exact solution for the problem of diffusive binding to a spherical macromole... more We have presented an exact solution for the problem of diffusive binding to a spherical macromolecule with two axially symmetric active patches. A highly accurate approximate formula for an effective steric factor has been suggested. This model solution may serve as a benchmark for further studies of diffusive interaction in more realistic models of anisotropic reactivity.
The rate constant that describes the diffusive encounter/reaction between a particle and a large ... more The rate constant that describes the diffusive encounter/reaction between a particle and a large sphere can be computed easily by solving the stationary diffusion (i.e. Laplace) equation for the particle density with appropriate boundary conditions imposed on the surface of the sphere. In one classic, textbook example, this calculation is used to estimate the binding rate constant of a ligand to a receptor-covered cell.
where and for the sake of simplicity we assume that ( ) ( 0, 0, Ω = ∞ × ∞) ( ) t φ is a continuou... more where and for the sake of simplicity we assume that ( ) ( 0, 0, Ω = ∞ × ∞) ( ) t φ is a continuous function and ( ) ( ) , k x t C α ∞ ∈ Ω 0,1,2 k ( = ) with ( ) 2 0, 0 t α ≠ . It is well known that the exact analytical solution to the problem (1), (2) is often difficult if not impossible to obtain. However, particularly for diffusion-limited reactions, the value of the local flux on the boundary proportional to the function ( ) 0 x x j t u ∞ = = −∂ is of main
We present here a review of existing analytical methods to solve boundary value problems of diffu... more We present here a review of existing analytical methods to solve boundary value problems of diffusion in media containing N non-overlapping inclusions.
We apply the generalized method of separation of variables (GMSV) to solve boundary value problem... more 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.
Physical chemistry chemical physics : PCCP, Jan 29, 2016
Correction for 'Theory of diffusion-influenced reactions in complex geometries' by Marta ... more Correction for 'Theory of diffusion-influenced reactions in complex geometries' by Marta Galanti et al., Phys. Chem. Chem. Phys., 2016, DOI: .
Zhurnal Eksperimental'noi i Teroreticheskoi Fiziki
The diffusion of highly dispersed aerosol particles in a turbulent medium with low intensity velo... more The diffusion of highly dispersed aerosol particles in a turbulent medium with low intensity velocity fluctuations and a temperature gradient is considered. The fluctuations of the velocity field are assumed to be Gaussian with a uniform pair correlation function. An expression for the effective diffusion tensor is found, taking into account interaction between thermophoresis and turbulent particle migration. It is shown that the effective thermophoresis velocity is always smaller than the regular velocity.
Physica A: Statistical Mechanics and its Applications, 1998
Within the generating functional approach an equation is derived for the coagulation of the ensem... more Within the generating functional approach an equation is derived for the coagulation of the ensemble average of the distribution function for ÿnely dispersed particles in an incompressible Gaussian stochastic velocity ÿeld of low intensity. We assume that particles may be treated as a passive admixture in a uctuating carrying medium. As against to our previous work the relevant pair correlation function is assumed to be of general form. Furthermore, the presence of particles depletion due to a volume sink has been taken into account.
In the papers by Elperin and Krasovitov 1–3 the authors claim that they have suggested a new meth... more In the papers by Elperin and Krasovitov 1–3 the authors claim that they have suggested a new method called “modified method of irreducible multipoles” in order to solve the quasi steady state heat and mass transfer equations for systems with many interacting burning particles. “A method of solution of the Laplace equation in a region exterior to N arbitrarily located spheres of different radii is suggested. The method is based on the expansion of the solution into irreducible multipoles.” (p. 79) 1. “The present study extends the modified method of irreducible multipoles expansion, suggested by Elperin and Krasovitov (1994) to combustion of random char/carbon particles of different radii.” (p. 167) 2. “The method of expansion into irreducible multipoles which was developed in our previous works (Elperin and Krasovitov, 1994a, 1994b) is applicable to more realistic problems and is particularly suitable for dense random clusters of droplets (particles).” (p. 288) 3. It is important to note here that mathematically the irreducible tensors approach gives the possibility of finding the solution of linear boundary-value problems (BVP) for the Laplace equation in a three-dimensional multi-connected domain.
A simple analytic approximation for the time-dependent rate constant in the diffusion-controlled ... more A simple analytic approximation for the time-dependent rate constant in the diffusion-controlled reactions between particles with isotropic reactivity and axially symmetric active site has been derived. It is shown that in the general case the definition of the effective steric factor depends on time and the size of active site.
The problem of diffusion to a reactive sphere inside a cavity through a hole is solved for arbitr... more The problem of diffusion to a reactive sphere inside a cavity through a hole is solved for arbitrary aperture values.
This paper reports on some results concerning the binding of diffusing ligands in a spherical 3D ... more This paper reports on some results concerning the binding of diffusing ligands in a spherical 3D region randomly filled by free receptors. It is shown that commonly accepted mean-field theory which is successfully used for bulk diffusion-controlled reactions cannot describe the behavior of ligand concentration in the diffusion layer close to the region boundary. To eliminate this drawback of the theory, we introduce a new complementary diffusion equation in the boundary layer with an appropriate matching condition. Using this equation, we find the characteristic ligand penetration length and total time-dependent flux of ligand binding to free receptors randomly distributed in a spherical region.
Using an approach based on the diffusion analog of the Cattaneo–Vernotte differential model, we f... more Using an approach based on the diffusion analog of the Cattaneo–Vernotte differential model, we find the exact analytical solution to the corresponding time-dependent linear hyperbolic initial boundary value problem, describing irreversible diffusion-controlled reactions under Smoluchowski’s boundary condition on a spherical sink. By means of this solution, we extend exact analytical calculations for the time-dependent classical Smoluchowski rate coefficient to the case that includes the so-called inertial effects, occurring in the host media with finite relaxation times. We also present a brief survey of Smoluchowski’s theory and its various subsequent refinements, including works devoted to the description of the short-time behavior of Brownian particles. In this paper, we managed to show that a known Rice’s formula, commonly recognized earlier as an exact reaction rate coefficient for the case of hyperbolic diffusion, turned out to be only its approximation being a uniform upper ...
The Brownian coagulation of highly dispersed aerosol particles in a stochastic medium with small ... more The Brownian coagulation of highly dispersed aerosol particles in a stochastic medium with small velocity fluctuations is analyzed. The fluctuations of the velocity field are assumed to be Gaussian with a uniform pair correlation function. Exact equations are found for the mean field of the nonuniform size distribution function of the particles. An effective Brownian-coagulation equation is constructed for a velocity correlation function of a specific type. The relationship between the turbulent diffusion coefficient and the effective coagulation kernel is determined.
The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable t... more The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudinal 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 longitudinal 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.
We have presented an exact solution for the problem of diffusive binding to a spherical macromole... more We have presented an exact solution for the problem of diffusive binding to a spherical macromolecule with two axially symmetric active patches. A highly accurate approximate formula for an effective steric factor has been suggested. This model solution may serve as a benchmark for further studies of diffusive interaction in more realistic models of anisotropic reactivity.
The rate constant that describes the diffusive encounter/reaction between a particle and a large ... more The rate constant that describes the diffusive encounter/reaction between a particle and a large sphere can be computed easily by solving the stationary diffusion (i.e. Laplace) equation for the particle density with appropriate boundary conditions imposed on the surface of the sphere. In one classic, textbook example, this calculation is used to estimate the binding rate constant of a ligand to a receptor-covered cell.
where and for the sake of simplicity we assume that ( ) ( 0, 0, Ω = ∞ × ∞) ( ) t φ is a continuou... more where and for the sake of simplicity we assume that ( ) ( 0, 0, Ω = ∞ × ∞) ( ) t φ is a continuous function and ( ) ( ) , k x t C α ∞ ∈ Ω 0,1,2 k ( = ) with ( ) 2 0, 0 t α ≠ . It is well known that the exact analytical solution to the problem (1), (2) is often difficult if not impossible to obtain. However, particularly for diffusion-limited reactions, the value of the local flux on the boundary proportional to the function ( ) 0 x x j t u ∞ = = −∂ is of main
We present here a review of existing analytical methods to solve boundary value problems of diffu... more We present here a review of existing analytical methods to solve boundary value problems of diffusion in media containing N non-overlapping inclusions.
We apply the generalized method of separation of variables (GMSV) to solve boundary value problem... more 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.
Physical chemistry chemical physics : PCCP, Jan 29, 2016
Correction for 'Theory of diffusion-influenced reactions in complex geometries' by Marta ... more Correction for 'Theory of diffusion-influenced reactions in complex geometries' by Marta Galanti et al., Phys. Chem. Chem. Phys., 2016, DOI: .
Zhurnal Eksperimental'noi i Teroreticheskoi Fiziki
The diffusion of highly dispersed aerosol particles in a turbulent medium with low intensity velo... more The diffusion of highly dispersed aerosol particles in a turbulent medium with low intensity velocity fluctuations and a temperature gradient is considered. The fluctuations of the velocity field are assumed to be Gaussian with a uniform pair correlation function. An expression for the effective diffusion tensor is found, taking into account interaction between thermophoresis and turbulent particle migration. It is shown that the effective thermophoresis velocity is always smaller than the regular velocity.
Physica A: Statistical Mechanics and its Applications, 1998
Within the generating functional approach an equation is derived for the coagulation of the ensem... more Within the generating functional approach an equation is derived for the coagulation of the ensemble average of the distribution function for ÿnely dispersed particles in an incompressible Gaussian stochastic velocity ÿeld of low intensity. We assume that particles may be treated as a passive admixture in a uctuating carrying medium. As against to our previous work the relevant pair correlation function is assumed to be of general form. Furthermore, the presence of particles depletion due to a volume sink has been taken into account.
In the papers by Elperin and Krasovitov 1–3 the authors claim that they have suggested a new meth... more In the papers by Elperin and Krasovitov 1–3 the authors claim that they have suggested a new method called “modified method of irreducible multipoles” in order to solve the quasi steady state heat and mass transfer equations for systems with many interacting burning particles. “A method of solution of the Laplace equation in a region exterior to N arbitrarily located spheres of different radii is suggested. The method is based on the expansion of the solution into irreducible multipoles.” (p. 79) 1. “The present study extends the modified method of irreducible multipoles expansion, suggested by Elperin and Krasovitov (1994) to combustion of random char/carbon particles of different radii.” (p. 167) 2. “The method of expansion into irreducible multipoles which was developed in our previous works (Elperin and Krasovitov, 1994a, 1994b) is applicable to more realistic problems and is particularly suitable for dense random clusters of droplets (particles).” (p. 288) 3. It is important to note here that mathematically the irreducible tensors approach gives the possibility of finding the solution of linear boundary-value problems (BVP) for the Laplace equation in a three-dimensional multi-connected domain.
A simple analytic approximation for the time-dependent rate constant in the diffusion-controlled ... more A simple analytic approximation for the time-dependent rate constant in the diffusion-controlled reactions between particles with isotropic reactivity and axially symmetric active site has been derived. It is shown that in the general case the definition of the effective steric factor depends on time and the size of active site.
The problem of diffusion to a reactive sphere inside a cavity through a hole is solved for arbitr... more The problem of diffusion to a reactive sphere inside a cavity through a hole is solved for arbitrary aperture values.
This paper reports on some results concerning the binding of diffusing ligands in a spherical 3D ... more This paper reports on some results concerning the binding of diffusing ligands in a spherical 3D region randomly filled by free receptors. It is shown that commonly accepted mean-field theory which is successfully used for bulk diffusion-controlled reactions cannot describe the behavior of ligand concentration in the diffusion layer close to the region boundary. To eliminate this drawback of the theory, we introduce a new complementary diffusion equation in the boundary layer with an appropriate matching condition. Using this equation, we find the characteristic ligand penetration length and total time-dependent flux of ligand binding to free receptors randomly distributed in a spherical region.
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Papers by Sergey Traytak