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This paper focuses on trapping of diffusing particles by a sphere with an absorbing cap of arbitrary size on the otherwise reflecting surface. We approach the problem using boundary homogenization which is an approximate replacement of non-uniform boundary conditions on the surface of the sphere by an effective uniform boundary condition with appropriately chosen effective trapping rate. One of the main results of our analysis is an expression for the effective trapping rate as a function of the surface fraction occupied by the absorbing cap. As the cap surface fraction increases from zero to unity, the effective trapping rate increases from that for a small absorbing disk on the otherwise reflecting sphere to infinity which corresponds to a perfectly absorbing sphere. The obtained expression for the effective trapping rate is applied to find the rate constant describing trapping of diffusing particles by an absorbing cap on the surface of the sphere. Finally, we find the capacitance of a metal cap of arbitrary size on a dielectric sphere using the relation between the capacitance and the rate constant of the corresponding diffusion-limited reaction. The relative error of our approximate expressions for the rate constant and the capacitance is less than 5% over the entire range of the cap surface fraction from zero to unity. [ABSTRACT FROM AUTHOR]

This paper is devoted to particle transport in a tube formed by alternating wide and narrow sections, in the presence of an external biasing force. The focus is on the effective transport coefficients-- mobility and diffusivity, as functions of the biasing force and the geometric parameters of the tube. Dependences of the effective mobility and diffusivity on the tube geometric parameters are known in the limiting cases of no bias and strong bias. The approximations used to obtain these results are inapplicable at intermediate values of the biasing force. To bridge the two limits Brownian dynamics simulations were run to determine the transport coefficients at intermediate values of the force. The simulations were performed for a representative set of tube geometries over a wide range of the biasing force. They revealed that there is a range of the narrow section length, where the force dependence of the mobility has a maximum. In contrast, the diffusivity is a monotonically increasing function of the force. A simple formula is proposed, which reduces to the known dependences of the diffusivity on the tube geometric parameters in both limits of zero and strong bias. At intermediate values of the biasing force, the formula catches the diffusivity dependence on the narrow section length, if the radius of these sections is not too small. [ABSTRACT FROM AUTHOR]

This paper is devoted to the effective transport coefficients of a particle in a tube of alternating diameter. Analytical expressions are derived for the effective mobility and diffusivity under strong bias conditions, i.e., in the limiting case where the external biasing force tends to infinity. The expressions give the transport coefficients as functions of the geometric parameters of the tube and the external force. They show that the effective diffusivity is a linear function of the square of the external force, whereas the effective mobility is independent of the force. The problem of finding effective transport coefficients in a tube of alternating diameter is too complex to be analyzed by conventional methods. Therefore, the expressions are derived in the framework of an intuition-based approach and validated by Brownian dynamics simulations. The obtained results extend a short list of available analytical expressions for the effective transport coefficients. [ABSTRACT FROM AUTHOR]

We investigate transport of point Brownian particles in a tube formed by identical periodic compartments of varying diameter, focusing on the effects due to the compartment asymmetry. The paper contains two parts. First, we study the force-dependent mobility of the particle. The mobility is a symmetric non-monotonic function of the driving force, F, when the compartment is symmetric. Compartment asymmetry gives rise to an asymmetric force-dependent mobility, which remains non-monotonic when the compartment asymmetry is not too high. The F-dependence of the mobility becomes monotonic in tubes formed by highly asymmetric compartments. The transition of the F-dependence of the mobility from non-monotonic to monotonic behavior results in important consequences for the particle motion under the action of a time-periodic force with zero mean, which are discussed in the second part of the paper: In a tube formed by moderately asymmetric compartments, the particle under the action of such a force moves with an effective drift velocity that vanishes at small and large values of the force amplitude having a maximum in between. In a tube formed by highly asymmetric compartments, the effective drift velocity monotonically increases with the amplitude of the driving force and becomes unboundedly large as the amplitude tends to infinity. [ABSTRACT FROM AUTHOR]

The generalized Fick-Jacobs equation is widely used to study diffusion of Brownian particles in three-dimensional tubes and quasi-two-dimensional channels of varying constraint geometry. We show how this equation can be applied to study the slowdown of unconstrained diffusion in the presence of obstacles. Specifically, we study diffusion of a point Brownian particle in the presence of identical cylindrical obstacles arranged in a square lattice. The focus is on the effective diffusion coefficient of the particle in the plane perpendicular to the cylinder axes, as a function of the cylinder radii. As radii vary from zero to one half of the lattice period, the effective diffusion coefficient decreases from its value in the obstacle free space to zero. Using different versions of the generalized Fick-Jacobs equation, we derive simple approximate formulas, which give the effective diffusion coefficient as a function of the cylinder radii, and compare their predictions with the values of the effective diffusion coefficient obtained from Brownian dynamics simulations. We find that both Reguera-Rubi and Kalinay-Percus versions of the generalized Fick-Jacobs equation lead to quite accurate predictions of the effective diffusion coefficient (with maximum relative errors below 4% and 7%, respectively) over the entire range of the cylinder radii from zero to one half of the lattice period. [ABSTRACT FROM AUTHOR]

We show that the effect of driving force F on the effective mobility and diffusion coefficient of a particle in a tube formed by identical compartments may be qualitatively different depending on the compartment shape. In tubes formed by cylindrical (spherical) compartments the mobility monotonically decreases (increases) with F and the diffusion coefficient diverges (remains finite) as F tends to infinity. In tubes formed by cylindrical compartments, at large F there is intermittency in the particle transitions between openings connecting neighboring compartments. [ABSTRACT FROM AUTHOR]

Reduction of three-dimensional (3D) description of diffusion in a tube of variable cross section to an approximate one-dimensional (1D) description has been studied in detail previously only in tubes of slowly varying diameter. Here we discuss an effective 1D description in the opposite limiting case when the tube diameter changes abruptly, i.e., in a tube composed of any number of cylindrical sections of different diameters. The key step of our approach is an approximate description of the particle transitions between the wide and narrow parts of the tube as trapping by partially absorbing boundaries with appropriately chosen trapping rates. Boundary homogenization is used to determine the trapping rate for transitions from the wide part of the tube to the narrow one. This trapping rate is then used in combination with the condition of detailed balance to find the trapping rate for transitions in the opposite direction, from the narrow part of the tube to the wide one. Comparison with numerical solution of the 3D diffusion equation allows us to test the approximate 1D description and to establish the conditions of its applicability. We find that suggested 1D description works quite well when the wide part of the tube is not too short, whereas the length of the narrow part can be arbitrary. Taking advantage of this description in the problem of escape of diffusing particle from a cylindrical cavity through a cylindrical tunnel we can lift restricting assumptions accepted in earlier theories: We can consider the particle motion in the tunnel and in the cavity on an equal footing, i.e., we can relax the assumption of fast intracavity relaxation used in all earlier theories. As a consequence, the dependence of the escape kinetics on the particle initial position in the system can be analyzed. Moreover, using the 1D description we can analyze the escape kinetics at an arbitrary tunnel radius, whereas all earlier theories are based on the assumption that the tunnel is narrow. [ABSTRACT FROM AUTHOR]

The paper deals with a diffusing particle that escapes from a cavity to the outer world through a narrow cylindrical tunnel. We derive expressions for the Laplace transforms of the particle survival probability, its lifetime probability density, and the mean lifetime. These results show how the quantities of interest depend on the geometric parameters (the cavity volume and the tunnel length and radius) and the particle diffusion coefficients in the cavity and in the tunnel. Earlier suggested expressions for the mean lifetime, which correspond to different escape scenarios, are contained in our result as special cases. In contrast to these expressions, our formula predicts correct asymptotic behavior of the mean lifetime in the absence of the cavity or tunnel. To test the accuracy of our approximate theory we compare the mean lifetime, the lifetime probability density, and the survival probability (the latter two are obtained by inverting their Laplace transforms numerically) with corresponding quantities found by solving numerically the three-dimensional diffusion equation, assuming that the cavity is a sphere and that the particle has the same diffusion coefficient in the cavity and in the tunnel. Comparison shows excellent agreement between the analytical and numerical results over a broad range of the geometric parameters of the problem. [ABSTRACT FROM AUTHOR]

Diffusion of a spherical particle of radius r in a tube with identical periodic dead ends is analyzed. It is shown that the effective diffusion constant follows the Stokes–Einstein relation, Deff(r)∝1/r, only when r is larger or much smaller than the radius of the dead end entrance. In between, Deff(r) not only deviates from the 1/r behavior but may also even become a nonmonotonic function, which increases with the particle radius for a certain range of r. [ABSTRACT FROM AUTHOR]

We analyze the relaxation of the particle number fluctuations in a membrane channel at arbitrary particle-channel interaction and derive general expressions for the relaxation time and low-frequency limit of the power spectral density. These expressions simplify significantly when the channel is symmetric. For a square-well potential of mean force that occupies the entire channel, we verify the accuracy of the analytical predictions by Brownian dynamics simulations. For such a channel we show that as the depth of the well increases, the familiar scaling of the relaxation time with the channel length squared is transformed into a linear dependence on the length. [ABSTRACT FROM AUTHOR]

A particle diffusing in a tube with dead ends, from time to time enters a dead end, spends some time in the dead end, and then comes back to the tube. As a result, the particle spends in the tube only a part of the entire observation time that leads to slowdown of its diffusion along the tube. We study the transient diffusion in a tube with periodic identical dead ends formed by cavities of volume Vcav connected to the tube by cylindrical channels of length L and radius a, which is assumed to be much smaller than the tube radius R and the distance l between neighboring dead ends. Assuming that the particle initial position is uniformly distributed over the tube, we analyze the monotonic decrease of the particle diffusion coefficient D(t) from its initial value D(0)=D, which characterizes diffusion in the tube without dead ends, to its asymptotic long-time value D(∞)=Deff4 realizations of the particle trajectory. The time-dependent mean squared displacement found in simulations is compared with that obtained by numerically inverting the Laplace transform of the mean squared displacement predicted by the theory, which is given by 2D⁁(s)/s. Comparison shows excellent agreement between the two time dependences that support the approximations used when developing the theory. [ABSTRACT FROM AUTHOR]

To escape from a cavity through a small window the particle has to overcome a high entropy barrier to find the exit. As a consequence, its survival probability in the cavity decays as a single exponential and is characterized by the only parameter, the rate constant. We use simulations to study escape of Langevin particles from a cubic cavity through a small round window in the center of one of the cavity walls with the goal of analyzing the friction dependence of the escape rate. We find that the rate constant shows the turnover behavior as a function of the friction constant, ζ: The rate constant grows at very small ζ, reaches a maximum value which is given by the transition-state theory (TST), and then decreases approaching zero as ζ→∞. Based on the results found in simulations and some general arguments we suggest a formula for the rate constant that predicts a turnover of the escape rate for ergodic cavities in which collisions of the particle with the cavity walls are defocusing. At intermediate-to-high friction the formula describes transition between two known results for the rate constant: the TST estimation and the high friction limiting behavior that characterizes escape of diffusing particles. In this range of friction the rate constants predicted by the formula are in good agreement with those found in simulations. At very low friction the rate constants found in simulations are noticeably smaller than those predicted by the formula. This happens because the simulations were run in the cubic cavity which is not ergodic. [ABSTRACT FROM AUTHOR]

We analyze trapping of diffusing particles by nonoverlapping partially absorbing disks randomly located on a reflecting surface, the problem that arises in many branches of chemical and biological physics. We approach the problem by replacing the heterogeneous boundary condition on the patchy surface by the homogenized partially absorbing boundary condition, which is uniform over the surface. The latter can be used to analyze any problem (internal and external, steady state, and time dependent) in which diffusing particles are trapped by the surface. Our main result is an expression for the effective trapping rate of the homogenized boundary as a function of the fraction of the surface covered by the disks, the disk radius and trapping efficiency, and the particle diffusion constant. We demonstrate excellent accuracy of this expression by testing it against the results of Brownian dynamics simulations. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Electrical conductivity of potassium chloride solutions containing polyethylene glycol (PEG) of different molecular mass was measured in a wide range of the polymer concentration up to 33 wt. % for PEG 300, 600, 2000, 4600, and 10 000. The data were used to find the dependence of microviscosity, η[sub micro], which characterizes the decrease of the ion mobility compared to that in the polymer-free solution, on the polymer volume fraction, [lowercase_phi_synonym]. We find that the dependence is well approximated by a simple relation η[sub micro]/η[sub 0]=exp[k[lowercase_phi_synonym]/(1-[lowercase_phi_synonym])], where η[sub 0] is viscosity of the polymer-free solution and k is a fitting parameter. Parameter k weakly depends on the polymer molecular mass growing from 2.5 for PEG 300 to its limiting value close to 2.9 for long chains. Using the [lowercase_phi_synonym]-dependence of microviscosity, we give a practical formula for the conductivity of PEG-containing electrolyte solutions. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We study the time dependence of the survival probability of a Brownian particle that escapes from a cavity through a round hole. When the hole is small the escape is controlled by an entropy barrier and the survival probability decays as a single exponential. We argue that the rate constant is given by k=4Da/V, where a and V are the hole radius and the cavity volume and D is the diffusion constant of the particle. Brownian dynamics simulations for spherical and cubic cavities confirmed both the exponential decay of the survival probability and the expression for the rate constant for sufficiently small values of a. [ABSTRACT FROM AUTHOR]

The Fokker–Planck–Kramers equation for a system composed by a reactive coordinate x coupled to a solvent coordinate y is employed to study the effect of additional degrees of freedom on the dynamics of reactive events. The system is studied numerically in the diffusional regimes of both coordinates, for different topologies of the bistable potential function and anisotropies of friction. The eigenvalue spectrum is evaluated by representing the time evolution operator over a basis set of orthonormal functions. A detailed analysis of the effect of the explicit consideration of the slow nonreactive mode is carried on to show that a variation of qualitative picture (scenario) of the reaction dynamics occurs when friction along different directions is strongly anisotropic, depending also on the structure of the two-dimensional potential surface. The numerical study supports both the qualitative picture of the reaction dynamics and the rate constant expressions obtained analytically. For those cases where the Langer theory has a restricted range of applicability because of the change in the reaction dynamics scenario, this fact has been numerically demonstrated. Here the Langer expression for the rate constant is replaced by the one obtained as a result of the consideration of the effective one-dimensional problem along the solvent coordinate, characterized by a smaller activation energy than that in the initial problem. All of these facts were confirmed by the numerical test, which shows a satisfactory agreement with the analytical results. © 1996 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The thermally activated escape of a Brownian particle over an arbitrarily shaped potential barrier is considered. Based on an approximate solution of the corresponding Fokker-Planck equation a rate expression is given. It agrees in the limiting case of high friction with the rate following from the corresponding Smoluchowski equation and, in the limit of weak friction with the rate obtained from transition state theory. For a parabolic barrier the approximate rate expression deviates less than 16% from the known result. The results for cusp shaped and quartic barriers agree with known expressions which have been obtained by other means. Estimates of the rates from numerical simulations are compared with the approximate rate expressions for the cusp and quartic barrier. [ABSTRACT FROM AUTHOR]

The variational transition state theory approach for dissipative systems is extended in a new direction. An explicit solution is provided for the optimal planar dividing surface for multidimensional dissipative systems whose equations of motion are given in terms of coupled generalized Langevin equations. In addition to the usual dependence on friction, the optimal planar dividing surface is temperature dependent. This temperature dependence leads to a temperature dependent barrier frequency whose zero temperature limit in the one dimensional case is just the usual Kramers–Grote–Hynes reactive frequency. In this way, the Kramers–Grote–Hynes equation for the barrier frequency is generalized to include the effect of nonlinearities in the system potential. Consideration of the optimal planar dividing surface leads to a unified treatment of a variety of problems. These are (a) extension of the Kramers–Grote–Hynes theory for the transmission coefficient to include finite barrier heights, (b) generalization of Langer’s theory for multidimensional systems to include both memory friction and finite barrier height corrections, (c) Langer’s equation for the reactive frequency in the multidimensional case is generalized to include the dependence on friction and the nonlinearity of the multidimensional potential, (d) derivation of the non-Kramers limit for the transmission coefficient in the case of anisotropic friction, (e) the generalized theory allows for the possibility of a shift of the optimal planar dividing surface away from the saddle point, this shift is friction and temperature dependent, (f) a perturbative solution of the generalized equations is presented for the one and two dimensional cases and applied to cubic and quartic potentials. [ABSTRACT FROM AUTHOR]

Using Brownian dynamics simulations, we study the effective mobility and diffusion coefficient of a point particle in a tube formed from identical compartments of varying diameter, as functions of the driving force applied along the tube axis. Our primary focus is on how the driving force dependences of these transport coefficients are modified by the changes in the compartment shape. In addition to monotonically increasing or decreasing behavior of the effective mobility in periodic entropy potentials reported earlier, we now show that the effective mobility can even be nonmonotonic in the driving force. [ABSTRACT FROM AUTHOR]

Diffusion of a solute in a periodic porous solid is analyzed. An expression for the effective diffusion coefficient is derived for a solute diffusing in a porous medium formed by a simple cubic lattice of spherical cavities connected by narrow tubes. This expression shows how the effective diffusion coefficient depends on microgeometry of the porous material. Generalizations to nonspherical cavities, other lattices, and nonequal diffusion coefficients in the cavities and in the tubes are discussed. © 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]