Vladimir Yu. Zitserman

Kinetics of escape through a small hole

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.

Activated rate processes : generalization of the Kramers-Grote-Hynes and Langer theories

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 ...

Boundary homogenization for trapping by patchy surfaces

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. (c) 2004 American Institute of Physics.

Communications: Drift and diffusion in a tube of periodically varying diameter. Driving force induced intermittency

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.

Diffusivity in periodic arrays of spherical cavities

We derive an expression for the effective diffusivity in a model porous medium formed by a periodic array of touching spherical cavities. Our result explicitly links the effective diffusion constant to the microgeometry of the porous material.

Effective diffusivity in periodic porous materials

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.

TRANSIENT DIFFUSION IN A TUBE WITH DEAD ENDS

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 V(cav) 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(infinity)=D(eff)<D. We derive an expression for the Laplace transform of D(t), denoted by D(s), where s is the Laplace parameter. Although the expression is too complicated to be inverted analytically, we use it to find the relaxation time of the process as a function of the geometric parameters of the system mentioned above. To check the accuracy of our results, we ran Brownian dynamics simulations and found the mean squared displacement of the particle as a function of time by averaging over 5x10(4) 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.

Conductivity and microviscosity of electrolyte solutions containing polyethylene glycols

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, ηmicro, which characterizes the decrease of the ion mobility compared to that in the polymer-free solution, on the polymer volume fraction, φ. We find that the dependence is well approximated by a simple relation ηmicro/η0=exp[kφ/(1−φ)], where η0 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 φ-dependence of microviscosity, we give a practical formula for the conductivity of PEG-containing electrolyte solutions.

Communication: Turnover behavior of effective mobility in a tube with periodic entropy potential

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.

One-dimensional description of diffusion in a tube of abruptly changing diameter: Boundary homogenization based approach

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.