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.
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28 May 2012
Research Article|
May 24 2012
Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized Fick-Jacobs equation
Leonardo Dagdug;
Leonardo Dagdug
1Departamento de Fisica,
Universidad Autonoma Metropolitana-Iztapalapa
, Mexico, 09340 Distrito Federal, Mexico
2Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology,
National Institutes of Health
, Bethesda, Maryland 20892, USA
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Marco-Vinicio Vazquez;
Marco-Vinicio Vazquez
1Departamento de Fisica,
Universidad Autonoma Metropolitana-Iztapalapa
, Mexico, 09340 Distrito Federal, Mexico
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Alexander M. Berezhkovskii;
Alexander M. Berezhkovskii
2Mathematical and Statistical Computing Laboratory, Division of Computational Bioscience, Center for Information Technology,
National Institutes of Health
, Bethesda, Maryland 20892, USA
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Vladimir Yu. Zitserman;
Vladimir Yu. Zitserman
3Joint Institute for High Temperatures,
Russian Academy of Sciences
, Izhorskaya 13, Bldg. 2, Moscow 125412, Russia
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Sergey M. Bezrukov
Sergey M. Bezrukov
4Program in Physical Biology, Eunice Kennedy Shriver National Institute of Child Health and Human Development,
National Institutes of Health
, Bethesda, Maryland 20892, USA
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J. Chem. Phys. 136, 204106 (2012)
Article history
Received:
March 01 2012
Accepted:
April 27 2012
Citation
Leonardo Dagdug, Marco-Vinicio Vazquez, Alexander M. Berezhkovskii, Vladimir Yu. Zitserman, Sergey M. Bezrukov; Diffusion in the presence of cylindrical obstacles arranged in a square lattice analyzed with generalized Fick-Jacobs equation. J. Chem. Phys. 28 May 2012; 136 (20): 204106. https://doi.org/10.1063/1.4720385
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