J. Phys. Chem. C 2007, 111, 15995-15997
15995
Pore-Size Dependence of Quasi-One-Dimensional Single-File Diffusion Mobility† K. K. Mon*,‡ and J. K. Percus§ Department of Physics and Astronomy, and Nanoscale Science and Engineering Center, UniVersity of Georgia, Athens, Georgia 30602, and Courant Institute of Mathematical Sciences and Department of Physics, New York UniVersity, New York, New York 10012 ReceiVed: May 18, 2007; In Final Form: August 20, 2007
We use kinetic Monte Carlo simulation to study the pore-size dependence of the anomalous single-file diffusion mobility of a hard disc fluid confined within a narrow channel. The channel has a half-width of Rp in units of the hard disk diameter. We present numerical data on the crossover from the exact 1D result with Rp ) (1/2) to the quasi-1D regime for (1/2) < Rp < 1. Our data are significant because exact results in the quasi-1D regime are very limited. A scaling dependence of (Rp - (1/2))4.65 is consistent with the measured increase in the mobility for almost seven decades.
I. Introduction Fluids in the presence of stochastic forces execute singlefile diffusion (SFD) when their motions are restricted by a confining geometry that forbids the passing of one fluid particle by another. This produces anomalous diffusion,1-12 with the growth of the mean-squared displacement increasing as the square root of time, and was demonstrated by an exact solution of the 1D model of the hard rod fluid.3 The long-time meansquared displacement was shown to be
〈(∆x)2〉 ) (l - a)〈|∆x|〉o
(1)
〈|∆x|〉o is the mean absolute displacement of noninteracting particles in the same restricted confinement under the same stochastic forces, and 〈|∆x|〉o ≈ xt. Because l is the average length per rod and a is the rod’s diameter, (l - a) is the mean distance between the ends of the rods. A fundamental measure of SFD is the mobility F, which is defined as
F)
〈(∆x)2〉
(2)
(2xt)
The mobility for a hard rod is
F)
(l - a)〈|∆x|〉o (2xt)
≈
l-a 2
(3)
F vanishes at the close-packed limit of l ) a and increases with l for l > a. Simulations and experiments confirm the expectation that anomalous diffusion survives the crossover from 1D to quasi1D confinements.1-2,4-12 An extension of the 1D exact solution to the quasi-1D regime (single-file fluids of hard discs and hard spheres within pores with a half-width of Rp > a/2) has defied intense efforts for decades. As a consequence, there are very †
Part of the “Keith E. Gubbins Festschrift”. * Corresponding author. E-mail:
[email protected]. ‡ University of Georgia. § New York University.
few exact results for quasi-1D SFD. In particular, one expects mobility F to increase with increasing pore width, but by how much? For a fixed average longitudinal separation between hard particles of l, the mobility should increase with enlarged halfwidth Rp. One can surmise a dependence on Rp of
F(Rp) ) Fo + f()
(4)
where ≡ Rp - (1/2). The unit of length is the hard disk diameter. Fo ≡ F(Rp ) (1/2)) is the 1D hard rod mobility. The increase in mobility from 1D (Fo) to quasi-1D (F(Rp > (1/2))) is then given by f() ) F(Rp) - Fo. Very little that is rigorous is known about this crossover function f() for the hard sphere fluids. Some molecular dynamic simulations of model liquids under random forces have been reported and are consistent with little or no dependence of the SFD on the radius of the cylindrical pore.10 Here we go much further and obtain a striking analytical fit that supplies a severe test of any future analytical theory. It has been shown that a kinetic Monte Carlo (random walk) type of stochastic simulation is adequate to capture the influence of the random force for self-diffusion in single-file formation.11 Monte Carlo simulations of hard sphere fluids in single-file geometry exhibit anomalous self-diffusion12 and are consistent with some dependence on the pore size. In this article, we employ kinetic Monte Carlo simulation to study, in more detail, f() for a model hard disk fluid confined within a hard channel. Our mobility data indicate that f() is indeed small over most of the quasi-1D regime and scales as (Rp - (1/2)),4.65 representing the increase in mobility over almost seven decades as Rp ranges from 0.52 to 0.98 in units of the hard disk diameter. Note that this is a very broad range. It encompasses the crossing from contact with nearest neighbors alone to include next-nearestneighbor contact single-file formation at Rp g (1/2) + (x3/4) Finally, the limit of single-file formation is reached when Rp ) 1 is approached. In the next section, we describe the simulation of hard disk diffusion in a channel. In section III, the simulation results are presented, and the article concludes with some remarks in section IV.
10.1021/jp0738558 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/06/2007
15996 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Mon and Percus
II. Monte Carlo Methods To study the mobility of the single-file self-diffusion of hard disk fluids with stochastic dynamics, we consider a constant volume Monte Carlo simulation.13 The hard discs have a diameter of a in a hard wall parallel channel of half-width Rp. All lengths will be in units of a. The length of the channel is L and the two ends obey periodic boundary conditions, but we follow only a tagged particle for a small distance compared with L/ . This implies that to study long times with high mobility a 2 large system is needed. For long times with low mobility, a large system is not needed. We use standard anisotropic Monte Carlo moves with the same step size of δ ) 0.1 in the transverse or longitudinal direction with equal probability. Isotropic composite moves are also used to move in both the transverse and longitudinal directions with uniform weights. One thousand particles are used with up to around 10 million Monte Carlo steps per particle. We have also used more particles to determine whether the system size is sufficient. Starting with an initial and reference configuration that has reached equilibrium, we monitor the mean-square deviation (MSD) in the longitudinal direction of the pore and plot it versus the square root of the simulation time step to extract the mobility from the slope. This is standard practice, and more information is given in ref 12. Block averaging and different runs are used to estimate statistical errors.
Figure 1. log-log plot of the Monte Carlo results for the crossover function of the single-file diffusion mobility f() versus ) (Rp - (1/ 2)), the difference in the pore size from the 1D limit. See eq 4. Rp is the channel half-width. The simulations use 1000 hard disks at an average longitudinal length per disk of l ) 1.0. A Monte Carlo step size of δ ) 0.1 is used for both anisotropic (×, longitudinal or transverse) and composite (0, longitudinal and transverse) types of Monte Carlo moves. The unit of length is the hard disk diameter a, and the unit of time is 1 Monte Carlo step per particle. The dotted lines indicate a power-law fit of 0.24.65(1.0. The statistical errors are approximately equal to or less than the size of the symbols. The solid lines are guides for the eye.
III. Results We use Monte Carlo simulations to study the crossover function, f() ) F(Rp) - Fo with ≡ Rp - (1/2). Most of the simulations are at a longitudinal density of 1 disk per unit length of the hard disk diameter a. This density, when ) 0, represents the close-packed limit with Fo ) 0. Results for lower density at 1.3639 disk per unit length are similiar. We note that for l ) 1 the variation of F is relative to the zero mobility of the 1D close-packed limit and this provides a computational advantage that permits us to study f over a range of seven orders of magnitude. At this density, one can study very small mobility by averaging over very long simulation. If the longitudinal density is less than 1 and Fo * 0, one must use much longer simulation to measure f() accurately. The results for the two different types of Monte Carlo moves over the range of Rp ) 0.52-0.98 are given in Figure 1 for l ) 1. The data exhibit power-law scaling with an estimated exponent of 4.65 ( 1.0 and can be fitted as ∼0.24.65. The estimated statistical error is approximately equal to or smaller than the size of the plotting symbols. The difference in the two types of Monte Carlo moves appears to be within the error of the numerical estimates. We have also some data at a lower density of l ) 1.3639. See Figure 2. These data are for a more limited range of pore sizes but exhibit scaling of ∼0.0754.65, consistent with the data for l ) 1 presented in Figure 1. Observe that the amplitude of the scaling law appears to decrease with decreasing density. A detailed study of this density dependence is in progress. It involves lower density simulations that are computational intensive. The limitation on lower density simulations is imposed by the greater demand on the length of the simulation. The mobility (F) is extracted from the slope of the mean-square displacement versus the square root of time plot. The measured Fmeasured contains statistical errors (δ), Fmeasured ) Ftrue ( δ ) Fo + f(). Thus, f() ) Fmeasured - Fo ) (Ftrue - Fo) ( δ, and f() can be measured accurately only for (Ftrue - Fo) . δ. Clearly, there is an advantage in picking a density with a
Figure 2. log-log plot of the Monte Carlo results for the crossover function of the single-file diffusion mobility f() versus ) (Rp - (1/ 2)), the difference in the pore size from the 1D limit. See eq 4. Rp is the channel half-width. The simulations use 1000 hard disks at an average longitudinal length per disk of l ) 1.3639. A Monte Carlo step size of δ ) 0.1 is used for anisotropic (×, longitudinal or transverse)-type Monte Carlo moves. The unit of length is the hard disk diameter a, and the unit of time is 1 Monte Carlo step per particle. The dotted lines indicate a power-law fit of 0.0754.65(1.0. The statistical errors are approximately equal to or less than the size of the symbols. The solid lines are guides for the eye.
vanishingly small Fo. At the highest density with l ) 1, Fo ) 0. At lower density, one must perform much longer simulations with a large system. Our data indicate that the variations of the mobility (f()) are
